∫ ∫ Application of Hydroacoustic Radiators for the Generation of Seismic Waves

Home , Sound speed gradient

Acoustical Physics, Vol. 48, No. 2, 2002, pp. 121Ð127. Translated from Akusticheskiœ Zhurnal, Vol. 48, No. 2, 2002, pp. 149Ð155. Original Russian Text Copyright © 2002 by Averbakh, Bogolyubov, Dubovoœ, Zaslavskiœ, Lebedev, Maryshev, Nazarov, Pigalov, Talanov.

Application of Hydroacoustic Radiators for the Generation of Seismic Waves V. S. Averbakh, B. N. Bogolyubov, Yu. A. Dubovo œ, Yu. M. Zaslavski œ, A. V. Lebedev, A. P. Maryshev, V. E. Nazarov, K. E. Pigalov, and V. I. Talanov Institute of Applied Physics, Russian Academy of Sciences, ul. Ul’yanova 46, Nizhni Novgorod, 603950 Russia e-mail: [email protected] Received January 31, 2001

Abstract—To excite seismic waves with a high coherence, powerful hydroacoustic radiators placed in a natural reservoir were used. Theoretical estimates and the test data demonstrate a high efﬁciency of the proposed method of seismic wave excitation. The calculations are in good agreement with the results of measurements. The results of phasing the radiation with the use of two monopole sources separated by a quarter-wave distance are presented. It is shown that the use of the proposed scheme of excitation makes it possible to control the radiation pattern while obtaining a high coherence of seismic waves. © 2002 MAIK “Nauka/Interperiodica”.

INTRODUCTION the predicted and measured values of the velocity of In an earlier publication [1], the authors proposed vibration of the Earth’s surface, which makes it possi- using the powerful hydroacoustic radiator developed at ble to evaluate the efficiency of radiation. In the follow- the Institute of Applied Physics of the Russian Acad- ing section, the possibility of controlling the radiation emy of Sciences [2] as a source of seismic waves. of two phased sources is studied, and the last section Owing to the relatively high frequency (about 200 Hz) summarizes the main results. and coherence of the radiation produced by such sources, they can be used for the investigation of near- THE BASIC THEORETICAL MODEL surface layers with a high spatial resolution and for the detection of local inhomogeneities [3]. Let us consider the following problem. A monopole source is located in a layer of liquid. This layer is The cited paper [1] and the report at a Meeting of the bounded by an air halfspace (an acoustically soft Acoustical Society of America [4] were devoted to the boundary) and by an elastic halfspace. It is necessary to theoretical evaluation of the efficiency of radiation of find the displacements at large distances from the body seismic waves. It was shown that a radiator source and the power of the source radiation. mounted in the upper part of a rigid tube filled with water and buried in the ground can provide a useful We combine the axis of symmetry of a cylindrical radiation power comparable with the acoustic power coordinate system with the gravity vector passing generated in a water medium. However, the suggested through the point where the source is positioned, the scheme of excitation has pronounced resonance proper- coordinates of this point being (0, –h). We represent the ties, which leads to the necessity of matching its param- displacements in the form of scalar (φ, ϕ) and vector eters. This causes problems in its practical realization. (y) potentials [7]: In this paper, an alternative scheme of excitation is theoretically studied, and the results of its field tests [5, 6] ∇φ, ÐH ≤≤z 0 (liquid layer) U =  (1) are compared with the theoretical estimates. ∇ϕ+ ∇ × y, z > 0 (ground). Seismically active regions are of most interest for seismology. These regions are often located near rivers Due to the axial symmetry of the problem, the poten- and lakes. It is of interest to use these natural reservoirs tials can be represented as the following integrals (y × for matching a hydroacoustic radiator with the ground. r = y × z = 0): In this case, for an accurate calculation of the radiation efficiency, it is necessary to take into account a great φ(), 1 φˆ ()κ, ()1 ()κκκ rz = ------∫ z H0 r d , number of details (the bottom profile, the contour of the 4π shore, and so on), which strongly complicates such cal- C culations. Therefore, it is necessary to use idealizations 1 ()1 ϕ()rz, = ------ϕˆ ()κ, z H ()κκκr d , (2) simplifying the task. Below, we consider a simple the- 4π∫ 0 oretical model. Then, in the next section, we compare C

122 AVERBAKH et al.

() where τ κ k γ c c p c c , m ρ ρ x ψ(), 1 ψ()κ, 1 ()κκκ = / 2, = 2/ 1, = 2/ 0 = / 0, = rz = ------∫ ˆ z H1 r d , 4π 2 τ2 2 τ2 C k2h p Ð , and y = k2H p Ð . ()1 We point out the following features of Eqs. (4): where H j is a Hankel function of the first kind of order j (the time dependence of all quantities is 1. The radiation of S-waves is absent in the direction ω assumed to be ~exp(–i t)). The path of integration is θ c2 θ γ 2 2 θ s = arcsin---- , because F( ) ~ Ð sin . This chosen to satisfy the causality principle. The solution c1 for the potentials is obtained by the method of sewing angle corresponds to a total reflection (without trans- together partial domains by using the continuity of the formation of the type of wave) of the transverse wave displacement and stress fields. In the solution obtained, arriving from the domain z > 0. Therefore, it is impos- it is possible to identify the following terms [9]: sible to excite a shear wave in the solid by a sound wave (i) body waves with a spherical divergence—the propagating in the liquid layer and incident on the contribution of the points of stationary phase; boundary between the two media at the angle corre- θ (ii) Rayleigh and Stoneley waves—the contribution sponding to s in the elastic halfspace. of the poles; and 2. The important kinematic relations should be (iii) one or two lateral waves (for c1 or c2 > c0)—the noted: contribution of the branch points (c1 and c2 are the velocities of longitudinal and shear waves in the 2.1. If c1 and c2 are greater than c0 (fast waves), the ground, and c is the sound velocity in the liquid layer). radiation of P- and S-waves is determined by the rays in 0 θ ≤ θ We will restrict our analysis to the body waves, the liquid within a cone with the apex angle * = because they are of most interest for tomography of the arcsin(c /c ). One can expect that, for c > c , only 1 0 1, 2 1, 2 0 Earth’s interior. a small area bounded by the θ -rays will be responsible The stationary points are determined by the condi- * κ θ ω ω for the excitation of body waves (Fig. 1). This property tions = k1, 2sin , where k1 = /c1 and k2 = /c2 are the can be used for suppressing a certain type of waves by wave numbers of longitudinal and transverse waves in using the directivity of the source. the ground, the angle θ is counted from the z axis of Ӷ symmetry of the coordinate system and corresponds to 2.2. For water-saturated rock, for which c2/c0 1, the direction of the radius vector from the source to the the radiation of S-waves is possible only in a narrow point of observation range of angles θ, and the longitudinal P-waves give the main contribution to the total power of the body wave E()γθsin k cosθ radiation (Fig. 3). ϕ()R, θ = Ði------1 exp()ik R , 2πR 1 (iii) The denominator of the first of Eqs. (4) deter- (3) ()θ θ mines the normal modes of the liquid layer ψ(), θ F sin k2 cos () R = Ð------exp ik2 R , 2πR 2 sin()y γ2 Ð τ2 + iycos()mp2 Ð12τ2(()Ð τ2 where R is the distance from the source. Equations (3) (5) ӷ τ2 ()γτ2 ())2 τ2 are valid on the condition that k1, 2R 1. The func- +4 1 Ð Ð = 0. tions E(á) and F(á) determining the spectral amplitudes of longitudinal and shear waves have the form Solving Eq. (5), it is possible to determine the projections of the wave vector on the z axis (kz) in the ()τ Q ()τ2 ()[ γ 2 τ2 () waveguide formed by the boundaries of the liquid E = ω------12Ð sin yxÐ / Ð sin y k2 layer. The following limiting cases can be distinguished: m 0 (an acoustically soft boundary, soft 2 2 2 2 π π ∞ + im p Ð12τ (()Ð τ rock) for which kzH = + n, n = 0, 1, …, and m π π (4) (a hard boundary, hard rock) for which kzH = /2 + n. +4τ2 ()γ1 Ð τ2 ())2 Ð τ2 cos()y ], As is seen from Eqs. (4), the maximal radiation of P- and S-waves occurs in the vicinity of the critical fre- τγ2 Ð τ2 quencies of the waveguide. In the general case, when F()τ = 2i------E()τ , 0 < m < ∞, the critical frequency of the lowest mode τ2 12Ð will be within c0/4H < f < c0/2H.

1 If the inverse reaction of radiation to the source characteristics is Now we evaluate the efficiency of excitation of seis- of little interest or weak, the near-field analysis is unnecessary. mic waves. The total power is determined by the This is justified when the response is weakly frequency depen- expression dent and slight changes in the resonance frequency of the source do not lead to considerable changes in the characteristics of the 1 whole radiating system. This situation takes place in many cases {}σ W = Ð---Re ∫ ikv k*dsi , (6) that can be of interest (Figs. 2b, 2c). 2

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 APPLICATION OF HYDROACOUSTIC RADIATORS 123 σ where ik is the stress tensor, v is the particle velocity in a solid in the wave ﬁeld, and (á)* denotes complex conjugation. Radiator ӷ θ By using Eq. (1) with the condition k1R, k2R 1, we * obtain WW= + W , 1 2 Elastic halfspace π/2 πρω3 ϕ 2 2 θ θ Fig. 1. Scheme of rays exciting fast seismic waves. W1 = ∫ k1 R sin d , (7) 0

π/2 W1, 2/W0 πρω3 ψ 2 2 θ θ 0 W2 = ∫ k2 R sin d , 10 (‡) 0 10–1 where W1 and W2 are the powers of compressional and shear waves, respectively, with the potentials ϕ and ψ determined by Eqs. (3). 10–2 Figures 2aÐ2c show the dependences of W and W 1 2 100 on the parameter k0H. The values of W1 and W2 are nor- ρ ω 2 π (b) malized to W0 = 0 k0Q /8 (the acoustic power of a monopole source in a boundless liquid). The calcula- 10–1 tions were performed for: –2 (a) granite with the parameters c1 = 5000 m/s, c2 = 10 2800 m/s, and m = 2.5; 0 (b) clay with the parameters c1 = 2000 m/s, c2 = 10 1000 m/s, and m = 2; and (c) –1 (c) water-saturated loam with the parameters c1 = 10 1800 m/s, c2 = 200 m/s, and m = 2 [8]. In all cases, we used h/H = 0.5 and the sound veloc- 10–2 ity in liquid was assumed to be c = 1500 m/s. The ver- 0 π π 0123456 tical dashed lines in Fig. 2a correspond to k0H = /2, . k0H The total power W1 + W2 is less than W0, because part of the radiation energy is transferred by the Ray- Fig. 2. Dependence of the power of body waves on the leigh and Stoneley surface waves and, in addition, a dimensionless frequency. The thick line corresponds to sound propagation occurs in the liquid layer. Therefore, compressional waves and the thin line corresponds to shear the maximums of the total power are reached near the waves. Additional comments are in the text. critical frequencies of the waveguide. With an increase in frequency, an increasing number of modes is excited in the liquid layer, and the values of the radiation max- ESTIMATION OF RADIATION EFFICIENCY. imums in Fig. 2 decrease. TEST DATA Thus, the proposed scheme of excitation of seismic The experiments were carried out on the bank of the waves by a powerful hydroacoustic radiator seems Trotsa River (Nizhni Novgorod region, Russia). This attractive, because it is possible to achieve a radiation minor forest river has a depth of 4Ð6 m near the bank level comparable with W0 in the case of high coherence. (H = 4.5 m, h = 2 m) where the measurements were Since sharp peaks in the frequency dependence of radi- made. A detailed description of the measurements and ated power Wj are absent for soft rock with c2 < c0, it is the equipment can be found in papers [5, 6]. Figure 4 not necessary to adjust the operating frequency. Only shows the schematic view of the measurements. The for hard rock, e.g., granite, such an optimization may be hydroacoustic transducer used in the experiments had needed. However, in this case, the operating frequency an acoustic power of W0 = 400 W [1, 2]. is determined simply as f = c0/4H. It was established [6] that the measured vertical It should be noted that, for moderate values of k0H, velocity components are determined mainly by the the radiation pattern also does not have any sharp reflected body waves (the amplitude of displacements peaks, and the maximal level of radiation of longitudi- is inversely proportional to the distance). The reflecting θ nal waves is reached in the direction = 0 (Fig. 3). This horizon was located at the depth Hb = 30 m and did not property is very important for insonification with the have any essential slope. The analysis of the times of use of longitudinal waves. arrival of radiated pulses as a function of distance

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 124 AVERBAKH et al. ( V1, 2/V0 arcsin c0 /c1) and the velocity of the longitudinal wave propagation in the bottom c Ӎ 1700 m/s. 1.0 1 By using the data presented above, we can estimate 0.8 the efﬁciency of radiation. The acoustic power (W0) of the transducer was 400 W at the operating frequency f = 226 Hz (the consumed electric power was 700 W 0.6 [2]). The volume velocity Q is determined by the expression 0.4 2c W 1 0 0 Ӎ 3 Q = ------πρ 0.09 m /s. (8) 0.2 f 0

The liquid particle velocity (see caption to Fig. 3) 0 30 60 90 θ equals

Fig. 3. Directivity of the body wave radiation for the fre- fQ 1 W quency used in the measurements (k H = 4.25). The param- V ==------0 , (9) 0 0 2c R R 2πρ c eters m, c1, and c2 correspond to the data of Fig. 2c. The 0 0 0 thick line corresponds to the angular distribution of the compressional wave amplitude and the thin line corre- 2 2 sponds to the shear wave. The displacements are normalized where R = d + 4H is the distance between the radia- ω π to U0 = V0/ , where V0 = k0Q/4 R is the particle velocity in tor and the receiver and d = |SP| (the thin line in Fig. 4). a boundless liquid in the field of a monopole source with strength Q. Figure 5 shows the dependence of the normal component (Vn) of the vibration velocity of the Earth’s surface on the distance. The points correspond to the mea- between the source and the receivers showed that c1 = sured values of Vn, and the solid line corresponds to the 1300Ð1400 and 1800Ð1950 m/s in the upper layer and computed values. The theoretical dependence was con- at a depth of more than 30 m, respectively. The analysis structed in the following way: by the SASW method [10] with the use of an additional shock exciter showed that the velocity of Rayleigh waves (1) The angle of incidence on the boundary (Hb) was determined as (measured with a good accuracy) was VR = 190 m/s. The Ӎ value obtained for VR corresponds to c2 200 m/s in the θ () upper layer. = arctan d/2Hb . (10) | | Additional hydroacoustical measurements were per- (2) By using the data presented in Fig. 3 with V0 θ formed for the determination of the bottom parameters. determined by Eq. (9), we calculated the value of V1( ). It was shown that the first waveguide mode was excited. (3) This quantity was multiplied by the reflection The increase in pressure near the bottom at a distance of coefficient |ν|. The reflection coefficient was calculated about 4 m from the source was observed. This allowed us for two liquid layers [11] (because of the absence of to estimate the angle of total internal reflection θ* = reliable data on the velocity of shear waves in a layer

V, 105 m/s S 20 + B 10 C Landing stage 5 A d 2 012345... P D 1 Positions of the sensors 0.7 the points are spaced at 3 m ( ) 50 100 150 d, m Fig. 5. Computed and measured amplitudes of the vertical Fig. 4. Schematic view of the measurements at the Trotsa projection of the velocity of surface vibrations along the river. The point P is separated from the radiator by 42 m. measurement track.

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 APPLICATION OF HYDROACOUSTIC RADIATORS 125 below the 30-m horizon, it was impossible to make 1.0 The level of 0.99 more accurate calculations): 0.8 mcosθ Ð n2 Ð sin2 θ νθ()= ------, (11) 0.6 θ 2 2 θ mcos + n Ð sin 0.4 where m = ρ(H + 0)/ρ(H – 0) and n = c (H – b νb 1 b Coherence 0.2 0)/c1(Hb + 0). The quantity was calculated for m = 1 and n = 1350/1850. 0 226 Hz | θ ν θ | θ 100 150 200 250 f, Hz (4) The value of V1( ) ( ) was multiplied by cos to find the normal component of the velocity near the Fig. 6. Coherence of radiation. Averaging was made over free surface (Vn). 50 pulses of radiation. The dashed lines indicate the fre- (5) Since the sensors detect also the waves reflected quency band corresponding to 90% of the radiation power ∆ from the surface, the value of Vn was doubled. The fac- ( f = 2/T). tor “2” corresponds to the free boundary provided that c Ӷ c (c = 1350 m/s, c = 200 m/s). 2 1 1 2 the reference signal was more than 99% in the vicinity (6) The measurements were performed using a set of eight sensors connected in phase. Therefore, the value of the carrier frequency, which is seen clearly in Fig. 6. of V should be multiplied by The received signal level as a function of the delay and n phase difference of radiators is represented in Fig. 7. sin[]()θπfL/c sin The calculations were made as follows. We assumed S()θ = ------1 -, (12) ()θπfL/c sin that the mutual influence of radiators is compensated 1 and the field generated by each radiator is described by Eqs. (3). The phase factors exp(ik1R) were brought to where L = 10 m, f = 226 Hz, and c1 = 1350 m/s. Thus, the solid line in Fig. 5 represents the product the midpoint of the system of two radiators. The synthe- |2V (θ)ν(θ)S(θ)cosθ|. sized temporal response was determined as a superpo- 0 sition of contributions of spectral components with the As is seen from Fig. 5, a good qualitative agreement is observed between the calculated and measured data. amplitudes proportional to expressions (3). The transfer The relatively wide scatter of experimental data is functions of the radiators were also taken into account caused, on the one hand, by the superposition of multiply in the calculations. The inclusion of the transfer func- reflected pulses and, on the other hand, by the possible tions made it possible to describe the observed spread- ing of the radiated pulses and the absence of a sharp variations of the layer parameters (z < Hb and z > Hb) along the measurement track (a change in the water sat- leading edge of the signal. In constructing the synthe- uration with increasing distance from the shore and so sized response, we took into account a single reflection on). Thus, the theoretical model proposed in the previous of a compressional wave from the boundary Hb = 30 m section adequately describes the observed magnitudes of (see above). The shear waves, converted waves, and displacements, which allows one to reliably evaluate the multiple reflections were not considered. The scheme power of radiation of P-waves. By using the plot shown of calculations differed from the scheme used in con- in Fig. 2c, we obtain W1 = 0.4W0 = 160 W. structing Fig. 5 only in the replacement of the factor S(θ) by unity, because, in the measurements with THE USE OF A SYSTEM OF PHASED phased radiators, a compact group of receivers was RADIATORS used. We used two radiators of the same type placed at the One can easily see the qualitative agreement same point as in Fig. 4. The positions relative to the between the results of measurements and the numerical water surface differed in that the radiators were at the modeling. The maximal level of the received signal depth H – h = 1.77 m and were separated by 3.7 m in observed in the experiment was reached for the phase the direction parallel to the measurement track (Fig. 4). difference equal to +90°. Along with the reflected com- The receiver of the seismic signal was located at point D pressional waves, which were taken into account in (Fig. 4), at a distance of 120 m from the bank. constructing the synthesized response, the shear waves, The frequency of radiation was f = 226 Hz, and the converted waves, and multiple reflections can also be pulse duration was T Ӎ 44.2 ms (ten periods of the car- observed. Presumably, the longer duration of the rier frequency). The number of pulses used in the response in Fig. 7a can be determined by these factors. coherent summation was 50. The phase difference of the carrier frequency of the radiators was changed at Figure 8 shows the radiation patterns of the group 30° intervals from −180° to +180°. The normalized of two sources. The unit level corresponds to the in- coherence function of the received seismic signal and phase radiation in the direction θ = 0. One can see that,

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 126 AVERBAKH et al.

∆Φ° (a) ∆Φ° (b) 180 dB 180 dB 150 –32 150 –32 –35 –35 120 120 –40 –40 90 –45 90 –45 60 –50 60 –50 –55 –55 30 30 –60 –60 0 –65 0 –65 –30 –70 –30 –70 –60 –75 –60 –75 –80 –80 –90 –90 –85 –85 –120 –90 –120 –90 –180 –140 –180 –140 0 200 400 600 800 0 200 400 600 800 Delay, ms Fig. 7. Received signal level as a function of the propagation delay and the phase difference between the signals supplied to the two radiators: (a) measurements and (b) calculation.

Level ∆Φ = –90° 1.0 ∆Φ = 0° ∆Φ = +90° 0.8 ∆Φ = 180

0.6

0.4

0.2

0 –90 –60 –30 0 30 60 90 θ

Fig. 8. Directivity of radiation for various phase relations between the sources in the group (calculation). varying the phase difference between the sources, it is hard bottom) and always can be determined experimen- possible to control effectively the directivity of the tally. For the operating frequency f ~ 100 Hz, the opti- radiation. mal depth of a reservoir is from 3 to 6 m, depending on the bottom characteristics.

CONCLUSION The measured values of the vertical displacements of the Earth’s surface are in a good agreement with the The analysis of the scheme of excitation of seismic results of calculations using the proposed computa- waves by a hydroacoustic radiator is performed. The tional model. This conclusion is important, because the scheme is convenient for practical applications. Unlike experiment was carried out near the bank of a real nat- the previous results [1], the proposed scheme of excita- ural reservoir with a variable depth, while, in the model, tion makes it possible to reach high radiation levels in a a boundless liquid layer of constant thickness is consid- wide frequency range. The maximal efﬁciency of radi- ered. The calculated value of the useful power is about ation is reached in the vicinity of the critical frequency 160 W and may be as high as 400 W in the case of the of the lowest mode of the liquid layer. This frequency optimization of the operating frequency (Fig. 2). For π π changes from k0H = /2 (a soft bottom) to k0H = (a comparison, we note that the Vibroseis seismic exciter,

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 APPLICATION OF HYDROACOUSTIC RADIATORS 127 which develops the force of 20 t, radiates approxi- 2. M. M. Slavinsky and B. N. Bogolubov, Evaluation of mately W1 = 14 W and W2 = 430 W at a frequency of Electromagnetic Source for Ocean Climate Acoustic 100 Hz [1]. The hydroacoustic transducer excites Thermometry at Lake Seneca, Technical Report (Woods mainly compressional waves (for water-saturated Hole Oceanographic Institution, 1993). rock), or compressional and shear waves of approxi- 3. C. H. Frazier, N. Cadalli, D. C. Munson, Jr., and mately equal powers for hard rock. At present, com- W. D. O’Brien, Jr., J. Acoust. Soc. Am. 108, 147 (2000). pressional waves are used in most applications, and the 4. A. V. Lebedev and A. M. Sutin, in Proceedings of ASA proposed scheme of excitation is best suited for the 130th Meeting (St. Louis, Missouri, USA, 1995). generation of waves of this type. Owing to the stability 5. B. N. Bogolyubov, Yu. A. Dubovoœ, V. Yu. Zaœtsev, et al., of the radiator operation and the constancy of the bot- Preprint No. 380, IPF RAN (Inst. of Applied Physics, tom parameters, it is possible to use long-term storage Nizhni Novgorod, Russian Academy of Sciences, 1995). and coherent data processing [6]. 6. V. S. Averbakh, B. N. Bogolyubov, Yu. M. Zaslavskiœ, et al., Akust. Zh. 45, 1 (1999) [Acoust. Phys. 45, 1 The high degree of coherence of the radiation pro- (1999)]. duced by the sources together with the simplicity of the 7. L. D. Landau and E. M. Lifshits, Course of Theoretical electronic control of their excitation makes it possible Physics, Vol. 7: Theory of Elasticity, 4th ed. (Nauka, to create phased sources on their basis with a desired Moscow, 1987; Pergamon, New York, 1986). directivity of radiation. 8. N. N. Goryainov and F. M. Lyakhovitskiœ, Seismic Meth- ods in Engineering Geology (Nedra, Moscow, 1979), Table 5. ACKNOWLEDGMENTS 9. K. Aki and P. G. Richards, Quantitative Seismology, This work was supported by the Russian Foundation Theory and Methods (Freeman, San Francisco, 1980; for Basic Research, project nos. 00-15-96741 and Mir, Moscow, 1983), Vols. 1, 2. 99-02-16957. We are grateful to Prof. A.M. Sutin for 10. K. H. Stokoe, G. R. Rix, and S. Nazarian, in Proceedings helpful discussions and to chief programmer A.V. Tsi- of the 12th International Conference on Soil Mechanics berev for preliminary processing of the experimental and Foundation Engineering (Rio de Janeiro, 1989), data. Vol. 1, pp. 331Ð334. 11. L. M. Brekhovskikh and O. A. Godin, Acoustics of Lay- ered Media (Nauka, Moscow, 1989; Springer, New York, REFERENCES 1990). 1. A. V. Lebedev and A. M. Sutin, Akust. Zh. 42, 812 (1996) [Acoust. Phys. 42, 716 (1996)]. Translated by A. Svechnikov

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002

Factorization of Integro-Differential Equations of the Acoustics of Dispersive Viscoelastic Anisotropic Media and the Tensor Integral Operators of Wave Packet Velocities L. M. Barkovsky and A. N. Furs Belarussian State University, pr. F. Skoriny 4, Minsk, 220080 Belarus e-mail: [email protected] Received June 13, 2000

Abstract—Factorization of tensor integro-differential wave equations of the acoustics of dispersive viscoelastic anisotropic media is performed for the one-dimensional case. The resulting ﬁrst-order partial differential equations include integral tensor functionals of the sound velocities of polarized plane wave pulses. The velocity operators Vˆ are expressed in a coordinate-free compact form. They are determined by the kernels of the integral representations and provide a general description of the kinematics and dynamics of wave packets for arbitrary propagation directions in anisotropic viscoelastic media. © 2002 MAIK “Nauka/Interperiodica”.

In monograph [1], Whitham gave some examples of axes). The sets of eigenwaves are characterized by the the factorization of several nonlinear partial differential spectra of three-dimensional evolution operators (also wave equations. The factorization reduces the initial called Caushy operators or propagators) for the wave equations to the first-order equations, which can be equations of crystal acoustics. solved by the existing methods of characteristics [2]. One of the methods simplifying the solution of wave Various aspects of the factorization approach were also equations consists in reducing the latter to first-order discussed in [3Ð6]. For systems of equations, the factor- equations (the factorization method) [1], which can be ization is complicated because of the necessity to solved using the known techniques [2]. In the first- include the commutations of their matrix coefficients. order scalar equations for nondispersive media consid- The problem of factorization is still more complicated ered in [1, 10], the wave velocity appears as the factor in the case of integro-differential equations [7, 8] when of the spatial derivative. In the case of a system of equa- the polarization and dispersion of wave packets are tions, the matrix of the velocities (the velocity tensor) taken into account. At the moment, the theory of sound plays the role of this factor. The tensor dispersion rela- beams and pulses is not sufficiently advanced to sys- tions are inherent in acoustic materials regardless of the tematically describe their polarization and possible complexity of their properties (such as linearity, nonlin- transformations, especially for spatially bounded plane earity, isotropy, anisotropy, and gyrotropy [11Ð14]). and solid sound beams in inhomogeneous anisotropic complex media [9]. A finite-duration shear disturbance Below, we consider polarized disturbances with the represented by a set of waves of different frequencies is wave normals of all their components being parallel to characterized by a polarization-dependent velocity and a fixed direction n. We perform the factorization of the can, to one or another measure, be transformed to a lon- one-dimensional tensor differential wave equation that gitudinal or quasi-longitudinal wave in the presence of describes the behavior of such disturbances in linear some inhomogeneities (e.g., interfaces) or external anisotropic viscoelastic media. On the basis of the fac- actions (as in parametric acoustics or electroacoustics). torization of tensor wave equations, we propose an If the phase velocity of this disturbance depends on fre- operator description of polarized acoustic packets. quency, the medium is dispersive, and the velocity of Malus was the first to show that nonpolarized light the disturbance as a whole will differ from the veloci- packets can become polarized at reflection. This fact is ties of its individual wave components. From the infi- indicative of a limited applicability of the scalar nite set of wave states, only three states are stable (trire- approach. For acoustics, the importance of this infer- fringence in crystal acoustics). These states correspond ence follows from the existence of Brewster’s angles to the eigenwaves (isonormal wave types). Any other for shear waves [15]. We consider the evolution tensor polarization types that are allowed by the isotropic solutions, for which the field at the initial point at the medium will be decomposed into eigenwaves when the initial moment is assumed to be given. In so doing, we wave is transmitted into the anisotropic medium, even simultaneously take into account the variations of the if the propagation direction n remains intact, with the wave field in the longitudinal and transverse subspaces exception of some special directions n (the acoustic in the process of propagation. Below, we show how the

FACTORIZATION OF INTEGRO-DIFFERENTIAL EQUATIONS 129 ∂ ∂ ∂ ∂ζ factorization method is used to reduce the second-order tives / xk by nk / , we represent Eq. (1) in the index- tensor wave equation of crystal acoustics of linear free form anisotropic viscoelastic media to a first-order integro- t differential equation. This equation involves the sec- ∂2u()ζ, t ∂2u()ζ, t ∂2u()ζ, t′ ------= Λ------+ ∫ dt′Φ()ttÐ ′ ------, ond-rank velocity tensor Vˆ , which is expressed as a ∂t2 ∂ζ2 ∂ζ2 (2) function of the elasticity tensor, the viscosity tensor, Ð∞ and the wave normal vector n. From our consideration, where the second-rank tensors Λ and Φ are determined according to the formulas it follows that the velocity tensor Vˆ is simply a mathematical consequence of the Christoffel equations [16]. Λ 1 Φ 1 im ==ρ---ciklmnknl, im ρ---Ciklmnknl. (3) Consider the factorization of the wave equation of acoustics of viscoelastic anisotropic media. To factorize the wave equation (2) of the acoustics of viscoelastic anisotropic media, we represent the vector The equations of motion of an elastic medium have field of the displacements of points of the medium the form [16] u(ζ, t) in the form of the Fourier expansion ∞ 2 ∂ u ()r, t ∂σ ()r, t 1 Ðiωt ρ------i - = ------ik , u()ζ, t = ------dωu()ζω, e , 2 ∂ π ∫ ∂ xk 2 t Ð∞ ∞ (4) where u r t u r t is the displacement vector of ω ( , ) = ( i( , )) ()ζω, ()ζ, i t points of the medium as a function of the radius vector u = ∫ dtu t e , σ of the observation point r and time t, ik is the stress Ð∞ ρ tensor, and is the density of the medium. Summation and substitute Eqs. (4) in Eq. (2): is performed over repetitive indices from 1 to 3. Below, ∞ we consider anisotropic viscoelastic media with mem- 2 Ðiωt 2 ∂ u()ζω, ory, for which the Hooke law can be written in the inte- ω  ω ()Λζω, ∫ d e Ð u Ð ------2 - gral form:  ∂ζ Ð∞ ∞ (5) 2 σ = c γ + Cˆ γ ∂ u()ζω, iωt′  ik iklm lm iklm lm ′Φ()′  Ð ∫dt t ------2 -e = 0. t ∂ζ  0 = c γ ()r, t + dt′C ()γttÐ ′ ()r, t′ , iklm lm ∫ iklm lm Introducing the tensor Φ(ω) (the Fourier transform of Ð∞ the kernel Φ(t)) ∞ ∞ where ciklm is the elasticity tensor characterizing the ω ′ ω ′ instantaneous response of the medium to an external Φω()= ∫dt′Φ()t′ ei t ≡ ∫ dt′Φ()t′ ei t (6) ∞ action, Cˆ iklm is the integral operator describing the 0 Ð ′ aftereffect of the medium, Ciklm(t – t ) is the kernel of and equating the expression in braces in Eq. (5) to zero, γ this operator, and lm is the strain tensor. Note that, by we obtain ′ virtue of causality, Ciklm(t – t ) = 0 for negative argu- 2 2 ∂ u()ζω, ments t – t′. Taking into account that the strain tensor Ðω u()ΛΦωζω, Ð0.[]+ ()------= γ 2 lm(r, t) can be expressed through the displacements ∂ζ γ ∂ ∂ ∂ ∂ ui(r, t) as lm = ( ul/ xm + um/ xl)/2 and using the sym- Then, we have metry of the tensors ciklm and Ciklm with respect to the ∂ ∂ permutation of the second and fourth indices, we obtain ωΛΦω() ωΛΦω() Ði +Ð+ ∂ζ------i Ð + ∂ζ------∂2u ()r, t ∂2u ()r, t ρ------i - = c ------m ∂ 2 iklm ∂x ∂x × ()ζω, ω ΛΦω() ∂t k l u = Ð i Ð + ∂ζ------(7) (1) t ∂2 (), ′ ′ ()′ um r t ∂ + ∫ dt Ciklm ttÐ ------∂ ∂ . × ÐiωΛΦω+ + ()------u()ζω, = 0, xk xl ∂ζ Ð∞ Let a wave packet propagate in the direction speci- where ΛΦω+ () is the square root of the operator Λ + fied by the unit vector n. Introducing the coordinate ζ = Φ(ω) (about taking the root of operators, see, e.g., nr along this direction and replacing the spatial deriva- [17]). The tensor differential operators appearing in

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 130 BARKOVSKY, FURS square brackets in Eqs. (7) are commuting, which The wave equation of crystal acoustics, Eq. (2), gen- allows us to change from Eqs. (7) to the first-order dif- eralizes the scalar equation ferential equations for the vector field of displacements u(ζ, ω): ϕ 2ϕ tt Ð0,v xx = (13) ∂ ()ζω, ω ()ΛΦωζω, ± ()u Ði u + ------∂ζ = 0. (8) which is encountered in different fields of physics [1]. ω Obvious complications in Eq. (2) are caused by the vis- Now, we multiply Eq. (8) by e–i t/(2π), integrate with coelasticity of the medium and by the fact that different respect to ω from –∞ to +∞, and use transformations (4) components of the displacement vector u are mutually to change from the Fourier transform u(ζ, ω) to u(ζ, t). related even in the boundary conditions, the tensor As a result, we obtain the first-order tensor integro-dif- nature of Eq. (2) being essentially caused by polariza- ferential equation in u(ζ, t): tion of the elastic waves. ∂u ∂u ± Vˆ = 0, (9) It is obvious that Eq. (2) cannot be reduced to Eq. (13) ∂t ∂ζ in the case of isotropic nonviscous media as well; i.e., it has a tensor (nonscalar) structure even for isotropic or media. We generalize the velocities of elastic waves to ∞ tensor quantities for reasons related to the contents of ∂u()ζ, t ∂u()ζ, t′ ------± dt′Vt()Ð t′ ------= 0, the known monographs by Whitham [1] and Lighthill ∂t ∫ ∂ζ [20], in which the problem on the velocities (especially, Ð∞ the group velocities) of linear and nonlinear waves of ∞ (10) ∂u()ζ, t ∂u()ζ, ttÐ ′ different nature occupies one of the most important ------± dt′Vt()′ ------= 0, places. The group velocity is used for constructing ∂t ∫ ∂ζ Ð∞ mathematical models of nonlinear equations for different complicated cases of anisotropic nonlinear disper- where the kernel V(t) of the integral operator Vˆ is given sive media. Whitham (see Ch. 5 in monograph [1]) clas- by the formula sified wave systems on the basis of linear, quasi-linear, ∞ and nonlinear first-order equations, as well as systems 1 Ðiωt of such equations. The factorization of Eq. (13) leads to Vt()= ------dωe ΛΦω+.() (11) 2π ∫ the equations Ð∞ ϕ ± ϕ t v x = 0, (14) The operator Vˆ has the dimension of velocity, and we will call it the operator of phase velocities of elastic whose solutions are waves in viscoelastic media. The tensor nature of this operator follows from the anisotropy of the medium, ϕ ==fx()Ð vt , ϕ gx()+ vt , (15) and its integral nature is caused by the dispersion of + Ð waves, which is a consequence of the viscosity of the where f and g are arbitrary functions. It is Eq. (14) medium. Only when the viscosity vanishes, Eqs. (10) rather than Eq. (13) that forms the basis for construct- become local: ing nonlinear model equations, the simplest of which ∂ ()ζ, ∂ ()ζ, has the form u t ± Λ u t ------∂ ------∂ζ - = 0. (12) t ϕ ()ϕϕ t +0,v x = (16) From Eq. (6), it follows that the kernel Φ(t) is related to the Fourier transform Φ(ω) by the formula where the propagation velocity v(ϕ) is a function of the ∞ local disturbance ϕ. Equation (16) is called the quasi- Φ() 1 ω ÐiωtΦω() linear equation, because it is nonlinear in ϕ, but linear t = ------∫ d e . ϕ ϕ 2π in the derivatives t and x. A general-form nonlinear Ð∞ equation in the function ϕ(x, t) allows an arbitrary func- ϕ ϕ ϕ Since Φ(t) = 0 for negative t (due to causality), the ten- tional relationship between , t, and x. The key to sor function Φ(ω) has a regular analytic continuation to solving equations like Eq. (16) is the method of charac- the upper half-plane of the complex plane ω [18]. The teristics in the (x, t) plane. Along every characteristic, a poles of the function Φ(ω) always lie in the lower half- partial differential equation is reduced to an ordinary plane of the ω plane. The kernel Φ(t) appearing in differential equation. In certain cases, this fact offers an Eq. (2) has the meaning of a response function. The analytical solution; in other cases, the partial differen- general properties of the response functions, including tial equation is reduced to a system of ordinary differ- those following from the causality requirements, were ential equations that can be solved using the procedures discussed in detail in the literature [19]. of step-by-step numerical integration [1].

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 FACTORIZATION OF INTEGRO-DIFFERENTIAL EQUATIONS 131

For nonviscous anisotropic elastic media, the evolu- According to relationship (6), the Fourier transform tion of the vector ﬁeld of displacements u in space and of the kernel Φ(t) is time is described by the formulas 2 ib ()ζ, ()ζ Λ Φω() τ------j - u+ t = F Ð t u0+, = ∑ jω . + i/t0 (17) j = 1 u ()ζ, t = G()ζ + t Λ u , Ð 0Ð Φ ω ω The function ( ) has the pole = –i/t0 in the lower which are the solutions to Eqs. (12). Here, u0± are the half-plane of the complex ω plane. ζ displacement vectors at the initial point 0 at the initial Using Eq. (11), we determine the kernel V(t) of the moment t and the functions F and G are arbitrary func- 0 phase velocity integral operator Vˆ . In view of proper- ζ Λ ζ Λ tions of the tensor arguments – t and + t , ties (19) of the projective tensors τ and τ , the square ζ 1 2 which are linear functions of and t. Evolution solu- root of the tensor Λ + Φ(ω) is tions (17) generalize the d’Alambert solutions (15). Solutions (17) are the evolution ones, because they 2 ΛΦω() τ ibj imply that the operator-valued functions F and G are + = ∑ j a j +.------ω - + i/t0 matrix-type operators (second-rank tensors) acting on j = 1 the initial vectors u0±, which are assumed to be given. Each of solutions (17) corresponds to stable (retaining To calculate integral (11), we expand the integrand in their envelope) polarized plane wave packets propagat- the series ing along the ζ axis in either positive or negative direc- ∞ ib ()Ði n()2n Ð 3 !!b n tion. a + ------j - = a 1 Ð,∑ ------j- j ω + i/t j ()ω n 2a In what follows, we apply the above factorization 0 n! + i/t0 j method to the wave equation with an exponential kernel n = 1 Ciklm(t) for a viscoelastic isotropic medium. where n!! =1 × 3 × 5 × … × n and (Ð1)!! ≡ 1. Closing In an isotropic medium, the elasticity tensor has the the integration path by the semicircle of inﬁnite radius form [16] in the lower half-plane of the complex plane ω for t > 0 ()δδ ()δ δ δ δ and in the upper half-plane for t < 0, we obtain ciklm = c11 Ð 2c44 ik lm + c44 il km + im kl , ∞ δ Ðiωt n where is the Kronecker delta, c = c , and c = dωe ()Ði n Ð 1 Ðt/t0 ik 11 1111 44 ------= 2π------t e θ()t , n ≥ 1. c . We assume that the response of the medium is ∫ n 2323 ()ω + i/t ()n Ð 1 ! described by an exponential function. The tensor Ð∞ 0 Ciklm(t) is symmetric with respect to permutations of the In addition, it is known that indices i, k and l, m and pairs of indices ik and lm. In the case under consideration, this tensor has the general ∞ ω form ∫ dωeÐi t = 2πδ()t , Ðt/t () [()1δ δ Ð∞ Ciklm t = C11 Ð 2C44 e ik lm where δ(t) is the Dirac delta function. Then, the kernel Ðt/t2()]θδ δ δ δ () + C44e il km + im kl t , V(t) of the phase velocity operator is given by the for- θ mula where t1 and t2 are the relaxation times and (t) is the Heaviside step function (θ(t) = 1 for t > 0 and θ(t) = 0 2 ≤ Ðt/t b for t 0). To simplify the calculations, we assume that () δ() θ() 0 j ≡ Vt = ∑ a j t + t e ------∫ the relaxation times are equal: t1 = t2 t0. In this case, 2a j the tensors Λ and Φ(t) given by Eqs. (3) take the form j = 1 2 2 ∞ n Ð 1 n Ð 1 Ðt/t0 ()Ð1 ()2n Ð 3 !!b t Λ ==a τ , Φ()t b τ e θ()t . (18) × j τ ∑ j j ∑ j j ∑ ------() -------j, n Ð 1 !n! 2a j j = 1 j = 1 n = 1 Here, a = c /ρ, a = c /ρ, b = C /ρ, b = C /ρ, and τ τ1 11 2 44 1 11 2 44 and the factorized equation (9) takes the form 1 and 2 are the projective tensors (operators) 2 ∞ ×2 ′ τ ⊗ τ ⊗ ∂ ()ζ, ∂u()ζ, t b Ðt /t0 1 ===nn, 2 1 Ð nn Ðn , u t ± Λ j ′ ------∂ ------∂ζ - + ∑ ------∫dt e ⊗ t 2 a where n n is the dyad (the direct product of vectors), j = 1 j 0 × (20) n is the tensor dual with the vector n [16], and 1 is the ∞ τ τ ()n Ð 1()′ n Ð 1 ∂ ()ζ, ′ unit tensor. The tensors 1 and 2 satisfy the conditions × Ð1 2n Ð 3 !!b jt τ u ttÐ ∑ ------() -------j------∂ζ = 0. τ2 τ τ2 τ τ τ τ τ n Ð 1 !n! 2a j 1 ==1, 2 2, 1 2 ==2 1 0. (19) n = 1

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 132 BARKOVSKY, FURS In the case of an isotropic medium with a small viscos- REFERENCES Ӷ ity (bjt0/aj 1, j = 1, 2), we can neglect all terms of the 1. G. B. Whitham, Linear and Nonlinear Waves (Wiley, series expansion in n in Eq. (20) except for the first New York, 1974; Mir, Moscow, 1977). term. As a result, we obtain the integro-differential equation 2. E. Kamke, Gewöhnliche Differentialgleichungen (Acad- emie, Leipzig, 1959; Nauka, Moscow, 1966). ∂u()ζ, t ∂u()ζ, t 3. P. M. Morse and H. Feshbach, Methods of Theoretical ------± Λ------∫ Physics (McGraw Hill, New York, 1953; Inostrannaya ∂t ∂ζ Literatura, Moscow, 1958, 1959), Vols. 1, 2. (21) 4. S. P. Tsarev, Teor. Mat. Fiz. 122 (1), 144 (2000). 2 ∞ b Ðt′/t ∂ ()ζ, ′ j 0τ u ttÐ 5. T. Grava, Teor. Mat. Fiz. 122 (1), 59 (2000). + ∑ ------∫dt'e j------= 0. 2 a ∂ζ 6. L. Bers, F. Jonh, and M. Schechter, Partial Differential j = 1 j 0 Equations (Interscience, New York, 1964; Mir, Moscow, For nonviscous media, Eqs. (20) and (21) are reduced 1966). to Eq. (12) with tensor Λ given by Eq. (18). 7. V. Volterra, Theory of Functions and of Integral and τ Integro-Differential Equations (Dover, New York, 1958; In Eqs. (20) and (21), the projective operators 1 and τ Nauka, Moscow, 1982). 2 take into account the variations of the propagating wave field in the longitudinal and transverse subspaces; 8. Yu. N. Rabotnov, Mechanics of a Deformed Solid (Nauka, the operator τ can be represented as τ e ⊗ e e ⊗ Moscow, 1988); Yu. N. Rabotnov, Elements of Hereditary 2 2 = 1 1 + 2 Mechanics of Solids (Nauka, Moscow, 1977). e , where e and e are arbitrary, mutually perpendicu- 2 1 2 9. F. Meseguer, M. Holgado, D. Caballero, et al., Phys. lar, unit vectors in the plane of the wave front. In the Rev. B 59 (19), 12169 (1999); M. Torres, F. R. Montero case of an anisotropic viscoelastic medium, the phase de Espinosa, D. Garcia-Pablos, and N. Garcia, Phys. velocity operator Vˆ is represented as the expansion in Rev. Lett. 82 (15), 3054 (1999). three projective operators of the eigenwaves with a unit 10. B. B. Kadomtsev and V. I. Karpman, Usp. Fiz. Nauk 103, trace (see, e.g., [12]). 193 (1971) [Sov. Phys. Usp. 14, 40 (1971)]. In conclusion, we note that modern acoustics brings 11. L. M. Barkovsky and F. T. N. Khang, Akust. Zh. 36, 550 into existence the problem of a high-precision descrip- (1990) [Sov. Phys. Acoust. 36, 306 (1990)]; Akust. Zh. tion of tensor acoustic fields in space and time, as well 37, 222 (1991) [Sov. Phys. Acoust. 37, 111 (1991)]. as the determination and quantitative evaluation of the 12. L. M. Barkovsky and F. I. Fedorov, Dokl. Akad. Nauk role of polarization in the focusing, transformation, and SSSR 218, 1313 (1974) [Sov. Phys. Dokl. 19, 688 filtering of sound beams. The velocity tensors given by (1974)]; Dokl. Akad. Nauk BSSR 19, 1070 (1975); Izv. Eqs. (9)Ð(11) allow one to progress toward the tensor Akad. Nauk BSSR, Ser. Fiz.ÐMat. Nauk, No. 2, 34 Fourier acoustics of wave beams in dispersive and vis- (1975); Izv. Akad. Nauk BSSR, Ser. Fiz.ÐMat. Nauk, coelastic anisotropic media. The initial scalarization of No. 2, 43 (1979). the problem on the ultrasonic propagation may lead to 13. L. M. Barkovsky and I. V. Mishenev, Akust. Zh. 40, 24 wrong results because of the possible transformations (1994) [Acoust. Phys. 40, 19 (1994)]. of longitudinal waves into transverse waves and vice 14. L. M. Barkovsky and A. N. Furs, J. Phys. A 33 (16), 3241 versa and because of the existence of the Brewster (2000). angles [15]. In the actual propagation conditions, the 15. K. S. Aleksandrov, in Problems of Modern Crystallogra- beam of nonpolarized shear waves can become polar- phy, Ed. by B. K. Vaœnshteœn and A. A. Chernov (Nauka, ized, thus acquiring new physical properties that were Moscow, 1975). neglected at the beginning. This fact and other circum- 16. F. I. Fedorov, Theory of Elastic Waves in Crystals stances should stimulate researchers to use the method (Nauka, Moscow, 1965; Plenum, New York, 1968). of phase and velocity tensors, which are a direct conse- 17. L. M. Barkovsky, G. N. Borzdov, and A. V. Lavrinenko, quence of the fundamental tensor equations of motion J. Phys. A 20, 1095 (1987). (Christoffel’s equations). 18. H. M. Nussenzveig, Causality and Dispersion Relations The factorization of tensor wave equations opens (Academic, New York, 1972; Mir, Moscow, 1976). the way to an extension of mathematical physics by the 19. D. A. Kirzhnits, Usp. Fiz. Nauk 152 (3), 399 (1987) introduction of nonlinear tensor equations, which take [Sov. Phys. Usp. 30, 575 (1987)]; O. V. Dolgov and into account the fact that a system can have spin E. G. Maksimov, Usp. Fiz. Nauk 135 (3), 441 (1981). degrees of freedom and generalize the known scalar 20. J. Lighthill, Waves in Fluids (Cambridge Univ. Press, equations (such as the KortewegÐde Vries and the Due- Cambridge, 1978; Mir, Moscow, 1981). ffing equations). The derivation of the velocity tensors and their application to some nonlinear dynamic systems will be the subject of a separate paper. Translated by A. Vinogradov

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Acoustical Physics, Vol. 48, No. 2, 2002, pp. 133Ð136. Translated from Akusticheskiœ Zhurnal, Vol. 48, No. 2, 2002, pp. 162Ð165. Original Russian Text Copyright © 2002 by Bel’kovich, Grigor’ev, Katsnel’son, Petnikov.

On the Utilization of Acoustic Diffraction in Monitoring Cetaceans V. M. Bel’kovich*, V. A. Grigor’ev**, B. G. Katsnel’son**, and V. G. Petnikov*** * Shirshov Oceanology Institute, Russian Academy of Sciences, Nakhimovskiœ pr. 36, Moscow, 117851 Russia ** Voronezh State University, Universitetskaya pl. 1, Voronezh, 394693 Russia *** Wave Research Center, General Physics Institute, Russian Academy of Sciences, ul. Vavilova 38, Moscow, 119991 Russia e-mail: [email protected] Received March 19, 2001

Abstract—An active acoustic technique for monitoring the whales is proposed. The technique allows one to monitor the whales’ crossing of a conventional borderline extending for several tens of kilometers in a shallow-water area. The potentialities of the technique are demonstrated in the framework of a numerical experiment by solving the problem of diffraction by model scatterers in an acoustic waveguide. The scatterers are selected in the form of soft spheroids with dimensions characteristic of various kinds of cetaceans. © 2002 MAIK “Nauka/Interperiodica”.

The necessity of remotely monitoring the migration sity ρ = 1 g/cm3, and sound velocity c = 1480 m/s. The of cetaceans arose in connection with the sharp source emits a continuous tone signal at the frequency decrease in the population of these large marine mam- f = 300 Hz. The reception is performed by a vertical mals in recent years (see, e.g., [1]). This problem is chain of three receivers positioned at a distance of 10 km especially important for shallow-water regions at the sea shelf, where the existence of cetaceans is threatened from the source. The receiver depths are 5, 20, and 40 m. by industrial activities. This paper proposes an active The waveguide is bounded from above by a free surface acoustic technique for monitoring cetaceans by way of and from below, by an absorbing liquid bottom. The monitoring their crossing a certain conventional bor- bottom parameters are as follows: the sound velocity ρ 3 derline extending for several tens of kilometers through c1 = 1780 m/s; the density 1 = 1.8 g/cm ; and the wave a shallow-water region. From the point of view of phys- π α 2 f α ical acoustics, such a monitoring implied the detection number k1 = ------(1 + i --- ), where = 0.015. The sound of the sound field perturbations caused by the diffrac- c1 2 tion of sound waves by the body of an animal that field perturbations ∆P produced by a soft spheroid crosses a stationary acoustic track between a stationary crossing a stationary acoustic track at various angles γˆ sound source and a receiver. To demonstrate the feasi- were simulated. The intersection point was located in bility of this technique, we present the results of a numerical experiment in the framework of which we the middle between the sound source and the receiver. solve the problem of the diffraction by spherical bodies [We note that, in this case, the perturbations of the in an oceanic-type waveguide. The dimensions of these sound field are on the average minimal, and they bodies and their acoustic parameters (the sound veloc- increase when the intersection point moves closer to the ity and the density) are close to the dimensions and source or to the receiver (for more details see [5]).] The parameters of various species of cetaceans [2, 3]. We note that the methods of solving the diffraction problems in a waveguide are well known (see, e.g., [4]). S However, the existing results refer to perfectly rigid r bodies of revolution or opaque screens, whereas in this s paper, we consider a soft spheroid with the parameters β β E 1 2 R close to the parameters of the medium in the γ waveguide (the seawater). x r0s ˆ r – r0s y The scheme of the numerical experiment is shown in Fig. 1. It is assumed that a point source of sound with Fig. 1. Scheme of the numerical experiment. Letters E, R, the radiation power W0 = 500 W is located at the bottom and S indicate the positions of the source, the receiver, and of a waveguide with the constant depth H = 40 m, den- the scatterer, respectively.

134 BEL’KOVICH et al. ± ± spheroid dimensions and its acoustic parameters were amplitude of the body, and km and kµ are the wave selected as follows: vectors of the incident with the index m and scattered (a) the large axis L = 25 m and the small axis D = 3 m with the index µ waves corresponding to the waveguide [typical dimensions of blue whales (Balaenoptera mus- modes. culus)]; The scattering amplitude is expressed through the (b) L = 10 m and D = 3 m [gray whales (Eschrich- angles θ and ϕ in a spherical coordinate system con- tius) and small rorquals (B. acuto-rostrata)]; nected with the scatterer, and these angles determine (c) L = 5 m and D = 1.2 m [white and killer whales the directions of the wave vectors of the incident and ± ± ≡ θ± ϕ± θ± (Delphinapterus leucas and Orcinus orca)]. scattered plane waves, F(,km ks ) F(,m m , µ , The sound velocity in a spheroid is c = 1540 m/s, ± s ϕµ ). In the case of the selected geometry of the numer- the spheroid density is ρ = 1.05 g/cm3, and the submer- s ical experiment (see Fig. 1), these angles are sion depth is zs = 20 m. ()γ β A mode description of the acoustic field was used to + Ð q cos ˆ + θ ==θ arccos ------m 1 , calculate the perturbations in a model waveguide. In m m k this case, the sound pressure P0 produced at the recep- r ()γ β tion point with the coordinates ( , z) in the cylindrical + Ð qµ cos ˆ Ð 2 θµ ==θµ arccos ------, coordinate system by a sound source located at the k point (0, z0) can be written in the form (5) ± σ P ()rz, ϕ π +− ------m 0 m = arctan()γ β , qm sin ˆ + 1 exp()iq r (1) ψ ()ψ() m ()γ σ = A∑ m z0 m z ------exp Ð mr/2 . ± − µ q r ϕµ = π + arctan ------, m m ()γ β qµ sin ˆ Ð 2 ψ ξ Here, m(z) and m are the eigenfunctions and eigen- β where 1, 2 are the azimuth angles determining the posi- values of the SturmÐLiouville boundary-value problem tion of a scatterer with respect to the source and the π i--- receiver (Fig. 1). According to the formulas [7] for the ξ γ ≡ | | ρ 4 ( m = qm + i m/2), r r , A = cW0e , and W0 is the selected model scatterer (a soft spheroid), the scattering source power. In the presence of a scattering body, the amplitude can be written in the form complex amplitude of the sound field P at the reception ∞ ∞ point was represented as a sum of the direct P0 and scat- ()± , ± 2i ε χθ, ± χθ, ± F km kµ = ----- ∑ ∑ nSnl()cos m Snl()cos µ tered Ps fields: P = P0 + Ps. The sound field perturba- k tions were calculated by the formula n = 0 ln= () () () () ∆ mÐ1 R 1 ()χϑ, R 1 '()χ , ϑ Ð R 1 '()χϑ, R 1 ()χ , ϑ PPP= Ð 0 . (2) × ------s nl nl s nl nl s Ð1 ()1 ' χ , ϑ ()3 χϑ, ()3 ' χϑ, ()1 χ , ϑ (6) To determine the scattered field, we used the approach ms Rnl ()s Rnl ()Ð Rnl ()Rnl ()s [5, 6] based on the representation of the scattering × []()ϕ± ϕ± matrix of the waveguide modes with the help of the cos n m Ð µ , scattering amplitude of the body in a free space: 1, n = 0 where ε =  is the Neumann symbol; S , n ≠ nl (), i ψ ()ψ()ψ() 2, n 0 Ps rz = A ------∑Sµ, m m z0 m zs µ z 8π ()1 ()3 µ, R , and R are the angular and radial prolate sphe- m (3) nl nl roidal functions of the first and third kinds (the primes exp[]iq()r + qµ rrÐ γ r + γ µ rrÐ × ------m s s exp Ð.------m s s- marking the symbols of the radial functions means the q r qµ rrÐ 2 m s s ϑ χ ks 2 2 derivatives with respect to ); s = ---- L Ð D (ks = Here, (rs, zs) are the coordinates of the scatterer posi- 2 ≡ |r | tion, rs s , and Sµm is the scattering matrix π ρ ρ χ 2 2 ϑ 2 f/cs), ms = s/ , = (k/2) L Ð D ; and = π[ + + ()+ , Ð + Ð ()+ , + Sµm = 4 amaµF km kµ + amaµF km kµ 2 2 (4) L/ L Ð D . The asymptotic formulas for spheroidal Ð + Ð Ð Ð Ð Ð + + a aµF()k , kµ + a aµF()k , kµ ], functions and the technique for calculating them m m m m numerically are given in [8]. ψ ± 1 ψ ()± d m/dz σ 2 2 The results of the numerical experiment for the where am = --- m z ------, m = k Ð qm , σ velocity of a model scatterer vs = 1 m/s are shown in 2 i m zz= s ± ± Fig. 2. The moment t = 0 corresponds to the situation k is the wave number, F(,km kµ ) is the scattering when a scatterer crosses the stationary acoustic track.

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 ON THE UTILIZATION OF ACOUSTIC DIFFRACTION IN MONITORING CETACEANS 135 in the = 60° ˆ γ e m (blue whales), 3 × 10000 25 0 (c) –1000 ptimal matched processing with a subsequent storage to the scatterer dimensions (a) 10000 0 (b) Time, s –1000 m (white and killer whales). 1.2 × 5 10000 0 (a) at the reception point a depth of 20 m in case scatterer motion across stationary acoustic track angl P ∆ –1000 0 0 0

1.0 0.8 0.6 0.4 0.2

0.05 0.05 P , Pa , –0.05 ∆ –0.05 m (gray whales and small rorquals), (c) 3 × Sound ﬁeld perturbations 10 (b) absence of acoustic noise (the upper series of plots) and in their presence (the middle series of plots). The result of quasi-o absence of acoustic noise (the upper series plots) and in their presence middle plots). The three columns of plots correspond series of plots. in the lower at the depths 5, 20, and 40 m are given for three receivers Fig. 2. Fig.

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 136 BEL’KOVICH et al.

As one can see from the figures, the motion of a soft medium (internal and surface waves). However, these spheroid causes small characteristic perturbations of fluctuations, as well as the scattered signals, are directly the acoustic field at the reception point. It is natural that proportional to the amplitude of emitted signals. The the perturbation magnitude depends on the scatterer possibility of detecting the scattered signals against size. However this dependence is complex because of their background does not depend on the radiation interference. We note that, in the general case, these intensity. We should also note that the hydrodynamic perturbations are asymmetric with respect to the variability of the medium and, first of all, the field of moment t = 0 when the stationary track is crossed at an internal waves have a considerable spatial anisotropy arbitrary angle. The symmetry takes place when the and pronounced geographic features, i.e., it depends on track is crossed at a right angle or when a scatterer the vertical profile of the medium density, the bottom crosses the track exactly in the middle, while moving at relief, and the amplitude of the tidal wave. [For exam- an arbitrary angle, and the perturbations are detected at ple, in the absence of horizontal stratification, internal the same depth as the depth of the sound source (z = z0). waves are completely absent (just this case was ana- (See [9] for more details.) lyzed in the numerical experiment).] In this connection, a detailed analysis of the prospects of an acoustic mon- In the actual conditions of a shallow sea, these per- itoring in a specific region of a shallow sea must be con- turbations can be concealed because of their smallness ducted with allowance for the possible fluctuations of by the sound field fluctuations due to the ambient noise. the “direct” signal and the acoustic noise, and both Indeed, the inclusion of this noise with a typical radia- ≅ interfering factors included in the analysis must be typ- tion level of 80 dB (in a frequency band of 1 Hz) in ical of this region. the described numerical experiment makes these perturbations visually indistinguishable. (See the middle series of plots in Fig. 2.) In such a situation, it is neces- ACKNOWLEDGMENTS sary to use special methods of signal processing to detect the fact of crossing the stationary track. An This work was supported by the Russian Foundation example of the application of one such method [10], for Basic Research, project no. 99-02-17671. which is based on matched filtering of the indicated perturbations with subsequent storage for all receivers REFERENCES used in the numerical experiment, is given in Fig. 2. It is necessary to note that we used matched quasi-opti- 1. J. Gedamke, D. Costa, and A. Dunstan, J. Acoust. Soc. mal filters for signals with an unknown initial phase. Am. 109, 3038 (2001). We also used the calculated perturbations shown in 2. V. M. Bel’kovich and A. V. Yablokov, Whales and Dol- Fig. 2 (the upper series of plots) as reference signals. As phins (Nauka, Moscow, 1972). one can see from Fig. 2 (the lower series of plots), the 3. L. Bergmann, Der Ultraschall und Seine Anwendung in utilization of special methods of signal processing pro- Wissenschaft und Technik (Hirzel, Zürich, 1954; Inos- vides an opportunity to detect the moment of crossing trannaya Literatura, Moscow, 1957). the stationary track even by the smallest cetaceans 4. S. M. Gorskiœ, V. A. Zverev, and A. I. Khil’ko, in Forma- (white and killer whales). tion of Acoustic Fields in Oceanic Waveguides, Ed. by V. A. Zverev (Inst. Prikl. Fiz., Akad. Nauk SSSR, Nizhni In conclusion, we note that the results obtained in Novgorod, 1991), pp. 82Ð114. this study demonstrate a possibility for designing the 5. V. A. Grigor’ev, B. G. Kantsel’son, V. M. Kuz’kin, and acoustic monitoring systems for various-size cetaceans V. G. Petnikov, Akust. Zh. 47, 44 (2001) [Acoust. Phys. with the use of relatively low-intensity and, therefore, 47, 35 (2001)]. ecologically safe sound sources. This is essential, 6. A. Sarkissian, J. Acoust. Soc. Am. 102, 825 (1997). because the acoustic characteristics of the bodies of 7. É. P. Babaœlov and V. A. Kanevskiœ, Akust. Zh. 34, 19 these animals are close to the parameters of seawater (1988) [Sov. Phys. Acoust. 34, 11 (1988)]. and, therefore, their acoustic scattering cross-section is 8. I. V. Komarov, L. I. Ponomarev, and S. Yu. Slavyanov, small. On the other hand, the required small radiation Spheroidal and Coulomb Spheroidal Functions (Nauka, intensity provides an opportunity to design autono- Moscow, 1976). mous, long-lived, and relatively cheap sourses. Cer- 9. V. M. Kuz’kin, Akust. Zh. (in press). tainly, the possibility of detecting the signals scattered 10. V. A. Zverev, A. L. Matveev, and V. V. Mityugov, Akust. by cetaceans depends not only on the level of the ambi- Zh. 41, 591 (1995) [Acoust. Phys. 41, 518 (1995)]. ent sea noise but also on the fluctuations of the “direct” signal detected by the receiver. These fluctuations are caused by the hydrodynamic variability of the water Translated by M. Lyamshev

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002

Effect of the Layer Inhomogeneity on the Particle Displacements in a Rayleigh-Type Wave V. S. Borshchan, T. A. Viksne, and O. D. Sivkova Novomoskovsk Institute of the Mendeleev University of Chemistry and Technology, ul. Druzhby 8, Novomoskovsk, Tula region, 301665 Russia e-mail: [email protected] Received June 17, 1999

Abstract—The structure of the displacements in a Rayleigh-type wave that propagates in an isotropic solid covered with a liquid layer with exponentially varying density and sound velocity is investigated. © 2002 MAIK “Nauka/Interperiodica”.

An inhomogeneous layer on the surface of a solid is layer inhomogeneity on the particle displacements in widely encountered in practice. The layers formed on the layer is urgent. The data on the distribution of the the surface of a metal in the processes of rolling and displacement amplitudes in depth are also needed, e.g., cold-work hardening are inhomogeneous [1]. On the for calculating the integral of the Rayleigh wave energy surface of a solid exposed to the vapor of its solvent, a flux over depth and for solving the problems on the layer with inhomogeneous physical properties is excitation of Rayleigh and StoneleyÐScholte waves at formed due to sorption processes and dissolution [2]. the interface or on the Rayleigh wave scattering by sur- Similar processes take place in gas sensors whose sens- face and subsurface defects. ing element is a sorbent film deposited on a solid substrate [3]. Layers with vertical gradients of sound The problem on the distribution of the particle dis- velocity exist in sea and ocean waters. In this case, the placements in a SAW field for the case of a homoge- stratification of the sound velocity essentially affects neous layer overlying an elastic halfspace is considered the structure of the sound field generated by a point in many papers, the main results of which are general- source [4Ð7], as well as the propagation of normal ized in monographs [14, 15]. It is shown that the pres- waves in the underwater sound channel [8Ð11]. In all ence of a solid or liquid layer on the surface of an elastic cases, the inhomogeneity of a layer on the surface of a halfspace leads to changes in the phase velocity and in solid affects the characteristics of surface acoustic the distribution of the amplitudes of particle displace- waves (SAW). This provides the possibility of obtain- ments. For the case of a homogeneous liquid layer, the ing the information on the gradients of the layer param- dispersion equation and the expressions for the dis- eters from acoustic measurements [12]. placement components in a Rayleigh-type wave (RTW) are given in [14]. It is shown that the displacement com- However, in determining the parameters of a strati- ponents vary in the layer by the cosine law, whereas, in fied layer, it is not sufficient to measure the phase and the halfspace, they vary in the same way as in a Ray- group velocities of SAW, because these quantities leigh wave, but the depth of the wave localization in the depend on the layer thickness, the laws governing the halfspace decreases with an increase in the layer thick- variations of the layer parameters (which, as a rule, are ness. The profiles of the amplitudes of the displacement unknown), and the values of these parameters at one of components in a homogeneous solid layer and in a half- the layer boundaries. In the simplest case of a liquid space are presented in [15] for SAW with vertical and layer with a linear distribution of its parameters, it is horizontal polarizations, for layers of various thickness. necessary to determine five quantities: the thickness, The case of a stratified layer can be theoretically inves- the density, and the sound velocity at the layer bound- tigated only for some specific laws of the parameter dis- ary and two coefficients characterizing the stratification tribution, for which the wave equation of the inhomo- of these parameters. In this connection, for an unambig- geneous medium has analytical solutions [16]. The dis- uous determination of the structure of the inhomoge- persion equations of SAW for some specific laws of the neous layer, it is necessary either to measure the SAW layer parameter variations are obtained in papers [12, velocities at several frequencies or to investigate the 17, 18], and the effect of the layer inhomogeneity on the distribution of the displacement amplitudes in the layer distribution of the particle displacements was not con- by an optical probe [13]; then, solving the inverse prob- sidered. The objective of this paper is to study the effect lem, one can determine the layer parameters and their of the stratification of the layer parameters on the dis- gradients. Therefore, the problem of the effect of the tribution of the displacement components in RTW.

138 BORSHCHAN et al. The horizontal and vertical displacement components in the liquid are determined by the expressions h [19] 1 ∂P 1 ∂P x W ==------and W ------, (7) x ω2ρ∂x z ω2ρ∂z respectively. Substituting Eq. (2) into Eqs. (7) and taking into account the relations between the arbitrary constants from [18], we perform some transformations to obtain z 2 kqkt D1 W x = A------Fig. 1. Halfspace with a layer. ()k2 + s2 D (8) ×αexp()Ð z/2 expi()ωtkxÐ Ð π/2 , A stratiﬁed layer is supposed to be liquid with the 2 qkt D2 ()α ()ω density and sound velocity varying with depth accord- Wz = A------exp Ð z/2 expi tkxÐ , (9) ing to the laws ()k2 + s2 D ρρ ()α ()β where ==0 exp z , c c0 exp z , (1) Nν()Y Jν()Y Ð Jν()Y Nν()Y and the halfspace is supposed to be isotropic (Fig. 1). 1 1 D1 = ------()() ()()-, (10) Let an RTW propagate in the positive direction along Jν Y0 Nν Y1 Ð Jν Y1 Nν Y0 the x axis, the liquid layer occupying the domain –h < α D = --- + βν D z < 0, and the halfspace occupying the domain z > 0. 2 2 1 Then, in the case of a harmonic dependence on x and t (11) for α ≠ 0 and β ≠ 0, the expression for the sound pres- () () () () β Jν Y1 Nν Ð 1 Y Ð Nν Y1 Jν Ð 1 Y sure in the liquid has the form [18] + Y------()() ()()-. Jν Y0 Nν Y1 Ð Jν Y1 Nν Y0 []() () PC= 1 Jν Y + C2 Nν Y These expressions cannot be analyzed in analytical (2) ×αexp()Ð z/2 expi()ωtkxÐ , form. Therefore, their analysis was performed on the basis of the numerical calculations with the use of the where Jν(Y) and Nν(Y) are the Bessel and Neumann following relations between the parameters of the solid functions, C and C are arbitrary constants, and 2 1 2 ρ k k α and the layer: ----- = 2.5; ----t = 1/3; ----l = 1/27; and ------= k ω ρ 2 0 β 0 k0 k k0 Y = -----β- exp(– z), k0 = ---- . (3) 0 c0 β ------= 1/6. The chosen value of α corresponds to a The corresponding dispersion equation [18] relating the k0 wave number k of the RTW to the layer thickness h can change in the layer parameters by about a factor of be written as λ λ three at the distance 0 ( 0 is the wavelength in the liq- 4 uid layer at the halfspace boundary). The dependences 2 ρ qk 2 ()2 2 0 t of the amplitudes of the vertical W and horizontal W 4k qsÐ k + s = ------ρ -, (4) z0 x0 D displacement components in the layer and the halfspace where on the z coordinate for various values of the layer thickness were calculated. For the layer, we used the ampli- α D = --- + νβ tudes given by Eqs. (8) and (9) and for the halfspace, 2 the relations derived in [14] (5) βk Nν()Y Jν ()Y Ð Jν()Y Nν ()Y 2qs ------0 ------1 Ð 1 0 1 Ð 1 0 U = Akexp()Ð qz Ð,------exp()Ðsz (12) Ð β ()() ()(), x0 2 2 Jν Y0 Nν Y1 Ð Jν Y1 Nν Y0 k + s 2 2 2 2k k0 4k + α () () ()β ν Uz0 = ÐAqqzexp Ð Ð,------exp Ðsz (13) Y0 ==------, Y1 Y0 exp h , =------, (6) 2 2 β 4β2 k + s where the wave number k is determined from the dis- 2 2 2 2 ρ ρ persion equation (4). q = k Ð kl , s = k Ð kt , and 0 are the densities of the halfspace material and the liquid at the liquidÐhalf- For the comparison between the amplitudes of displacements with and without the layer, we introduced the space boundary, and kl and kt are the wave numbers of longitudinal and shear waves in the isotropic medium. relative amplitudes: W x0 = Wx 0/Uz0R, Ux0 = Ux0/Uz0R,

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 EFFECT OF THE LAYER INHOMOGENEITY ON THE PARTICLE DISPLACEMENTS 139

(a) (b)

Wz0 Uz0 Wz0 Uz0 2 2 λ λ h/ 0 = 0.2 1 h/ 0 = 0.3 1 4 2 3 3

λ –0.3 0 0.4 z/ 0

–4 λ 5 –0.2 0 0.4 z/ 0 4

Wz0 Uz0 Wz0 Uz0 2 λ 2 8 λ h/ 0 = 0.4 h/ 0 = 0.5 1 4 1 4 3 3 –0.5 5 5 λ λ 6 0 0.4 z/ 0 –0.4 4 0 0.4 z/ 0 4 –2 6 –2

λ Fig. 2. Dependences of the relative amplitudes in the layer, Wz0 , and in the halfspace, Uz0 , on the relative coordinate z/ 0 for various values of the layer thickness and for the cases (1, 4) α = 0, (2, 5) α > 0, and (3, 6) α < 0.

Wz0 = Wz0/Uz0R, and Uz0 = Uz0/Uz0R, where Uz0R are tudes for greater layer thickness are not presented, displacement amplitudes (12) and (13) in the Rayleigh because, in this case, according to Eqs. (1), the param- wave in the plane z = 0 at h = 0 and k = kR (kR is the solu- eters of the liquid at the free layer boundary, reach val- tion to Eq. (4) with the right-hand side equal to zero). ues that are noncharacteristic for the liquid. The vertical coordinate and the layer thickness were λ λ It is seen that, in the first normal wave (curves 1Ð3), also expressed in relative units: z/ 0 and h/ 0. For com- α parison, we plotted the displacement profiles for the Wz0 increases for > 0 (curve 2), as compared to the α case of a homogeneous layer (α = β = 0) with the den- homogeneous layer (curve 1), and decreases for < 0 sity ρ and the sound velocity c . The corresponding (curve 3). This behavior of the amplitudes is caused by 0 0 a change in the wave resistance (ρc) of the layer: for expressions for the displacement amplitudes take the α form [14] > 0, the wave resistance decreases in the direction of the free boundary of the layer, which leads to an 2 []() increase in the amplitude, as compared to a homoge- qkt sin gz+ h W x0 = Ak------, neous layer. In the case of α < 0, an increase in ρc and gk()2 + s2 cos()gh (14) a decrease in the amplitude takes place. Within the []() layer, the difference in the run of curves 1–3 increases qkt cos gz+ h Wz0 = Akt ------, as the free layer boundary is approached and reaches ()k2 + s2 cos()gh the maximal value Wz0 (–h) at this boundary. The com- 2 2 parison of Figs. 2aÐ2d shows that, with the increase in where g = k0 Ð k . the layer thickness, the value of Wz0 (–h) increases, as Figure 2 shows the dependences of the relative compared to a homogeneous layer for α > 0 and amplitudes of the vertical displacement components in decreases for α < 0, and further remains constant (Fig. 3). α the layer Wz0 and in the halfspace Uz0 on the relative From Figs. 2aÐ2d, it is also seen that, for > 0, the λ wave energy concentrates near the free boundary of the coordinate z/ 0 for different values of the relative layer λ λ thickness h/ 0. The profiles of the displacement ampli- layer: at h/ 0 = 0.5 (Fig. 2d, curve 2), Wz0 is close to

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 140 BORSHCHAN et al.

– for α = 0 (curve 4), α > 0 (curve 5), and α < 0 (curve 6). Wz0(–h) Near the point of origination of this wave (Fig. 2b) in 10 the case α > 0, the displacement amplitudes in the layer 8 2 are smaller than in the ﬁrst normal wave, whereas, in half-space, they are greater. This character of displace- 6 ments in the second normal wave results from the fact 1 that the phase velocity of this wave lies between the 4 3 velocities of the shear and Rayleigh waves in the elastic 2 halfspace; i.e., the wave is close to a bulk wave and has a greater localization depth in the halfspace. Since, in 0.1 0.2 0.3 0.4 0.5 this case, the effect of the layer on the propagation of λ h/ 0 SAW is weak, the inhomogeneity affects the value of Wz0 to a lesser extent (the difference in the run of

Fig. 3. Dependence of the relative amplitude Wz0 (–h) at curves 4–6 is smaller than that of curves 1–3). λ the free layer boundary on the relative thickness h/ 0 for the cases (1) α = 0, (2) α > 0, and (3) α < 0. As is seen from Fig. 4, the horizontal displacement components Wx0 and Ux0 vary, depending on the layer thickness and the sign of the parameter gradients, in the λ zero in the region 0.15 < z/ 0 < 0 and then rises sharply. same way as the corresponding vertical components. This testiﬁes to the fact that the inhomogeneity leads to The main difference is the zero value of Wx0 at the free a redistribution of the wave energy within the layer. layer boundary and its discontinuity at the layerÐhalfs- λ ≥ pace boundary, which is determined by the boundary For h/ 0 0.3, a second normal wave exists. Its dis- conditions and occurs for both a homogeneous and an placement amplitudes Wz0 are also shown in Figs. 2bÐ2d inhomogeneous layer.

(a) (b) – – – – Wx0 Ux0 Wx0 Ux0

2 1.2 h/λ = 0.2 h/λ = 0.3 0 1 0 4 3 0.8 1 2 0.4 3 λ –0.3 5 0 0.4 z/ 0 4 λ –0.2 0 0.4 z/ 0 –4

5 0 5 λ –0.4 0.4 z/ 0 –0.5 0.4 z/λ 4 6 –2 4 6 –2 0

λ Fig. 4. Dependences of the relative amplitudes in the layer, W x0 , and in the halfspace, Ux0 , on the relative coordinate z/ 0 for various values of the layer thickness for the cases (1, 4) α = 0, (2, 5) α > 0, and (3, 6) α < 0.

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 EFFECT OF THE LAYER INHOMOGENEITY ON THE PARTICLE DISPLACEMENTS 141

As indicated by Figs. 2 and 4, the amplitudes Ux0 2. E. V. Khamskiœ, Crystalline Materials and Products (Khimiya, Moscow, 1986). and Uz0 of the displacement components in the halfspace vary insignificantly due to the stratification of the 3. G. Fischerauer, F. L. Dickert, and R. Sikorsky, in Pro- ceedings of IEEE International Frequency Control Sym- layer. This results from the fact that the parameters of posium (Pasadena, USA, 1998), p. 608. the liquid at the halfspace boundary have the same values in all cases considered above. The corresponding 4. A. Maguer, Warren L. J. Fox, H. Schmidt, et al., J. Acoust. changes grow with the increase in the layer thickness Soc. Am. 107, 1215 (2000). λ 5. C. W. Holland and J. Osler, J. Acoust. Soc. Am. 107, and reach 20% for h/ 0 = 0.5 in the plane z = 0. Conse- quently, the inhomogeneity of the layer leads to a redis- 1263 (2000). tribution of energy between the layer and the substrate. 6. O. P. Galkin and S. D. Pankova, Akust. Zh. 44, 57 (1998) α [Acoust. Phys. 44, 44 (1998)]. For > 0, the displacement components Ux and Uz decrease, as compared to a homogeneous layer (the 7. E. A. Borodina and Yu. V. Petukhov, Akust. Zh. 46, 437 energy is transferred from the substrate to the layer), (2000) [Acoust. Phys. 46, 373 (2000)]. and for α < 0, these components increase (the energy is 8. T. C. Yang, J. Acoust. Soc. Am. 74, 232 (1983). transferred from the layer to the substrate). The layer 9. Z. Ye, J. Appl. Phys. 78, 6389 (1995). inhomogeneity practically does not affect the depth of 10. R. A. Vadov, Akust. Zh. 44, 749 (1998) [Acoust. Phys. the wave penetration into the substrate. 44, 651 (1998)]. Thus, the inhomogeneity of a layer on the surface of 11. N. V. Studenichnik, Akust. Zh. 44, 659 (1998) [Acoust. a solid essentially affects the amplitudes of the compo- Phys. 44, 571 (1998)]. nents of particle displacements in an RTW. A redistri- 12. P. Kielczynski and W. Pajewsky, Appl. Phys. A 48 (5), bution of the wave energy takes place between the layer 423 (1989). and the substrate, as well as within the layer, and this 13. I. B. Yakovkin and D. V. Petrov, Diffraction of Light by redistribution depends on the layer thickness and the Acoustic Surface Waves (Nauka, Novosibirsk, 1979). magnitude and sign of the gradients of the layer param- 14. I. A. Viktorov, Acoustic Surface Waves in Solids (Nauka, eters. By changing the layer thickness and the degree of Moscow, 1981). its inhomogeneity, it is possible to control the velocity 15. Acoustic Surface Waves, Ed. by A. A. Oliner (Springer, and the structure of surface waves. This result can find Berlin, 1978). application in acoustoelectronic signal processors. 16. L. M. Brekhovskikh and O. A. Godin, Acoustics of Lay- ered Media (Nauka, Moscow, 1989; Springer, New York, ACKNOWLEDGMENTS 1990). 17. V. Yu. Zavadskiœ, Calculation of Wave Fields in Open We are grateful to O.Yu. Serdobol’skaya for discuss- Spaces and in Waveguides (Nauka, Moscow, 1972). ing the purpose and the results of this study. 18. V. S. Borshchan and O. D. Sivkova, Akust. Zh. 40, 139 (1994) [Acoust. Phys. 40, 124 (1994)]. REFERENCES 19. M. A. Isakovich, General Acoustics (Nauka, Moscow, 1973). 1. B. P. Rykovskiœ, V. A. Smirnov, and G. M. Shchetinin, Local Hardening of Workpieces by Surface Cold-Work (Mashinostroenie, Moscow, 1985). Translated by A. Svechnikov

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002

Long-Range Sound Propagation in the Shallow-Water Part of the Sea of Okhotsk R. A. Vadov Andreev Acoustics Institute, Russian Academy of Sciences, ul. Shvernika 4, Moscow, 117036 Russia e-mail: [email protected] Received February 14, 2001

Abstract—Experimental data on the long-range propagation of explosion-generated sound signals in the shallow-water northern part of the Sea of Okhotsk are analyzed. The propagation conditions in this region are characterized by a fully-developed underwater sound channel that captures the rays crossing the channel axis at angles lower than 3°. The experimental data reveal a small increase in the duration of the sound signal in pro- portion to the range with the proportionality factor lower than 0.00025 s/km. The frequency dependence of attenuation exhibits a pronounced minimum whose position on the frequency axis is close to the critical frequency of the first “water” mode (about 160 Hz). The increase in the attenuation coefficient at lower frequencies is confirmed by the field calculations performed with the wave-field computer code and is explained by the sound energy loss in the bottom sediments. At frequencies higher than 200 Hz, as in the Baltic Sea, the most probable reason for the attenuation to exceed the absorption in sea water is sound scattering by internal waves. © 2002 MAIK “Nauka/Interperiodica”.

For years, the researchers of the Acoustics Institute reaches Ð1.7 to Ð1.5°C. Near the bottom (at a sea have repeatedly studied long-range propagation of depth of about 150 m), the temperature also remains explosion-generated sound signals in the underwater negative (Ð1.4 to Ð1.6°C). sound channel (USC) of the Sea of Okhotsk. The exper- The experimental studies of long-range sound prop- iments were carried out in both deep-water [1] and shal- agation were carried out in the shallow-water region of low-water (125Ð155 m) parts of the sea. the Sea of Okhotsk in summertime (August). The prop- For the Sea of Okhotsk, a significant spatial variabil- agation path was 335 km in length, with the sea depth ity of the oceanographic parameters is characteristic [2, changing from 125 to 155 m along the path. Prior to 3]. The northern and north western parts of the sea are experimenting, six sets of hydrological measurements represented by shallow-water areas. The central part were performed at different distances from the receiv- (about 70% of the total sea area) has typical depths of ing vessel. By continuously lowering and lifting the 800Ð1200 m. On the south, the Kuril Hollow is situated Istok-3 device, the following water parameters were with sea depths of 3300Ð3400 m. measured every 1.5 s: the temperature, the electric conductivity (salinity), and the hydrostatic pressure. For The bottom sediments of the Sea of Okhotsk are of a each set of data, the values of these parameters were terrigenic nature. For the region of the experiment in the recalculated to the sound speeds according to the for- northern shallow-water part of the sea, aleurite diatoma- mula of Wilson [5]. ceous clays and diatomaceous silts are typical [4]. Figure 1 shows the vertical sound speed profiles The waters of the Sea of Okhotsk undergo a measured at both ends and at the middle of the path. At cyclonic circulation. Through the northern straits of the some fractions of the path, multilayer water structures Kurile Island system, warm surface waters of the were observed with different temperatures in individual Pacific Ocean arrive, changing their properties as they layers. This phenomenon leads to “tongs” in the c(z) pass into the interior areas of the sea. Through the profiles with the tong thickness less than 5–8 m and the southern straits, cold waters move from the sea to the differences in the sound speeds up to 0.4Ð0.6 m/s. The ocean. The Tsushima current introduces warm salt depth of the USC axis changed from 30 to 60 m along waters through the La Perouse Strait. the path. The sound speed varied slightly at the axis Several stratification types of water masses can be depth. The difference in the sound speeds near the bot- distinguished in the northern part of the Sea of tom and at the channel axis was about 1.5 m/s, and the Okhotsk. The surface waters (the 30Ð60-m layer) are corresponding difference between the surface and the cooled in winter down to the temperature of freezing. axis reached 55 m/s. In the discontinuity layer, the In summer, the near-surface layer is heated up to 6–10°ë, sound speed gradient changed from 0.5 to 2 1/s. Below and, under the mixed layer, an undersurface water the USC axis (at depths greater than 50 m), the gradient mass forms. In its core, the minimal temperature was 0.02 1/s, which is close to the hydrostatic value. To

LONG-RANGE SOUND PROPAGATION 143 make the profile c(z) more evident, we choose different c, m/s scales of the abscissa in Fig. 1 for undersurface waters 1441 1442 1443 1450 1500 and those at horizons deeper than 25Ð30 m. 0 To illustrate the variability of the hydrological parameters along the path at the depths 30Ð125 m (within the 100-m layer under the temperature discontinuity layer), we present the field of sound speeds (Fig. 2) calculated from the data of measurements. The step in the sound speed values is 0.2 m/s between the adjacent isospeed curves. The numbers near the curves indicate the excess of the sound speed above 1440 m/s. In this picture (at 50 the bottom), the data obtained from echo-sounding performed during the experimentation are presented (the scales in distance and depth are the same as at the top of the picture). Up to a distance of 150Ð200 km along 1 the path, a monotone bottom rise took place, and the sea 2 depth decreased from 150 to 125 m. At longer ranges, 3 the depth rather sharply increased to 155 m (in a region of about 50 km in length) and then decreased to 138 m. 100 Two vessels were used in the experiment. The receiving vessel drifted approximately 95 nautic miles south of Magadan. The transmitting vessel took a head- ing of 270° (in the western direction relative to the reception point) at 10Ð11 knots, along the 58° latitude. From the transmitting vessel, small explosive charges were dropped and exploded at a depth of 50 m (near the z, m USC axis) with the use of pressure-sensitive detonators. Fig. 1. Vertical profiles of the sound speed c(z). Data of A total of 50 charges were exploded. The time interval measurements supporting the acoustic experiment: (1) near between successive explosions varied depending on the the receiving vessel, (2) at a distance of 155 km, and (3) at distance. At points nearest to the receiver, it changed a distance of 320 km. from 2 to 10 min; at distances longer than 50 km, it was equal to 30 min. At the moment of each charge drop, the distance between the vessels was determined from the of the explosion-generated signal can be used: the max- propagation time of the acoustic signal and then was imal value of this function is as high as 0.95Ð0.98. refined according to observations periodically per- At shorter distances, time delays between the sig- formed with the use of the satellite navigation systems nals separately arriving at the receiver are not longer of both vessels. The explosion-generated signals were than the period of the first oscillation (45 ms). At the received by omnidirectional hydrophones at depths 10, distances 190Ð200 km from the source, they have the 50, and 120 m. same order of magnitude as this period. Thus, for the During experimenting, there was fog, swell of temporal elongation of the signal in the region at hand, Beoufort 3, and wind of less than 6 m/s (at about 40°). the proportionality factor does not exceed 0.00025 s/km, which is in a good agreement with the calculation per- The explosion-generated signals that were received formed with the simplified formulas [6] obtained for a within the frequency band 10Ð20 Hz to 1Ð2 kHz from bilinear c(z) profile. the distance 10Ð20 km or more were single-ray arrivals and had the shape of two short pulses of less than 1-ms On the basis of the experimental data analyzed duration. These pulses formed by the shock wave and within a frequency band of 40Ð1000 Hz, the sound the first gas-bubble oscillation were equal in their val- attenuation coefficients were estimated. These coeffi- ues and had the same signs. The time separation of the cients were determined by comparing the experimental pulses corresponds to the period of the first bubble decay of the sound field with the cylindrical law of geo- oscillation (T = 45 ms in our case). metrical spread. As a characteristic that is equivalent to the energy of the explosion-generated signal within the In the case of a multiray reception, each ray in the frequency band ∆f, the following value was used: time structure of the explosion-generated signal is represented by its own pair of pulses. The time structure of T the sound field has a twofold nature: each pulse of the 2 () E f = ∫ p f t dt, shock wave is followed by that of the gas-bubble oscillation. The first and second halves of the received mul- 0 tiray signal are identical. To confirm this fact, the cross- where T is the duration of the explosion-generated sig- correlation function between the first and second parts nal and pf(t) is the signal sound pressure normalized to

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144 VADOV

Sound speed ﬁeld 0.03 0.6 1.2 0.8 1.6 1.4 1.0 0.04 1.0 1.2 0.6 0.05 1.2 0.06 0.8

0.07 1.0

0.08 1.2 0.09 , km Z 0.10 1.4

0.11 2.2 1.8 1.6 2.0 2.2 0.12

0.13

0.14

0.15

0 50 100 150 200 250 300 R, km

Fig. 2. Sound speed field along the path at depths of 30–125 m. Adjacent isospeed curves differ by 0.2 m/s, the numbers near the curves indicate the excess of the sound speed over 1440 m/s. The lower part of the figure presents the bottom relief measured by echo-sounding. the frequency band ∆f. To obtain the power spectrum of At a frequency of about 200 Hz, a pronounced min- the explosion-generated signal, a computer code was imum can be noticed in the frequency dependence of used that was earlier developed to analyze the experi- attenuation. Earlier, in analyzing the data of similar mental decays of the sound field and to estimate the studies [7] in the Baltic Sea, such a minimum was attenuation coefficients at individual frequencies. attributed to the critical frequency of the “water” modes. This conclusion was attained, because the Bal- At the reception depth of 20 m, the signals suffi- tic-Sea studies were carried out in spring and summer ciently exceeded the noise level at distances shorter when the conditions in the propagation channel consid- than 17 km. For these signals, the loss die to bottom erably varied: the critical frequency of the first mode reflections is significant. At the two other reception varied by a factor of two to three and the position of the horizons, all recorded signals are above the noise level minimum in the frequency dependence of attenuation within the frequency band 40Ð1000 Hz. also changed. As a result of processing the signals received at For the experiment in the Sea of Okhotsk, the calcu- depths of 50 and 120 m, the values of the attenuation lated critical frequency of the first water mode is about coefficients were obtained (Fig. 3). There is nearly no 160 Hz, which does not significantly differ from the fre- difference in these values for depths of 50 and 120 m. quency of the attenuation minimum. To verify the con- The values obtained for the attenuation coefficients are cept of the relation between the attenuation minimum much higher than the sound absorption coefficients in and the critical frequency of the water mode, we carried the sea water, which are presented in [1] for the Sea of out a set of calculations with the use of the computer Okhotsk. code by Avilov [8], which accounts for changes in the

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002

LONG-RANGE SOUND PROPAGATION 145 propagation conditions along the path, namely: the β, dB/km changes in the sound speed profiles measured at differ- 0.4 ent parts of the path and the data of echo-sounding. The calculations were performed for the frequency band 0.2 40Ð800 Hz, with the noise-like signals in 1/3-octave bands. A relatively simple model of the bottom was used: a liquid layer (with a sound speed of 1600 m/s, a 0.1 loss factor of 0.015, and a density of 1.8 g/cm3) overly- 0.08 ing a liquid halfspace (2600 m/s, 0.001, and 4.0 g/cm3, 0.06 respectively). A good quantitative agreement was 0.04 obtained between the calculated and observed low-frequency attenuation coefficients. 0.02 Figure 4 shows the results of calculations for the fre- 20 40 60 80100 200 400 600 1000 quencies 50, 100, 200, and 400 Hz. At frequencies f, Hz higher than 300Ð500 Hz, the calculated decays of the sound field (without the absorption in the water) follow Fig. 3. Sound attenuation in the shallow-water (125Ð155 m) the cylindrical law of the geometrical spread for the dis- northern part of the Sea of Okhotsk: (᭺ ᭺ ᭺ ᭺) the experimental attenuation coefficients determined from the decay tances from the source, which exceed 50 km: the devi- •••• ations of the average decay curves from the cylindrical of the sound field at a depth of 50 m, ( ) similar values for a depth of 120 m, (᭝ ᭝ ᭝ ᭝) the calculation with Avilov’s law are no higher than 0.5 dB for the entire path. At fre- computer code, and (_ _ _ _ _) the frequency dependence of quencies lower than 300Ð500 Hz, the difference attenuation obtained by the formula β = 0.205f 1.13. between the calculated and cylindrical decays increases as the frequency decreases: the additional loss caused by sound absorption in the bottom sediments becomes south western part of this sea (the measurements were higher for lower frequencies and reaches 0.014 dB/km performed in a narrow shelf patch near the south-west- at 50 Hz. The latter value agrees well with the experi- ern coast of Sakhalin) and, hence, cannot be extended mental data. It is worth mentioning that the objective of to the large shelf zone of the northern part of the Sea of the calculations was not to develop a bottom model for Okhotsk. At the same time, the Baltic Sea and the Sea the specific region. Instead, the calculations were per- of Okhotsk have a clearly defined layer of temperature formed to estimate the contribution of the bottom sedi- discontinuity (at a depth of ~25 m) and comparable sea ments to the sound attenuation at frequencies lower depths (80Ð100 m in the Baltic Sea and 125Ð155 m in than the critical ones and to verify the existence of the the Sea of Okhotsk). These similarities allows one to proposed mechanism responsible for the formation of assume that, in the experimental region of the Sea of the minimum in the frequency dependence of attenua- Okhotsk, short-period internal waves also exist and tion. their properties are similar to those of the Baltic Sea. At frequencies higher than 200 Hz, the frequency dependence of the sound attenuation in the shallow- Using the known limiting angles, at which the water water part of the Sea of Okhots can be expressed in the rays cross the USC axis, and the known full cycle form: lengths of these rays, and assuming a similarity of nature and amplitude of internal waves in the two seas, β 1.13 = 0.205f dB/km, one can estimate the difference in attenuation, which is where the frequency is specified in kilohertz. At such caused by the sound scattering from the internal waves. frequencies, the attenuation coefficients in the Sea of For the Baltic Sea, the angles at which the water rays Okhotsk are, by a factor of 3Ð5, higher than those cross the USC axis are Ð(6°Ð8°) < θ < +(6°Ð8°), and the obtained in the Baltic Sea. The shapes of the decay lengh of the full cycle is D = 1.5Ð2.5 km for these rays. curves are also somewhat different for these seas. In For the Sea of Okhotsk, the corresponding values are analyzing the experimental data of the Baltic Sea [7], a –2.6° < θ < +2.6° and D = 4Ð6 km. The loss caused by conclusion was made that the sound scattering by inter- scattering is proportional to the cycle length of the nal waves was the most probable factor responsible for water rays (the internal wave is associated with the tem- the observed high attenuation in that region. The analy- perature discontinuity layer, and the scattering takes sis included the data of the internal wave observation in place within the upper half-cycle) and to the squared the studies of sound attenuation in the Baltic Sea, as limiting angle of their crossing the USC axis (see, e.g., well as the results of modeling the off-channel sound [10, 11]). From these considerations, the ratio of the scattering by internal waves. attenuation coefficients can be estimated for the Baltic During the experiments in the Sea of Okhotsk, no Sea and the Sea of Okhotsk. The calculated ratio proves observations of internal waves were carried out. As far to be approximately equal to three, which well agrees as the author knows, only publication [9] exists on mea- with the experimental estimate (3Ð5). Of course, this suring internal waves in a coastal zone of the Sea of agreement means nothing more than indirect evidence Okhotsk. Specifically, the published data referred to the in favor of the hypothesis that the sound scattering by

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146 VADOV

pR× and the first gas-bubble oscillation are well resolved, each of them individually reproducing the time structure of the propagating sound field. β = 0.0035 dB/km a (ii) For the conditions of the experiment, a small increase in the duration of the explosion-generated signal is observed; the proportionality factor between the duration and the distance is about 0.00025 s/km (upon β = 0.0215 dB/km subtracting the period of the bubble oscillation). b (iii) The experimental data on the sound attenuation within the frequency band 40Ð1000 Hz are obtained. 1 The attenuation is much higher than the sound absorption in the sea medium. The frequency dependence of 2 c attenuation has a minimum at a frequency of about 200 Hz, which is close to the critical frequency of the β = 0.055 dB/km first water mode (about 160 Hz). (iv) The mechanism responsible for the minimum in the frequency dependence of sound attenuation in the USC of a shallow sea is revealed. As the frequency decreases starting from the critical one (for the water modes), the sound attenuation increases because of the acoustic energy loss in the bottom sediments. At fre-

30 dB β = 0.15 quencies higher than the critical one, the most probable dB/km reason for the attenuation to exceed the absorption is d the sound scattering by short-period internal waves.

REFERENCES 0 50 100 150 200 250 300 R, km 1. R. A. Vadov, Akust. Zh. 45, 174 (1999) [Acoust. Phys. Fig. 4. Results of the sound ﬁeld calculations by Avilov’s 45, 143 (1999)]. computer code at the frequencies (a) 400, (b) 200, (c) 100, 2. V. F. Sukhoveœ, Seas of the Ocean (Gidrometeoizdat, and (d) 50 Hz in 1/3-octave bands: (1) the range dependence Leningrad, 1986). of the sound level corrected for the cylindrical law of geometrical spread and (2) the linearly approximated decay of 3. T. A. Bernikova, Hydrology and Industrial Oceanology the sound level corrected for the cylindrical law (the linear (Pishchevaya Promyshlennost’, Moscow, 1980). approximation by the least-squares method for the part of 4. P. L. Bezrukov, Tr. Inst. Okeanolog. im. P. P. Shirshova, the path 50Ð335 km). At each frequency, near the corresponding curve, the attenuation coefﬁcient is indicated, as Akad. Nauk SSSR 32, 15 (1960). determined from the inclination of the line approximating 5. W. D. Wilson, J. Acoust. Soc. Am. 34, 866 (1962). the sound level decay corrected for the cylindrical law. 6. L. M. Brekhovskikh and Yu. P. Lysanov, Fundamentals of Ocean Acoustics (Gidrometeoizdat, Leningrad, 1982; internal waves is the most probable attenuation mecha- Springer, New York, 1991). nism (at frequencies higher than the critical one) for 7. R. A. Vadov, Akust. Zh. 47, 189 (2001) [Acoust. Phys. shallow-water sound channels. The real validation of 47, 150 (2001)]. this hypothesis requires the development of a rigorous 8. K. V. Avilov, Akust. Zh. 41, 5 (1995) [Acoust. Phys. 41, theory of sound scattering by internal waves and a spe- 1 (1995)]. cial-purpose acoustic experiment including the mea- 9. A. P. Nagovitsyn and E. N. Pelinovskiœ, Meteorol. surements of the internal waves. Gidrol. 4, 124 (1988). In conclusion, we formulate the main results of the 10. R. H. Mellen, D. G. Browning, and L. Goodman, experiment on long-range propagation of the explosion- J. Acoust. Soc. Am. 60, 1053 (1976). generated signals in the shallow-water (125Ð155 m) 11. Sound Transmission through a Fluctuating Ocean, Ed. region of the Sea of Okhotsk with a fully-developed USC. by S. Flatte (Cambridge Univ. Press, Cambridge, UK, 1979; Mir, Moscow, 1982). (i) In the time structure of the multiray explosion- generated signal received at the USC axis, at the distances up to 250Ð300 km, the pulses of the shock wave Translated by E. Kopyl

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002

Effect of the Interaction of Spatial Modes on the Nonlinear Dynamics of Sound Beams I. S. Vereshchagina* and A. A. Perelomova** * Mathematical Faculty, Kaliningrad State University, ul. Aleksandra Nevskogo 14, Kaliningrad, 236041 Russia e-mail: [email protected] ** Faculty of Engineering Physics and Applied Mathematics, Gdansk Technical University, Gdansk, 80952 Poland e-mail: [email protected] Received July 31, 2000

Abstract—Propagation of an acoustic beam in a medium with a combined second- and third-order nonlinearity is studied. The derivation of the dynamics equations and the determination of modes is performed using the orthogonal-projection operator technique. The problem on the beam evolution considered with allowance for weak nonlinearity, diffraction, and dissipation leads to a set of equations describing the interaction of directed waves and a quasi-stationary (thermal) mode. In the conditions of a directed beam, the inclusion of the interaction leads to a modiﬁed KhokhlovÐZabolotskayaÐKuznetsov equation with quadratic and cubic nonlinearities. The solutions to the problem are obtained in the region near the beam axis, in the form of series expansions in the transverse coordinate up to the focal point. The results of calculations are represented in graphical form for different nonlinearity combinations. © 2002 MAIK “Nauka/Interperiodica”.

One of the major problems of the theory of nonlin- an incorrect value of the time-average flux of angular ear acoustic waves is related to an adequate description momentum. This points to the necessity of taking into of the distortions suffered by an acoustic signal in the account the mode interaction. In the cited review [4], course of its propagation in a medium. Any perturba- attention is given to the generation of the thermal mode tion of a medium can be represented as a superposition due to the dissipation of the acoustic mode related to of modes characteristic of a given problem, which the viscosity of the medium or the attenuation at the determine the basic possible types of motion. Taking shock wave front. We note that the nonisentropic (ther- into account the nonlinearity of a beam leads to the mal) mode must be taken into account even in the interaction of modes and to their self-action. A correct absence of viscosity and heat conductivity of the consideration of the mode interaction is most simple medium. This is essential for linear problems with non- when the separated modes are independent (i.e., they isentropic initial conditions and in studying the nonlin- are separated by the projectors that are orthogonal in ear dynamics, because a nonlinear generation and the the linear approximation). The modes, as a basis, must interaction of all modes take place. The thermal mode describe all possible types of motion, i.e., the set of pro- determined here as one of the principal modes is not jection operators must be complete. necessarily the secondary mode caused by the attenuation of an acoustic wave. Traditionally, the thermal The basic equations of the theory describing the mode is treated as a secondary one [4], which evidently propagation of nonlinear acoustic beams are the makes it impossible to consider the problems with KhokhlovÐZabolotskaya (KhZ) and KhokhlovÐZabo- essentially nonisentropic initial conditions. lotskayaÐKuznetsov (KhZK) equations, which take into account the viscosity of the medium. These equa- In this paper, we consider the problem of the propa- tions are obtained by the method of a slowly varying gation and focusing of a nonlinear acoustic beam. For profile, when the evolution of the wave profile is stud- the derivation of the equations describing the wave ied in the x–ct coordinate system moving with the dynamics, we propose the projection operator method. velocity of the wave propagation in the linear approxi- In the framework of the proposed approach, we obtain mation [1Ð3]. a set of coupled equations describing the interaction of two acoustic modes (the right and left beams) and the The KhZ and KhZK equations do not take into quasi-stationary thermal mode. These equations take account the effect of the other two modes: the station- into account the weak nonlinearity (quadratic and ary mode and the backward wave. The necessity to take cubic) of the perturbation, the diffraction, and the dissi- into account the backward wave and the thermal mode pation of the beam. We note that, here, we do not use in the one-dimensional dynamics is indicated in the the assumption concerning the potential character of review [4]. There, it is also demonstrated that the the flow. In particular, this approach provides a more approximation of one directed acoustic mode leads to accurate expression for the nonlinear constants in the

148 VERESHCHAGINA, PERELOMOVA KhZK equation (for example, for the right beam) for case, the contribution of dissipative terms is (as usual the terms with quadratic and cubic nonlinearities. The [2]) much smaller than that of nonlinear terms. calculated coefficients take into account the nonlinear ∂ρ ∂() ρV generation of other modes (the left beam and the sta- +0,------= tionary mode) and their inverse influence on the princi- ∂t ∂x pal right wave. ∂ ∂ ∂ ∂2 ρ V ρ V p 4ηζ V ++V Ð --- + ------= 0,(1) Approximate solutions to the equation derived for ∂t ∂x ∂x 3 ∂x2 the right beam are determined, including the solutions in the near-focal region, and the plots illustrating the ∂e ∂e ∂V ∂2T ρ ++ρV p Ð χ------= 0. beam behavior at different distances from the focus and ∂t ∂x ∂x ∂x2 the effect of the third-order nonlinearity on the profile dynamics are presented. Here, p is pressure, T is temperature, ζ and η are the coefficients of volume and shear viscosities, respectively, and χ is the coefficient of heat conductivity. The PROJECTION OPERATORS. coefficients of viscosity and heat conductivity are MODE INTERACTION IN A ONE-DIMENSIONAL assumed to be constant, and the background is assumed PROBLEM WITH VISCOSITY to be homogeneous. To make system (1) complete, it is AND HEAT CONDUCTIVITY necessary to add two more equations. For this purpose, we take the caloric and thermal equations of state: e = ρ ρ Starting from the paper by Kovasznay and Boa The- e( p, ) and T = T( p, ). These equations have the fol- Chu [5], in hydrodynamics it is common to single out lowing form for an ideal gas: three modes (the vortex mode, the sound pressure p p e ==------, T ------, (2) mode, and the entropy mode), which separate the types ργ()Ð 1 ργ()Ð 1 C of possible perturbations according to the character of v where Cv is the specific heat of the gas at constant vol- motion and dissipation, in studying perturbations in a γ viscous heat-conducting medium. In the linear approx- ume and = Cp/Cv is the adiabatic constant. Here, we imation, these modes do not interact and a perturbation consider the case of an ideal gas, although the caloric is generally represented as a sum of three modes. We and thermal equations of state in the general form allow also suggest the introduction of three modes in the lin- one to study the dynamics of any liquid or gas. To do ear approximation. However, in contrast to [5], we sep- this, it is necessary to expand the small perturbations of arate the perturbations according to the basic types of internal energy and temperature into a Taylor series in motion, which are determined by the dispersion rela- the vicinity of the background values of pressure and tion in the linear approximation, rather than according density with the required accuracy. Let us change to to the character of motion and dissipation. This allows new variables: us to determine the modes using specific relations, ppÐ ρρÐ namely, through the eigenvectors of the linear problem. p' ==------0, ρ' ------0, ε 2ρ ερ Here, we restrict our consideration to a relatively sim- c 0 0 ple case of a homogeneous medium in the absence of V x ct flows as a background. However, since it is easy to for- V ' ==ε----- , x' λ---, and t' =----λ . mulate the proposed method as an algorithm, it is also c easy to consider stratified media with flows. It is impor- Here, the background parameters are denoted by the tant that the modes determined in this way are mutually index 0, λ is the characteristic wavelength, c is the adi- orthogonal and form a complete set of basis vectors, abatic velocity of sound, and ε is the small amplitude and the relations obtained in the linear approximation parameter. Below, we omit the primes marking the are also valid for a nonlinear case. Let us briefly dimensionless quantities. Eliminating T and e from the describe the idea of the proposed method of mode sep- initial set of equations and ignoring the viscosity and aration. heat conductivity, we represent Eqs. (1) in the linear approximation in the form At the first stage, we consider the problem of the ∂ϕ propagation of an acoustic beam in the linear approxi------+0,Lϕ = mation. As the initial equations, we take the system of ∂t hydrodynamic equations for the density ρ, mass veloc- ∂ ity V along the x axis, and the internal energy of a unit 0 ------0  ∂x mass e with the equation of motion in the NavierÐ V (3) Stokes form. In the equation of energy balance, we ∂ ϕ ==p , L ------00 . ignore the nonlinear viscous cross terms (which is a  ∂x ρ common approach in acoustics: see, e.g., [2, 4]). This  ∂ approximation assumes the amplitude of the acoustic ------00 disturbance and the thermoviscosity to be small. In this ∂x

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EFFECT OF THE INTERACTION OF SPATIAL MODES ON THE NONLINEAR DYNAMICS 149

Let us consider plane waves representing the mass where the function ϕ˜ on the right-hand side of the ω velocity as V = Vk exp(i t – ikx) and the quantities p equation takes into account the nonlinear terms and has and ρ in an analogous form. The dispersion relation the form ω ± ω for Eqs. (3) has three roots 1, 2 = k and 3 = 0, which correspond to three independent branches of the disper- ω ∂V ∂p sion relation. In this case, 1 = k describes the wave ÐV + ρ ω ∂ ∂ propagating to the right, 2 = –k describes the left wave, x x ω and 3 = 0 describes the stationary addition to the back- ∂ ∂ ground. ϕ˜ = εp γ V + O()ε2 . ÐV∂ Ð p∂ The eigenvectors corresponding to these three solu- x x T T  tions have the forms e1 = (1, 1, 1) , e2 = (–1, 1, 1) , and ∂ρ ∂V T ÐV Ð ρ e3 = (0, 0, 1) , whence we have linearly independent ∂x ∂x ϕ ρ T ϕ solutions for each branch: 1 = (V1, p1, 1) , 2 = (–V2, ρ T ϕ ρ T ρ ρ ρ p2, 2) , and 3 = (0, 0, 3) . The corresponding opera- Let us change to the new variables 1, 2, and 3 and tors of projection onto each branch can be represented assume that, for the eigenvectors, the constraint equa- in the explicit forms tions that are determined by the linear problem are ρ ρ ρ ρ  valid: V = V1 + V2 = 1 – 2, p = p1 + p2 = 1 + 2, and ± ρ ρ ρ ρ 1/2 1/2 0 000 = 1 + 2 + 3. Applying the projection operators to P , ==± , P . (4) both sides of Eq. (5), we obtain a set of nonlinearly cou- 12 1/2 1/2 0 3 000 ±1/2 1/2 0 01Ð1 pled equations for the density perturbations of each mode: It is very important that the projectors determined by Eqs. (4) are mutually orthogonal. This allows a correct ∂ρ ∂ρ ε 3 ∂ n n n ρ ρ ()ε2 separation of the initial disturbance into independent ------++cn------∑ Yim, i------m + O = 0,(6) (in the linear approximation) modes. The components ∂t ∂x 2 ∂x im, = 1 of the eigenvectors are evidently related as follows: V1 = ρ ρ ± 1 = p1, V2 = – 2 = –p2, and p3 = V3 = 0. The projection where n = 1, 2, 3; c1, 2 = 1, and c3 = 0. The indices 1, 2, operators, as usual, satisfy the requirements of normal- and 3 correspond to directed and stationary modes, ˆ ˆ ization and orthogonality: P1 + P2 + P3 = I , PkPn = 0 , which in the problem of nonlinear dynamics are appro- ≠ ˆ ˆ priate to be called quasi-directed and quasi-stationary k n, and PkPk = Pk, where I and 0 are the unit and null k matrices, respectively. modes. The coefficients Yim, are determined in the tables The fact that it is possible to represent the projection operators in an explicit form provides an opportunity to 1 m 123 separate any disturbance into the fields of the right and Yim, left waves and the stationary mode. For this purpose, it is sufficient to apply a corresponding projection opera- i ϕ 1 γ + 1 –(γ + 1) 0 tor to the vector k = Pkϕ. It is important that the suggested separation into 2 γ – 3 –(γ + 1) 0 modes does not imply the separation of disturbances 3–1–10 into vortex and vortex-free ones. Moreover, the stationary mode is not necessarily secondary, i.e., generated by an acoustic wave, as in [4]. This provides an oppor- 2 Yim, m 123 tunity to extend the class of the examined problems, e.g., to the problems with essentially nonisentropic ini- i tial conditions. According to our definition, the thermal 1 γ + 1 –(γ – 3) 0 mode is one of the basic types of motion even in media γ γ without viscosity and heat conductivity, in contrast to 2 + 1 –( + 1) 0 the secondary thermal mode caused by the attenuation 3 110 of an acoustic wave [4]. It should also be noted that, in the case of a stratified medium, the elements of the pro- 3 jection operators are integro-differential operators [6]. Yim, m 123 The initial set of hydrodynamic equations (1) neglecting the viscosity and heat conductivity can be i written in the form 1 –2(γ – 1) 2(γ – 1) 2 ∂ϕ 2 –2(γ – 1) 2(γ – 1) –2 ------+ Lϕ = ϕ˜ , (5) ∂t 32–20

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 150 VERESHCHAGINA, PERELOMOVA In particular, for the right-propagating wave with NONLINEAR DYNAMICS OF SOUND BEAMS allowance for self-action, from Eqs. (6) we obtain the INTERACTING WITH THE THERMAL MODE known evolution equation [1, 2]: Let us consider the problem of the propagation of a ∂ρ ∂ρ γ ∂ three-dimensional acoustic beam with a weak diver- 1 1 ε + 1ρ ρ ()ε2 ------++------1------1 +O = 0. (7) gence along the y and z directions, which are orthogo- ∂t ∂x 2 ∂x nal to the propagation direction. As earlier, we assume the dissipative and nonlinear terms to be small. As Assuming the terms connected with viscosity and heat above, at the first stage we consider the linear problem conductivity in Eqs. (1) to be small, as well as the non- without taking into account the heat conductivity and linear terms, we represent Eqs. (1) in terms of dimen- viscosity: sionless variables in the form of Eq. (5) with a new vector on the right-hand side: ∂ϕ ------1 +0.L ϕ = (10) ∂t 1 1 ∂V ∂p ÐV + ρ ϕ ρ T ∂x ∂x ∂ Here, 1 = (Vx, Vy, Vz, p, ) and Vx, Vy, and Vz are the ρ2 p velocity components. Following the concept of a Ð ∂ ∂ ∂ 2x ϕ˜ = εp γ V + ε slowly varying profile (the transverse velocities change ÐV∂ Ð p∂  slower) [1, 2, 7], we introduce the dimensionless vari- x x 0  ables: V' = V /εc, V' = V /εc, V' = V /εc, y' = y β /λ, ∂ρ ∂V 0 x x y y z z ÐV Ð ρ ∂ ∂ and z z β λ, where β is the small parameter charac- x x (8) ' = / terizing the beam divergence. Below, we omit the ∂2 δ V primes marking the dimensionless variables. The 1------∂x2 matrix in Eq. (10) is of the fifth order: 2 ()ε3, εδ + δ ∂ ()γp Ð ρ + O . ∂ ------2 ------ γ 2 00 0 ∂------0 Ð 1 ∂x x  ∂ 0 00 0 β----- 0 ∂y Here, ∂ L1 = 00 0 β----- 0 . η ∂z δ ζ 4 1 δ χ1 1 1  1 ==+ ------ρ------λ-, 2 ------Ð ------ρ------λ-, ∂ ∂ ∂ 3 0c Cv C p 0c ------β----- β----- 00 ∂x ∂y ∂z δδδ  and = 1 + 2 ∂ ∂ ∂ ------β----- β----- 00 are the small dimensionless parameters taking into ∂x ∂y ∂z account the contributions of viscosity and heat conduc- The corresponding dispersion relation has five roots. tivity. Applying the operator P1 to Eq. (5) with the right- hand side in the form of Eq. (8), we obtain the Burgers One of the roots ω = 0 corresponds to the stationary equation for the right mode with allowance for the self- mode. Four other roots correspond to acoustic modes action: and are determined by the independent combinations of ω , kx, ky, and kz with different signs, which satisfy the ω2 2 β 2 2 2 condition = (kx) + [(ky) + (kz) ]. Two roots corre- ∂ρ ∂ρ γ + 1 ∂ δ ∂ 2 ------1 ++------1 ε------ρ ------ρ Ð ------ρ +O()ε , εδ = 0.spond to the right acoustic mode and two roots, to the ∂ ∂ 1∂ 1 2 1 t x 2 x 2∂x (9) left one. We assume that the motion is quasi-plane and weakly linear, which allows us to treat the correspond- It is necessary to note that, in Eq. (8), the coefficients ing terms in the initial set of equations as small addi- before the terms that correspond to the quadratic non- tions. Taking into account the small nonlinear and ther- linearity and thermoviscosity differ from the expres- moviscous terms leads to the following set of equa- sions given in [1Ð3, 7], where the stationary mode is not tions: considered and the constraint equations correspond only to the acoustic modes: p = ρ. It is evident that, with 0 ∂∂ ∂2 ∂2  such a description, the derived equations are unsuitable ----- + L ϕβ= ------+ ------p + ϕ˜ . (11) ∂t∂t ∂ 2 ∂ 2  1 for investigating the complete nonlinear dynamics, z y ρ because they lead to an incorrect determination of the k ϕ matrix elements Yim, , which are responsible for the The operators L and have the same form as in the case ϕ ∂ϕ ∂ mode interaction. of the one-dimensional problem, ˜ 1 = ˜ / t. Applying

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 EFFECT OF THE INTERACTION OF SPATIAL MODES ON THE NONLINEAR DYNAMICS 151 one-dimensional projection operators to both parts of strate that a linear boundary-value problem in a semi- the set of equations (11), we obtain coupled equations infinite space x > 0 (initially unperturbed) corresponds to a strict realization of the relations for the right mode ∂ρ ∂ρ 3 ∂ n n ε n ∂ at the boundary and in the whole semi-infinite space. ------++c ------∑ Y , ρ ------ρ ∂t ∂t n ∂x 2 im i∂x m The generation of other modes by the principal mode im, = 1 and their evolution is described by Eqs. (12). We use the nonsingular theory of perturbations to study the gener- ε2 3 ∂ 3 ∂2  n ρ ρ ()ρ ρ 1 n ρ ation of two other modes and their reciprocal effect on + ---- ∑ T im, i m 1 Ð 2 + ---∑ Bi ------i 2 ∂x 2 ∂x2  the first one, as in [8, 9]. We assume that the nonlinear- im, = 1 i = 1 (12) ity prevails over the viscosity and beam divergence. We 3 δ β ε2 β ∂2 ∂2 also assume that , ~ . The evolution of the density Θnρ ()ε3,,εβ εδ disturbance of the first mode with allowance for self- Ð ---------+ ------∑ i i + O = 0. 2 ∂ 2 ∂ 2 action is determined by the equation y z i = 1 ± ()0 ()0 ()0 2 ()0 Here, as before, c1, 2 = 1; c3 = 0; and the coefficients ∂ ∂ρ ∂ρ ε ∂ρ ε ∂ρ  1 1 ()ργ ()0 1 ρ()0 2 1 n ------++------+ 1 1 ------+---- 1 ------Yim, , which take into account the quadratic nonlinear- ∂t ∂t ∂x 2 ∂x 2 ∂x ity, are determined earlier in the tables of Eq. (6). The () (14) ∂2ρ 0  2 2 Θn n n δ 1 β∂ ∂ ()0 coefficients i , Bi , and T im, , which determine the ------ --- ρ Ð 2 Ð0.------2 + ------2 1 = contributions of the beam divergence, thermal conduc- 2 ∂x  2 ∂y ∂z tivity, and cubic nonlinearity, respectively, are given in the tables: For the second and third modes in the first approximation with allowance for their nonlinear generation, we have n Θn Bi n 12 3 i n 123 ()1 ()0 ε i ρ ()ρxyzt,,, = Ð ---()γ + 1 i 2 2 2 δ δ δ δ ()γ , , 1 Ð 1 Ð 2 2/ Ð 1 1 110 t ∂ 2 δ Ð δ Ðδ δ /()γ Ð 1 2 110 ×ρ()0 ()τ,,,τ ρ ()0 ()ττ,,,τ 1 2 2 ∫ 1 xt+ Ð yz ∂------1 xt+ Ð yz d , δ δ δ ()γ x 322 2 2 Ð2 2/ Ð 1 3 000 0 1 2 3 ρ ()1 ()ρ,,, ()0 εγ() T im, = T im, = 1, and T im, = 0 for all i, m. 3 xyzt = 3 + Ð 1 t The set of equations (12) describes the interaction of ∂ three independent modes, i.e., the right, left, and ther- ×ρ()0 (),,,τ ρ ()0 ()τ,,,τ ∫ 1 xyz ------1 xyz d . mal modes. In particular, Eq. (12) yields the known ∂x KhZK equation taking into account the propagation of 0 only the right wave: ρ()0 The solutions to the equations for 2 (x, y, z, t) and () () 2 ρ 0 ρ 0 ∂ ∂ρ γ + 1 ∂ δ ∂ 3 (x, y, z, t) are evidently zero: 2 (x, y, z, t) = 0 and ------1 + ε------ρ ----- ρ Ð ------ρ ∂τ∂x 2 1∂τ 1 2∂τ2 1 ρ()0 (13) 3 (x, y, z, t) = 0. Integrating Eq. (14) with respect to 2 2 β∂ ∂ 2 time, we obtain --- ρ ()ε ,,εδ εβ Ð ------2 + ------2 1 +0.O = 2 ∂ ∂ ∂ () ∂ () y z ρ 0 (),,, ρ 0 (),,, ()ε ∂----- 1 xyzt + ∂------1 xyzt = O . (15) The latter equation is transformed into the KhZ equa- t x tion when δ = 0. Taking this equation into account, we can write for the next approximation: () THE MODIFIED ρ 1 ()xyzt,,, KHOKHLOVÐZABOLOTSKAYAÐKUZNETSOV 2 (16) εγ() EQUATION + 1 []()ρ()0 (),,, 2 ()ρ ()0 (),,, 2 = ------1 xyzt Ð 1 xtyz+ 0 , Let us assume that one of the modes, the right one 8 for definiteness, prevails, i.e., it is generated predomi- () ρ 1 ()xyzt,,, nantly. This corresponds to the realization of linear 3 (17) relations (probably, approximate) for this mode. Prob- εγ()Ð 1 ()0 2 ()0 2 = Ð------[]()ρ ()xyzt,,, Ð ()ρ ()xyz,,,0 . lems with initial conditions and boundary-value prob- 2 1 1 lems can be considered in the framework of the indicated approximation (in this case, the correspondence Taking into account the reciprocal effect of the second must exist at the boundary). It is possible to demon- and third modes, according to Eqs. (12), (16), and (17)

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 152 VERESHCHAGINA, PERELOMOVA we obtain an evolution equation of the next order for described near the beam axis only at small distances the first mode: from the source, x Ӷ ωa2/2c. Here, ω is the fundamental frequency of the source, a is the beam radius, and c ∂ ∂ρ()1 ∂ρ()1 is the sound velocity. A signal cannot be treated as a ------1 + ------1 - ∂t ∂t ∂x quasi-harmonic one, and it is necessary to expand the total field of acoustic pressure as a power series in a small parameter. The inclusion of higher-order correc- ∂ρ()1 ∂ρ()1 ε ()1 ()1 tions leads to incorrect results, which makes it impossi- + ---ρ ()γ + 1 ------1 - + ε()ρA + 1 ------1 - (18) 2 1 ∂x 1 1 ∂x ble in the framework of this approach to describe the most interesting region where the focusing of a finite- 2 ()1 amplitude sound beam takes place. In acoustics, the ∂ ρ β ∂2 ∂2 1δ 1 ρ()1 paraxial approach was first used by Rudenko, Soluyan, Ð ------Ð0,---------+ ------1 = 2 ∂x2 2 ∂y2 ∂z2 and Khokhlov [10]. An analytical method for describing a beam in the where paraxial region was developed recently [11, 12]. This 1 1 1 m method makes it possible to study also the motion in the A = --- ∑ ()Y , /2 + Y , Y , /()c Ð c 1 2 m 1 1 m 11 1 m focal plane. There are also efficient numerical methods m = 23, for solving the KhZ equation [13].

1 2 We follow the basic concepts [11, 12] in the analyt- = Ð---()γ + 1 Ð 1. 8 ical description of the beam evolution. We assume the problem to be cylindrically symmetric and represent The new coefficient A1 before the cubic term corre- Eq. (18) in cylindrical coordinates: sponds to the correction that takes into account the ∂ ∂ ∂ ∂ P P εα 2 P reciprocal influence of two other modes on the first one, ∂τ ∂σ Ð NP∂τ Ð MNP ∂τ 1 while T , = 1 in the same term takes into account the (19) 11 2 2 1 ∂ P 1 ∂P self-action of the order of ε for the principal mode. = ------+,--- 4G ∂R2 R∂R THE MODIFIED KHOKHLOVÐZABOLOTSKAYA where P = p/pp. Here, p is the dimensional pressure, pp EQUATION. AN ANALYTICAL SOLUTION is the peak value of the sound pressure at the source, σ = x/d, τ = ω(t – x/c), R = r/a, x is the axial coordinate, The set of equations (12) and the approximate evo- d is the focal length, and r is the transverse coordinate. lution equation (18) are too complicated to obtain a The quantities r, x, t, c, and ω are dimensional; G = x /d general solution, even if we ignore the thermoviscous 0 ˜ 3ρ ε γ ω terms. Equation (18) differs from the KhZK equation in is the dimensionless constant; x = 2c 0/ ( + 1) pp is the cubic nonlinear term taking into account the effect the characteristic distance within which the shock wave ω 2 β of the left beam and quasi-stationary mode (which are is formed; and x0 = a /2 c is the characteristic average generated in the course of the propagation of the right diffraction length for an acoustic beam with the fre- acoustic beam) on the right wave. quency ω. The quantity N = d/x˜ characterizes the qua- Let us recall briefly the advances in the study of the dratic nonlinearity of the beam, and G characterizes the KhZK equation within the last few years. The study of beam focusing. the KhZ and KhZK equations started in the 1970s. The This equation differs from the classical KhZ equa- method of nonlinear geometric acoustics for high-fre- tion by the presence of the term corresponding to cubic quency sources, λ/a Ӷ 1 (a is the characteristic size of nonlinearity with the coefficients the source), was used at that time [3]. The method 1 allows one to describe the evolution up to the focal p A + T , ()γ + 1 M ===------p , α ------1 11 Ð------. region, but not within the region itself, because the 2 1 c ρ 8 expressions become singular. Simultaneously, in 0 Y11, optics, the paraxial approximation was successfully Note that the sign of α calculated for an ideal gas with used. This approximation provided excellent results in allowance for the reciprocal effect of modes differs from nonlinear optics and laser physics for describing the α 1 1 γ self-focusing of a light beam. The successful applica- the sign of the parameter 0 = T 11, /Y11, = 1/( + 1) when tion of the paraxial approach is related to the consider- the calculation takes into account only the self-action of ation of narrow-band quasi-harmonic signals and, the right beam. This fact demonstrates the importance therefore, with the possibility of an independent sepa- of taking into account the resonance interaction of all ration of the amplitude and the eikonal [3]. However, as modes. applied to nonlinear acoustics, the approach imposes We try the a solution to the evolution equation (19) restrictions on the domain of applicability of the results (as in [11, 12]) in the form of a series expansion in pow- obtained with its help: the solution is adequately ers of the parameter R = r/a:

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 EFFECT OF THE INTERACTION OF SPATIAL MODES ON THE NONLINEAR DYNAMICS 153

p 14p 14

1 12 12

8 8

2 6 6

4 4 3 2 2

–3 –2 –1 001234 12345 τ –2 –2

–4 –4

–6 –6

Fig. 1. Evolution of the disturbance proﬁle for α = Ð0.3 and σ = (1) 1.01, (2) 1.2, and (3) 2.0 (the solid lines); the crosses represent the case without the cubic nonlinearity.

(),,σθ ()σθ, Ð1 PR = P0 f ()η = ()cos() η+ Gsin()η , R2 R4 (20) η2 2 ()η 1/2 ----- ()σθ, ----- ()…σθ, ()η + ln f + P2 ++P4 . g = N------, 2 4 1 + GÐ2 Then, performing calculations analogous to [11, 12], η ()η but taking into account the contribution of the cubic ζη() G + ln f = arctan------η ()η- , term, we obtain a solution for P0 in the form Ð Gfln 2 2 1 g()η δη() N G ηη()η P ()ητ, = ------sin()ηθÐ + ------sin()ηζηÐ () = ------ln cos + Gsin . 0 f ()η 0 2 f ()η 21()+ G2 η (21) Now, at ε = 0 (ignoring the cubic nonlinearity), it is αNMG g()η' Ð ------∫------sin()ηηη' Ð ζη()' sin()Ð ' dη' easy to plot the signal profiles for different values of the 2 f ()η' parameters N and G characterizing the focusing and the 0 diffraction, respectively. The simplest way to do this is (here, we use the same notations as in [11]). In contrast to treat Eqs. (21) and (22) as a system of equations in θ η τ τ θ η to the solutions given in [11, 12], the solution given by the form P0 = P0( 0, ), = ( 0, ), which determines Eq. (21) takes into account the contributions of both (at a fixed value of η) the function P = P (τ) in a para- θ 0 0 quadratic and cubic nonlinearities and also the interac- metric form. In this case, 0 plays the role of a parameter. tion between different modes. If necessary, it is possible To plot a signal profile at ε ≠ 0, it is necessary first to to obtain a solution for P2 [the next order relative to calculate the integral on the right-hand side of Eq. (22). Eq. (21)] with the help of the relations similar to those Regretfully, it is impossible to calculate it analytically in [11, 12]. in the general form suitable for analysis. However, at θ τ η Here, 0( , ) is determined from the solution of a fixed values of N, G, and η, it can be calculated as a transcendental equation: θ function of 0, e.g., with the help of the MAPLE soft- τθ ()η () θ ζη() α ware package. In this case, it is possible to obtain the = 0 Ð g sin 0 + Ð NMG η perturbation profiles in a parametric form with the help 2 (22) ×θ()η 1 ()η () η ζη() η δη() of the same software. ∫ sin 0 + ' + ---g ' sin ' Ð ' d ' Ð . 2 Figure 1 demonstrates the profiles of the beam at 0 various distances from the source σ with allowance for The following notations are introduced in Eqs. (21) the contribution of the cubic nonlinearity. The values of and (22): other parameters determining the contribution of non-

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 154 VERESHCHAGINA, PERELOMOVA

p 14

12 1 10

2 8

3 4

–1 0 1 2 345τ –2

–4

–6

Fig. 2. Evolution of the disturbance profile for α = 0.3 and M = (1) 0.05, (2) 0.5, and (3) 1.0 (the solid lines); the crosses represent the case without the cubic nonlinearity. linearity are selected as follows: N = 0.3 and G = 10. 2. O. V. Rudenko and S. I. Soluyan, Theoretical Founda- The value of the parameter α for an ideal diatomic gas tions of Nonlinear Acoustics (Nauka, Moscow, 1975; is α = Ð0.3. The profiles obtained without taking into Consultants Bureau, New York, 1977). account the cubic contribution are shown in the same 3. M. B. Vinogradova, O. V. Rudenko, and A. P. Sukho- figure by crosses, and they closely coincide with the rukov, The Theory of Waves, 2nd ed. (Nauka, Moscow, plots presented in [11, 12]. 1990). Figure 2 shows the profiles of the beam at α = –0.3, 4. S. Makarov and M. Ochmann, Acustica 82, 579 (1996). N = 0.3, and G = 10 for different values of the nonlinearity parameter: M = 0.05, 0.35, and 1.0. The profile 5. Boa Teh-Chu and L. S. Kovasznay, J. Fluid Mech. 3, 494 obtained without taking into account the contribution of (1958). the cubic nonlinear term is represented by crosses (it 6. A. A. Perelomova, Acta Acust. (China) 84 (6), 1002 coincides with the calculations from [10, 11]). (1998). In Fig. 2, the plots obtained with allowance for the 7. V. P. Kuznetsov, Akust. Zh. 16, 548 (1970) [Sov. Phys. cubic nonlinearity lie below the plots taking into account Acoust. 16, 467 (1971)]. only the quadratic term and have sharper peaks. As the 8. S. B. Leble, Nonlinear Waves in Waveguides with Strat- nonlinearity parameter M grows, the wave amplitude ification (Springer, Berlin, 1990). decreases, the peak is shifted to the left, and a loop-type singularity is formed in the vicinity of the pressure peak. 9. S. P. Kshevetskiœ and S. B. Leble, Izv. Akad. Nauk SSSR, In the figure, the loops are marked with dots. Basically, Mekh. Zhidk. Gaza 3, 1169 (1988). the loop-type singularity can be eliminated analogously 10. O. V. Rudenko, S. I. Soluyan, and R. V. Khokhlov, Dokl. to the “backlash” singularity in the Riemann wave in the Akad. Nauk SSSR 225 (5), 1053 (1975) [Sov. Phys. theory of weak shock waves [2]. Dokl. 20, 836 (1975)]. 11. M. F. Hamilton, V. A. Khokhlova, and O. V. Rudenko, ACKNOWLEDGMENTS J. Acoust. Soc. Am. 101, 1298 (1997). We are grateful to Professor S.B. Leble for the inter- 12. M. F. Hamilton, O. V. Rudenko, and V. A. Khokhlova, est taken in our work and for valuable comments. Akust. Zh. 43, 48 (1997) [Acoust. Phys. 43, 39 (1997)]. 13. J. N. Tjotta, S. Tjotta, and E. H. Vefring, J. Acoust. Soc. Am. 88, 2859 (1990). REFERENCES 1. E. A. Zabolotskaya and R. V. Khokhlov, Akust. Zh. 15, 40 (1969) [Sov. Phys. Acoust. 15, 35 (1969)]. Translated by M. Lyamshev

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002

Acoustical Physics, Vol. 48, No. 2, 2002, pp. 155Ð161. Translated from Akusticheskiœ Zhurnal, Vol. 48, No. 2, 2002, pp. 187Ð194. Original Russian Text Copyright © 2002 by Grigor’ev, Tolstikov, Navrotskaya.

Transfer Constant of a Multilayer Acoustic Transducer at the Direct and Inverse Transformation M. A. Grigor’ev, A. V. Tolstikov, and Yu. N. Navrotskaya Chernyshevski State University, ul. Astrakhanskaya 83, Saratov, 410026 Russia e-mail: [email protected] Received May 28, 2001

Abstract—The algorithms for calculating the direct and inverse transfer constants of an acoustic transducer with an arbitrary number of intermediate layers between the piezoelectric layer, the acoustic duct, and the rear acoustic load are described. The results of a numerical analysis are presented and discussed. As an illustration, ° a 100-MHz transducer formed by a (Y + 36 )-cut LiNbO3 plate ﬁxed on a fused-quartz acoustic duct with the help of ﬁve metal layers is considered. The other side of the plate carries two metal layers and a rear load. The phaseÐfrequency characteristics and the transformation loss as a function of frequency are analyzed for the cases of direct and inverse transformation under the assumption that the signal is supplied and retrieved by a two-wire line. © 2002 MAIK “Nauka/Interperiodica”.

The well-known works (e.g., [1Ð7]) on the analysis and the inverse transfer constants of an acoustic trans- of acoustic transducers for volume waves considered ducer for volume waves with an arbitrary number of the direct transformation of the electromagnetic energy intermediate layers and with a rear acoustic load. The to the acoustic one. The sought-for quantity was the electromagnetic signal is supplied to or retrieved from ratio of powers of the transformed waves, which is usu- the transducer by a two-wire line with a given wave ally called the conversion coefficient. The phaseÐfre- resistance. A parasitic shunt capacitance and a connect- quency characteristic was not considered in the above- ing-wire inductance with an active loss resistance are mentioned works. It is useful to find out, what the also assumed to be connected with the piezoelectric phaseÐfrequency and the amplitudeÐfrequency charac- element in parallel and in series, respectively. teristics of a complex multilayer piezotransducer are and how can they be controlled. This information is important, e.g., for the formation of an optimal impulse THE MODEL UNDER ANALYSIS response of a transducer operating in the mode of emis- AND THE ASSUMPTIONS sion and reception of short videopulses in a pulsed acoustic microscope [8, 9]. The means that influence to Figure 1 demonstrates schematically the transducer some extent the phaseÐfrequency and amplitudeÐfre- under analysis. Between the piezoelectric layer 1 and quency properties of a transducer could be intermediate the acoustic duct 2 there are M intermediate layers (sublayers introduced between the piezoelectric, the acoustic duct, and the rear acoustic load. The latter can also be considered as a structure component intended for h31 h3n h3N h1 h2M h2m h21 correcting the above-mentioned characteristics. The quantity that most fully characterizes a trans- 3 31 3n 3N 1 2M 2m 21 2 ducer is the transfer function or transfer constant, which is the ratio of the complex amplitudes of the b b transformed waves. Knowing this constant allows one to determine the above-mentioned conversion coeffi- C cient and the phase difference between the output and Rlos sh input signals. L It is well known that the reciprocity theorem holds for a piezoelectric transducer [10]. However, it would aa be interesting to compare the direct and the inverse Fig. 1. The model of a piezoelectric transducer: (1) piezo- transfer constant for a specific example of a complex electric, (2) acoustic duct, and (3) rear acoustic load; the multilayer piezoelectric element used at high and indices 2m and 3n represent the sublayer and superlayer microwave frequencies. numbers, respectively; Csh is the parasitic shunt capacitance and Rlos and L are the active resistance and the inductance The purpose of this paper is to develop the algo- of the conductor connecting the transducer with the trans- rithms and the programs for calculating both the direct mission line.

156 GRIGOR’EV et al. layers) with the common index 2 and individual num- conductivity of the piezoelectric, are negligibly small; bers m increasing from 1 to M towards the piezoelectric in addition, all sublayers, superlayers, and the rear load layer. On the other side, between the piezoelectric and are assumed to be isotropic; and the piezoelectric and the rear load 3, there are also N media (superlayers) the acoustic duct are oriented in such a way that the ten- with the common index 3 and numbers n changing from sor equations describing the electromechanical pro- 1 to N in the direction from the rear load to the piezo- cesses in the media under consideration are reduced to electric. The sublayers and superlayers are short acous- scalar equations. tic waveguides with given lengths h2m and h3n, acoustic velocities v and v , and acoustic wave resistances 2m 3n GENERAL RELATIONS Z2m0 and Z3n0. We take that the acoustic duct and the rear load have infinite lengths and the given acoustic wave The equations relating the electric and mechanical resistances Z20 and Z30, respectively. The sublayer and quantities in the piezoelectric have the form the superlayer with the numbers 2M and 3N, respec- ∂ ∂ tively, are metal electrodes. In the diagram, they are T 1 = c1 u1/ zeEÐ , (3) connected with the terminals bb, which in their turn are De= ∂u /∂z + εE, (4) connected with the parasitic shunt capacitance Csh. The 1 two-wire transmission line with the wave resistance Z0 ∂ ∂ is connected with the terminals aa. The latter, in their D/ z = 0, (5) turn, are linked up to the terminals bb through the where T1 is elastic stress; u1 is the elastic strain; D is the inductance L and the loss resistance Rlos. electric displacement (induction); E is the alternating ε In the case of direct transformation, the direct elec- electric field strength; c1, e, and are the elastic con- tromagnetic (EM) wave arrives at the electric side of stant, the piezoelectric modulus, and the dielectric con- the transducer through the transmission line. The EM stant, respectively. wave creates a complex voltage amplitude Vaa+ across As follows from Eq. (5), D is coordinate indepen- the terminals aa, which we take to be the input one. In dent and, consequently, is a function of time only and the case of the inverse transformation, a voltage appears jωt has the form D = D0e , where the amplitude D0 can across the same terminals, and the complex amplitude generally be considered a complex quantity. of this voltage Vaa+ is taken to be the output one. In the In the piezoelectric layer, both the direct and inverse first case, the generator producing the signal is assumed waves of the elastic strain exist simultaneously: to be matched with the transmission line, i.e., its inter- ()β ()β nal resistance is equal to Z0. In the second case, the line u1 = u1+exp Ð j 1z + u1Ðexpj 1z . (6) is loaded by the matched resistance Z . 0 β On the acoustic side, it is the acoustic duct cross sec- Here, 1, u1+, and u1– are the propagation constant and tion at the boundary with the first sublayer (m = 1) that the complex amplitudes of these waves, respectively; z is taken as the output or input, depending on the trans- is the coordinate with the positive direction from the formation direction. The output signal or the input one power source to the load. The coordinate origin is at the is assumed to be the elastic stress of the acoustic wave piezoelectric layer boundary closest to the aforemen- outgoing or incoming through the duct with the com- tioned source. plex amplitude T2+ at the aforesaid boundary. To determine the voltage across the piezoelectric The desired transfer constants can be defined as fol- layer (see the terminals bb in Fig. 1) in the quasistatic lows: approximation (rotE = 0), we use Eq. (4) and obtain h () 1 K+ = Z0s/Z20 T 2+/V aa+ (1) V bb = ∫ E1dz in the case of the direct transformation, and 0 (7) D h Ð jβ h jβ h K = Z /()Z s ()V /T (2) 0 1 e[]()1 1 ()1 1 Ð 20 0 aa+ 2+ = ------ε - Ð --ε u1+ e Ð 1 + u1Ð e Ð 1 . in the case of the inverse one. Here, s is the area of the piezoelectric element. The complex quantities D0, u1+, and u1– are determined Defined in this way, the aforesaid coefficients are by the continuity conditions for the acoustic stress and dimensionless quantities, and their absolute values the current (or impedance) at the piezoelectric bound- squared are equal to the well-known transformation aries and also by the parameters of the electric circuit η η connected to the terminals bb. coefficients + = Pac/PEM+ and – = PEM/Pac+, where Pac and PEM are the acoustic and electromagnetic powers, The electric current Ie flowing in the external circuit respectively. equals the displacement current in the piezoelectric: The problem will be solved in the one-dimensional I ==J s ()∂D/∂t sj =ωDs, (8) approximation (∂/∂x = ∂/∂y = 0) on the assumption that e e the attenuation and diffraction of waves, as well as the where Je is the displacement current density.

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 TRANSFER CONSTANT OF A MULTILAYER ACOUSTIC TRANSDUCER 157

Using Eqs. (3) and (6), we represent the acoustic Tm, in and current Im, in with the output ones Tm, out and stress and the acoustic current in the piezoelectric Im, out. The transfer matrix for a section of the uniform layer as waveguide has the form [11] β β Ð j 1z j 1z e T ()z = c*Ð()jβ u e + jβ u e Ð -- D , (9) 1 1 1+ 1 1Ð ε 0 cosθ jZ()θ/s sin []a = 2m 2m0 2m , (11) Ð jβ z jβ z ()θ θ ()() ∂ ∂ ω ()1 1 jssin 2m /Z2m0 cos 2m I1 z = Ð u1/ t sj= Ð su1+e + u1Ðe . (10) 2 Here, c* = c1(1 + k ) is the elastic constant renormal- where m is the layer number; θ = ωh /v . 2 2 ε m 2m 2m ized due to the piezoelectric effect, k = e /c1 is the electromechanical coupling factor squared. The acoustic wave resistance of the rear load Z30 is The sublayers connecting the piezoelectric layer transformed by the series of N superlayers into some with the acoustic duct are a cascade connection of impedance ZN in the input plane of the superlayer acoustic waveguide sections. Each of them is a linear immediately adjacent to the piezoelectric. The value of two-port network connecting the input acoustic stress ZN can be found by applying the well-known expression

()ω ()ω Z3, n Ð 1 cos h3n/v 3n + jZ3n0 sin h3n/v 3n Z3n = Z3n0------()ω ()ω (12) Z3n0 cos h3n/v 3n + jZ3, n Ð 1 sin h3n/v 3n to each superlayer in turn, from n = 1 to n = N, by taking ues found are the input values for the sublayer chain. Z3, n – 1 = Z30 at n = 1. Here, Z3n and Z3, n – 1 are the acous- Using the transfer matrix (11) and applying it in turn tic impedances at the input boundaries of the superlay- to all sublayers, from m = M to m = 1, one can find the ers with numbers n and (n Ð 1). acoustic stress and current at the output boundary of the last sublayer. The resulting value T21, out = T2+ is just the desired acoustic stress at the output of the trans- DIRECT TRANSFORMATION ducer (at the input of the acoustic duct). One has but to Let the input signal be the voltage with a complex find the transfer constant according to definition (1). amplitude Vaa+ induced across the terminals aa by the direct wave in the transmission line. The piezoelectric To put the above algorithm into effect, it is useful to element is electrically characterized (at the terminals bb) introduce the dimensionless displacements u1' ± = by the so-called radiation impedance Zrad. Taking into u1±/N0, where N0 is a norm defined by the formula account the shunt capacitance Csh along with the series inductance L and resistance R , we obtain the imped- εω los N0 = eD0/Z10 . ance across the terminals aa: Z = R ++jωLZ/1()+ jωC Z . (13) Then, the continuity conditions for the acoustic imped- aa los rad sh rad ances at the piezoelectric boundaries take the form This impedance loads the transmission line with the wave resistance Z . The reflection coefficient for the Ð jβ h Z jβ h Z 0 ' 1 1M ' 1 1M wave of voltage across the aforesaid terminals is u1+e ------Ð 1 + u1Ðe ------+ 1 = Ð j, (16) Z10 Z10 Γ ()() E = Zaa Ð Z0 / Zaa + Z0 . (14) Z Z Evidently, ' N ' N u1+------+ 1 + u1Ð------Ð 1 = j. (17) ()Γ Z10 Z10 V aa+ = V aa/1+ E , (15) where Vaa is the complex amplitude of the total voltage Here, ZN and ZM are the acoustic impedances in the across the terminals aa, Vaa = Vaa+ + Vaa–, and Vaa– is its superlayer at n = N and in the sublayer at m = M at the part corresponding to the reflected wave. piezoelectric boundaries. The first impedance is calculated by a successive application of formula (12), by The voltage Vaa induced by the source of the signal causes the appearance of voltage across the piezoelec- varying n from 1 to N. The second impedance is found tric element (the bb terminals). As a consequence, a sta- by the same formula, replacing n by m and varying m tionary wave of elastic strain with complex amplitudes from m = 1 to m = M with taking Z2, m – 1 = Z20 at m = 1. u1+ and u1– is excited in the piezoelectric. The waves of The solution of the system of Eqs. (16), (17) gives the acoustic stress and acoustic current appear. The latter complex dimensionless strain amplitudes u1+' and u1Ð' . two have complex amplitudes T1(h1) and I1(h1) at the sublayer boundary and are calculated according to The electric radiation impedance of the piezoelec- Eqs. (9) and (10), if u1+, u1–, and D0 are known. The val- tric element can be found by dividing voltage (7) by

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 158 GRIGOR’EV et al. current (8). Changing to the dimensionless quantities, the transducer output. Then the sought-for transfer con- we obtain stant K+ can be determined by Eq. (1). The transformation coefficient is Z k2 1 ------rad = Ð jj+ ------X ()2 ()β η = K 2, (24) 0 1 + k 1h1 (18) + + Ð jβ h jβ h × []u' ()e 1 1 Ð 1 + u' ()e 1 1 Ð 1 , and the phase difference between the excited acoustic 1+ 1Ð wave and the direct EM wave supplied to the trans- ω where X0 = 1/ C0 is the capacitive resistance of the ducer is piezoelectric element and C = εs/h is its static capaci- 0 1 () tance. ϕ Im K+ + = arctan------(). (25) The electric impedance Z across the terminals aa Re K+ aa Γ and the reflection coefficient E are determined by Eqs. (13) and (14), respectively. Taking that the electric displacement D is predetermined and using the Kirch- INVERSE TRANSFORMATION hoff rules for the electric circuit containing the piezo- Let an acoustic wave be supplied through the acous- electric element with the impedance Z , the shunt rad tic duct to the transducer and let this wave have a given capacitance C , the inductance L, and the resistance sh complex strain amplitude u at the boundary with the R , one easily obtains 2+ los first sublayer. Then, at this boundary, the acoustic stress ω ()ω V aa = j D0s 1 + j CshZrad Zaa. and current of the wave have the amplitudes ω Substituting this formula into Eq. (15) and expressing T 2+ = Ð jZ20 u2+, (26) D through the norm N , we obtain the complex ampli- 0 0 ω tude of the voltage at the input of the transducer: I2+ = Ð j u2+s. (27) V ()ε1 + k2 ε Z The reflected waves are combined with the incident ------aa+ = jω2s ------0 r 10 ones and give rise to the complex amplitudes of acous- N v 0 1 (19) tic stress and current at the boundary: Z × ()ω aa T = Ð jZ ωu ()1 + Γ , (28) 1 + j CshZrad ()------Γ -. 2 20 2+ T 1 + E ω ()Γ Introducing the dimensionless strains in Eqs. (9) and I2 = Ð j u2+s 1 Ð T , (29) (10) and taking z = h , we find the complex amplitudes Γ 1 where T is the reflection coefficient of the elastic- of acoustic stress and current at the output of the piezo- stress wave, which can be easily found, if the acoustic electric to the sublayer with the number M: impedance of the piezoelectric element Z2 is known. In T ()zh= Ð jβ h jβ h the case of the inverse transformation, this impedance 1 1 ω[]1 1 1 1 ------= jZ10 Ðu1+' e ++u1Ð' e j , (20) characterizes the load of the acoustic duct. A technique N0 Γ for calculating Z2 and then T will be shown below [see Eq. (42)]. I ()zh= Ð jβ h jβ h 1 1 ω ()' 1 1 ' 1 1 ------= Ð j su1+e + u1Ðe . (21) Suppose that we found T and I , which are the input N0 2 2 quantities T21, in and I21, in for the series of sublayers. The values obtained are the input ones for the cas- Considering the system of sublayers as a cascade con- cade connection of the two-port networks represented nection of linear two-port networks characterized by by the media between the piezoelectric and the acoustic the transfer matrix (11), one can determine the corre- duct. Using the transfer matrix (11), one can write the sponding amplitudes T2M, out and I2M, out at the output of equations that provide the output stress and current the entire system. The calculations are performed by amplitudes for each layer, if the input ones are given: Eqs. (22) and (23). After their M-fold application, we obtain the acoustic stress and current amplitudes T1 and ()θ Z2m0()θ T 2m, out = cos 2m T 2m, in Ð j------sin 2m I2m, in, (22) I1 at the input of the piezoelectric at z = 0. In the piezo- s electric, we have the strain waves with the amplitudes u and u and the current density J , which are related ssinθ 1+ 1– e 2m ()θ to the values of T and I determined above by the for- I2m, out = Ð j------T 2m, in + cos 2m I2m, in. (23) 1 1 Z2m0 mulas The output values, T and I , for the layer with the 2m, out 2m, out 2 v number 2m are the input ones, T and I , ω()1 k Z10 1 2(m Ð 1), in 2(m Ð 1), in Ð jZ10 u1+ Ð u1Ð + jJe------= T 1, (30) for the layer 2(m Ð 1). Performing the calculations by ω 1 + k2 ε Eqs. (22) and (23) with m changing from M to 1, we ω () obtain the complex amplitude T2+ of acoustic stress at Ð j su1+ + u1Ð = I1. (31)

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 TRANSFER CONSTANT OF A MULTILAYER ACOUSTIC TRANSDUCER 159

Knowing Je, one can also easily calculate the voltage At the other boundary, at z = 0, we cannot write a amplitude Vaa+ at the output of the transducer (the ter- similar condition, because, in this case, the acoustic minals aa). impedance at this boundary is the sought-for quantity Equations (30) and (31) involve three unknown depending on the electric load of the piezoelectric element. Therefore, as the second equation, we use the interrelated quantities u1+, u1–, and Je. To find the last one, it is necessary to obtain one more equation. For condition of the existence of stationary oscillations this purpose, one can use the condition for the existence written for the terminals bb. Using the dimensionless of stationary oscillations at the terminals bb: strain amplitudes, we obtain β β Z +0,Z = ()Ð j 1h1 ()j 1h1 piez L (32) u1+' e Ð 1 + u1Ð' e Ð 1 where Z is the electric impedance of the piezoelec- 2 piez ()β h ()1 + k Z (37) tric element in the case of inverse transformation, and = ------1 1 - j-----L- + 1 . 2 X ZL is the load impedance at the terminals bb. The first k 0 quantity can be calculated by dividing voltage (7) across the piezoelectric element by current (8). The sec- The normalized acoustic strain and current in the piezo- ond quantity can be found by the formula electric at the boundary z = 0 (i.e., at the output of the sublayer 2M) are Z ++R jωL 0 los ω[]()' ' ZL = ------ω ()ω . (33) T 2M, out/N0 = Z10 j Ðu1+ + u1Ð Ð 1 , (38) 1 + j Csh Z0 ++Rlos j L ω()' ' As a result, we obtain from Eq. (32): I2M, out/N0 = Ð j u1+ + u1Ð s. (39) β β ()Ð j 1h1 ()j 1h1 Using the transfer matrix (11), we can write the equa- u1+ 1 Ð e + u1Ð 1 Ð e tions allowing one to determine the input stress and cur- 2 ε (34) rent amplitudes for each sublayer, if the output ones are ()1 + k known: + JesZL Ð jX0 ------2 ------0.= k Z10v 1 Z2m, 0 T , = ()cosθ T , + j------()sinθ I , , (40) From Eqs. (30), (31), and (34), one can easily deter- 2m in 2m 2m out s 2m 2m out mine numerically the complex amplitude of the density J of electric current flowing through the piezoelectric θ e ssin 2m ()θ element and, then, calculate the amplitude of the volt- I2m, in = j------T 2m, out + cos 2m I2m, out. (41) Z , age across the terminals aa: 2m 0 Taking the stress and the current at the M-sublayer J sZ Z 2 e L 0 output to be equal to the values from Eqs. (38) and (39) V aa+ = ------ω -. (35) Z0 ++Rlos j L and then applying successively Eqs. (40) and (41) for m Turn now to the beginning of this section and dem- varying from M to 1, one can calculate T21, in and I21, in, Γ whose ratio gives the acoustic impedance Z2. Then, the onstrate how the reflection coefficient T is found. Out- line first the way of solving this problem. First, it is nec- sought-for reflection coefficient will be essary to calculate the dimensionless amplitudes of the Γ Z2 Ð Z20 direct and inverse waves u' and u' in the piezoelec- T = ------. (42) 1+ 1Ð Z2 + Z20 tric. Then, we find the amplitudes of the normalized Γ acoustic stress T /N and current I /N in the piezoelec- It is this value T that should be substituted into Eqs. (28) 1 0 1 0 and (29). tric at the sublayer boundary z = h1. Further, using the transfer matrix (11), these quantities should be recalcu- The transfer constant K– is calculated according to definition (2). In this case, the amplitude of the output lated to the acoustic duct output, i.e., T21, in/N0 and voltage V is determined by Eq. (33), and the ampli- I21, in/N0 should be found, their ratio being the acoustic aa+ tude of the input elastic stress T2+ is found by Eq. (26). impedance Z2 in the output plane of the acoustic duct. It is the latter quantity that determines the sought-for The coefficient for the inverse (acoustoelectric) trans- Γ reflection coefficient T. formation is calculated by Eq. (24), and the phase difference between the output and input signals is calculated The continuity condition for the impedance at the by Eq. (25) with a change of indices from + to Ð. boundary z = h1 between the piezoelectric and the adjacent superlayer, when written for the dimensionless displacement amplitudes, has the form COMPARISON OF THE DIRECT AND INVERSE β β TRANSFER CONSTANTS Z Ð j 1h1 Z j 1h1 u' ------3N Ð 1 e + u' ------3N + 1 e = Ð j. (36) 1+Z 1ÐZ On the basis of the above algorithms, two indepen- 10 10 dent PC programs with the outer frequency loop were Here, Z3N is calculated by Eq. (12). developed.

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 160 GRIGOR’EV et al.

f, GHz The model analyzed was the structure shown in 0.2 0.4 Fig. 1. The piezoelectric layer was taken to be connected with the acoustic duct by ﬁve intermediate metal 0.4 –5 sublayers, and the other side of the piezoelectric was 0.3 loaded with a rear absorbing load with the help of two –10 η 0.2 ϕ ϕ 0.1 metal superlayers. It was assumed that, depending on π / , dB –15 0 the transformation direction, the line was either + ϕ

η –0.1 matched with the generator or loaded by a matched –20 –0.2 load. The transformation coefﬁcient and the phase dif- η –25 –0.3 ference between the output and the input signals were –0.4 calculated by Eqs. (24) and (25). –30 The analysis included wide-range variations of the layers thickness, the acoustic wave resistance of layers Fig. 2. Frequency dependences of the transformation coef- ﬁcient η( f ) and the phase shift ϕ( f ) between the output and and the sound velocity in them, the inductance and the input signals. The piezoelectric is (Y + 36°)-cut lithium nio- active resistance of the connecting wire, the wave resis- bate (32 µm). The acoustic duct is fused quartz. The rear tance of the transmission line, the acoustic wave resis- load is epoxy resin. The sublayers (beginning from the piezo- tances of the acoustic duct and the rear absorbing load, electric layer) are chromium (0.04 µm), copper (0.2 µm), indium (1 µm), copper (0.2 µm), and chromium (0.04 µm). the parasitic shunt capacitance, and the area and thick- The superlayers are (in the same order) chromium (0.04 µm) ness of the piezoelectric element. and copper (0.2 µm). L = 6 nH, R = 0.5 Ω, C = 0, Z = Ω los sh 0 In all cases, the direct and the inverse transfer con- 50 . The diameter of the transducer is 5 mm. ϕ stants differed only in sign: K+ = –K–, while tan and, consequently, the angle ϕ were the same. The fre- ϕ ϕ f, GHz quency dependences of the angles + and – were 0.2 0.4 almost linear with negative derivatives ∂ϕ/∂f in the vicinity of the maximal values of η or η . It is well 0.4 + – –5 0.3 known that the linear frequency dependence of the out- putÐinput phase shift takes place but in the case of the –10 0.2 0.1 transmission line section without dispersion.

–15 ϕ 0 π /

, dB Figures 2 and 3 illustrate the results of calculating + –0.1 ϕ η ϕ η –20 + and + as functions of frequency for two specific η –0.2 transducers. Similar curves are obtained in the case of –25 –0.3 the inverse transformation. The transducers in question –0.4 –30 were structures consisting of a (Y + 36°)-cut lithium niobate plate, an acoustic duct made of fused quartz, a Fig. 3. The same as in Fig. 2 for the thickness of the indium rear acoustic load of epoxy resin, five sublayers, and layer 6.5 µm. two superlayers. The materials and the thicknesses of the sublayers and superlayers, as well as the necessary geometric and electric parameters of the transducers, are For the case of the direct transformation, the pro- indicated in the cutline of Fig. 2. The results presented in gram successively executes the inner loops of calculat- Figs. 2 and 3 were obtained for the 10Ð450 MHz fre- η ing the values of ZM and ZN by Eq. (12); the simulta- quency range. The dependence ( f ) has two maximums neous solution of Eqs. (16) and (17); the calculations by in the aforementioned range. One is near 100 MHz, and Eqs. (18), (13), (14), and (19)Ð(21); the inner loop of the other is at about 320 MHz. The second maximum calculation by Eqs. (22) and (23); and, finally, the cal- relates to the excitation of the so-called third harmonic. culation of K , η , and ϕ by Eqs. (1), (24), and (25). The calculation at higher frequencies reveals other + + + maximums at frequencies of ~560 MHz (the fifth har- For the inverse transformation, the program suc- monic), ~790 MHz (the seventh harmonic), and so on. cessively executes the inner loop of calculating ZN by The effectiveness of excitation decreases compared to Eq. (12); the simultaneous solution of Eqs. (36) and the previous maximum by ~10, ~5, ~3 dB, etc., respec- (37); calculations by Eqs. (38) and (39); the inner loop tively. The nonsymmetric nature of the curves is caused of calculation by Eqs. (40) and (41); calculations by by the influence of intermediate layers. Variations of their thickness can lead to a considerable change in the Eqs. (28), (29), and (42); the M-fold loop of calculation curves’ shapes. It can be seen that, for a thin indium by Eqs. (22) and (23); calculations by Eq. (33); the layer (1 µm), the dependence η( f ) (see Fig. 2) is nar- simultaneous solution of Eqs. (30), (31), and (34); cal- rower, though higher, than for a thick layer (6.5 µm) culations by Eq. (35); and, finally, the calculation of K–, (see Fig. 3). As for the phase shift ϕ, its frequency η ϕ –, and – by Eqs. (2), (24), and (25) after changing the dependence obtained for the thin indium layer is less indices + to Ð in Eqs. (24) and (25). steep and deviates more widely from a linear one. The

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 TRANSFER CONSTANT OF A MULTILAYER ACOUSTIC TRANSDUCER 161 observed jumps of the phase angle are exactly equal to 2. T. M. Reeder and D. K. Winslow, IEEE Trans. Micro- π, and the derivative ∂ϕ/∂f at the jump points remains wave Theory Tech. 17 (11), 927 (1969). continuous. 3. J. D. Larson, T. M. Reeder, and D. K. Winslow, IEEE It should be noted that, in the frequency interval Trans. Microwave Theory Tech. 18 (9), 602 (1970). from ~0.18 to ~0.25 GHz, the phase angle exhibits a smooth curve similar to an upturned letter N with a 4. Yu. A. Zyuryukin, V. I. Nayanov, and V. A. Polotnyagin, ∂ϕ ∂ Radiotekh. Élektron. (Moscow) 15, 797 (1970); ibid. 15, region where the derivative is / f > 0. This is 1059 (1970). observed near the point, where the transformation coef- ﬁcient drops to zero. 5. V. A. Polotnyagin and V. N. Shevchik, Radiotekh. Élek- tron. (Moscow) 17, 1260 (1972). CONCLUSION 6. M. A. Grigor’ev, S. S. Kuryshov, and A. V. Tolstikov, Akust. Zh. 36, 255 (1990) [Sov. Phys. Acoust. 36, 139 In this paper, on the basis of the wave approach, we (1990)]. developed algorithms for calculating the direct and inverse transfer constants of a piezoelectric transducer 7. M. A. Grigor’ev, V. V. Petrov, and A. V. Tolstikov, with an arbitrary number of intermediate layers and Radiotekh. Élektron. (Moscow) 35, 1977 (1990). with a rear load. We also developed the programs for a 8. Isao Ishikawa, Kageyoshi Katakura, and Yukio Ogura, PC and applied them to diverse variants of transducers, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 46, 41 which allowed us to demonstrate that the direct and the (1999). inverse transfer constants differ only in sign. The 9. D. A. Knapik, B. Starkoski, Ch. J. Pavlin, and F. S. Fos- amplitudeÐfrequency and phaseÐfrequency character- ter, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 47, istics obtained for the direct transformation and the 1540 (2000). inverse one fully coincide. The algorithms developed for calculating the transfer constants can be used in the 10. V. V. Furduev, The Reciprocal Theorems in Mechanical, Acoustic, and Electromechanical Quadripoles (GITTL, analysis of the pulse responses of complex multilayer Moscow, 1948). piezoelectric transducers. 11. A. L. Fel’dshteœn, L. R. Yavich, and V. P. Smirnov, Com- ponents of Waveguide Technology: A Handbook REFERENCES (Sovetskoe Radio, Moscow, 1967). 1. N. F. Foster, G. A. Coquin, G. A. Rozgonyi, and F. A. Vannata, IEEE Trans. Sonics Ultrason. 15 (1), 28 (1968). Translated by A. Kruglov

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002

Reflection of Normal Acoustic Waves in Thin Plates from a Grating with a Periodically Distributed Mechanical Load S. G. Joshi*, B. D. Za œtsev**, and I. E. Kuznetsova** * Department of Electrical and Computer Engineering, Marquette University, Milwaukee, WI 53201-1881, USA ** Institute of Radio Engineering and Electronics, Russian Academy of Sciences, Saratov Branch, ul. Zelenaya 38, Saratov, 410019 Russia e-mail: [email protected] Received December 29, 1999

Abstract—Reflection of zero-order normal acoustic waves excited in a thin piezoelectric plate from a set of conducting strips of a finite thickness is studied both theoretically and experimentally. The analysis shows that the effects produced by the short-circuiting of the plate surface and by the elastic load on the impedance ratio of adjacent plate segments are in opposition to each other. These effects can be commensurable, and, hence, for each wave type, there is a certain value of the strip thickness at which the reflection coefficient becomes equal to zero. The experimental results obtained for a shear horizontal normal wave (an SH0 wave) propagating in a lithium niobate plate are in good agreement with the theory and justify the use of the equivalent-circuit model in analyzing the properties of reflectors of the type under study. © 2002 MAIK “Nauka/Interperiodica”.

A reflector for normal acoustic waves is a useful, To calculate the characteristics of such a reflector, and often indispensable, component for many practical we used an equivalent circuit consisting of alternating applications [1, 2]. Such reflectors are used in resona- segments of sound channels with different wave imped- tors, unilateral SAW transducers, delay lines with low- ances. The theoretical analysis showed that an efficient level spurious signals, directional couplers, various reflection of zero-order normal acoustic waves can be kinds of filters, etc. Several types of reflection gratings achieved by using a grating with a relatively small num- based on the use of electric and mechanical loads, as well ber of elements. The experimental study of an SH0 wave as etched grooves on a substrate surface, have been suc- reflector that was made on the basis of silver strips cessfully implemented in the design of SAW reflectors. deposited on a thin lithium niobate plate showed a good Recently, normal acoustic waves in thin plates have agreement with the theoretical results. This fact justi- attracted considerable interest because of their unique fied the use of the equivalent-circuit model in analyzing properties, which are promising for designing new sen- the reflection of acoustic waves in piezoelectric plates sors and data processing devices and also for studying under the effect of a purely mechanical load. the properties of materials [3]. Evidently, the aforemen- Let us consider the theoretical model of a reflector tioned types of SAW reflectors can also be used for the with a mechanical load. The structure of such a reflec- reflection of acoustic waves in plates. The reflection of tor is shown in Fig. 1. It consists of a grating formed by a zero-order normal acoustic wave with the shear hori- finite-thickness strips deposited on the surface of a zontal polarization (an SH0 wave) from a set of thin conducting strips arranged on the YÐX surface of lith- plate. Adjacent plate segments have different acoustic ium niobate was studied in [4]. The results of this study impedances, and, therefore, a propagating wave is showed that the reflection of normal acoustic waves in reflected from every strip and gives rise to a resulting plates was more efficient than the reflection of SAW reflected wave. The operation of this reflector with propagating in the same material. This result is allowance for multiple internal reflections can be ana- explained by the stronger electromechanical coupling lyzed with the help of the equivalent circuit convention- of waves in plates, as compared to SAW. From physical ally used for SAW [5]. The circuit was modified for considerations, one would expect that gratings with a describing the reflection of acoustic waves in plates periodic mechanical load must also be more effective from a set of thin conducting electrodes [4]. The modi- for waves in plates. In this paper, we present the first fied circuit contains no electromechanical transducers, results obtained by studying the reflection of zero-order because the aforementioned electrodes are not con- normal acoustic waves from a grating with a periodic nected with each other. The modified circuit used in our mechanical load. study is shown in Fig. 2. Each grating element is repre-

REFLECTION OF NORMAL ACOUSTIC WAVES IN THIN PLATES 163 sented by a T-section, and the elements of the circuit are a b characterized by the quantities () Z1 = iZtan kb/2 , (1) 1 2 N d () Z2 = ÐiZ/sin kb , (2) () h Z1m = iZm tan kma/2 , (3) () Z2m = ÐiZm/sin kma . (4) Fig. 1. Structure of a reflector for waves in a plate. The Here, a and b are the lengths of the reflector seg- reflector contains N reflecting elements. ments with and without mechanical load, respectively (see Fig. 1), and Z = vρS and k are the total mechanical impedance and the wave number in the plate without A Z1m Z1m Z1 Z1 Z1m Z1m load, where ρ is the density, v is the wave velocity, and S is the cross-sectional area of the acoustic waveguide. v ρ The quantities Zm = m mSm and km are the mechanical Z Z Z Z impedances and the wave numbers for the mechani- 2m 2 2m ρ cally loaded segments, where m and Sm are the effective densities and cross-sectional areas of the two-layer B waveguide. The meaning of these two quantities will be explained below. The total reflection coefficient R of Fig. 2. Equivalent circuit of a reflector with a mechanical the whole grating can be determined from the relation load. R = (Z – Zin)/(Z + Zin), where Zin is the total input impedance of the whole reflector. The equivalent circuit Reflection coefficient shown in Fig. 2 allows one to calculate the reflection 1.0 4 coefficient as a function of the frequency and the number of reflecting elements for different values of the 0.8 3 2 impedance ratio of adjacent segments Zm/Z. Evidently, 0.6 the reflection of the acoustic wave is caused by a jump in the wave parameters of the grating elements, namely, 0.4 1 ≠ when Zm /Z 1. The wider the ratio Zm/Z differs from 0.2 unity, the greater the reflection coefficient is. Figure 3 0 shows the characteristic dependences of the reflection 0.7 0.8 0.9 1.0 1.1 Zm/Z –0.2 coefficient on the ratio Zm/Z at the central frequency for different numbers of strips in the grating. –0.4 1 Now, let us estimate the impedance ratio Zm/Z for a –0.6 reflector with a mechanical load. In the case of thin con- 2 –0.8 ducting electrodes whose mass is neglected, we have 3 Zm/Z = vm/v [4]. In the case of a grating with a periodic –1.0 4 mechanical load, the situation becomes more complicated, because the adjacent segments differ not only in Fig. 3. Dependence of the reflection coefficient on the velocity, but also in density and in cross-sectional area. impedance ratio Zm/Z at the central frequency for different numbers of strips in the reflection grating: N = (1) 2, (2) 5, As the effective density for a mechanically loaded seg- (3) 8, and (4) 12. ment of the grating, we used the density value averaged over the cross section of the waveguide; i.e., the den- ρ ρ ρ sity ratio was determined by the formula m/ = [ h + one shear horizontal normal wave (the SH wave). For ρ ρ ρ 0 1d)]/ (h + d), where 1 and d are the density and the each of these waves, the cut of the plate, the propaga- thickness of the layer, respectively. The ratio of the cross- tion direction, and the plate thickness were selected so sectional areas was determined as Sm/S = (h + d)/h. as to obtain the maximal coefficient of electromechan- To determine the velocities v and vm, we performed ical coupling K. For lithium niobate, the corresponding a rigorous analysis of zero-order normal acoustic waves parameters for each of the aforementioned modes are as propagating in single-layer and two-layer waveguides follows [6]: for the A0 wave, the Y + 128° cut, the X based on lithium niobate. We assumed that the substrate λ thickness was much smaller than the wavelength λ, i.e., direction of propagation, and h/ = 0.25; for the S0 h/λ Ӷ 1. In this case, only three types of waves can wave, the Y cut, the X + 50° direction of propagation, λ propagate in the plate: two Lamb normal waves (the and h/ = 0.1; and for the SH0 wave, the Y cut, the X λ antisymmetric A0 wave and the symmetric S0 wave) and direction of propagation, and h/ = 0.1.

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 164 JOSHI et al.

Zm/Z both the electric potential and the normal component of 1.3 the electric displacement and the zero stress value at the 1 1 (a) 2 piezoelectricÐvacuum boundary; and (iii) the zero stress 1.2 3 value at the layerÐvacuum boundary. From the theo- 4 2 retical analysis, we determined the dependence of the velocity on the normalized thickness h/λ for a given 1.1 3 m ratio d/h. Then, to determine the velocity vm and the 4 λ normalized plate thickness h/ m, we used the condi- 1.0 tion ()λ ()λ 0.9 h/ m v m = h1/ 1 v , (5)

1.2 1 which meant that the frequency of the propagating (b) wave was the same within loaded and unloaded seg- 4 2 ments. 1.1 3 The calculated dependences of the impedance ratio 1 on d/h are shown in Figs. 4aÐ4c for the A , S , and SH 1.0 0 0 0 2 waves, respectively. The mechanical load was made of 3 the following materials: gold, silver, titanium, and 0.9 4 aluminum with the densities 19.31, 10.5, 4.5, and 2.7 g/cm3, respectively. The density of lithium niobate 0.8 was 4.628 g/cm3. Figure 4 presents the results for two situations: the regions between the strips are electri- 1.2 1 cally free (the solid lines) and short-circuited by a thin conducting film (the dashed lines). (c) 2 1.1 3 In the first case, two factors are present: a mechani- 4 cal load on the surface and the short-circuiting of the 1.0 surface. The latter factor reduces the acoustic imped- 1 ance, whereas the mechanical load causes an increase 2 in the acoustic impedance with increasing layer thick- 0.9 3 4 ness. Thus, the aforementioned factors “work” in opposite directions; i.e., for certain values of d/h, the 0.8 acoustic impedance ratio becomes equal to unity. This –3 –2 –1 10 10 10 occurs when the electric and mechanical effects com- d/h pensate each other and the total reflection coefficient is equal to zero. Fig. 4. Dependence of the impedance ratio Z /Z on d/h for m When the regions between the strips are short-cir- the (a) A0, (b) S0, and (c) SH0 waves. The layers are made of (1) gold, (2) silver, (3) titanium, and (4) aluminum. The cuited, the jump in the electric boundary conditions is surface areas between the strips are electrically free (the absent, and the reflection is caused by the mechanical solid lines) and short-circuited by a thin conducting film load only. (the dashed lines). The aforementioned dependences allow one to optimize the situation when the impedance ratio is suffi- The mechanical loads were metal layers, which cient to obtain an efficient reflector governed exclu- were easy to deposit and allowed a choice of acoustic sively by the mechanical load. impedance in a wide range of values. At first, we deter- In line with the results presented above, we per- mined the wave velocity v for a given plate thickness formed an experimental study of the reflection of an h/λ. For this purpose, we used the method described in SH0 wave from a grating with a purely mechanical load. The acoustic waveguide was made on the basis of a [7]. To determine the velocity vm, we analyzed the propagation of acoustic waves in a two-layer structure Y-cut lithium niobate plate with the direction of propa- containing a piezoelectric substrate and a perfectly con- gation along the X axis. To excite an acoustic wave at an ducting elastic layer of finite thickness. We used the operating frequency of 3 MHz, we used an interdigital transducer with a period of 1.2 mm and with two finger standard equations of motion for both elastic media, the pairs. To reduce the reflection of the acoustic wave from Laplace equation for the piezoelectric plate and the the transducer and to reduce the level of the multiple vacuum, and the corresponding equations of state. The reflection signal, the transducer was made on the basis mechanical and electric boundary conditions were as of split electrodes. The capacitance of the transducer follows: (i) the stress and displacement continuity and was compensated by a corresponding inductance, and the zero value of the electric potential at the boundary the transducer was matched with the transmission line between the two elastic media; (ii) the continuity of through a matching transformer. The reflector consisted

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 REFLECTION OF NORMAL ACOUSTIC WAVES IN THIN PLATES 165 of five silver strips 10 µm thick, which were vacuum Reflection coefficient deposited with a period of 0.6 mm on the plate surface. 0.7 To obtain a good adhesion of the reflecting strips, we first deposited a thin chromium film. This film short- 0.6 circuited the tangential components of the electric field accompanying the wave within the unloaded segments, and, thus, the reflection occurred due to the mechanical 0.5 load only. The amplitudes of the incident and reflected waves, Ai and Ar, were measured in a pulsed mode 0.4 using an electrostatic probe [4]. The latter was a piece of a coaxial line whose central lead ended in a thin tungsten needle 100 µm in diameter. The opposite end 0.3 of the coaxial line was cable-connected with the measuring oscilloscope. The tungsten needle of the probe 0.2 touched the surface of the piezoelectric and could be moved smoothly from the transducer to the reflector by 0.1 a precision mechanism. With the propagation of an 100 200 300 400 500 600 acoustic wave pulse, an electric signal proportional to Plate thickness, µm the strength of the piezoelectric field accompanying the wave appeared at the probe output. By changing the Fig. 5. Dependence of the reflection coefficient on the plate probe position, it was possible to separate in time the thickness for an SH0 wave propagating in a Y-cut lithium pulses corresponding to the incident and reflected niobate plate along the X axis: the theoretical dependence waves on the screen of the oscilloscope. After the (the solid line) and the experimental data (the squares). amplitudes of these waves were measured, the reflection coefficient R was calculated as R = Ar/Ai. To study equivalent circuits is suitable for analyzing the reflec- the reflection of acoustic waves at different values of tors under discussion. d/h, the plate thickness was decreased by polishing the back side of the plate. For this purpose, we used a special glass unit intended for fastening the plate, as ACKNOWLEDGMENTS described in [8]. This method allowed us to reduce the This work was supported by the US National Sci- plate thickness from its initial value of 500 µm to a ter- ence Foundation. minal value of 100 µm and to measure the characteristics of the reflector for several intermediate values of d/h. Figure 5 shows the reflection coefficient obtained REFERENCES from the experiment as a function of the plate thickness 1. C. K. Campbell, Surface Acoustic Wave Devices for along with the corresponding theoretical dependence Mobile and Wireless Communications (Academic, San calculated with allowance for the short-circuiting of the Diego, 1998). surface. One can see a good agreement between the 2. R. C. Williamson, in Surface Wave Filters, Ed. by experimental data and the theoretical curve. H. Matthews (Wiley, New York, 1977). 3. D. S. Ballantine, R. M. White, S. J. Martin, A. J. Ricco, Thus, the theoretical analysis shows that, for a E. T. Zellers, G. C. Frye, and H. Wohltjen, Acoustic Wave reflector based on a set of conducting strips of finite Sensors (Academic, San Diego, 1997). thickness, the effect of short-circuiting of the plate sur- 4. B. D. Zaitsev and S. G. Joshi, IEEE Trans. Ultrason. Fer- face and the effect of an elastic load on the impedance roelectr. Freq. Control 46 (6), 1539 (1999). ratio of adjacent plate segments are in opposition to 5. W. R. Smith, in Physical Acoustics, Ed. by W. P. Mason each other. For the efficiency of a reflector of acoustic (Academic, New York, 1981), Vol. 15. waves in plates, the effects of a mechanical load and the 6. I. A. Borodina, S. G. Joshi, B. D. Zaœtsev, and I. E. Kuz- short-circuiting of the plate surface are commensura- netsova, Akust. Zh. 46, 42 (2000) [Acoust. Phys. 46, 33 ble. Therefore, for each type of waves, there exists a (2000)]. certain value of the strip thickness at which the reflec- 7. S. G. Joshi and Y. Jin, J. Appl. Phys. 70, 4113 (1991). tion coefficient becomes zero. This conclusion is of 8. B. D. Zaitsev, S. G. Joshi, and I. E. Kuznetsova, IEEE practical importance for the development of a reflector Trans. Ultrason. Ferroelectr. Freq. Control 46 (5), 1298 (1999). with a controlled reflection coefficient. The experimental results presented above are in good agreement with the theory and testify that the use of the model based on Translated by E. Golyamina

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002

Sound Scattering from Small Compact Inhomogeneities in a Sea Waveguide: The Inverse Problem A. D. Zakharenko Paciﬁc Oceanological Institute, Russian Academy of Sciences, Far East Division, ul. Baltiœskaya 43, Vladivostok, 690041 Russia e-mail: [email protected] Received April 4, 2001

Abstract—An approach based on the redistribution of acoustic energy between waveguide modes is used to state the inverse problem of sound scattering from small compact inhomogeneities in a shallow-water waveguide. Numerical results are presented. © 2002 MAIK “Nauka/Interperiodica”.

As shown in earlier publications [1, 2], three-dimen- tion. For simplicity, we consider only the case of a sur- sional problems of sound scattering in a shallow-water face inhomogeneity of the sea bottom. The efficiency of waveguide can be successfully approached from the the approach is illustrated by the results of model cal- point of view of the energy redistribution between culations. waveguide modes. Weston and Fawcett [2] considered the sound scattering by the bottom inhomogeneities and derived simple formulas for the mode conversion STATEMENT OF THE INVERSE PROBLEM coefficients, Smn, which determine the amount of the We consider a stationary sound field of circular fre- acoustic energy scattered into the nth mode from the quency ω, which is described by the complex sound mth mode of the incident field. Fawcett [3] applied pressure u in the region –H ≤ z ≤ 0, where z is the ver- these formulas to reconstruct the form of axially sym- tical coordinate. The horizontal coordinates are denoted metric inhomogeneities of the bottom by remote sens- by x and y. ing. Similar formulas were derived by Fawcett [4] for ρ acoustic scattering by three-dimensional axially sym- We assume that the density and the sound velocity metric inhomogeneities of the density and sound veloc- Ò are piecewise continuous functions with discontinui- ity. The representation of the sound field as the super- ties at the surface z = h(x, y), at which the following position of modes was used to state inverse problems of conditions are fulfilled scattering and minimization [5]. 1∂u 1∂u In the previous paper [6], we obtained new formulas u+ ==uÐ, ρ---∂----- ρ---∂----- , n + n Ð for the mode conversion coefficients, Smn, from small compact inhomogeneities of the density, the sound where the symbols + and Ð denote the limits of the velocity, and the internal boundaries of the sea bottom. variables at a considered point z = z0 from above, i.e., In these formulas, the inhomogeneities are represented ∂ ∂ at z > z0, and from below, respectively, and / n denotes as the Fourier and FourierÐBessel series in angular and the derivative with respect to the normal. radial coordinates, respectively. This leads to a more detailed analytical description of acoustic scattering. Next, we assume that the sound velocity and the density above and below the interface depend only on In this paper, on the basis of these formulas, we state z, and a function describing this interface can be repre- the problem of the reconstruction of an inhomogeneity sented as h(x, y) = h + h (x, y), where h is a constant from the observation data obtained in the far field. The 0 1 0 and h1 is a small quantity, as compared to the typical problem is formulated as a linear operator equation of π ω wavelength L = 2 c0/ (c0 is the typical sound veloc- the first kind, which can be solved by the method of sin- ity), and is zero outside a limited region Ω. gular value decomposition. Cutting off the series representing inhomogeneities gives a regularizing effect, In the Born approximation, the acoustic pressure is which complements the mechanism of the regulariza- represented by u = uinc + usc, where uinc is the incident tion on the basis of the selection of singular vectors field and usc is the principal term of the scattered field. As with small singular values. Such a statement, contrary the incident field, we consider the field of a vertically dis- to that used in [5], allows one to solve problems of tributed source (a vertical array) located at the point with higher dimension with a smaller amount of computa- the horizontal coordinates x0, y0. Such a field can be rep-

SOUND SCATTERING FROM SMALL COMPACT INHOMOGENEITIES IN A SEA WAVEGUIDE 167 resented as a linear combination of modes: uinc = where () 1 2 2 mp π exp()ik r Ð iπ/4 im()αÐ s ∑a H (k ξ)φ (z), where ξ = ()xxÐ + ()yyÐ i 2 n r 0 j 0 j j 0 0 Aˆ jns = Ð------e . φ 4 and j and kj are determined by the eigenvalue problem knrr dφ ω2 { ()1 ()mp ρ d 1 j φ 2φ φ 2G1 HmsÐ k jr0 Fs + ikjknG2 -----------+ -----2- j ==k j j, j z = 0 0, dz ρ dz () 2iα () c × []1 () 0 1 ()mp } HmsÐ2+ k jr0 + e HmsÐ k jr0 Fs Ð 1 , dφ ------j = 0, L γ m mp  dz zH= Ð F = J -----p r J ()k r J ()k r rrd , (3) s ∫ mL msÐ j s n with the internal boundary conditions 0 1dφ 1dφ ()ρ ρ φ φ φ ==φ , ------j ------j Ð Ð + d n d j j+ jÐ ρ ρ G1 = ------dz + dz Ð ρ2 dz dz  + + + at z = h0 and with the eigenfunction normalization  0 ω21 1 φ φ  1 2 -+ ------Ð ------n j , ---φ dz = 1. ρ 2 ρ 2  ∫ ρ j +c+ ÐcÐ zh= 0 ÐH 1 1 The scattered field can also be represented as a lin- G = φ φ ----- Ð ----- . φ 2 n jρ ρ ear combination of the functions j with range-depen- + Ð zh= 0 dent coefficients. In particular, if the incident field con- In these formulas, it is assumed that rr is large enough ()1 inc ξ φ () sists of a single mode u = H0 (kj ) j(z), the scattered 1 to replace the Hankel functions Hµ (knrr) by the prin- field can be represented as ∞ ciple terms of their asymptotics at rr . We note sc ()φ, , () u = ∑S jn xy; x0 y0 n z , that no similar limitations are imposed on the source coordinates. where the coefficients S , which are called the mode jn Formulas (1) and (2) determine the linear transfor- conversion coefficients, describe the scattering from the mations from the data specifying the inhomogeneity to jth mode of the incident field to the nth scattering mode. the scattering data: In practice, these coefficients are easily obtained from ˆ {}˜ {}˜ (), α the analysis of the mode composition of the sound field A: h1mp S jns rr; r0 0 , measured by vertical or horizontal arrays. {}˜ {}(), α , α The expressions for the mode conversion coeffi- A: h1mp S jn rr r; r0 0 , cients Sjn derived in [6] are used to represent h1 as a where –∞ < m < ∞, 1 ≤ p < ∞, –∞ < s < ∞, and 0 < j, double Fourier and FourierÐBessel series (see, e.g., [8]) n ≤ M, where M is the number of the modes under con- in angular and radial coordinates, respectively, in the sideration; (r , α ) ∈ Er and (r , α ) ∈ Es; Er and Es are α r r 0 0 cylindrical coordinate system (r, , z): the sets of the horizontal coordinates of the points ∞ ∞ where the sources and receivers are located. The matrix (), α ()α ˜ ()γ h1 r = ∑ ∑ exp im h1mpJm r p/L , elements of the operator Aˆ in the key case of a single m = Ð∞ p = 1 source and a single receiver are given by Eq. (3). where L is such that the region Ω is within the circle of The inverse problem consists in the inversion of γ radius L with the center at the coordinate origin, p are these operators on the measured scattering data, i.e., in the positive roots of the equation Jm(r) = 0, and Jm are solving the operator equations of the first kind. The use the Bessel functions. Denoting the horizontal coordi- ˆ α α of the scattering data {S jns (rr; r0, 0)} is natural, if nates of the receiver and the source by (rr , r) and α measurements were carried out on the circles of suffi- (r0 , 0), respectively, we write the expressions for Sjn ciently large radius. The typical finite-dimension from [6] as the Fourier series approximation of this problem is obtained by limiting ∞ the number of the reconstructed coefficients h˜ and (), α , α ˆ (), α ()α 1mp S jn rr r; r0 0 = ∑ S jns rr; r0 0 exp is r , the number of the Fourier harmonics in the mode con- s = Ð∞ (1) version coefficients and by presetting the finite sets Er s where the coefficients Sˆ jns are given by the formula and E . In this case, we do not require that the number of the reconstructed coefficients coincide with the ∞ ∞ dimension of the space of scattering data. In this situa- mp ˜ Sˆ jns = ∑ ∑ Aˆ jnsh1mp, (2) tion, we deal with a pseudoinversion of the operator A, m = Ð∞ p = 1 and in this paper, we suggest using the method of

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 168 ZAKHARENKO pseudoinversion based on the singular value decompo- The inverse problem was solved with the use of the sition [7]. This method includes the regularization pro- ˆ α scattering data {S jns (rr; r0, 0)}. The calculations were cedure, which consists in projecting the solution on the carried out in the MATLAB system. In this system, the halfspace that is orthogonal to singular vectors with ⑀ ⑀ ⑀ method of pseudoinversion on the basis of the singular singular values lying in the interval (– , ), where is value decomposition was represented by the pinv pro- the regularization parameter. cedure. For the calculations, the scattering data were

prepared in the following way: the exact values Sˆ jns NUMERICAL RESULTS were supplemented by a Gaussian white noise with a

Calculations were carried out for a two-layered normal deviation of 3% from the mean value of Sˆ jns . medium (water/bottom) with a constant density and a The calculations were carried out for several realiza- constant velocity for each layer. For the water layer tions of noise, with two and four sources [in Figs. 1Ð3, with a thickness of 200 m, the density and the sound versions (a) and (b), respectively]. Because of symme- 3 velocity were taken to be 1000 kg/m and 1470 m/s, try, we used only the mode conversion coefficients Sjn respectively; for the bottom layer with a thickness of that satisfy the condition n ≥ j, so that their number 500 m, these parameters were 1150 kg/m3 and 2000 m/s. was equal to 10, and the number of angular harmonics The sound frequency was 20 Hz. For these conditions, in each coefficient was 21. The number of the FourierÐ there are four propagating modes, which persist when Bessel coefficients in the reconstructed function h1 the thickness of the bottom layer tends to infinity. Only was eight, the number of the angular Fourier harmon- these four modes were used in the calculations. ics varied. The test inhomogeneities of the sea bottom were Figures 1Ð4 show the results of calculations for M = M 2 σ2 given by the functions 2. In Figs. 1Ð3, the exact harmonic AMr exp(–r / ) is ˆ h ()r, α = A rM cos()Mα exp()Ðr2/σ2 , compared with the functions Reh1M (r) reconstructed 1 M (4) using eight terms in the FourierÐBessel series whose M = 0123.,,, coefficients were obtained by the pseudoinversion of σ × the operator Aˆ for three different realizations of ran- The parameter was taken equal to 30, A0 = 1, A1 = 8 –2 × –3 × –4 dom noise. 10 , A2 = 3 10 , and A3 = 1 10 . Thus, each inhomogeneity was described by only one angular cosine Figures 1 and 2 show the effect of the method of the pseudoinversion based on the singular value decompo- ˆ M 2 σ2 Fourier harmonic Reh1M (r) = AMr exp(–r / ). In the sition on the results of calculating the regularization argument r, the functions (4) are well approximated by parameter. The regularizing effect of cutting off the the FourierÐBessel series of eight terms. These data series representing the inhomogeneities is shown in were used for calculating the scattering coefficients Sjn, Fig. 3. In all calculations, the mean square deviations of where j, n = 1, …, 4, for the sources with the coordi- the harmonics that are not shown in the figures from α ° ° ° ° nates r0 = 500 m, 0 = 0 , 90 , 180 , and 270 . zero in terms of the mean-square norm are the same as the deviations of the reconstructed fundamental har- In this case, for the values of r0 that are large enough ()1 monic from the exact one. This is clearly seen in Fig. 4, to replace the Hankel function Hs (kjr0) by its asymp- which exhibits the form of the surface of the initial totics, the coefficients Sjn have closed expressions function and the surface reconstructed by 17 Fourier × –6 ()() harmonics with the regularization parameter 5 10 exp iknrr + k jr0 ÐiMπ/2 for four sources (see Fig. 1b). S jn = Ð------e knrr k jr0 The calculations carried out for all remaining func- × cos()M()ψαÐ A ()G + k k G cos()α Ð α tions of the form of Eq. (4) for M = 0, 1, 3 provide, in r M 1 j n 2 0 r general, the same results. κMσ2()M + 1 κ2σ2 The amount of computation, which can be estimated ×  ------expÐ------, as a moderate one, is mainly related to constructing the 2M + 1 4 matrix of the operator Aˆ . Note that, in this case, the where basic properties of the sound waveguide are also κ 2 2 ()α α encoded in the matrix. The amount of calculation = k j ++kn 2k jkn cos 0 Ð r , related to the solution of the operator equation (the Ðk sin()α Ð α pseudoinversion) is relatively insignificant. ψ = arctan ------j 0 r - . k + k cos()α Ð α Thus, the model calculations show that the state- n j 0 r ment of the inverse problem as a linear operator equa- These expressions were used for testing the accuracy of tion on the basis of the formulas for solving the direct the representations of the functions of inhomogeneities problem from the previous paper [6] provides the nec- by their FourierÐBessel series. essary regularization and allows an efficient solution of

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 SOUND SCATTERING FROM SMALL COMPACT INHOMOGENEITIES IN A SEA WAVEGUIDE 169

(a) (b) 1.0

0.8

0.6

0.4

0.2

–0.2 0 50 100 0 50 100

Fig. 1. Reconstruction of the inhomogeneity function (3) for M = 2 with allowance for 17 angular Fourier harmonics: the initial function (the solid line) and the solutions to the inverse problem (the dashed lines) for three realizations of 3% noise. (a) Two sources at the α ° ° α ° ° ° ° ⑀ × –6 angles 0 = 0 and 90 ; (b) four sources at the angles 0 = 0 , 90 , 180 , and 270 . The regularization parameter is = 5 10 .

(a) (b) 1.0

0.8

0.6

0.4

0.2

–0.2 0 50 100 0 50 100

Fig. 2. Reconstruction of the inhomogeneity function (4) for M = 2 with allowance for 17 angular Fourier harmonics: the initial function (the solid line) and the solutions to the inverse problem (the dashed lines) for three realizations of 3% noise. Versions (a) and (b) are the same as in Fig. 1. The regularization parameter is ⑀ = 1 × 10–6.

(a) (b) 1.0

0.8

0.6

0.4

0.2

–0.2 0 50 100 0 50 100

Fig. 3. Reconstruction of the inhomogeneity function (4) for M = 2 with allowance for seven angular Fourier harmonics: the initial function (the solid line) and the solutions to the inverse problem (the dashed lines) for three realizations of 3% noise. Versions (a) and (b) are the same as in Fig. 1. The regularization parameter is ⑀ = 5 × 10–6.

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 170 ZAKHARENKO

(a) (b)

1 1

0 0

–1 –1 100 100 100 100 0 0 0 0 –100 –100 –100 –100

Fig. 4. Surface of the sea bottom inhomogeneities described by function (4) for M = 2: (a) the initial surface and (b) the surface reconstructed on the basis of the data obtained from four sources (Fig. 1b). the inverse problem by conventional means. This 3. J. A. Fawcett, Inverse Probl. 6, 185 (1990). makes the proposed approach suitable for use in prac- 4. J. A. Fawcett, J. Acoust. Soc. Am. 102, 3387 (1997). tice for ﬁeld experiments. 5. R. P. Gilbert, T. Scotti, A. Wirgin, and Y. S. Xu, J. Acoust. The reviewer of this paper pointed to signiﬁcant Soc. Am. 103, 1320 (1998). coherent effects that accompany the backscattering of 6. A. D. Zakharenko, Akust. Zh. 46, 200 (2000) [Acoust. sound [9]. These effects cannot be taken into account in Phys. 46, 160 (2000)]. the Born approximation used in this paper. To include 7. A. E. Albert, Regression and the MooreÐPenrose them in the consideration, the formalism must be Pseudoinverse (Academic, New York, 1971; Nauka, extended, which will be the subject of the following Moscow, 1977). studies. 8. H. Bateman and A. Erdélyi, Higher Transcendental Functions (McGraw-Hill, New York, 1953; Nauka, Mos- cow, 1966), Vol. 2. REFERENCES 9. Yu. A. Kravtsov and A. I. Saichev, Usp. Fiz. Nauk 137 1. F. Ingenito, J. Acoust. Soc. Am. 82, 2051 (1987). (3), 501 (1982) [Sov. Phys. Usp. 25, 494 (1982)]. 2. B. T. R. Wetton and J. A. Fawcett, J. Acoust. Soc. Am. 85, 1482 (1989). Translated by Yu. Lysanov

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002

Konenkov’s Waves in Anisotropic Layered Plates D. D. Zakharov Institute for Problems of Mechanics, Russian Academy of Sciences, pr. Vernadskogo 101-1, Moscow, 117526 Russia e-mail: [email protected] Received April 11, 2001

Abstract—Rayleigh-type ﬂexural waves localized near the edge of a thin anisotropic layered plate are investigated. The following effects are revealed: a wave attenuation with oscillations, a change of sign of the energy ﬂux for certain types of anisotropy, and the appearance of stationary waves. © 2002 MAIK “Nauka/Interperiodica”.

To all appearance, flexural waves of the Rayleigh Consider a thin packet with a symmetric structure, type were first investigated in [1] for the case of aniso- which consists of anisotropic elastic layers. Denote the tropic media. Such waves must be taken into account at total thickness of the packet by 2h, use the dimension- the excitation of a free edge, of the edge contact of dif- less orthogonal coordinates x1, x2 normalized to h, and ferent materials, etc. [2Ð4]. However, unlike plane Ray- take the internal stress-strained state of the layered leigh waves, which are basic in a number of technolo- packet to be a long-wave one, i.e., satisfying the classi- gies used for nondestructive testing, seismic monitor- cal relations of the theory of layered plates [7]. The ing, and the like (the bibliography for the case of interlayer contact is supposed to be an ideal one. All the anisotropic media is presented, e.g., in [5, 6]), flexural elastic constants below are taken to be normalized to waves are not so famous. The reason is the compara- the maximal value (out of the packet layers) of the tively small damping coefficient of a flexural wave (of Young modulus, and the densities are normalized simi- the order of ν4, where ν is the Poisson ratio) in an iso- larly. tropic medium. The appearance and widespread use of promising composite materials raises the following nat- Consider now harmonic oscillations of a semi-infi- ural questions: nite packet characterized by the flexural stiffness 3 Ω ≥ matrix D = d pq and occupying the region : x2 0, (i) do the Rayleigh flexural waves exist in media ∞ ∞ – < x1 < . The packet edge x2 = 0 is taken to be stress- with a rather general type of anisotropy; ω free. Then, the normal deflection w = w∗(x1, x2)exp(i t), (ii) if so, what are their properties (the energy, the normalized to the half-thickness h, satisfies the equation damping factors, etc.); {}()ρω∂ , ∂ 2 (iii) how are these properties affected by the stack- L3 1 2 Ð w* = 0, ing asymmetry in layered media (the correlation ()∂ , ∂ ≡ 3 ∂4 3 ∂3∂ between flexure and the plane strained state). L3 1 2 d11 1 + 4d16 1 2 (1) In spite of the interest manifested by some authors ()∂3 3 2∂2 3 ∂ ∂3 3 ∂4 [4], no detailed investigations of this problem were +2 d12 + 2d66 1 2 ++4d26 1 2 d22 2, published. This paper gives the answers to the first two questions. and the free-edge boundary conditions

()∂ , ∂ ()∂ , ∂ M 1 2 w* ==0, F 1 2 w* 0, (2)

d3 ∂2 ++d3 ∂ ∂ d3 ∂2 M = Ð 12 1 26 1 2 22 2 , F 3 ∂3 ()∂3 3 2∂ 3 ∂ ∂2 3 ∂3 2d16 1 +++d12 + 4d66 1 2 4d26 1 2 d22 2

ω ρ where is the cyclic frequency, is the dimensionless ing to the normal moment M22 and the normal intersect- ∂ ∂ integral density, and M and F are operators corresponding Kirchhoff force P2z = 2 1M12 + 2M22.

172 ZAKHAROV

Let us analyze the possibilities for the existence of roots ξ: Reξ = 0; otherwise, the roots are real or com- wave solutions propagating along the edge and expo- plex ones. nentially decaying inside the packet, i.e., the Rayleigh- Statement 2. All the complex roots of Eq. (3) are type solutions conjugate; two pairs of complex conjugate roots (along () with the flexural wave of the Rayleigh type) can exist ik1 x1 + k2 x2 w ==Ae , A const, Im k < 0. 4 * 2 only in the range s < s∗, s = inf L()1, ξ . We use the designations * ξ ∈ R Indeed, the fourth-order characteristic polynomial 3 d ρω2 k L(1, ξ) is positive definite in the space of real numbers d ===------pq, s4 ------, ξ ----2, pq d 4 k ξ. At s = s∗, due to the polynomial smoothness, Eq. (3) dk1 1 has at least one real root with a twofold multiplicity. For where d is some stiffness value chosen for the normal- s > s∗, the number of real roots is no less than two. ≥ ization. Taking for definiteness k1 > 0, we obtain from Thus, for s s∗, there is no more than one pair of com- Eq. (1) a characteristic equation with constant coeffi- plex conjugate roots, which is insufficient to satisfy two cients for the parameter ξ: boundary conditions (2) with a simultaneous exponential decay along the x axis. (), ξ 4 ≡ ξ 2 L 1 Ð s d11 + 4d16 Statement 3. The phase velocity V of the Rayleigh- (3) R ()ξ2 ξ3 ξ4 4 type flexural wave is bounded from above +2 d12 + 2d66 ++4d26 d22 Ð s = 0. < 2 ρ The following statements are evident. V R V*, V* = s*k1 d/ . Statement 1. In the absence of mixed stiffness con- Denote the sought-for pair of complex roots by ξ ξ 1, 2 stants (d16 = d26 = 0), Eq. (3) can have purely imaginary (Im 1, 2 > 0). The edge conditions (2) take the form

det∆()s = 0,

d ++2d ξ d ξ2 d ++2d ξ d ξ2 (4) ∆()s = 12 26 1 22 1 12 26 2 22 2 , ()ξξ2 ξ3 ()ξξ2 ξ3 2d16 +++2d12 + 4d66 1 4d26 1 d22 1 d16 +++d12 + 4d66 2 4d26 2 d22 2

2 A d ++2d ξ d ξ iξ k x iξ k x ik x ------2 = Ð------12 26 1 22 -1, w ()x , x = {}A e 1 1 2 + A e 2 1 2 e 1 1, (5) A ξ ξ2 * 1 2 1 2 1 d12 ++2d26 2 d22 2 i.e., the problem of the existence of the desired waves is 1/4 2 2 2 1 1 reduced to analyzing the roots s of Eq. (4) on the s = ÐDCE+ 23Ð22a + a Ð --- + --- , branches ξ (s) and ξ (s). 2 2 1 2 (7) In the specific case of an orthotropic medium with 2 E the main axes coinciding with x1 and x2, the situation is a = ----, considerably simplified. The mixed stiffness constants C are d16 = d26 = 0, and the characteristic equation (3) has purely imaginary branches of the roots ξ (s) and ξ (s). where the positive definiteness of the radicals follows 1 2 from that of the elastic constants tensor. It is easily seen Choosing the normalizing coefficient d = d22, we obtain that the amplitude ratio (5) is also real. It is the real part iωt 1/2 of the solution Re{w∗(x1, x2)e } that has a physical ξ {}− 4 2 d11 12, ==iC+ D+ s , DCÐ ------, meaning, and the angular displacements of cross sec- d22 θ θ tions 1, 2 and the longitudinal displacements u1, u2 are d + 2d 2d given by the formulas C ==------12 66-, E ------66, iωt d22 d22 θ ∂ {}, θ ()α , α = ÐαRe w*()x1 x2 e , uα = z α = 1 2 2 1/2 (6) 4 4 () EDs+ + CDsÐ + fs ==------------1, Ðk ξ x A Ðk ξ x i()ωtk+ x 4 4 1 1 2 2 1 2 2 1 1 EDsÐ + CDs+ + u1 = ÐzReik1 A1 e + ------e e , A1 fs()==.1⇔ det ∆()s 0 4 2 2 When s ∈ [E – D, C – D], the function f(s) varies Ðk ξ x A Ðk ξ x i()ωtk+ x ξ 1 1 2 ξ 2 1 2 2 1 1 from 0 to +∞, i.e., the real root s of Eq. (6) always exists u2 = zRek1A1 1 e + 2 ------e e . A1 and is expressed as

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 KONENKOV’S WAVES IN ANISOTROPIC LAYERED PLATES 173

Hence, at the real amplitude A , the displacements u , u 2 θ θ 1 1 2 (and the angles 1, 2) follow the harmonic law with a 1 phase difference of –π/2. In one period of oscillations, an arbitrary point (x1, x2) describes an ellipse by moving in the positive direction, and the semi-axes of the ellipse exponentially decrease with distance from the free edge. 3 In the speciﬁc case of an isotropic (and transversely 1 isotropic) medium, Eq. (7) leads to the well-known 4 relation derived by Yu.K. Konenkov [1]:

1/4 s = {}()1 Ð ν ()3ν Ð2121 + Ð2ν + ν2 , 2 where ν is the Poisson ratio. ϕ It is also of interest to analyze the qualitative behav- 0 1 ior of the phase velocity VR (its ratio to the velocity VB of a common ﬂexural wave with the wave vector k) and Fig. 1. Velocity ratio VR/VB in polar coordinates for the the period-average value of the power ﬂow ᑣ materials (1) CP/0, (2) CP/90, (3) OP/0, and (4) OP/90. ω ω2 L ()cosϕ, sinϕ 1/4 V ==------3 , π π B k ρ s = 0.99769 (for OP/0) and 0.91085 (for OP/90), ξ ξ min(Im 1, Im 2) = 0.0444 (for OP/0) and 0.03616 ω2 3 1/4 ω d22 (for OP/90), V R ==---- s ------ρ , ᑣ ω | |2 k1 / d k1A1 = Ð56.6055 (for OP/0) and Ð48.26285 (for OP/90). 1/4 V d3 The curves for the velocity ratio are presented in r()ϕ ==------R s------22 , Fig. 1. V L ()cosϕ, sinϕ B 3 Thus, for the case of orthotropic materials with stan-

2π dard orientation, we can make the following conclu------ω +∞ sions: ω ᑣ = Ð------dt {}Reθú 1ReM + RewúReP dx , (i) the flexural wave of the Rayleigh type exists at all 2π ∫ ∫ 11 1z 2 times; 0 0 ξ ξ (ii) the damping factor min(Im 1, Im 2) proves to where the dot means the time derivative and M11 and be higher than that for an isotropic medium, where the ∂ ∂ P1z = 2 2M12 + 1M11 are the normal moment and the corresponding value does not exceed 0.01; intersecting Kirchhoff force in the packet cross section (iii) the phase velocity of the Rayleigh flexural wave perpendicular to the x1 axis. is not minimal among all possible flexural waves, as it With a view of a numerical illustration, we take two is in the case of an isotropic medium. ≠ types of an orthotropic material: the carbon-filled plas- In a more general case of anisotropy (d16, d26 0), it tic (CP) with the Young moduli E1 = 12800 and E2 = is impossible to find analytical expressions for the roots 2 ξ ξ 840, the shear modulus G12 = 460 kg/mm , the Poisson 1(s), 2(s), and one has to resort to numerical analysis. ratio ν = 0.37, and the density ρ = 1.5 g/cm3; and the The search procedure is as follows: the parameter s is 12 ξ ξ organic plastic (OP) with the respective constants E1 = specified, and two solutions 1(s) and 2(s) are deter- ν ρ 2600, E2 = 1800, G12 = 230, 12 = 0.14, and = 1.4. The mined from Eq. (3), then Eq. (4) is checked (the real and main orthotropic axes either coincide with the reference imaginary parts of det∆(s)). Let us model the situation axes (CP/0) or are rotated through the right angle using the examples of the T material (ρ = 1.58 g/cm3, 2 ν (CP/90); the thickness is taken to be 2h = 1 mm and d = E1 = 13000, E2 = 975, G12 = 600 kg/mm , and 12 = ρ max(d11, d22). 0.27), and the E material ( = 2, E1 = 4500, E2 = 1300, ν The typical values of s and the normalized values of G12 = 440, and 12 = 0.29) whose principal axes x1' , x2' the radiation power of the wave under study are as fol- are rotated through the angle 0 < ψ < π/2 with respect lows: to the axes x1 and x2. The curves for the typical values π π ψ ξ ψ ξ ψ s = 0.99982 (for CP/0) and 0.506225 (for CP/90), of s( ), the branches 1( ), 2( ), and the amplitude ξ ξ min(Im 1, Im 2) = 0.0362 (for CP/0) and 0.00949 ratio (5) are presented in Figs. 2Ð5. (for CP/90), The behavior of the radiation power ᑣ(ψ) is also of ᑣ ω | |2 / d k1A1 = Ð61.0861 (for CP/0) and Ð15.38159 interest (the curves for the normalized values of this (for CP/90); function are presented in Fig. 6). When k1 > 0, the

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 174 ZAKHAROV

s(ψ) transfer. Further more, at the next critical angle, ψ∗∗ = 1.0 0.22862π (T) and 0.29861π (E), the wave again becomes stationary, and at ψ > ψ∗∗ the negative sign of the power flux is restored. This fact is new and peculiar exclusively to media with a sufficiently general anisot- ≠ ropy (d16, d26 0). This fact was not noted earlier for the anisotropic media, and it is fundamentally absent in the case of isotropic media or conventionally oriented orthotropic ones. E To clarify the effect, consider the behavior of the T generalized acoustic impedances M I P k2 I 0.5 I* ==------11------m , I* ==------1z ------1 p, II =+ I , m θ. p . m p 0 0.5 1 V R w V R ψ/π θ . ω 1 ==Ðik1w, w i w, Fig. 2. Characteristic value curves for T and E. ∞ ω2 2+ ᑣ k1 2 () 1 = Ð------∫ w Re Im + I p dx2. 1 2V R 0 Following the elementary analogy for the damped har- ψ/π monic oscillator 0 0.5 .. . my ++by cy ω c I* ==------. - imÐ ω---- + b, (8) Reξ y –1 –Imξ 2 one can expect a positive value of Re(Im + Ip) and a possible change of sign of the imaginary part. Figures 7Ð9 –2 demonstrate the curves for the real and imaginary parts of the generalized impedances Im, Ip, Im + Ip for three values of the angle ψ: beyond the interval [ψ∗, ψ∗∗] (Figs. 7, 9) and within the interval (Fig. 8). It can be ξ ψ ξ ψ Fig. 3. Roots 1( ) and 2( ) for T. seen that, beyond the interval [ψ∗, ψ∗∗], the qualitative analogy with Eq. (8) is valid, while at the intermediate power flux should seemingly be negative. For both value of ψ, an active edge zone with the opposite direc- materials, however, one can find the critical value of the tion of the energy flow is present, which leads to the orientation angle, ψ∗ = 0.07647π (T) and 0.10492π (E), change of sign of the integral ᑣ. This situation takes when the sought-for wave turns into a stationary one, place at any intermediate value of ψ. In particular, if one and then (at ψ > ψ∗) changes the direction of energy considers the value of Re(Im + Ip) at one point x1 = 0 as

1 1

ψ/π 0 0.5 ReA2/A1 0.1 ImA2/A1 Reξ 2 2 –Imξ –1 0 0.5 ψ/π

–2 Fig. 5. Amplitude ratio A /A as a function of ψ for the (1) ξ ψ ξ ψ 2 1 Fig. 4. Roots 1( ) and 2( ) for E. T and (2) E materials.

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 KONENKOV’S WAVES IN ANISOTROPIC LAYERED PLATES 175

ℑ ω 2 Re / d k1A1 1

ψ/π

0 k1x2 0.5 0 2 4 6 8 10

E Ip + Im I T p ψ π –1 –1 = Im 25

Fig. 6. Normalized radiation powers. Im

Fig. 7. Normalized impedances.

Re Re

0 2 4 6 8 10 Ip + Im k x Ip Ip + Im 1 1 0.1 I Ip π m ψ = 27 Im 200

–2 ψ 2π = 5 k1x2 0 Im Im 2 4 6 8 10

Fig. 8. Normalized impedances. Fig. 9. Normalized impedances. a function of ψ, this function will change its sign from 2. Yu. I. Bobrovnitskiœ, M. D. Genkin, V. P. Maslov, and positive to negative and back, at certain values to the A. V. Rimskiœ-Korsakov, Wave Propagation in Struc- left and to the right of the interval of critical angles, tures Made of Thin Rods and Plates (Nauka, Moscow, respectively. Between these values, the width of the 1974). opposite-energy-flow zone is finite, while between the 3. A. S. Zil’bergleœt and I. B. Suslova, Akust. Zh. 29, 186 critical angles it is this zone that dominates the radia- (1983) [Sov. Phys. Acoust. 29, 108 (1983)]. tion power integral. Thus, we can state that the Rayleigh flexural waves 4. M. V. Belubekyan and I. A. Engibaryan, Izv. Ross. Akad. in anisotropic media essentially and qualitatively differ Nauk, Mekh. Tverd. Tela, No. 6, 139 (1996). from both Rayleigh compression-tension-shear waves [5, 6] and flexural waves in isotropic media. 5. P. Chadwick and G. D. Smith, in Advances in Applied Mechanics (Academic, New York, 1977), Vol. 17, pp. 303Ð376. ACKNOWLEDGMENTS 6. S. A. Kaptsov and S. V. Kuznetsov, C. R. Acad. Sci. This work was partially supported by the INTAS, (Paris), Ser. IIb 327, 1123 (1999). project no. 96-2306. 7. S. G. Lekhnitskiœ, Anisotropic Plates (Gostekhizdat, Moscow, 1957). REFERENCES 1. Yu. K. Konenkov, Akust. Zh. 6, 124 (1960) [Sov. Phys. Acoust. 6, 122 (1960)]. Translated by A. Kruglov

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002

Interference Immunity of a Focused Antenna V. A. Zverev Institute of Applied Physics, Russian Academy of Sciences, ul. Ul’yanova 46, Nizhni Novgorod, 603600 Russia e-mail: [email protected] Received January 31, 2001

Abstract—The interference immunity of a focused antenna in the ﬁeld of interference produced by discrete wideband sources is considered. It is shown that the focused antenna that processes the signal by the dark-ﬁeld method provides a high interference immunity. Calculations show that this signal processing technique can provide acoustic monitoring in the presence of intense interference produced by ships. © 2002 MAIK “Nauka/Interperiodica”.

Interest in issues concerned with signal extraction in we extend the dark-field method used with the focused the presence of interferences persists (see, e.g., [1, 2]). antenna to the case of a wideband signal. These issues are topical in acoustics, because, in many In solving this problem, we limit our consideration applications, the interference is not a uniformly distrib- to the minimal number of sources—to just two sources, uted noise, but exhibits a significant regularity. A clas- one of which is stronger than the other. We will show sical example of such an interference is the noise pro- that, in this case, the coordinates and the intensity of the duced by ships, which prevails at low frequencies (from weaker source can be determined in the presence of the 10 to 200Ð300 Hz) [3]. Such interferences can be strong source, the only limiting factor being the distrib- rejected using long antennas with signal processing that uted noise rather than the intensity of the strong source. is capable of eliminating or substantially suppressing The problem is solved by linear filtering. Therefore, if the effect of noise produced by discrete sources. it can be solved in this simplified form, one can expect Among the methods that efficiently reject such interfer- to succeed in the case of multiple sources. The supposi- ences, adaptive methods [4] are most popular. How- tion that there are only two sources does not present a ever, in acoustic communications through a medium fundamental limitation. It allows us to focus on the with a complex irregular structure, the efficiency of method and results, leaving the difficulties (mostly these methods diminishes [5, 6]. It should be noted that mathematical) associated with multiple sources to the requirements of regularity of the medium that pro- future analysis. vide the efficiency of the adaptive methods become much more stringent with an increase in the number of Let two sources (a strong source and a weak one) the antenna array elements and in the antenna length. with a continuous spectrum be present. Let the sources Increasing the antenna aperture is useful and necessary be uncorrelated. This assumption is adopted only to for suppressing interferences produced by scattering make the problem definite. The problem is solved iden- from a rough water surface typical of acoustics. tically for uncorrelated and for correlated (completely or in part) sources. We emphasize that the adaptive The proven method referred to as the shadow or the methods use uncorrelated signals. The antenna consists dark-field method has long been known in optics [7]. of a number of hydrophones uniformly spaced a distance This method is capable of rejecting (darkening) the s apart in the horizontal direction. The antenna aperture interference sources in the presence of irregular aberra- is sufficient to be focused at all operating frequencies. tions produced by optical elements. A distinctive fea- These conditions are thoroughly studied in [8]. ture of this method is that, with an increase in lens aperture, it significantly relaxes the requirements imposed Let us determine the coordinates of the strong on the regularity of the wave field. A similar method source with the help of the focused antenna. We assume can also be applied in acoustics [8]. As shown in [8], the that the signal produced by the strong source on the acoustic dark-field method used for processing signals receiver antenna exceeds the signal produced by the weak received by a sufficiently long (focused) antenna array source enough (in the numerical example, by 40 dB) for basically differs from the signal processing in adaptive the radiation from the weak source to be neglected. The arrays. The dark-field method applied to the focused frequency spectrum of the spherical wave field pro- antenna is conceptually close to the cepstral analysis duced by the wideband source and incident on the [9], which is widely used in acoustics. hydrophone number n of the linear antenna can be represented as A focused antenna operating with a monochromatic acoustic field has been considered in [8]. In this paper, P()ω, n = G()ω exp[]iωTn(). (1)

INTERFERENCE IMMUNITY OF A FOCUSED ANTENNA 177

Here, G(ω) is the frequency spectrum of the ﬁeld produced by the source, ω is the frequency, and T(n) is the delay of the source signal incident on the hydrophone number n. If the wave is spherical and the source resides on the perpendicular erected to the antenna at its central element, we have

s 2 2 Tn()= -- R +,()nmÐ (2) c where c is the light velocity, R is the distance to the source normalized by s, and m is the index of the central hydrophone. The source coordinates can be found from the maximal response of the antenna, which occurs when the spherical wave front is compensated. This is a maximum of the following expression in the variables q and r:

2 Fig. 1. Fragment of the ﬁeld of view of the focused antenna. (), ()ω, []ω (),, ω The axes represent the direction towards the source and the Uqr = ∫ ∑P n exp Ði Mqrn d . (3) distance to it. The vertical axis represents the sum of inten- ω n sities of the received signal over all frequencies of the antenna frequency range. Here,

s 2 2 Mqr(), = -- r +.()nqÐ (4) written so as to take into account its shift by the angle c ϑ from the normal to the antenna: Expression (3) reaches its maximum when the ()ω, ()ω {}ω[]() spherical wave front is compensated completely. This P2 n = G2 exp i T2 n + hn , (5) occurs when r = R and q = m in formula (4), which where T 2(n) is given by Eq. (2) at R2 = 64 and h = yields the source coordinates. s -- sin(ϑ). The spectrum of the signal at the hydrophones The above search procedure was simulated using c the following parameters. The antenna consisted of will be 256 hydrophones. Both sources emitted uncorrelated Gaussian random signals with realizations being 32 ele- PC()ω, n = P1()ω, n + 10Ð2P2()ω, n . (6) ments long. In accordance with results obtained in [10, 11], the spectrum of the emitted signals was supple- First of all, let us compensate the front curvature of mented with zeroes to extend the length of the realiza- the stronger wave. The parameters of this wave are tion by a factor of eight. This operation was necessary, obtained by the signal processing described above. To because, in numerical calculations, the delay given by compensate the front curvature of the stronger wave, Eqs. (2) and (4) can take only discrete values. A.A. Pav- one should apply the following operation lenko showed that the additional zeroes in the spectrum significantly reduce the effect of delay quantization W()ω, n = PC()ω, n exp[]ÐiωT1()n . (7) errors. The distance was R = 128 (half-length of the Then we should obtain the spatial spectrum of the antenna), and m = 128. signal at the antenna for each frequency ω by applying The surface determined by Eq. (3) is illustrated in the Fourier transform to Eq. (7) with respect to n. Let us Fig. 1 by a function of r and q. This surface can be used denote the spatial frequency corresponding to the vari- to find the curvature of the incident wave front. Unfor- able n as ν. To visualize this spatial spectrum, we inte- tunately, this processing procedure does not comply grate its squared magnitude with respect to the fre- with the dark-field method. The radiation from the quency ω: source can be screened in its most intense part, but this action will not extract the field of the weaker source, ()ν Φ[]()ω, 2 ω S = ∫ n W n d . (8) because Eq. (3) involves an irreversible operation (summation over frequencies). ω Φ To apply the dark-field method, we should eliminate Here, x[Z(x)] means the Fourier transform of the the summation over frequencies, which makes Eq. (3) function Z(x) with respect to x. irreversible. The function S(ν) defined by Eq. (8) is illustrated in We write the spectrum of the stronger signal [let this Fig. 2a. Both signals are seen in the figure: the strong be P1(ω, n)] using Eq. (1) at R = 256 (the antenna one as a discrete spectral line and the weak one as a length). The spectrum of the second signal should be blurred curve, because the curvature of the weak sig-

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 178 ZVEREV

0 (a) –20 –40 –60 –80

0 (b) –20 –40 –60 –80

0 (c) –20 –40 –60 –80 –300 –200 –100 0 100 200 300

Fig. 2. The intensity integrated over all signal frequencies in decibels with respect to the maximum of the focused signal of the strong source versus the angle of arrival of the incident wave normalized by the antenna resolution. The antenna is focused at the strong source, and the source signal is (a) not darkened and (b) darkened. Figure 2c refers to the antenna focused at the weak source with the strong source darkened.

Φ ω nal’s front is greater, which considerably broadens its exclusively for the function n[W( , n)] rather than for spatial spectrum. S(ν). After that, we integrate the result with respect to ω To implement the dark-field method, one must the frequency as in Eq. (8) to obtain the function block the domain (reduce the signal in it to zero) occu- shown in Fig. 2b. pied by the spatial spectrum of the strong signal Now, only the weak signal is present. To find its shown in Fig. 2a. This should be done necessarily and parameters, it is necessary to compensate the curvature of its wave front. To this end, we go over from the spectrum to the signal by applying the inverse Fourier transform with respect to frequency ν and multiply the result by the factor E()ω, n = exp[]iωT1()n exp[]ÐiωMqr(), . (9) The first exponential here is necessary for compen- sating the factor in the weak signal that removes the wave front curvature of the strong signal. The second exponential serves for removing the curvature of the wave front of the weak signal by choosing the appropri- ate q and r. The result of this procedure is shown in Fig. 2c. The process of fitting the parameters for removing the curvature of the small-signal wave front is illustrated in Fig. 3. The effect of the dark-field method is clearly seen in Fig. 4. The field of the strong source is not darkened (the spatial spectrum in the corresponding domain is Fig. 3. Fragment of the field of view of the focused antenna multiplied by unity rather than by zero). As a result, the near the weak source with the strong source darkened. The field of the strong source covers the field of the weak axes are labeled as in Fig. 1. source completely, although the antenna is focused at

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 INTERFERENCE IMMUNITY OF A FOCUSED ANTENNA 179

0 of dispersion due to the propagation of sound in a waveguide can be compensated by increasing the dark- –20 ened area of the focused antenna. The two most promising results were obtained in [11]. The first consists in –40 the consideration of the dark-field method that works in –60 the presence of high phase distortions in the antenna. The second consists in that it was demonstrated how to –80 determine the phase distortions in the antenna from the field of the strong source alone in order to compensate –100 them in the same way as the wave front curvature. –120 –300 –200 –100 0 100 200 300 ACKNOWLEDGMENTS Fig. 4. Field of view of the antenna focused at the weak I am grateful to the reviewer for comments which, source with the field produced by the strong source not being taken into account, improved the paper. darkened (the upper curve) and darkened (the lower curve). The axes are labeled as in Fig. 2. This work was supported by the Russian Foundation for Basic Research, project nos. 00-15-96741 and 99-02-164010. the weak source rather than at the strong one. The lower part of Fig. 4 shows the result when the field of the strong source is darkened. The difference in the resid- REFERENCES ual background levels is almost 70 dB, which illustrates 1. P. K. Willet, P. E. Swaszek, and R. S. Blum, IEEE Trans. the interference immunity of the focused antenna in this Signal Process. 48 (12), 3266 (2000). example. 2. Z. Kang, C. Chatterjee, and V. P. Roychowdhary, IEEE Basically, such a processing is applicable to the case Trans. Signal Process. 48 (12), 3328 (2000). of multiple sources. The number of sources that can 3. R. J. Urick, Principles of Underwater Sound (McGraw- effectively be darkened is determined by the number of Hill, New York, 1975; Sudostroenie, Leningrad, 1978). array elements. The point is that the darkening of each 4. R. A. Monzingo and T. W. Miller, Introduction to Adap- element reduces the number of antenna elements. tive Arrays (Wiley, New York, 1980; Radio i Svyaz’, In the above calculations, it is important that the Moscow, 1986). antenna is ideal. Both the spread in the positions of 5. V. A. Zverev, Akust. Zh. 40, 401 (1994) [Acoust. Phys. individual antenna elements and the random variations 40, 360 (1994)]. of the amplitude and phase of the field incident on them 6. A. B. Gershman, V. I. Turchin, and V. A. Zverev, IEEE are ignored. If the fact that the antenna is not ideal is Trans. Signal Process. 43 (10), 2249 (1995). taken into account, the result will change. The corre- 7. L. A. Vasil’ev, Schlieren Methods (Nauka, Moscow, sponding study requires additional calculations and, 1968; Israel Program for Scientific Translations, Jerusa- perhaps, experiments. lem, 1971). Here, we only note that the above method for pro- 8. V. A. Zverev, A. L. Matveev, M. M. Slavinskiœ, and cessing the signals received by a focused antenna can A. A. Stromkov, Akust. Zh. 43, 501 (1997) [Acoust. give a similar interference immunity in the presence of Phys. 43, 429 (1997)]. multipath propagation and when the parameters of the 9. A. V. Oppenheim, R. W. Schafer, and T. G. Stockham, antenna and the fluctuating medium are spread ran- IEEE Trans. Audio Electroacoust. 16 (3), 437 (1968). domly. In the presence of multipath propagation, a long 10. V. A. Zverev, A. A. Pavlenko, A. D. Sokolov, and focused antenna may resolve or not resolve the virtual G. A. Sharonov, Akust. Zh. 47, 76 (2001) [Acoust. Phys. sources that appear in this case. If the antenna does not 47, 62 (2001)]. resolve such sources, its interference immunity remains 11. V. A. Zverev and A. A. Pavlenko, Akust. Zh. 47, 355 the same as in a homogeneous unbounded space. If the (2001) [Acoust. Phys. 47, 297 (2001)]. antenna resolves these sources, they can be eliminated exactly as the real sources. This possibility has been demonstrated in [8], where it was shown that the effect Translated by A. Khzmalyan

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002

Application of a Noise Signal in Acoustic Tomography of the Arctic Basin V. M. Kudryashov Andreev Acoustics Institute, Russian Academy of Sciences, ul. Shvernika 4, Moscow, 117036 Russia e-mail: [email protected] Received April 4, 2001

Abstract—The time correlation function of a noise signal propagating in an arctic-type waveguide is considered. For a coherent signal, the time cross-correlation function is formed with the use of either the total signal, or a single selected mode, or a reference signal at one of the correlator inputs. The use of narrow-band signals is shown to be preferable, because the waveguide dispersion affects the waveguide response. It is demonstrated that, for tomographic investigations in a waveguide irregular along the path, it is expedient to correlate the signals received at two different points that are selected on the path so as to enclose the waveguide part the variability of whose parameters is the object of interest. © 2002 MAIK “Nauka/Interperiodica”.

Acoustic tomography is determined as the method crete spectrum [4]. The waves of the continuous spec- of reconstructing the local characteristics of an object trum (the continuum) are taken into account in the from a set of its integral characteristics (projections) by sound scattering from the bottom as a component subjecting them to special processing [1]. In a wider (along the horizontal coordinate r) of the mode attenu- sense, tomography can be interpreted as a method for ation (damping). These waves affect the imaginary part obtaining the information on the internal structure of ς of the longitudinal wave number m(r) of a mode when the object of interest from the characteristics of the the waveguide is irregular in r. We call a waveguide a probing signal (see [1], p. 7). For example, in an inho- plane-layered one, if its deterministic parameters do not mogeneous medium, one has to estimate the variation depend on r. The function Φ (z) is the eigenfunction of of the sound velocity proﬁle within a part of the path. In m this paper, the problem of determining the variations of the waveguide [2Ð5]. The z axis is directed downward, the waveguide parameters from the characteristics of and the value z = 0 corresponds to the mean level of free the received signal, i.e., the inverse problem, is not con- water. The eigenfunctions are orthonormalized by the sidered. This study is limited to a direct problem: we condition show that these variations can be revealed by the time correlation functions of a noise signal. In the cited pub- 1 for nm= {}Φ ()Φz ()z = (2) lication [1], the authors consider deterministic signals m n 0 for nm≠ . including harmonic and pulsed ones. Below, we consider an acoustic signal formed by a noise source, The braces denote the orthonormalization operation which, within the observed realization, possesses statis- whose form depends on the parameters of the bound- tically stationary and ergodic properties as a function of aries. In a liquid medium, this operation is reduced to time. an integral of the product of eigenfunctions with the The sound pressure produced by a narrow-band weighting function ρ(z), which is the density of the liq- point source in a waveguide can be represented in terms uid, with respect to z. The integration is performed over of a mode expansion [2, 3]: the whole liquid column [4, 5]. (),, p r zt In the narrow-band approximation, within the signal (1) frequency band, no considerable variations occur in the ∆ ()Φ() ()()ω = w fp∑ m r m z FtÐ tm exp Ði 0t . eigenfunctions of the waveguide and in the group m velocities vm of modes determined by the relationship µ Here, the constant w is expressed in Pa and corre- ∂ς v Ð1 m sponds to a frequency band of 1 Hz: 20 logw = W, m = ------∂ω , ωω where W characterizes the source power in decibels in = 0 a frequency band of 1 Hz at a distance of 1 m from the ω π ω π source; ∆f is the frequency band of the signal. The sum- where = 2 f, f is the frequency of sound, 0 = 2 f0, mation over m is performed over all modes of the dis- and f0 is the central frequency of the signal in hertz.

APPLICATION OF A NOISE SIGNAL IN ACOUSTIC TOMOGRAPHY OF THE ARCTIC BASIN 181

The function pm(r) characterizes the spatial variabil- sound pressure is nonzero. This deterministic quantity ity (in the distance r) of the sound field. For a coherent corresponds to the second statistical moment [7]. field in a plane-layered waveguide, we have The time cross-correlation function is determined by the expression () πΦ ()()1 ()ς pm r = i m z0 H0 mr , (3) (),,, τ (),, (),, τ Kc r1 r2 z1 z2; = p r1 z1 t p* r2 z2 t Ð . where z0 is the depth of the sound source. Similarly, the quantity pm(r) is calculated for the central frequency in Substituting Eq. (1) in this expression, we obtain the case of a narrow-band signal. (),,, τ 2∆ () () Φ Kc r1 r1 z1 z2; = w fp∑∑ m r1 pm r2 Thus, we calculate the quantities pm(r), m(z), and (4) ς m n m by the program for calculating the harmonic sound ×Φ ()Φ()()τ ()ω τ field at the frequency f0. Expression (1) describes a m z1 n z2 B Ð tm + tn exp Ði 0 . coherent field, if a stochastic scattering occurs in the waveguide along the sound propagation paths. If the We assume that, at the point of observation, the sig- waveguide parameters wary regularly with r, i.e., the nal is received by a vertical array with the sensitivity of variability can be described by deterministic functions its elements being determined according to the vertical variability of the Nth eigenfunction. Then, the Nth of r, the quantity pm(r) can be calculated either in the adiabatic approximation [3, 4] neglecting the regular mode can be singled out of the sound field. For this pur- Φ scattering or in the approximation taking into account pose, we multiply Eq. (4) by N(z2) and integrate the the regular scattering (transformation) of modes [6Ð8]. result with respect to z2, at least over the layer of the The regular scattering is related to the variations of the concentration of the Nth mode. Applying the condition eigenfunctions with r, which results in the violation of of orthogonality of modes, we obtain the orthogonality condition for different-number (),, τ 2∆ () () modes calculated for different r: K N r1 r2 z1; = w fp*N r2 ∑ pm r1 (5) m {}Φ () Φ () ≠ ≠ m z n z rr= + ∆r 0 for n m. ×Φ ()()τ ()ω τ rr= 1 1 m z1 B Ð tm + tn exp Ði 0 . In both approximations, the calculations are based This expression is analogous in its structure to the on the Pierce scheme known as the vertical-mode and pulsed waveguide response in which the role of the horizontal-ray approximation [1, 3, 4]. The sound field envelope of the transmitted signal is played by B(τ) and of a mode is calculated along the mode ray described in the time t is replaced by the delay τ. A similar result can the horizontal plane by an eikonal-type equation whose be obtained when the Nth mode is separated in its ≠ right-hand side contains the longitudinal wave number arrival time tN from the modes with m n. Manipulating of the mode instead of the wave number in water. The the delay at one of the correlator inputs, we can obtain calculation of the mode rays can be performed by the a pulsed-response-type expression. well proven programs for a two-dimensional waveguide. A widely used method [9, 10] is based on the cor- The coefficients pm(r) are calculated along the mode relation of the received signal with the reference one rays by using a transport equation or another scheme, F(t – τ)exp[iω (t – τ)]. Then, we obtain the expression e.g., of the type used in [6, 7]. In these calculations, at 0 (),,τ each step along the distance, one has to take into Pc r z account the mutual transformations of modes of the dis- (6) crete spectrum. ∆ ()Φ() ()τ ()ω τ = w fp∑ m r m z B Ð tm exp Ði 0 . In Eq. (1), t is time and tm is the propagation time m of the mth mode between the transmission and reception points spaced at a distance r. In a plane-layered This formula is analogous to the expression for the pulsed waveguide response in which the role of the waveguide, tm = r/vm. In the adiabatic approximation, τ pressure pulse is played by Pc(r, z, ). The advantage t = r dη/v ()η . In the regular scattering approxima- over the pulsed waveguide response caused by a pulsed m ∫0 tion, the calculation is more complicated, because new signal is that one can obtain an analog of the pulse at the modes can be formed in the course of the propagation. point of observation when the signal has a low power In addition, p (r) can be affected by the contributions near the source. The required energy is accumulated m because of the use of a signal of long duration. This made by other modes because of the scattering. way, one obtains a high signal-to-noise ratio due to the The noise character of the signal is determined by a effect of a quasi-coherent accumulation. The disadvan- random homogeneous ergodic function of time F(t). We tage of this approach is its high sensitivity to the effects assume that Ft()Ft()Ð τ = B(τ), where B(0) = 1 and related to the movements of the transmission and reception points and to the variations of the waveguide the overbar denotes averaging over time. Therefore, we parameters with time. An example is the Doppler effect, have p()r, z = 0. The time correlation function of the which less strongly affects the correlation function of

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 182 KUDRYASHOV the type given by Eq. (5), if it produces approximately arctic water layer is observed with the vertical sound equal effects on all modes forming the pulsed response. velocity gradient close to the hydrostatic one. The Expressions (4)–(6) correspond to a coherent field, waves refracted within this layer (the refraction is pos- i.e., a field averaged over an ensemble of realizations of itive) have the maximal group velocities and, hence, the stochastic scatterers. In calculating the variance of the minimal propagation times tm. These waves form the sound pressure of the total field, we use the transport leading edge of the acoustic pulse. The waves captured equation in the approximation of a multiple forward by the near-ice channel are the last to arrive. Figure 1a scattering and a single backward scattering [11, 12]. If shows the profile c(z) for the central region of the Arctic the parameters of the waveguide vary regularly with r basin. At a frequency of 20.1 Hz, for the free water sur- while the scale of these variations is (as usual) much face, the group velocities of the first three modes are greater than the horizontal correlation radii of stochas- equal to 1439.168, 1451.375, and 1455.384 m/s. Let us tic scatterers, the calculation of the mean intensity of consider an ice cover 65% of which is smooth ice and the total field is performed in the same way as the cal- the rest is hummocky ice. The longitudinal wave veloc- culation for a plane-layered (on the average over an ity cl in ice is 3500(1 Ð i0.004) m/s, the shear wave ensemble of scatterers) waveguide, but with allowance velocity is ct = 1800(1 Ð i0.04) m/s, and the ice density for the regular variability. If, in a plane-layered (on the is 0.91 g/cm3. According to the histogram of the lower average) waveguide, the mode transformation coeffi- surface of ice, the thickness of smooth ice is h1 = 2.6 m, cients [7, 11Ð13] determined by the stochastic scatter- the square root of the variance of the ice draught is σ ing are independent of r, then, in an irregular 1 = 1.8 m, and at the upper surface of the smooth ice, waveguide, one has to solve the differential transport σ ˜ 1 = 0.4 m. For hummocky ice, we have h2 = 6.6 m and equation with allowance for the variations of the aver- σ = 3.3 m. The horizontal correlation scales of rough- age (over the ensemble of the scatterer realizations) 2 mode transformation coefficients. ness are 120 and 44 m for smooth and hummocky ice, respectively. The expressions presented above for the time corre- The coefficients of sound reflection from the ice lation functions describe these functions in the complex cover were studied in [14, 15]. form. To adjust them to the real experiment, it is necessary to separate their real part, which, as a function of In the presence of the ice cover, the group velocities τ τ τ of the first three water modes are 1439.112, 1450.51, the delay , can be reduced to the form A( )cos(w0 ). The information is carried by the envelope of the corre- and 1455.211 m/s. From the comparison of these values lation function A(τ). Its structure depends on the form with the corresponding values for a free water surface, of the function B(τ), and the latter depends on the one can see that the ice cover has almost no effect on energy spectrum of the transmitted signal, as well as on the group velocities of normal waves. the characteristics of the receiving filter and, especially The ice cover affects the attenuation coefficients of on the frequency band ∆f of the signal transmitted normal waves. For an acoustically soft boundary, at a through the filter. In particular, the time correlation frequency of 21 Hz, the attenuation coefficients of the scale τ of the function B(τ) is proportional to 1/∆f. first three water modes are equal to 0.000055 dB/km. 0 ∆ τ Hence, by increasing f, we reduce 0 and, as a result, This value is determined by the sound absorption in obtain a narrower (in the delay) peak of the envelope of water. In the presence of the ice cover, we obtain the time correlation function corresponding to a spe- 0.01801, 0.00735, and 0.00503 dB/km. The calcula- cific mode. tions were performed as in the previous publications [14, 15]. The maximal spatial attenuation is observed The peak of the envelope of the mth mode corre- for the lowest-order modes. The first mode is captured sponds to the point τ t on the delay axis. Since t is = m m by the near-ice channel and has its maximum at a depth different for different modes, the received pulse is of 160 m, although this mode partially penetrates the spread. For the water modes propagating in an arctic Atlantic water layer. waveguide whose axis is adjacent to the ice cover, the minimal group velocity is characteristic of the modes Within the narrow-band approximation, a broaden- ∆ that are captured by the near-ice layer lying between the ing of the signal bandwidth f narrows the pulse peak ∆ ice cover and the Atlantic water layer. The thickness of of a mode. One would expect that the greater f, the the near-ice layer varies from 250 to 450 m with easier the identification of modes separated by a delay τ increasing distance from the Fram strait toward the time exceeding 0. However, this tendency is limited by coast of Canada. This is explained by the fact that the the waveguide dispersion of modes, which leads to Φ warm Atlantic waters arriving from the northern Atlan- changes in tm, m(z), and pm(r). For example, if tm var- ∆ tic become spread as they move further in the eastern ies by more than 1/ f, the mode peak is destroyed direction. In the Atlantic waters, the sound velocity gra- becoming broader, and the form of the envelope of the dient is close to zero, whereas in the near-ice layer, the time correlation function changes. The waveguide dis- gradient of c(z) is maximal, its average value being persion depends on the waveguide parameters. equal to 5Ð6 hydrostatic gradients. Under the Atlantic To switch to the broadband approximation, we use water layer, at depths greater than 750Ð10000 m, the the scheme described in [16]. We assume that, in the

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 APPLICATION OF A NOISE SIGNAL IN ACOUSTIC TOMOGRAPHY OF THE ARCTIC BASIN 183 frequency band ∆f, the energy spectrum of the signal is c (z), m/s c (z), m/s described by the function G(ω) normalized so as to sat- 1430 1450 1470 1490 1440 1450 1460 isfy the condition 0 0 ω ∆ω 0 + /2 ∫ G()ω = 1, ω ∆ω 0 Ð /2 where ∆ω = 2π∆f. We divide the frequency band ∆f ∆ 1 100 into intervals fj: J ∆ ∆ ∑ f j = f . j = 1 ∆ 12 We assume that, in the frequency band fj, the signal is a narrow-band one with f0j being the central frequency 2 200 ω π of the jth component; 0j = 2 f0j. We calculate the partial time correlation functions for each of the compo- ∆ nents in the frequency band fj and combine them with allowance for the phases and with the weighting factors (a) (b) ω G( 0j). As shown in [16], a broadband correlation function is characterized by changes in both its envelope and carrier, and the latter may not coincide with 3 300 ω τ cos( 0 ). z, km z, m Let us consider an example demonstrating the effect of the waveguide dispersion on the signal form. We Fig. 1. Sound velocity profiles c(z). assume that the sound velocity profile and the parameters of the bottom and the ice cover do not vary along the waveguide. The sound velocity profile is presented in mode dispersion. The higher-order water modes are the Fig. 1a. The waveguide depth (corresponding to the first to arrive. The contribution of the bottom-reflected waterÐground boundary) is 3000 m. The bottom is mod- waves is insignificant due to the high attenuation eled by a homogeneous elastic halfspace characterized by caused by the bottom reflection. The first water mode the longitudinal wave velocity cl = 1850(1 Ð i0.01) m/s, arrives last. In the narrow-band approximation, the the shear wave velocity c = 350(1 Ð i0.01) m/s, and den- width of each pulse corresponding to an individual t ∆ sity equal to 2 g/cm3. According to the histogram mode is proportional to 1/ f. In the broadband approx- obtained for the central Arctic region to the west of the Lomonosov ridge, 65% of the ice cover is smooth ice |P |, µPa with an average thickness h1 = 2.6 m, the average c drought (the square root of the roughness variance) 1000 σ 1 = 1.8 m, and the average height of roughness at the σ upper boundary 1/4.5. For the hummocky part of the 800 σ ice cover, the parameters are h2 = 6.6 m and 2 = 3.3 m. The ratio of the hummock height to its drought in water is 1/4. The density of ice is 0.91 g/cm3, and the 600 wave velocities are cl = 3500(1 Ð i0.04) m/s and ct = 1800(1 Ð i0.04) m/s. We consider the envelope of the pulsed response 400 τ of the waveguide Pc(r, z, ) for z = z0 = 60 m and W = 100 dB/Hz. The distance between the transmission and reception points is 900 km. The mean frequency 200 of sound is f0 = 21 Hz, and the signal bandwidth (resultant of the transmitted signal and the receiving 0 filter) is 7.5 Hz. 610 614 618 622 626 τ, s The result of the calculation in the narrow-band approximation is shown in Fig. 2. Figure 3 presents Fig. 2. Calculated envelope of the time correlation function the result obtained in the broadband approximation. In P (r, z, τ) in the narrow-band approximation for f = 21 Hz, ∆c 0 Fig. 2, one can notice the angular dispersion, i.e., the f = 7.5 Hz, r = 900 km, z = z0 = 60 m, and W = 100 dB/Hz.

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 184 KUDRYASHOV

| | µ | | µ Pc , Pa Pc , Pa

810 720 720 630 540 540

450 2 360 1 360

180 270 180 0 90 610 614 618 622 626 τ, s 0 610 614 618 622 626 SF

Fig. 3. Calculation for the same waveguide parameters as in Fig. 4. (1) Same as in Fig. 2, but for ∆f = 2 Hz; (2) the Fig. 1 in the broadband approximation. dashed curve corresponds to a 1.5 km shorter distance. imation, the time of the mode propagation within the i.e., a deterministic quantity, which is used in solving signal bandwidth varies considerably. This effect is the problems of monitoring where even the roll of the most pronounced for the second water mode, which is transmitting and receiving systems can deteriorate the transformed from a narrow pulse to a pulse spread in measurement results. The finiteness of the averaging the arrival time. For the first water mode, the waveguide interval T leads to the appearance of a fluctuation com- dispersion proves to be much weaker, although this ponent with a variance proportional to (∆fT)–1 at the mode exhibits a broadening of the peak of its envelope. correlator output. The fluctuation component masks the However, the position of the top of this peak is the same deterministic component and imposes a limitation on in both approximations. When the signal bandwidth is the measurement accuracy and on the very possibility ∆f = 2 Hz, the signal remains narrow-band within the of detecting the deterministic component. Hence, the whole 900-km-long propagation path. The correspond- stability of the parameters of the experiment in time is ing plot is shown in Fig. 4. It is similar to the plot pre- of fundamental importance. sented in Fig. 2 with some deviations in the amplitudes and widths of the peaks. The above consideration may have another application. Let us assume that the transmitterÐreceiver dis- In Fig. 4, peak 1 corresponds to the arrival of the first tance is unknown, but the direction of the signal arrival water mode at a distance of 900 km, and peak 2 corre- is known. For example, the signal is received by a hor- sponds to the arrival of the same mode at a 1.5 km izontal array or a set of receivers, which can be used to shorter distance. The delay between these two peaks is form a receiving system with spatial selectivity. By approximately equal to 1 s, which should be expected placing a test hydrophone at a known distance from the taking into account the sound velocity in water. The receiving system in the direction from which the signal supplement to the plot of the envelope of the signal cor- arrives, or introducing simple geometric corrections, relation function is made for the following reasons. The we determine the time delay between the signal arrivals expressions presented above for the time correlation at the receiver and at the test hydrophone. Knowing this functions imply the averaging over an infinite realiza- delay and the delays in the signal propagation times to tion. In reality, the time interval T, over which the aver- the points of observation from the transmitter, through aging is performed, is finite due to both technical fac- simple calculations we determine the distance between tors and time variations of the waveguide parameters, the transmitting and receiving systems. In our case, the which lead to the violation of the statistical stationary distance is about 900 km (which can be easily verified). state and ergodicity of the noise signal. In addition, the More precisely, the calculation yields approximately spatial positions of the transmitter and the receiver can 930 km with allowance for the accuracy of the determi- vary because of the drift of the ice fields to which the nation of the peak positions on the delay axis. The posi- transmitter and the receiver are attached. It is possible tion of a peak can be determined with higher accuracy, that the transmitter and the receiver are objects inde- if the peak is made sharper by using, e.g., receiving fil- pendently moving in water. The motion of the objects ters whose frequency characteristic provides sharper leads to changes in the form of the correlation function, peaks.

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 APPLICATION OF A NOISE SIGNAL IN ACOUSTIC TOMOGRAPHY OF THE ARCTIC BASIN 185

The problem of determining the distance becomes coastal zone, such systems can be cable-connected with more complicated when the waveguide parameters vary stationary coastal stations. along the sound propagation path, as well as in the pres- As for the observations (e.g., climate monitoring), ence of an additive noise caused by the medium and a they should be performed in the deep-water regions of multiplicative noise caused by the stochastic scattering the arctic ocean, because these regions are character- of sound. In the case under consideration, the latter can ized by a relative stability of the water column parame- be neglected, so that the signal is a coherent one. ters in depth. The coastal part of the slope and the shal- As the second example, we consider a path that is low-water regions are characterized by unstable acous- irregular in the distance r. The parameters of the path tic parameters of the water column because of such are taken from the experiment carried out by research- phenomena as tides, seasonal variability, and currents. ers from the Acoustics Institute in the 1980s [17]. The The variations of the sound velocity profile that occur path contains a shallow-water part whose parameters in the deep-water regions due to the variations of the correspond to the FranzÐVictoria trough. The length of salinity and temperature of the water layers are rela- the shallow-water part is 5 km. The slope of the bottom tively small. Hence, the variations caused by these phe- extends to a distance of 110 km where the depth reaches nomena in the mode propagation times are also small. a value of 3 km. At distances from 110 to 200 km, a Therefore, the perturbations introduced by the shallow- deep-water region was observed in the experiment. In water parts of the path can cause considerable varia- the calculations, we assume that this region extends to tions of the pulse structure of the signal. For example, a distance of 900 km in the direction of the general we note the following factor. In shallow-water regions decline of the waveguide depth. In the deep-water at high frequencies, the group velocities of modes usu- region, the sound velocity profile c(z) corresponds to ally decrease with increasing mode number. In deep- curve 2 in Fig. 1b. In the shallow-water region, the water regions, the situation is reversed. As a result, the sound velocity takes the values 1440, 1450.2, 1451.4, differences in the mode propagation times tm can vanish and 1444.5 m/s at the depths 0, 50, 140, 170, and 400 m, [10]. As the zero time of the mode propagation over an respectively. Recall that the initial, i.e., zero, depth in irregular path, it is convenient to take the instant of the the profiles c(z) corresponds to the free water level, mode transition from a surfaceÐbottom mode to the although the water layer begins below the ice cover, and stage of its separation from the bottom reflection, i.e., this fact is taken into account in the calculations by to a water mode. For example, on the path under study, recalculating the sound velocity profile according to the the first water mode, which is concentrated in a narrow position of the iceÐwater boundary. To describe the layer near the sound channel axis, at low frequencies variation of the sound velocity profile from shallow- (20 Hz and higher) becomes separated from the bottom water to deep-water one within the irregular region, we and passes to the adiabatic stage (for the coherent com- use linear interpolation. The parameters of ice in the ponent of the sound pressure) starting approximately deep-water region are the same as in the preceding from a distance of 10 km and beyond, when the sound example. In the shallow-water region, the ice cover is source is positioned in the shallow-water region. In the smooth everywhere. Its thickness is 1 m, the average experiment on the time monitoring of this mode, it is drought is σ = 0.6 m, and the correlation scale of the ice expedient to construct the time correlation function by roughness is 40 m. The ground is considered as an elas- formula (5), where N = 2 (the first mode at frequencies tic halfspace with Òl = 1800(1 – i0.005) m/s and Òt = of several tens of hertz is the flexural wave of the ice 350(1 – i0.005) m/s. The ground density is 2 g/cm3. cover, which rapidly attenuates with increasing dis- These parameters approximately correspond to the tance r); thus, N = 2 corresponds to the first water mode. shallow-water region and the initial part of the slope at Figure 5 presents the envelope of the time correla- low frequencies. We use the same parameters for the tion function (5) as a function of the delay for an irreg- ground in the deep-water region, because, at a distance ular path on the condition that the total coherent noise of 900 km, the contribution of the bottom reflections is signal received at a distance of 900 km correlates with insignificant, if the waves reflected from the bottom in the first water mode of the same signal received at a dis- the regular deep-water region of the waveguide are con- tance of 10 km. The envelope is similar in its shape to sidered. the pulse transmitted through a distance of 890 km. The The source of sound generates a noise signal with higher-order modes arrive first, forming a kind of a sin- the central frequency f0 = 21 Hz and is observed in the gle pulse. Then the pulse corresponding to the first frequency band ∆f = 2 Hz. The transmission depth is water mode arrives; this mode is the only one captured 40 m, and the reception depth is 50 m. by the near-ice water layer at a frequency of 21 Hz. The study of the sound propagation along a path that At the given transmitter and receiver depths, the first contains deep-water and shallow-water regions is not water mode prevails in the signal received at a distance only of scientific interest, but also of practical value. of 10 km. Therefore, if we construct a time correlation Large and heavy transmitting and receiving systems function for the same conditions, but by formula (4), for intended for long-term operation are rather installed in r2 = 10 km and r1 = 900 km, the envelope of the cross- shallow-water regions for technical reasons. In a correlation function Kc will take the form shown in

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 186 KUDRYASHOV

|K|, (µPa)2 propagation times are most pronounced for the modes of the first numbers, which are concentrated in the 27 water layers where the changes in the sound velocity profile are maximal. 24 Thus, with the above formulation of the problem, it 21 is possible to reveal a change in the sound velocity profile within a given part of the sound propagation path, 18 e.g., in the course of a long-term experiment, on condition that the path geometry and the transmitted signal 15 are invariable. 12 9 ACKNOWLEDGMENTS 6 This work was supported by the Russian Foundation for Basic Research, project no. 01-02-16636. 3 0 REFERENCES 610 614 618 622 626 SF 1. V. V. Goncharov, V. Yu. Zaœtsev, V. M. Kurtepov, Fig. 5. Envelope of the time correlation function given by A. G. Nechaev, and A. I. Khil’ko, Acoustic Tomography of the Ocean (IPF RAN, Nizhni Novgorod, 1997). Eq. (5) for an irregular propagation path; r1 = 900 km and r2 = 10 km. 2. L. M. Brekhovskikh, Waves in Layered Media, 2nd ed. (Nauka, Moscow, 1973; Academic, New York, 1960). | | µ 2 3. L. M. Brekhovskikh and Yu. P. Lysanov, Fundamentals K , ( Pa) of Ocean Acoustics (Gidrometeoizdat, Leningrad, 1982; Springer, New York, 1991). 62 4. B. G. Katsnel’son and V. G. Petnikov, Acoustics of a 56 Shallow Sea (Nauka, Moscow, 1997). 5. V. D. Krupin, Akust. Zh. 46, 789 (2000) [Acoust. Phys. 49 46, 692 (2000)]. 42 6. V. M. Kudryashov, Akust. Zh. 33, 55 (1987) [Sov. Phys. Acoust. 33, 32 (1987)]. 35 7. V. M. Kudryashov, Akust. Zh. 34, 117 (1988) [Sov. Phys. 28 Acoust. 34, 63 (1988)]. 8. E. C. Shang and Y. Y. Wang, J. Acoust. Soc. Am. 105, 21 1592 (1999). 14 9. P. N. Mikhalevsky, A. N. Gavrilov, and A. B. Baggeroer, IEEE J. Ocean Eng. 24 (2), 183 (1999). 7 10. P. N. Mikhalevsky, A. B. Baggeroer, A. Gavrilov, and 0 M. Slavinsky, EOS Trans. Am. Geophys. Union 76 (27), 610 614 618 622 626 SF 265, 268 (1995). 11. F. G. Bass and I. M. Fuks, Wave Scattering from Statisti- Fig. 6. Same as in Fig. 5, but for the sound velocity profile cally Rough Surfaces (Nauka, Moscow, 1972; Perga- corresponding to curve 1 from Fig. 1b. mon, Oxford, 1978). 12. V. M. Kudryashov, Matem. Probl. Geofiz. (Novosibirsk), No. 4, 256 (1973). Fig. 5 with a rather small difference between the enve- 13. V. M. Kudryashov, Akust. Zh. 42, 438 (1996) [Acoust. lopes of the time cross-correlation functions: the peak Phys. 42, 386 (1996)]. positions on the time delay axis are identical, but the 14. V. M. Kudryashov, Akust. Zh. 42, 247 (1996) [Acoust. maximal values are somewhat smaller in the second Phys. 42, 215 (1996)]. case, because the magnitude of the eigenfunction of the 15. V. M. Kudryashov, Akust. Zh. 45, 529 (1999) [Acoust. first water mode is less than unity. Phys. 45, 472 (1999)]. Figure 6 presents a plot for the case similar to that of 16. V. M. Kudryashov, Akust. Zh. 34, 1081 (1988) [Sov. Fig. 5. The only difference is that, for the deep-water Phys. Acoust. 34, 618 (1988)]. region, we used the sound velocity profile given by 17. V. M. Kudryashov, in Proceedings of All-Union Acousti- cal Conference (Moscow, 1991), Section D, p. 35. curve 1 from Fig. 1b. The change in the sound velocity profile caused a noticeable change in the time (delay) structure of the envelope. The changes in the mode Translated by E. Golyamina

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002

Effect of a Mudcake on the Propagation of Stoneley Waves in a Borehole G. A. Maximov and M. E. Merkulov Moscow State Engineering Physics Institute (Technical University), Kashirskoe sh. 31, Moscow, 115409 Russia e-mail: [email protected] Received March 19, 2001

Abstract—The problem of detecting a permeable stratum blocked by a mudcake with the help of acoustic measurements inside a borehole is considered. Different physical models of the mudcake are compared: in the form of a highly viscous liquid layer, in the form of a soft elastic shell, and in the form of an elastic shell ﬁxed in an arbitrary way to the borehole walls. Numerical calculations are presented for the wave ﬁeld in a borehole. © 2002 MAIK “Nauka/Interperiodica”.

In view of the numerous predictions concerning the elastic membrane, and other authors [6] consider the exhaustion of basic oil reserves, the problems of both mudcake as a layer of rock with a reduced permeability. prospecting for new deposits and efficient use of known In both cited publications, a complete wave problem for oil supplies become increasingly important. For years, the Biot model was solved for an arbitrary frequency studies had been carried out to determine the relation range, which considerably complicates the description between the properties of low-frequency Stoneley and the physical interpretation of the results. waves excited in a borehole and the permeability of the In this paper, we consider only the long-wave surrounding medium. approximation, because, in acoustic measurements, the Laboratory experiments [1] showed that the proper- characteristic wavelengths observed in both the sur- ties of Stoneley waves in a borehole surrounded by a rounding medium and the borehole fluid are usually permeable medium are adequately described by the much greater than the borehole diameter. Three differ- Biot theory [2, 3]. However, in actual conditions, the ent physical models of the mudcake are developed. In walls of the borehole can be covered with an almost the first model, the mudcake is described as a highly impermeable layer, which considerably reduces the viscous liquid layer formed in a permeable porous fluid flow from the reservoir to the well. The formation medium of the stratum. This representation can be eas- of such a layer, namely, a mudcake, can be caused, e.g., ily reduced to the model of a low-permeability layer [6] by the drilling mud that is deposited on the walls of the and, from this point of view, it is more general. In the borehole or penetrates under hydrostatic pressure into second model, the mudcake is considered as a free elas- the porous medium. On the other hand, experiments [4] tic shell situated near the wall of the borehole. The third showed that in the course of drilling, solid particles of model differs from the second in that the shell is fixed rock can penetrate into the pores of the borehole walls in an arbitrary way to the borehole walls. This model and form a thin (<0.5 mm) barrier, which is 100 to includes an additional assumption that the shell thick- 20000 times less permeable than the surrounding ness is much smaller than the borehole diameter. medium. It was shown that this process leads to a For each of the models, numerical calculations are decrease in the average permeability of rock by a factor performed to determine the wave field inside a borehole of 2Ð150. Thus, a mudcake considerably reduces the intersecting a permeable stratum that is blocked by a production of an oil pool and can even hinder its detec- mudcake. The calculations make it possible to draw the tion by a fluid flow. conclusions concerning the possibility of detecting a Currently, none of the existing technologies is able permeable stratum in the case of a wave and a mudcake to determine the position of a permeable stratum in the with characteristic parameters. presence of a mudcake in a borehole. The first step in The frequency range used in vertical seismic profil- the development of such technologies is a theoretical ing as a rule does not exceed several hundreds of hertz, study of the effect of a mudcake on the acoustic wave and the borehole diameter does not exceed 20 cm. field inside a borehole. For this purpose, it is necessary Hence, the characteristic wavelengths in both the bore- to develop a model that adequately describes the mud- hole fluid and the surrounding medium far exceed the cake. Active studies in this area of research are carried borehole diameter. Therefore, in deriving equations, we out in the United States [5, 6]. For example, some use the long-wave approximation λ ӷ b (b is the bore- authors [5] represent a mudcake as an impermeable thin hole radius).

188 MAXIMOV, MERKULOV The propagation of small-amplitude waves in a liq- or, in terms of the frequency representation, uid is described by the equations of motion, continuity, 0 2 2 ρ () and state: ∂ P f ω 2 f f ------+ ------P = iω------V ()rbz= ,,ω . (6) ∂ 2 2 f b r ∂ () z c0  ρ ()ρrzt,, +00 ∇V f ()rzt,, = ∂t f f When transverse filtration flows in the boreholeÐ  stratum system are absent, the fluid velocity near the  0 ∂ ()f ρ V = Ð∇P ()rzt,, (1) borehole boundary coincides with the velocity charac-  f ∂t f  terizing the displacement of the borehole walls (or the P ()rzt,, = c2ρ . inner surface of the mudcake). The fluid filtration f 0 f through the mud is neglected in this case. () Here, P (r, z, t), ρ (r, z, t), and V( f ) = {V f (r, z, t), 0, In the long-wave approximation, to obtain a closed f f r equation for the pressure in the borehole, we can use the ()f V z (r, z, t)} denote the deviations of the local pressure, quasistatic relation between the applied pressure and density, and mass velocity, respectively, from their the displacement of the borehole walls (the inner sur- ρ0 face of the mudcake), which can be determined from equilibrium values; f and c0 are the initial density of the static equations of the theory of elasticity [7Ð9]. the liquid and the sound velocity in it. Since we limit our consideration to the linear approxi- In the long-wave approximation, it is natural to mation, this relation will also be linear: characterize the sound field in the borehole by the ()f (),,ω dynamical quantities averaged over the borehole cross V r rbz= = APf + B. (7) section [7, 8]: Then, the closed equation for the pressure will have the b form 2π P ()zt, = ------P ()rzt,, rrd , ∂2P 2ρ0 f 2∫ f f 2()ω ω f πb ------+ K P f = i ------B, (8) 0 ∂z2 b b 2π where the wave number is determined by the expres- ρ ()zt, = ------ρ ()rzt,, rrd , (2) sion f π 2∫ f b 1 0 --- ω2 2ρ0 2 b K()ω = ------Ð iω------f A . (9) ()f 2π ()f 2 b V ()zt, = ------∫V ()rzt,, rrd . c0 πb2 0 This expression can also be represented in the form [5, 6] Here, b is the radius of the borehole. In Eqs. (1) and (2), i we took into account that, in the long-wave approxima- K()ω = ωS()ω 1 +,------(10) tion, the distributions of the dynamical quantities in the 2Q borehole are axially symmetric, so that averaging over ω –1 ω π where S( ) = c ( ) is the slowness, i.e., the reciprocal the angle yields the factor 2 . of the phase velocity of the tube wave, and Q is the According to Eqs. (2), after averaging the equation Q-factor. of continuity and the equation of motion (its projection The coefficients Ä and Ç in Eq. (7) depend on the on the borehole axis) over the borehole cross section, parameters of the borehole wall, the mud, and the we obtain the relationships fluid. For simple cases, these coefficients are well known. If the borehole is surrounded by a homoge- ∂ ()f 1 ∂ 2 ()f V ()zt, = Ð----- ρ ()zt, Ð ---V ()r= bzt,, , (3) neous elastic medium, these coefficients are A = ∂z z ρ0 ∂t f b r ω µ µ f i b/2 and B = –Pext A, where is the shear modulus of the elastic medium and Pext is the effective external () ∂V f ()zt, ∂P stress. Then, the wave number will be determined by ρ0 ------z = Ð------f . (4) the expression f ∂t ∂z 0 2 2 1 ρ Combining Eqs. (3) and (4), we obtain an inhomo- K ()ω = ω---- + -----f . (11) 2 µ geneous wave equation for the pressure field in a fluid- c0 filled borehole: If the elastic medium under consideration is charac- 2 2 0 ()f ∂ P 1 ∂ P 2ρ ∂V ()r= bzt,, terized by a nonzero permeability, the expression for ------f Ð ------f = ------f ------r , (5) the velocity will have an additional term associated ∂ 2 2 ∂ 2 b ∂t z c0 t with the transverse filtration flows in the boreholeÐstra-

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 EFFECT OF A MUDCAKE ON THE PROPAGATION OF STONELEY WAVES 189 tum system. In this case, the wave number will have the blocked by a mudcake. The main problem is that the form [8]: mudcake may exhibit different properties in different conditions. Therefore, we developed several models of ρ0 () 2 2 1 f 0 m0 1K1 z the mudcake [10]. In fact, only two fundamentally dif- K ()ω = ω---- ++----- 2 ρ ------, (12) 2 µ f () c K f z K0 z ferent ways are possible in describing this object: the 0 mudcake can be considered either as a highly viscous ηm liquid or as a solid shell. where the argument z = iω------0 b is determined by k0K f both the frequency and the combination of the parame- MODEL OF A MUDCAKE AS A HIGHLY ters characterizing the porosity m0, the permeability k0, VISCOUS LIQUID η and the dynamic viscosity of the fluid ; K0 and K1 are the Macdonald functions. This model represents the mudcake as a highly vis- In this paper, we determine the coefficients of the cous liquid layer formed in the porous medium of a per- quasistatic relation between the applied pressure and meable stratum. In the framework of this model, the the displacement of the inner surface of the mudcake. coefficients characterizing the relation between the For this purpose, we consider the following system: a velocity of the inner mud surface and the pressure have borehole in a permeable homogeneous elastic medium the form (see Appendix A):

I ()γ R K ()γ R a I ()γ R K ()γ R F()γ R ------0 1 1 + ------0 1 1 + ------1 ------1 1 1 Ð ------1 1 1 2 1 I ()γ R K ()γ R K I ()γ R K ()γ R ω ()γ ˜ ()γ ------1 1 0 1 1 0 f 1 1 1 0 1------1 0 - A1 = m0 i F 1 R0 F 1 R0 ()γ ()γ ()γ ()γ , (13) ()γ I0 1 R1 K0 1 R1 a1 I1 1 R1 K1 1 R1 F 2 R1 ------()γ Ð ------()γ + ------()γ + ------()γ I0 1 R0 K0 1 R0 K f 1 I0 1 R0 K0 1 R0

ext B1 = ÐP A1, (14) where R0 is the inner radius of the mud layer (in this model, it coincides with the borehole radius b); R1 is the outer 2 η γ ω 2 radius of the mud layer; a = k0Kf / m0; j = i /a j ; j = 1, 2, where 1 corresponds to the mud and 2 to the ﬂuid; K0, I0, K1, and I1 are zero-order and first-order modified Bessel functions; ()γ () ()γ ai K1 i R , ˜ () I1 z F i R ===------()γ -; and i 12; F z ------(). K fiK0 i R I0 z

Taking into account the elastic motion of the borehole ρ0 ρ0 2 2 1 2 f 1 walls, we obtain an expression for the wave number K = ω ---- ++-----f ------, (16) describing the wave propagation in the borehole: 2 µ b W ()ω + W ()ω c0 i p ρ0 ρ0 2 2 1 2 f A1 where K = ω ---- +.-----f Ð i------(15) c2 µ b ω η 0 ()ω ω() Wi = Ði R1 Ð R0 ------, kimp When the thickness of the mudcake is zero, i.e., R0 = R , this expression is reduced to Eq. (12). ()1 1 K˜ z˜ H ()z˜ iωm η In [6], the authors described the mudcake in the W ()ω ==Ð------0 -, z˜ ------0 b, p m b ()1 () ˜ framework of the Biot model as a thin layer of the sur- 0 H1 z˜ k0K rounding elastic medium with a reduced permeability ()1 ()1 kimp. Since permeability is involved in Eq. (13) only in H0 and H1 are the zero-order and first-order Hankel combination with the viscosity, k/η, our model can be ˜ easily reduced to the model described in [6]. For this functions of the first kind, and K is a fairly complex purpose, it is sufficient to assume that the compression combination of the porosity and the elastic moduli that characterize the skeleton of the porous medium and the modulus of the mud is equal to that of the fluid: Kf1 = K . Then, by varying the viscosity of the mud, we fluid (detailed formulas can be found in [5]). Perform- f2 ing the numerical comparison, we took into account the effectively vary the permeability of rock. The cited ˜ publication [6] presents the low-frequency approxima- difference between K and Kf by introducing a correction of the total result (see Eq. (13) in [6]): tion factor close to unity.

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 190 MAXIMOV, MERKULOV

Table 1. Parameters used in the calculations

ρ0 3 Fluid-ﬁlled borehole Fluid density f , kg/m 1000

Sound velocity in the ﬂuid c0, m/s 1500 Fluid viscosity η, Pa s 0.001 Borehole radius b, cm 14.5

Porous medium Longitudinal wave velocity cp, m/s 3735 Transverse wave velocity cs, m/s 2080 ρ 3 Density of rock b, kg/m 2337 Porosity m0 0.2 Permeability k0, D 0.1 Mudcake Sound velocity in the mud cCL, m/s 1500 η Mud viscosity CL, Pa s [0.001Ð0.1] ρ 3 Mud density CL, kg/m 1000 Mudcake thickness R1 Ð R0, mm 1

When the thickness of the mudcake is zero, Eq. (16) is necessary to perform the substitution ω –ω. Tak- is reduced to an expression equivalent to Eq. (12): ing into account the relation between the Hankel and π () ν + 1 1 () Macdonald functions, Kν(z) = --- i Hν (iz) [11], one ρ0 1 ()˜ 2 2 ω2 1 f ρ0 m0 1H1 z K = ---- +.-----Ð 2 f ------() - (17) can easily verify that Eqs. (12) and (17) are structurally 2 µ K˜ z˜ H 1 ()z˜ c0 0 identical. It should be noted that, to change to the frequency rep- To compare our model of a highly viscous liquid resentation, we used the Fourier transform with the core with the model of a low-permeability layer, we used Eq. eiωt, whereas in the aforementioned publications [5, 6], (16) and the parameters from [6] (Table 1). The results the core e–iωt was used. Therefore, to compare results, it of this comparison are presented in Fig. 1.

Normalized velocity Damping factor (1/Q) 1.0 1.0

1 D/Pa s 0.9 0.8 15 D/Pa s

0.8 0.6

0.7 0.4

15 D/Pa s 0.6 0.2

1 D/Pa s 0.5 0 200 400 600 800 1000 0 200 400 600 800 1000 Frequency, Hz

Fig. 1. Comparison of the frequency dependences of the phase velocity c(ω)/c (left) and the damping factor 1/Q (right) calculated 0 η using the model of a highly viscous liquid (the solid curves) for the parameters k0/ CL = 1 and 15 D/Pa s with the corresponding dependences obtained for the model of a low-permeability layer [5]. The dashed curves correspond to the calculations for an open borehole.

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 EFFECT OF A MUDCAKE ON THE PROPAGATION OF STONELEY WAVES 191

MODEL OF THE MUDCAKE AS AN ELASTIC and WMC is the membrane stiffness, this parameter CYLINDRICAL SHELL being artificially introduced in the boundary condition between the porous medium and the mudcake: In this model, the mudcake is represented as an elastic σ σ cylindrical shell, which has the inner radius R and the rr1(b) = P2(b) + WMC[Ur2(b) – ur2(b)]. Here, rr is the 0 radial component of the stress tensor at the surface of outer radius R1 and is situated near the inner surface of the borehole (see Appendix B). The coefficients charac- the mudcake; P2 is the fluid pressure in the pores; and terizing the quasistatic relation between the velocity of Ur2 and ur2 are the displacements of the fluid and the the inner mud surface and the pressure have the form skeleton of the porous medium, respectively. In the case W = ∞, this condition is reduced to the zero differ- 2 MC R0 Ψ()z []1 + d + 1 Ð ------1Ð ence between the displacements: Ur2(b) – ur2(b) = 0. In ω 1  µ i R0 R1 1 the case WMC = 0, we obtain the equality of pressures: A2 = ------, σ 2 R 2 R 2 rr1(b) = P2(b). d µ 1 Ð -----0 + Ψ()z 1 + d -----0 (18) 1 1R 1R In developing our model, we used the latter condi- 1 1 tion, i.e., the condition of equal pressures (see Appen- ω [] i R0 ÐPext 1 + d1 dix B). Therefore, to perform the comparison, we must B2 = ------, set W = 0 in Eq. (20). 2 R 2 R 2 MC d µ 1 Ð -----0 + Ψ()z 1 + d -----0 (19) 1 1R 1R Substituting A2 into Eq. (9), we obtain the wave 1 1 number of the tube wave in our model. If we separate where ρ0 λ µ µ the term f /Mf and use the formula d1 = ( 1 + 1)/ 1 Φ˜ ()z K zK ()z corresponding to the case of fixed ends (u = 0), the Ψ()z == µ------, Φ˜ ()z ------f ------0 , zz 2 µ Φ˜ () m K ()z wave number can be written in the form 2 2 + z 0 1 ρ0 ρ0 ()µ2 and the parameter d is determined by the following 2 2 1 2 f 1 Ð f c 2 K ()ω = ω---- ++------f ------, (21) expressions depending on the type of the stressed state: 2 M b ()XW+ ()ω H H c0 f p 3λ + 2µ d = ------when the shell ends are free (σ = 0) λ + 2µ zz λ µ µ µ 1 + 1 and where X = 2 2 1fc ------λ µ is the term that distin- 1 + 2 1 λµ + guishes our result from Eq. (20) at WMC = 0. d = ------when the shell ends are fixed (uzz = 0). µ When the membrane thickness is equal to zero, i.e., Index 1 corresponds to the mudcake, and index 2 to R1 – R0 = 0, we have fc = 0, which yields X = 0, and the the porous elastic medium. results coincide. In the other limiting case k0 0, we ∞ We note that, in this model, the coefficient B is not have Wp , and both Eqs. (20) and (21) are reduced related to A by Eq. (14), so that the velocity character- to the expression izing the displacement of the inner mud surface is not ρ0 proportional to the difference between the pressures 2()ω ω21 f K = ----2 + ------. (22) inside the borehole and at infinity. This is a conse- c M f quence of the fact that, under pressure, the mudcake is 0 not only displaced as a whole, but it is also deformed, Figure 2 shows the results of calculations by Eqs. (20) i.e., its thickness changes. and (21) for the parameters presented in Table 2 (the In [5], the mudcake was considered as an elastic mem- values of the parameters are taken from [5]). brane. The result obtained in this case in the low-frequency approximation is as follows (Eq. (58) from [5]): STATISTICAL MODEL K2()ω To compare the model of a highly viscous liquid with the cylindrical shell model, it is necessary to cal- ρ0 ρ0 ()µ2 2 1 f 2 1 Ð f c 2 (20) culate the wave field inside the borehole. In the follow- = ω ---- ++------f ------, 2 M []W + W ()ω b 2 ing section, we will show that the results obtained with c0 f MC p H these two models differ considerably. The calculations using the cylindrical shell model with the characteristic where parameters show that the mudcake practically does not µ []λ + 2µ + f ()λµ Ð µ ()+ µ affect the wave field (this conclusion was also made in M = ------2 1 1 c 1 2 1 1 ; f λ ++2µ f ()µ Ð µ [5]), whereas the calculations performed in the frame- 1 1 c 2 1 work of the model of a highly viscous liquid show con- 2 siderable changes in the wave field structure due to the R0 µ µ µ λ µ λ µ fc = 1 Ð ----- ; H = 2 Ð fc( 2 Ð 1)( 1 + 1)/( 1 + 2 1); mudcake. R1 ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 192 MAXIMOV, MERKULOV

Slowness, ms/m Damping factor (1/Q) 1.2 1.0

1.1 0.8

1.0 0.6

0.9 0.4

0.8 0.2

0.7 0 101 102 103 101 102 103 Frequency, Hz

Fig. 2. Comparison of the frequency dependences of the Stoneley wave slowness S(ω) (left) and the damping factor 1/Q (right) calculated using the model of an elastic membrane for the parameters h = 3 mm (the solid curves) and 6 mm (the dot-and-dash curves) with the corresponding dependences obtained for the membrane model [5]. The triangles refer to h = 3 mm and the diamonds refer to h = 6 mm. The dashed curves represent the calculations for an open borehole.

This discrepancy between the results called for the the case under consideration, the coefficients in Eq. (7) development of a more realistic model in which the have the form mudcake is considered as a thin cylindrical elastic shell fixed to the borehole walls in an arbitrary way (see iωb iω Appendix C). This model contains a random parame- A = ------+ ------, ter—the distance between the fixing points. Varying its 3 µ ˜ ()()() 2 W MC + K f zK0 z / m0bK1 z (23) average value, one can obtain the results similar to those derived from the two models considered above. In B3 = ÐPext A3,

Table 2. Parameters used in the calculations

ρ0 3 Fluid-filled borehole Fluid density f , kg/m 1000

Sound velocity in the fluid c0, m/s 1500 Fluid viscosity η, Pa s 0.001 Borehole radius b, cm 10.5

Porous medium Longitudinal wave velocity cp, m/s 3360

Transverse wave velocity cs, m/s 1675 ρ 3 Density of rock b, kg/m 2670

Porosity m0 0.2

Permeability k0, D 0.1 p Mudcake Longitudinal wave velocity in the mud cCL , m/s 1500

s Transverse wave velocity in the mud cCL , Pa s 320 ρ 3 Mud density CL, kg/m 1100

Mudcake thickness R1 Ð R0, mm [0Ð6]

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 EFFECT OF A MUDCAKE ON THE PROPAGATION OF STONELEY WAVES 193

Normalized velocity Damping factor (1/Q) 1.0 1.0

1 0.8 3 0.8 2 0.6

0.4 0.6

0.2 2 3

0.4 1 0200 400 600 800 1000 0 200 400 600 800 1000 Frequency, Hz

ω Fig. 3. Comparison of the frequency dependences of the phase velocity c( )/c0 (left) and the damping factor 1/Q (right) for different models of the mudcake: (1) the highly viscous liquid model, (2) the elastic shell model, and (3) the statistical model. The dashed curves correspond to an open borehole. where L = 4 mm and D = 13 mm2. These parameters were Ð1 selected so as to demonstrate most clearly the interme- 2 δ hb 25 δ hb Ð------------Ð------------diate position of the statistical model. Eh 1 + s4 3τ 14+ s4 3τ W˜ = ------2 e + e MC ()σ2 2 Below, we present examples of calculated seismo- 1 Ð b grams that illustrate the effect of the mudcake on the σ acoustic wave field generated in a borehole by a pulsed is the shell stiffness; h is the shell thickness; E and are source positioned inside the borehole. To obtain the the Young modulus and the Poisson ratio, respectively; seismograms, we used the TUBEWAVE software [7, 2 8], which allowed us to calculate the wave field in a τ D L = ---- ; s = ----- – 1; L is the average distance between borehole embedded in a stratified elastic medium. L D the fixing points; and D is its variance. The wave num- Figure 4a shows the simplest case of a borehole in a ber is expressed by the formula homogeneous elastic medium. In this case, a wave 2 propagates in the borehole without reflections. If a per- K (24) meable stratum with the same elastic parameters as those of the surrounding medium is introduced in the ρ0 ρ f 2 1 f 2 0 1 system, a difference will appear in the fluid velocities = ω ---- ++------. 2 µ ˜ near the borehole walls. Then, the wave numbers given c b W MC + K zK ()z /()m bK ()z 0 f 0 0 1 by Eqs. (11) and (12) will differ by only the second term We note that, in this model, the velocity of the inner in Eq. (12), i.e., the term associated with the permeabil- surface of the mudcake is proportional to the pressure ity. This may give rise to a reflected wave (Fig. 4b). difference Pf – Pext despite the fact that, as in the previ- In our calculations, we used the following parame- ous case, the mudcake is described as a solid body. This ρ0 3 is a consequence of the thin shell approximation. ters: the initial fluid density f = 1000 kg/m ; the initial η Figure 3 presents the comparison of the frequency fluid velocity cf = 1500 m/s; the fluid viscosity = dependences of the phase velocity and the damping fac- 0.001 Pa s; the borehole radius b = 10 cm; the density tor of a tube wave for the model of a highly viscous liq- of the surrounding elastic medium ρ = 2000 kg/m3; the uid, the elastic shell model, and the statistical model. longitudinal and transverse velocities of sound in this The calculations were performed using the parameters medium cp = 4500 m/s and cs = 2500 m/s, respectively; η from Table 2. The mud viscosity was CL = 0.1 Pa s; the the thickness of the permeable stratum (for the case cor- mudcake thickness was h = 5 mm; the average length responding to Fig. 4b) d = 4 m; the porosity of the and its variance in the statistical model were taken to be medium m0 = 15%; and the permeability of the medium

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 194 MAXIMOV, MERKULOV

(a) (b)

–50 –51 –58 –60 –62 –66 –70 –74 –78 –80 –82 –86 –90 Depth, m –91 –98 –102

Permeable stratum –106 –110 –114 –118 –122 –120

Fig. 4. Synthetic seismograms illustrating the propagation of Stoneley waves in a borehole (a) in the absence and (b) in the presence of a permeable stratum.

η CL/KCL ≈ k0 = 10 D. The permeability was chosen to be high to realized when ------η > 1. In the elastic shell model, make the effect under investigation more pronounced. f /K f µ µ Ӷ the stratum is visible when 1/ 2 1 and invisible Now, let us assume that the permeable stratum is µ µ ӷ when 1/ 2 1, and the situation is intermediate when blocked by mud. Depending on the mudcake and stra- µ µ ≈ tum parameters, three situations are possible: 1/ 2 1. In the framework of the statistical model, the µ (a) mud is amenable to varying pressure; then, the 1 hb Ӷ stratum is visible when µ------1 and invisible when seismogram has the form shown in Fig. 4a, i.e., the stra- 2 L tum is clearly “visible”; µ 1 hb ӷ (b) mud is resistant to varying pressure; then, the µ------1, and the intermediate situation corresponds 2 L seismogram has the form shown in Fig. 4b, i.e., the stra- µ tum is “invisible”; 1 hb ≈ to µ------1. (c) the intermediate situation. 2 L Thus, the effect of the mudcake on the wave field Calculations performed for the characteristic param- should manifest itself as a change in the amplitude ratio eters of both borehole and mud showed that, for the of the reflected and transmitted waves. The stronger the highly viscous liquid model, case (b) is realized (the stra- mudcake blocks the permeable stratum, the smaller the tum is invisible), and for the elastic shell model, case (a) amplitude of the reflected wave. takes place (the stratum is clearly visible). This is the dis- For each model of the mudcake and for a fixed mud- crepancy of the results that was mentioned above. In the cake thickness h = 1 cm and with fixed parameters for calculations, we used the following characteristic param- the borehole, we chose mud parameters that correspond eters of mud: the density ρ = 2000 kg/m3, the longitu- to each of the three aforementioned situations (Fig. 5). CL dinal sound velocity c p = 1000 m/s, the transverse sound In the model of a highly viscous liquid, the perme- CL η s CL/KCL velocity (for the elastic shell model) cCL = 500 m/s, and able stratum is visible at ------η = 1 and invisible the viscosity η = 1Ð100 Pa s. A similar calculation for f /K f CL η the statistical model showed that case (a) is most likely CL/KCL ӷ when ------η 1, and the intermediate situation is to be realized. f /K f ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 EFFECT OF A MUDCAKE ON THE PROPAGATION OF STONELEY WAVES 195

(a)(b) (c) –50 –51 –58 –60 –62 –66 –70 –74 –78 –80 –82 –86 Depth, m –90 –91 –98 –102 –106 Permeable stratum –110 –114 –118 –122 –120

Fig. 5. Synthetic seismograms illustrating the effect of the mudcake on the reﬂection of Stoneley waves and their transmission through a blocked permeable interval for the cases of (a) a weak, (b) intermediate, and (c) strong blockage. The parameters corresponding to these cases for (1) the model of a highly viscous liquid, (2) the elastic shell model, and (3) the statistical model are as η /K µ µ η /K µ µ CL CL 1 Ӷ 1 hb Ӷ CL CL ≈ 1 ≈ 1 hb ≈ follows: case (a): (1) ------η = 1, (2) µ------1, (3) µ------1; case (b): (1) ------η > 1, (2) µ------1, (3) µ------1; f /K f 2 2 L f /K f 2 2 L η µ µ CL/KCL ӷ 1 ӷ 1 hb ӷ case (c): (1) ------η 1, (2) µ------1, (3) µ------1. f /K f 2 2 L

Thus, in this paper, we discussed the problem of whereas, in this paper, we proposed a new method for detecting a blocked permeable stratum by means of its calculation. We showed that, in the low-frequency acoustic measurements in a borehole. In this context, limit, the results obtained in [5, 6] in the framework of we studied the response of such an object as a mudcake the Biot theory agree well with the results obtained by to the effect of an acoustic wave ﬁeld. Several ways of us. However, the results obtained in this paper are describing the mudcake were proposed, and it was derived in a much simpler way and have a clearer phys- found that its response to the effect of acoustic waves ical interpretation. strongly depends on the mudcake model. The calcula- In closing, we conclude that, currently, there exists tion performed in the framework of the model repre- a theoretical basis for the determination of mud-cake- senting the mudcake as a highly viscous liquid layer blocked permeable intervals in a borehole from the showed that, in the case of a viscosity characteristic of character of the propagation of Stoneley waves. How- a mud solution, a mudcake several millimeters thick ever, the choice of the most adequate model of a mud- completely blocks the permeable interval. A similar cake is hampered by the lack of experimental data. calculation using the elastic shell model yields the opposite result even for a one-centimeter-thick mudcake. A more realistic model representing the mudcake APPENDIX A as an elastic membrane ﬁxed in an arbitrary way to the borehole surface allowed us to combine the two previ- (THE HIGHLY VISCOUS LIQUID MODEL) ous results by averaging them with different weights. Formally, this procedure resulted in the introduction of In the framework of this model, a mudcake is considered as a highly viscous liquid layer formed in the ˜ the effective stiffness of the membrane W MC . In [5], the porous medium of a permeable stratum and character- membrane stiffness was introduced as a parameter, ized by the inner radius R0 and the outer radius R1.

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 196 MAXIMOV, MERKULOV

A laminar fluid flow in a porous medium is The general solution for R0 < r < R1 has the form described by the continuity equation and the Darcy law: ∂()mρ () f ()ρ f iω iω ------∂ -+0, div m f V = (), ω   t Pr = C1K0 -----2-r + C2 I0 -----2-r η (A.1) a a m0 ()f 1 1 gradPpor = Ð------V , k0 << where m and m0 are the current and initial porosities of for R0 rR1 and the medium, respectively; ρ and ρ0 are the current and f f iω iω (A.5) f (), ω V( ) Pr = C3K0------r + C4 I0------r initial densities of the fluid; is the mass velocity of 2 2 the fluid (the skeleton of the porous medium is assumed a2 a2 to be stationary); k0 is the permeability of the medium; η and is the dynamic viscosity of the fluid. <<∞ for R1 r , Setting m = m0 = const (i.e., neglecting in this way the contact compressibility) and taking into account the compressibility of the fluid according to the relation- where K0, I0, K1, and I1 are the zero-order and first- order modified Bessel functions [11]. The constants C 0 P i ship ρ = ρ 1 + ------por- , we linearize Eqs. (A.1): f f K are determined from the following boundary condi- f tions: at long distances from the borehole, the deviation

∂P ()f of the pressure from the equilibrium value should tend ------por- +0,K divV = to zero, i.e., P(r ∞, ω) 0. At the mud–fluid ∂t f (A.2) boundary in the porous medium, the conditions of m η () equal pressures and equal velocities must be satisfied, gradP = Ð------0 -V f . por k and at the borehole boundary, the mud pressure should 0 be equal to the fluid pressure in the borehole: Here, Kf is the bulk modulus of the fluid. From Eqs. (A.2), we easily obtain the filtration equation (), ω (), ω PR1 Ð 0 = PR1 + 0 ∂P  por 2∆ ()f ()f ------+0,a Ppor = (A.3) v ()R Ð 0, ω = v ()R + 0, ω (A.6) ∂t  1 1 (), ω ()ω 2 η ∆ PR0 = P f . where a = k0Kf/ m0 and is the Laplacian. We seek the solution in the form Ppor = Pext + P(r, t). In terms of the frequency representation with allow- Here, Pf represents the deviation of the pressure in ance for the cylindrical symmetry, the problem of filtra- the borehole from the equilibrium pressure. tion can be formulated as follows: Thus, substituting the general solutions (A.5) into  ω ∆ (), ω i (), ω << Eqs. (A.6), we obtain a set of equations for the determi-  rPr Ð0------Pr = for R0 rR1 2 nation of the constants Ci. Using the solution of the fil-  a1  (A.4) tration problem for R0 < r < R1 and Eqs. (A.2) relating  iω V( f ) and P, we obtain (through cumbersome transfor- ∆ Pr(), ω Ð0------Pr(), ω = R <

()γ ()γ ()γ ()γ ()γ I0 1 R1 K0 1 R1 a1 I1 1 R1 K1 1 R1 F 2 R1 ------+ ------+ ------Ð ------() I ()γ R K ()γ R K I ()γ R K ()γ R v f (), ω ω ()γ ˜ ()γ ------1 1 0 1 1 0 f 1 1 1 0 1------1 0 - R0 = P f m0 i F 1 R0 F 1 R0 ()γ ()γ ()γ ()γ , (A.7) ()γ I0 1 R1 K0 1 R1 a1 I1 1 R1 K1 1 R1 F 2 R1 ------()γ Ð ------()γ + ------()γ + ------()γ I0 1 R0 K0 1 R0 K f 1 I0 1 R0 K0 1 R0

iω γ One can easily verify that, at R R , i.e., when the mud where j = -----2- , j = 1, 2; 0 = 1 a j layer thickness is equal to zero, Eq. (A.7) is reduced to ()γ () the expression, which coincides with the result ()γ ai K1 i R , ˜ () I1 z obtained in [5] by solving a simple ﬁltration problem F i R ===------()γ -, i 12; F z ------(). K fiK0 i R I0 z without the mud layer:

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 EFFECT OF A MUDCAKE ON THE PROPAGATION OF STONELEY WAVES 197

()f ω ()γ v = P f m0 i F 2 R0 ()γ (A.8) ω a2 K1 2 R0 = P f m0 i ------()γ . K f 2 K0 2 R0 One can also show that Eq. (A.7) transforms to Eq. (A.8) when the parameters of the ﬂuid and the mud are equal, γ γ i.e., when 1 = 2.

APPENDIX B (THE CYLINDRICAL ELASTIC SHELL MODEL) In this model, the mudcake is represented as an elastic shell characterized by the inner radius R0 and the outer radius R1 and situated near the inner surface of the borehole with the radius R2 (Fig. 6). We assume that I only radial displacements take place. Then, from the conditions of equal pressures and displacements at the II layer boundaries, we obtain an expression for the dis- III IV placement velocity of the mudcake with allowance for the transverse flows in the boreholeÐstratum system. Fig. 6. Geometry of the problem for a borehole blocked by a mudcake: (I) the borehole fluid, (II) the mudcake, (III) the The equilibrium equation for a solid under a surface intermediate fluid layer, and (IV) the surrounding perme- force has the form [12] able porous medium.

21()Ð σ grad() divu Ð0,()12Ð σ curlcurlu = Using Eqs. (A.1)Ð(A.5) from Appendix A, we obtain where σ is the Poisson ratio. the expressions for the pressure and the fluid displacement in the porous medium: The deformation caused by a pressure that is uniform along the tube has the form of a radial displace- IV () ()ω 2 Ppor r = Pext + CK0 i /a r , (B.4) ment ur = u(r). In this case, we obtain IV () 1 1 ()ω 2 upor r = C------K1 i /a r . (B.5) grad() divu = 0, K 2 f iω/a 1∂()ru divu ==------∂ const, Now, we formulate the conditions of the equality of r r (B.1) pressures and displacements at the layer boundaries: b uar= + ---. σII () r rr R0 = ÐP f , σII ()R = σIII()R , (B.6) Using the Hooke law, we derive an expression for the rr 1 rr 1 radial component of the stress tensor: II() III() ur R1 = ur R1 . σ () µ µ b The displacement at the fluid–porous medium bound- rr r = 2a d Ð 2 ----2, (B.2) r ary (r = R2) is determined by the filtration of the fluid and the elastic displacement of the porous medium: λ µ 3 + 2 III() IV() IV () where d = ------when the ends of the shell are free ur R2 = ur R2 + m0urpor R2 , λ + 2µ (B.7) λµ σIII() σIV() IV () σ + rr R2 ==rr R2 ÐPpor R2 . ( zz = 0) and d = ------µ - when the ends are fixed (uzz = 0). The conditions at infinity have the form We will first consider the fluid in layer III as a solid σIV()∞ by formally introducing its shear modulus µ , and, in rr = ÐPext, f (B.8) the final formula, we will set µ = 0. Then, according to IV f P ()∞ = P . Eqs. (B.1), the displacement of the fluid has the form por ext Substituting Eqs. (B.1) and (B.2) into the boundary III() b f conditions (B.6)Ð(B.8), we obtain a set of linear equa- ur r = a f r + -----. (B.3) r tions in the unknowns a1, b1, bf, b2, and C:

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 198 MAXIMOV, MERKULOV

b 2a µ d Ð 2µ -----1 = ÐP  1 1 1 1 2 f  R0   µ µ b1 ()µ λ µ b f 2a1 1d1 Ð22 1----- = a f f + f Ð 2 f -----  R2 R2  1 1  b  ()ω 2 ()µ λ µ f ÐPext Ð2CK0 i /a R2 = a f f + f Ð 2 f -----2  R2  (B.9)  b1 b f a1 R1 + ----- = a f R1 + -----  R1 R1   µ b2 ()ω 2 ÐPext Ð 2 2----- = Ð Pext Ð CK0 i /a R2  R2  2  b f b2 1 ()ω 2 a f R2 + ----- = ----- + m0C------K1 i /a R2 .  R2 R2 ω 2 K f i /a µ Solving this set of equations and setting f = 0, we derive an equation for the displacement of the mudcake:

R 2 Ψ()z R 2 R 2Ψ()z R 2 P d + -----0 1 + ------1 Ð -----1 + -----1 ------1 Ð -----0 Ð P ()d + 1 b R f 1 R K R R µ R ext 1 uII()R ==a R + -----1 -----0 ------1 f 2 2 1 ------1 , r 0 1 0 R 2 2 Ψ() 2 2 2 0 µ R0 z R1 Ψ()R1 R0 1d11 Ð ----- 1 + ------1 Ð ----- + z ----- 1 + d1----- (B.10) R1 K f R2 R2 R1 K zK ()z ηm Φ˜ ()z Φ˜ ()z ==------f ------0 ; ziω------0 R ; Ψ()z = µ------. m K ()z k K 2 2 µ Φ˜ () 0 1 0 f 2 2 + z

The corresponding velocity in the frequency repre- APPENDIX C sentation is related to displacement (B.10) by the factor iω: v (ω) = iωu(ω). We limit our consideration to the (THE STATISTICAL MODEL) case R1 = R2, i.e., to the case of a mud layer immedi- In the two preceding appendixes, we considered the ately adjacent to the borehole wall. Then, we have model of a highly viscous liquid and the elastic shell ω model. These models yielded entirely different results. (), ω i R0 v CL R0 = ------In the case of a highly viscous liquid, the filtration flows 2 can be neglected. On the other hand, in the elastic shell (B.11) 2 model, a one-centimeter-thick mudcake practically R0 Ψ()z ()P Ð P []1 + d + P 1 Ð ------1Ð does not affect the velocity of the fluid near the bore- f ext 1 f R µ × 1 1 hole walls. ------2 2 -. µ R0 Ψ()R0 This discrepancy between the results required the d1 11 Ð ----- + z 1 + d1----- R1 R1 development of a more realistic model. In such a model, the mudcake is considered as a thin elastic For a shell of zero thickness (R0 = R1), Eq. (B.11) cylindrical shell fixed in an arbitrary way to the bore- takes the form hole walls. For simplicity, we assume that the positions of the fixing points obey cylindrical symmetry. The dis- P Ð P v ()R , ω = iωR ------f ext tance between the neighboring fixing points L is a ran- CL 0 0 2Ψ()z dom value whose distribution is described by some () (B.12) probability density. The probability density function is ω ∆ 1 m0 K1 z unknown, but we can assume that its behavior is similar = i R0 P ------µ +.------() 2 2 K f zK0 z to the Gaussian distribution, because L depends on many parameters. Then, the probability density func- Expression (B.12) coincides with Eq. (A.8) correct tion should have a dome-like shape characterized by a ω ∆ 2µ to the term i R0 P/ 2, which represents the velocity mean value and a variance of L; it should be defined determined by the elastic motion of the borehole walls. within the interval 0 < L < ∞ and normalized to unity.

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 EFFECT OF A MUDCAKE ON THE PROPAGATION OF STONELEY WAVES 199

In this paper, for simplicity and convenience, we use The boundary conditions have the form the probability density function in the form ∂ ξ(x) = 0 and ------ξ(x) = 0 at x = 0, L. 1 s ∂ GL()= ------L exp()ÐL/τ , (C.1) x τs + 1s! These boundary conditions allow us to consider the where s is a positive integer. plate fixed at many different points as a set of individual cells, which are fixed at their ends and have different lengths. We introduce the notation Averaging of the Mudcake Displacement within a Single Cell 4 12 k = ------2 -2 . (C.3) The equilibrium equation for a thin cylindrical plate h R in the absence of the dependence on the azimuth angle Evidently, a particular solution to inhomogeneous has the form [12] ξ ∆ 4 equation (C.2) is 1 = P/Dk , and the general solution ∂4 ∆ to the homogeneous equation can be easily determined ξ 12 ξ P ξ λ ------4 + ------2 -2 = ------, by applying the substitution = exp( x). From the con- ∂x R h D dition that the displacements be real and from the (C.2) 3 boundary conditions, we obtain the expression for the Eh D ≡ ------, displacement ξ of a cylindrical shell fixed at the edges. 12() 1 Ð σ2 Averaging this expression for ξ over x, we obtain where ξ is the radial displacement of the plate, E is the L σ 1 ∆P Young modulus, is the Poisson ratio, h is the plate ξ ==--- ∫ξ()x dx ------Fb(), (C.4) thickness, R is the borehole radius, and ∆P is the differ- L Dk4 ence between the pressures on the two sides of the 0 ∆ plate: P = Pf – Pout. where

2 sinh()2b + sin() 2b Ð 2[]sin()b cosh()b + sinh()b cos()b Fb()= 1 Ð ------(C.5) b cosh()2b + cos() 2b Ð 2 and b = kL/ 2 . other near the maximum of the function G(b(L)) given The function F(b) has the following asymptotics: by Eq. (C.1). In this case, we obtain b → 0 b → ∞ ∞ 4 4 s L F(b) b /180 and F(b) 1 – 2/b. Since the ∆P b L Ð---τ function F(b) has a fairly complex form, we select an 〈〉ξ = ------∫------e dL Dk4τs + 1s! ()b + δ 4 approximating function of a simpler form for the sub- 0 sequent calculations: ∞ (C.8) s + 1 s + 4 ∆Pα b Ðαb fb()= b4/()b + δ 4. (C.6) = ------∫------e db, Dk4s! ()b + δ 4 When b is small, the function f(b) behaves as (b/δ)4 0 and, when b is large, f(b) ≈ 1 – 4δ/b. By varying the where δ parameter , it is possible to obtain a coincidence of the α τ functions F(b) and f(b) for different values of b. = 2/ k. (C.9) This integral can be calculated. Then, the average displacement will have the form Averaging of the Mudcake Displacement ∆ over All Cells 〈〉ξ P Φαδ(), = ------4 s . Let the distribution of the cell length L be described Dk by the probability density G(L). Then, the procedure of Here, averaging over L has the form Φαδ(), s ∞ ∞ s + 1 s + 4 3 ∆ ()αδ ∂ ()αδ αδ P ()s ()αδ 〈〉ξ ==∫ ξ()L GL()dL ------∫GL()FbL()()dL. = ------Ð1 ------Ð------e Ei , Dk4 (C.7) s! ∂αδ()s + 4 6 0 0 Ðxt ∞ e Performing the integration in (C.7), we replace the where Ei(x) = ∫ ------dt. function F(b) in the integrand by the function f(b) given 1 t by Eq. (C.6). The parameter δ is chosen so as to make The analytical form of the function Φ(αδ, s) is rather the functions F(b) and f(b) as close as possible to each complicated. Since our aim is to obtain the most simple

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 200 MAXIMOV, MERKULOV function that is convenient for calculations and ade- plished by introducing an additional term in the expres- quately describes the main features of the behavior of sion for the velocity. The final result has the form the physical quantity, we again use an approximating v function. It has the form tot () ------= P f Ð Pext Φ ()αδ, iωR ' s (C.14) 1 1 1 A A (C.10) × ------+.------= --- exp Ð------1 αδ +,exp Ð------2 αδ 2µ ˜ ()()()  RWMC + K f zK0 z / m0K1 z 2 1 + B2s 1 + B2s When the thickness of the mudcake tends to zero, we have where the parameters A1, A2, B1, and B2 are chosen so ˜ as to obtain the closest approximation of the initial W MC 0, and Eq. (C.14) is reduced to Eq. (B.12). function. In the calculations, we used the values A1 = 2, A = 25, B = 1, and B = 4. 2 1 2 REFERENCES Substituting the initial parameters of the problem, we obtain 1. K. W. Winkler, H. Liu, and D. L. Johnson, Geophysics 54, 66 (1989). 2 ∆P()1 Ð σ 21 〈〉ξ = ------R --- 2. M. A. Biot, J. Acoust. Soc. Am. 28, 168 (1956). Eh 2 3. V. N. Nikolaevskiœ, Mechanics of Saturated Porous (C.11) Media (Nedra, Moscow, 1977). 2 δ hR 25 δ hR ∆P × exp Ð------+ exp Ð------≡ ------. 4. P. A. Francis, M. R. P. Eigner, I. T. M. Patey, and 4 4 ˜ 1 + s 3τ 14+ s 3τ W MC I. S. C. Spark, in Expanded Abstracts of the SPE Euro- pean Formation Damage Conference (Soc. Petrol. Eng., In this formula, the displacement is expressed The Hague, 1995), p. 101. through the pressure difference between the two sides 5. H. Liu and D. L. Johnson, J. Acoust. Soc. Am. 101, 3322 of the shell. Now, we introduce in our consideration the (1997). permeable stratum and the filtration flows from it (as in 6. B. W. Tichelaar, Hsui-Liu Liu, and D. L. Johnson, the previous model). Since we already averaged the dis- J. Acoust. Soc. Am. 105, 601 (1999). placement along the borehole, we now consider the 7. A. M. Ionov, O. V. Kozlov, and G. A. Maksimov, Akust. mudcake as an ordinary (not fixed) shell. Then, by anal- Zh. 41, 603 (1995) [Acoust. Phys. 41, 529 (1995)]. ogy with (B.1)Ð(B.8), we obtain the set of equations 8. A. M. Ionov and G. A. Maximov, Geophys. J. Int. 124 (3), 888 (1996).  〈〉ξ ≡ u = ()P Ð P /W˜  f out MC 9. J. E. White, Underground Sound. Application of Seismic P = P + CK ()z Waves (Elsevier, Amsterdam, 1983; Mir, Moscow,  out ext 0 (C.12) 1986).  R () 10. G. A. Maksimov and M. E. Merkulov, in Proceedings of um= 0C------K1 z .  K f z Scientific Session MIFI-99 (MIFI, Moscow, 1999), Vol. 1, p. 159. Solving this set, we obtain 11. Handbook of Mathematical Functions, Ed. by M. Abra- ()P Ð P mowitz and I. A. Stegun (Dover, New York, 1971; u = ------f ext . (C.13) Nauka, Moscow, 1979). ˜ ()()() W MC + K f zK0 z / m0 RK1 z 12. L. D. Landau and E. M. Lifshits, Course of Theoretical Physics, Vol. 7: Theory of Elasticity, 4th ed. (Nauka, The corresponding velocity in the frequency represen- Moscow, 1987; Pergamon, New York, 1986). tation is related to displacement (C.13) by the factor iω: v (ω) = iωu(ω). It is also necessary to take into account the elastic motion of the borehole walls. This is accom- Translated by E. Golyamina

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002

Fish Target Strength Estimation Using Multiple Echo Statistics1 M. Moszynski Technical University of Gdansk, ul. Narutowicza 11/12, 80-952 Gdansk, Poland e-mail: [email protected] Received June 26, 2001

Abstract—When fish strength is estimated indirectly from the sounder echo amplitudes, the inverse techniques of solving the so-called “single-beam integral equation” are quite satisfactorily used. This approach needs prior knowledge of the beam pattern PDF, as it represents the kernel of the integral equation to be solved and is usually calculated under the assumption of a uniform spatial distribution of fish. However, it may be shown that in some cases this assumption is not necessarily justified. For instance, when the density of fish increases, one receives multiple echoes from the same single fish in successive transmissions, which results in observing so-called fish echo traces. Typically used fish counting methods are either simple direct echo counting statistics or fish traces statistics [1]. Increased fish concentration is not only the reason of multiple echo formation resulting in the fish traces in consecutive pings. As it is easily seen from the geometry of the phenomenon, even a relatively low-density fish aggregation forms multiple echoes and, hence, fish traces if the vessel (or fish) relative speed is low enough and the beam pattern angular width (sampling volume) is large enough. In some situations, the uniform assumption works properly only for the cases of large numbers of samples. Taking into account this phenomenon, the accuracy of the solution can be improved by including the fish traces counting statistics in calculating the beam pattern PDF. In this paper, two different models of fish traces statistics are investigated: one assuming the vessel movement with stationary fish and the other with a stationary vessel and moving fish. Both approaches are modeled numerically and verified experimentally using the data obtained from a dual-beam system. The comparison of both approaches, i.e., for single echo traces and multiple echoes, is carried out using Windowed Singular Value Decomposition (WSVD) and Expectation Maximization and Smoothing (EMS) inverse techniques of fish target strength estimation in both the absolute domain (backscattering length estimation) and the logarithmic domain (target strength estimation). © 2002 MAIK “Nauka/Interperiodica”.

1 INTRODUCTION first one is used in the analysis of fish counts estimates [1], whereas the second one plays a crucial role in indi- Indirect methods of fish target strength estimation rect fish target strength estimation [2]. However, it using single beam echosounder systems fall into the appears that these two seemingly separate problems are category of inverse problems in which the probability closely related when one considers the PDF of the beam density function (PDF) of target strength is estimated pattern in the context of multiple echoes from individ- from fish echoes. Due to hydroacoustics system charac- ual fish. teristics, the reconstruction of the fish target strength PDF is based on incomplete data [2]. This kind of prob- The widely used assumption of a uniform spatial lem is an example of the statistical linear inverse prob- distribution of fish in the water column leads to a sine- lem, which is typically ill-conditioned and can be law distribution of the angular position of the fish. This solved using direct inverse techniques, based on regu- assumption is valid only for the case of single or non- larization or iterative techniques in which additional multiple echoes received from individual fish in con- constraints are specified. In most cases, the observed secutive pings. However, when acquiring actual data data are restricted to the certain echo amplitude dynamic from acoustic surveys, the multiple or correlated ech- range limited by the side-lobe level. This approach oes may be collected from the same fish forming the allows omission of the problem of ambiguity of the beam fish traces. pattern function [2]. However, to calculate this function, an additional assumption on the spatial distribution of In this paper, the analysis of two models of fish fish in the beam pattern volume is to be made. traces is presented and the PDF’s of the number of mul- The statistics of the so-called fish traces, which are tiple echoes occurring in fish traces and the angular represented by multiple echoes received from the same position of the fish are also derived. Later on, the beam fish in consecutive echosounder transmissions, and pattern PDF is calculated based on the same assump- beam pattern probability density function (PDF) seem tions as in the case of the fish traces statistics. Finally, to be two absolutely separate and unrelated issues. The the beam pattern PDF is used as the kernel of the “single beam integral equation” for reconstructing the fish 1 This article was submitted by the author in English. target strength estimate from the echo data.

202 MOSZYNSKI

θ z

ρ α π (a) Model 1— 1:U(0, r)R (b) Model 2— :U(0, /2)

zmax

rmax r ρ r r ρ ρ 1 t t α α

Fig. 1. Geometry of two models for the number of echoes in ﬁsh traces analysis.

FORMULATION OF THE PROBLEM The unknown statistics of the number of fish N1 can be derived from the geometrical equation: Two models of the fish traces statistics, the geometry of which is illustrated in Fig. 1, have been considered: 2 2 ρ 2 N1 = ∆------r Ð,1 (3) (1) a moving vessel model with stationary fish, d (2) a moving fish model with a stationary vessel. where ∆d represents the sampling distance between consecutive points. The random variable r represents In the first model, the uniform vessel movement ρ the radius of a circle, and the random variable 1 repre- with stationary fish is assumed as detailed in Fig. 1a. sents the distance between the center of that circle and The second model assumes a fish movement along an the trace of the fish. In this model, one assumes that a arbitrary path in the transducer beam pattern cross-sec- fish may appear in the circle in such a way that the dis- tion, as shown in Fig. 1b. tance from the centre to the trace of the fish is equally ρ Let us further assume that the distribution of the probable, so that, in other words, the distribution of 1 variable z representing the depth at which fish appears is uniform in a range interval (0, r). This allows us to ρ in the conical area defined by observation angle θ is treat the random variable 1 as a product of two random max ρ uniform, i.e., variables 1 = ru, where the variable u is represented by a normalized uniform distribution. Substituting this relation into Eq. (3), we obtain: () 1 pz z = ------, (1) zmax 2 2 N = ------r 1 Ð.u (4) 1 ∆d where zmax represents the maximum depth. Due to the linear relation between depth and radius of a circular Now we can treat again the number N of fish traces as θ a product of two random variables x r ∆d and slice z = r tan max , the distribution of the random vari- = 2 / y = able r also becomes uniform, i.e., (1 – u2)1/2 and calculate the probability distribution func-

tion as an integral equation pz(z) = ∫ px (x)py(z/x)/xdx, () 1 pr r = ------, (2) which gives the PDF of N as rmax 1 θ ∆d y du where rmax = zmax/ tan max is the maximum possible p ()N = ------, (5) N1 ∫ 2rmax 2 y radius of the circular cross-section of the beam pattern. ∆dN 1 Ð y ------1- 2rmax MOVING VESSEL which, in its turn, yields AND STATIONARY FISH MODEL ∆d π ∆dN p ()N = ------Ð arcsin------. (6) N1  In the first model, all fish traces represent parallel 2rmax 2 2rmax lines crossing every circular slice of the observation ∆ cone. Thus, the fish position in consecutive pings can be The expression d/(2rmax) in Eq. (6) may be treated as represented by equidistant points on parallel chords. the parameter of the data measurement system and can

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 FISH TARGET STRENGTH ESTIMATION USING MULTIPLE ECHO STATISTICS 203 be calculated from the relation for the mean value of the Moving vessel stationary fish pN(N) random variable: 0.20 1π N ∆ π () --- {} d pN N = ---Ð arcsin---- EN1 = ------. (7) a 2 a 2rmax 8 0.15 aπ The probability density function of the number of EN{}= ------a = 20 8 echoes in ﬁsh traces for this model is presented in Fig. 2. 0.10

MOVING FISH AND STATIONARY VESSEL MODEL 0.05 In the second model, one assumes that the fish crosses an arbitrary circular cross-section of the conical sampled volume with an equally probable angle α. 0 5 10 15 20 From geometrical relations, the number of fish traces N can be expressed as Fig. 2. Theoretical probability density function (PDF) of the 2 number N of echoes in fish traces for moving vessel and sta- N = ------r sinα, (8) 2 ∆d tionary fish model. where the random variable α represents the crossing Moving fish stationary vessel pN(N) angle. The unknown distribution of the number of fish 0.20 2 N2 echoes in the fish trace N2 can be derived again from the p ()N = ------arctanh 1 Ð ------N π 2 equation determining the PDF of the product of two a a random variables x = 2r/∆d and y = sinα. Assuming a 0.15 uniform distribution of the angle α, we obtain: {} a EN = π--- a = 20 1 0.10 ∆d 2 1 dy p ()N = ------, (9) N2 ∫ π 2rmax 2 y ∆dN 1 Ð y ------2- 0.05 2rmax which eventually leads to: 0 5 10 15 20 ∆d ∆d 2 N p ()N = ------arctanh 1 Ð.------N (10) N2 πr 2r 2 max max Fig. 3. Theoretical PDF of the number N of echoes in fish traces for moving fish and stationary vessel model. The mean value of the random variable N2 with a PDF in the form of Eq. (10) is given by: where the random variable z represents the fish depth ∆d 1 ρ EN{}= ------. (11) and the random variables R and represent the fish 2 π 2 ρ2 2rmax position coordinates related by the equation R = + z2. Let us also consider the random variable t called the The probability density function of the number of trace distance, which represents the distance of the fish echoes in fish traces for this model is presented in Fig. 3. from the crossing point of the circular slice. Assuming that the fish swims on the chord and is “sampled” uni- STATISTICS OF THE ANGULAR POSITION formly in the consecutive pings, we can treat its PDF as OF FISH FOR MULTIPLE ECHO TRACES uniform in a range (0, 2rsinα). Thus, the trace distance α Let us now consider the distribution of the angular random variable can be expressed as t = 2rsin u, position of fish θ in the transducer beam that is neces- where u again has a normalized uniform distribution. sary for calculating the beam pattern PDF. The random Taking into account the cosine law in the nonright- variable θ can be expressed as (see Fig. 4): angled triangle (Fig. 4) we obtain:

z 1 2 2 2 θ ==arccos--- arccos------, (12) ρ = r + t Ð 2rtsinα R 2 (13) ()ρ z 2 2 2 1 + / = r ()12Ð ()sinα ()uuÐ .

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 204 MOSZYNSKI

z R

r ρ t α

zmax

rmax

Fig. 4. Geometry of multiple echo traces of the ﬁsh.

π2 Substituting z r tanθ , which removes the z depen- where K(k) = ( 1 – k2sin2ϕ)–1/2dϕ represents a com- = max ∫0 dence from Eq. (5) and the r dependence in Eq. (6), we plete elliptic integral of the first kind. receive the equation for the angular position θ: Both distributions are illustrated in Fig. 5. It is worth 1 θ = arccos------. (14) to note that, as one could expect, there are more echoes 22Ð ()sinα 2()uuÐ 2 received from larger angles, which results in an increase in the distribution as compared to the sinelike Equation (7) shows that the distribution of the angular distribution known for the case of nonmultiple echoes position θ depends only on the distribution of the crossing angle α as random variable u represents uniform received from a single fish. distribution resulting in the PDF of variable u – u2 –1/2 expressed as p 2 (x) = (1/4 – x) . The distribution of uuÐ BEAM PATTERN PDF BASICS the variable α depends on the angular relations between fish and vessel movement [2] and can change from To derive the beam pattern PDF, let us first con- sine-law, when the stationary fish model is used, to uni- sider an ideal circular piston transducer in an infinite form distribution when the stationary vessel model is baffle, for which the one-way beam pattern function b used. Both models give the distribution of the variable 2 2 –1/2 is given by 4sin α as p 2 (x) = (4x – x ) /π for the sine-law 4sin α –1/2 model or p 2 α (x) = (4 – x) /4 for the uniform distri- 2J ()x 4sin b()θ = ------1 , (17) bution one. Finally, using the formulae for the PDF of x the product of random variables and transforming according to Eq. (7), we receive for the first and the sec- where x is defined by x = kasinθ, k is the wave number, ond model, respectively: a is the transducer radius, and Jn is the Bessel function 1 tanθ sinθ of the first kind of order n. The logarithmic form of the pθ ()θ = ------K ------, (15) 1 2 θ tanθ 3 θ two-way pattern in decibels is derived by the simple tan max max cos transform B(θ) = 10logb (θ)2 = 20logb (θ). θ ()θ 1 1 sin pθ2 = ------, (16) The beam pattern PDF p (B) represents the kernel 2 θ θ 2 3 θ B tan max tan cos function of the inverse problem for a two-way system 1 Ð ------θ - tan max and can be obtained from its absolute variable form

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 FISH TARGET STRENGTH ESTIMATION USING MULTIPLE ECHO STATISTICS 205

pθ1 Model 1 pθ2 Model 2 35

20 θ θ max = 6° max = 6° 15

0 2 4 6 0 2 4 6 θ°, deg

Fig. 5. Theoretical distribution of the angular position θ for the two analysed models.

pb, which can be expressed as a parametric function operating frequency and 0.4 ms pulse length. The cal- 2 θ θ θ pb(b) = (b ( ), pb( )) with the angle as a parameter: culation of the beam pattern is performed only for a narrow beam channel, as echo from this channel can be  2 used for inverting the target strength of the fish. The 2J ()x pθ()θ tan θ p ()b = ------1 , ------, (18) beam pattern was fitted using the following approxima- b x 8J ()x J ()x ------1 2 - tion proposed in [1]: x 1 ()θ --γ- where pθ is a probability density function of the random ()θ ()Ðγ 1 Ð cos b = 112Ð Ð ------θ , (21) angular position of fish. Then, using a logarithmic 1 Ð cos 3dB transform of variables B(b) = 20 logb , its PDF relation can be written as where the exponential coefficient γ = Ð0.1 was fitted numerically to the actual pattern. The logarithmic B B ------transform and the inclusion of nonmultiple echo statis- ln10 20 20 p ()B = ------10 p 10 . (19) tics, Eq. (20), leads to the equation B 20 b γB θ ------ln10 γ 1 Ð cos 20 A typical approach in the PDF calculation of the p ()B = ------3dB10 , (22) B Ðγ θ angular fish position pθ is based on the assumption of a 20 12Ð 1 Ð cos max uniform distribution of fish in the water column (further called nonmultiple echoes statistics), which gives the and the inclusion of multiple echo statistics, Eq. (15), sine-law distribution of the angular position θ [1]: leads to the equation 1 ln10 γ pθ()θ = ------sinθ, (20) () ------θ pB b = Ðγ 1 Ð cos max 20 12Ð where θ is the maximum angle of the beam pattern γB (23) max θ ------1 Ð cos tanθ 1 20 involved in the calculation. However, as will be shown × ------3dBK ------10 , 2 θ 3 in the next section, for datasets obtained during a sur- tan θ tan max cos θ vey where several echoes from one fish are present in max consecutive pings, a more accurate assumption should where θ can be calculated as the inverse of b(θ) from be made. Eq. (20): θ ()()0.05γB ()θ ()Ðγ BEAM PATTERN PDF FOR AN ACTUAL SYSTEM = arccos 1Ð 1Ð 10 1 Ð cos 3dB /1Ð 2 . The way of calculating the beam pattern PDF for a Figure 6 illustrates the approximation of the actual Biosonics’s dual-beam ESP (i.e., Echosounder Signal beam pattern of a narrow beam channel and two PDF Processor) system is presented. The ESP system uses a functions, one with nonmultiple echo assumption and dual-beam (6°/15°) digital echosounder of 420 kHz the other with multiple echo assumption.

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 206 MOSZYNSKI

B(θ), dB

0 (a) –3 dB –5

–10 –11 dB

–15 θ ()θ 10 ()()Ðγ 1 Ð cos B = -----γ-log112Ð Ð ------θ 1 Ð cos 3dB –20 θ γ 3dB = 3° = 0.1 –25 0 2 4 6 8 10 θ pB(B) °, deg 0.05 (b) (c) 0.04

0.03

0.02

0.01

0 –40 –30 –20 –10 0 –40 –30 –20 –10 0 B, dB

Fig. 6. (a) Beam pattern approximation, (b) beam pattern PDF for the nonmultiple echoes assumption, and (c) beam pattern PDF for the multiple echoes assumption.

SURVEY RESULTS ulations (mostly salmon and trout) in Coeur d’Alene Lake, To justify the correctness of the presented analysis and Idaho (provided by J.B. Hedgepeth, Biosonics Inc., Seat- validate its results, actual fish echo data was used. The data tle and E. Parkinson, University of Vancouver, Canada) was acquired from an acoustic survey on pelagic fish pop- using a dual-beam digital echosounder of 420 kHz operating frequency and 0.4 ms pulse length. There were processed records of over 6500 pings from which pN(N) 700 over 10000 fish echoes were extracted for analysis and, using software algorithms, 2009 fish were counted. The 600 distribution of the number N of multiple echoes in fish traces is shown in Fig. 7 in the form of a histogram. The 500 results match Model 2 of distribution presented in Fig. 5. However, it is also possible that it matches Model 1 due 400 to the border effect in obtaining the PDF estimate by the histogram technique (the range between N = 0 and N = 1 300 cumulates as only N = 1 has physical sense).

200 Numerical experiments conducted on survey data show good agreement with the presented models of fish 100 statistical behavior during measurements. The mean value of the distribution is equal to 5.3. It is worth noting that in practice it is possible that the complicated 0 5 10 15 20 25 30 fish behavior can be modeled by a mixture of Model N 1 and Model 2 due to the relative movement of the fish Fig. 7. Sample distribution of the number of multiple and the vessel. Model 2 with a uniform distribution of echoes in fish traces from survey. the crossing angle represents a more “random” case

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 FISH TARGET STRENGTH ESTIMATION USING MULTIPLE ECHO STATISTICS 207

WSVD algorithm EMS algorithm 1500

1000 (TS) TS p 500

0 –60 –50 –40 –30 –60 –50 –40 –30 1500

) 1000 BS L ( BS L

p 500

0 0.01 0.03 0.01 0.03 LBS

Fig. 8. Results of the ﬁsh target strength TS estimation and the backscattering length LBS estimation using nonmultiple echo statistics.

WSVD algorithm EMS algorithm 1500

1000 (TS) TS p 500

0 –60 –50 –40 –30 –60 –50 –40 –30 1500

) 1000 BS L ( BS L

p 500

0 0.01 0.03 0.01 0.03 LBS

Fig. 9. Results of the ﬁsh target strength TS estimation and the backscattering length LBS estimation using multiple echo statistics. than Model 1 with its sine-law distribution of the cross- position with nonnegative constraint, as a direct method ing angle. of calculation represents the classical approach to solving ill-conditioned integrals. The second technique is Two different inverse techniques were used to more sophisticated: Expectation, Maximization, and observe the difference in the results obtained by using Smoothing based on an iterative algorithm with statis- both assumptions concerning multiple echoes cases. tical constraints, which also seems to be the more The ﬁrst technique, Windowed Singular Value Decom- robust inverse technique [3]. The results of the typically

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 208 MOSZYNSKI used assumption of nonmultiple echoes statistics are traces. The results allow one to use multiple echo statis- presented in Fig. 8, and the results of using multiple tics for the beam pattern calculation [4]. echoes statistics are shown in Fig. 9. Figures 8 and 9 shows that both inverse techniques perform better when proper assumptions are made. It is worth noting that, when the kernel of the single-beam CONCLUSION integral is properly chosen, the large artefacts in estimates disappear. It is especially observed in the case of Two different approaches to the statistics of fish reconstructing the backscattering length, which repre- echo traces in the calculation of the beam pattern PDF sents numerically a more ill-posed situation. The esti- were analysed in this paper. When the data acquired mates presented in Fig. 8 suggested the existence of a during surveys contain multiple echoes from single large number of very small fishes, which after verifica- fish, it appears that the typically used approach is not tion using the dual-beam data appeared to be incorrect. adequate. The presence of a multiple number of echoes By contrast, the results presented in Fig. 9 clearly show in the fish traces can be verified numerically during data the existence of one group of large fish with the other postprocessing, which may show that the more ade- smaller group of smaller fish (using the WSVD quate approach needs to use the statistics of the number method). Note also that the EMS method may over- of multiple echoes in fish traces. The results presented smooth the PDF and create an impression of the exist- in the paper allow use of multiple echo statistics for the ence of only one group of fish. beam pattern PDF calculation. The results of using two different assumptions on REFERENCES the statistics of fish echo traces in the process of recon- 1. R. Kieser and T. J. Mulligan, Can. J. Fish. Aquat. Sci. 41, structing the target strength and backscattering length 451 (1984). of the fish population from single beam data are pre- 2. A. Stepnowski and M. Moszynski, J. Acoust. Soc. Am. sented in the paper. When the data acquired during 107, 2554 (2000). measurements contain multiple echoes from one fish, a 3. J. B. Hedgepeth, V. F. Gallucci, F. O’Sullivan, and more adequate approach is suggested. The presence of R. E. Thorne, ICES J. Mar. Sci. 56 (1), 36 (1999). a multiple number of fish echo traces was verified 4. M. Moszynski and A. Stepnowski, in Proceedings of the numerically during data postprocessing resulting in a Fifth European Conference on Underwater Acoustics distribution of the number of multiple echoes in fish (Lyon, France, 2000).

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002

Nonstationary Self-Focusing of Sound Beams in a Vibrationally Excited Molecular Gas N. E. Molevich Korolev State Aerospace University, Samara, Moskovskoe sh. 34, Samara, 443086 Russia e-mail: [email protected] Received November 30, 1999

Abstract—Mechanisms of the self-action of sound in quasi-stationary gases with nonequilibrium-excited vibrational states of molecules are considered. It is demonstrated that the observation of a self-focusing of sound is possible in such media. Two mechanisms of the self-action of sound are responsible for the self-focusing in an acoustically active medium: the cooling of gas by sound and the excitation of acoustic streamings in opposite directions. © 2002 MAIK “Nauka/Interperiodica”.

∆ τ 2 χ Various mechanisms of the self-action of sound that homogeneous within the time t < T = a / , where a is lead to its defocusing or self-focusing in heat-conduct- the characteristic dimension of the system and χ is the ing viscous media are considered in [1Ð11]. According thermal diffusivity. to the estimates [4], it was demonstrated that the contri- This paper considers both mechanisms of the self- bution to the self-action of a beam observed before the action of sound propagating in a quasi-stationary, shock formation can be made only by the mechanisms vibrationally excited gas. related to the heating of the liquid because of the sound The initial set of equations of gas dynamics has the absorption and to the excitation of a longitudinal acous- form tic streaming. In liquids and gases in thermodynamic equilibrium the second mechanism always leads to the dρ ------+0,ρdivV = defocusing of sound, because the direction of the dt acoustic streaming coincides with the propagation dV η direction of a sound beam. As for the thermal mecha- ρ------= Ð∇P + η∆V + ξ + --- ∇divV, nism, it leads to a self-focusing in the case of the sound dt 3 propagation in liquids (except for water) and to defocusing in the case of the sound propagation in a gaseous ρ dT dEv T d χ∆ medium, because in the latter case, the sound velocity CV∞------+ ------Ð ---ρ------= CP∞ T (1) increases with temperature. Therefore, the sound prop- dt dt dt agates faster in the central, hotter, part of a beam than η ∂ ∂ ∂ ∂ in the less heated peripheral region, and a diverging m V iV i V k 2δ V i + ------ρ------∂ ------∂ + ------∂ - Ð --- ik------∂ + Q, thermal lens is formed in the medium. xk xk xi 3 xi

It was demonstrated in [12] that the situation dEv Ee Ð Ev ρT changes in thermodynamically nonequilibrium media, ------= ------+ Q, P = ------. dt τv ()T, ρ m e.g., in a vibrationally excited molecular gas. The dynamic properties of such a medium are determined Here, d/dt = ∂/∂t + V∇; V, ρ, P, and T are the veloc- by the second viscosity whose sign depends on the ity in the gas, its density, pressure, and temperature; η degree of nonequilibrium [13, 14]. In media with a neg- is the shear-viscosity coefﬁcient; m is the molecular ative second viscosity, the dissipated energy ﬂux is mass; Ev and Ee are the vibrational energy per molecule directed from the medium to the wave, and such a and its equilibrium value; τv is the vibrational relax- medium acquires focusing properties. However, the ation time; and Q is the power of the energy source that strong inhomogeneity of nonequilibrium media pre- maintains the excitation of vibrational states. For sim- vents the observation of the self-focusing of sound. plicity, we assume that Q, χ, and η are independent of ρ From this point of view, nonstationary nonequilibrium T and ; CV` and CP` are the heat capacities of transla- media are of interest. On the one hand, these media stay tional and rotational degrees of freedom (and also of acoustically active, as was demonstrated in [15Ð17]. On equilibrium vibrational modes) at constant volume and the other hand, there is no need to provide for a special pressure. The equations are written in terms of energy heat sink in them, and such a medium can be fairly units. In the case of cylindrically symmetric beams, the

210 MOLEVICH vector V has two components: the longitudinal compo- gence, is the quantity of a higher degree of smallness nent U and the transverse component W. The transverse than U. velocity component W, which results from the diver- We try a solution to Eqs. (1) in the form

() UU= 0 ()rxt,, () U 1 + ------()rxt,, exp[]Ði∫ωdt + ikx + c.c., 2 (2) () ()0 (),, PP= 0 t + P rxt () P 1 + ------()rxt,, exp[]Ði ωdt + ikx + c.c., 2 ∫ and analogously, for other components, where ω and k The equation that determines the sound amplitude are the frequency and the wave vector of an acoustic variation has the form wave. In a quasi-stationary approximation, the parameters of a nonequilibrium medium must change slowly in ∂ ∂ 1 i ∆ ()1 comparison with ω, i.e., ------++------Ð ------⊥ g U ∂x U∞ ∂t 2k ∂P ∂T (4) 1 0 1 0 S Ӷ ω ()0 ()0 ------∂ - ==------∂ ------τ - , ik ()1 2U T P0 t T 0 t v CV∞ = Ð---- U ------+ ------. 2 U∞ T 0 1 ∂ρ ------0 ρ ∂ = 0, (1)| (1)|2 0 t This equation ignores the cubic terms ~U U , which correspond to an inertialess change of state [7, 8] 0 0 where S = (Ev Ð Ee )/T0 is the degree of nonequilibrium and, according to [4], make no significant contribution 0 0 to the self-action of sound at distances L < L , where L in the medium and E and E are the unperturbed val- p p v e is the distance of shock formation. If the characteristic ues of the vibrational energy and its equilibrium value. length of self-focusing is LF > Lp, self-focusing is also In addition, the length of an acoustic pulse must lie in possible [5, 18, 19] but difficult to describe analytically, ω–1 Ӷ τ the range ti < T. and this case is beyond our consideration. The terms on The acoustic streaming U(0) caused by an acoustic the right-hand side of Eq. (4) correspond to two mech- wave of frequency ω is assumed to be incompressible. anisms of self-action: the excitation of acoustic stream- We use the geometric-acoustics approximation, in ing and the heating of the medium by sound. The dissi- which U(0), U(1), P(0), P(1), and other complex ampli- pation coefficient is g = δ + g , where tudes are assumed to be functions slowly varying with V t, x, and r, which satisfy the condition ω2 η χ δ 4 ∂ ∂ = ------3 ------+ ------2 Ð2 3ρ C ∞ ------∼∼------()∇⊥ k ∼ ε, 2U∞ 0 V ω∂t k∂x where ε is the smallness parameter. is the coefficient of sound absorption in a heat-conduct- The geometric-acoustics approximation is applicable ing viscous medium, when the beam width is a > 2π/k. The complex ampli- ωτ ӷ ()γ ∞ Ð 2 S tudes for a high-frequency acoustic wave ( v 1) are α gV = ∞ + ------τ linearly related: 4CP∞U∞ v ()1 ()1 ()1 ()1 ρ T ==U U∞m/CP∞, P U U∞ 0, is the coefficient of the velocity perturbation growth in (3) ρ()1 ()1 ρ ()1 ∇ ()1 ω a quasi-stationary medium [15Ð17], and ==U 0/U∞, W U∞ ⊥U /i , ω2ξω() where ∇⊥ is the transverse gradient, U = γ ∞T /m is ` 0 α∞ = ------γ 3 ρ the velocity of high-frequency sound, and ` = CP`/CV`. 2U∞ 0 Substituting Eqs. (2) with the relations between the components given by Eqs. (3) into the set of equa- is the absorption (or amplification at ξ < 0) coefficient tions (1), we obtain a system of three reduced equations related to the presence of relaxation processes in the describing the self-action of sound. medium and to the second viscosity ξ formed by them.

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 NONSTATIONARY SELF-FOCUSING OF SOUND BEAMS 211 ωτ ӷ χ χ 0 The second-viscosity coefficient at v 1 has the where 0 = CP∞/.CP form [14] In Eq. (6), we ignore the term responsible for the ()2 2 0 ρ []()τ 2 ρ U∞ Ð U0 CV 0 CV + S vT Ð CV∞ U∞ 0 adiabatic mechanism of heating and the value of ξ ==------. ∂ (0) ∂ –1 ω2τ ω2τ P / t, which is admissible for ti > (4U`g) and v CV∞ v CP∞CV∞ a/U` [4]. Here, U = C0 T /C0 m is the velocity of low-fre- According to Eq. (6), an acoustically active medium 0 P 0 V in the field of an intense sound wave is not heated but, ωτ Ӷ 0 τ quency sound ( v 1); CP = CP` + Cv + S( vT + 1) on the contrary, cooled. 0 τ and CV = CV` + Cv + S vT are the low-frequency heat We can further simplify the set of equations (4)Ð(6) capacities at constant pressure and at constant volume, with the help of the known methods of nonlinear geo- τ ∂ τ ∂ metric optics (acoustics) [21]. For this purpose, we rep- vT = ln v/ lnT0, and Cv is the equilibrium vibrational heat capacity. resent the complex amplitude U(1) in the form The second viscosity (and α ) are negative when () τ ` U 1 = A()ζ,,rtexp[]ikψζ(),,rt , Cv + S( vT – CV`) < 0. This condition corresponds to the presence of a positive feedback between the acoustic where A is the real wave amplitude, ψ is the eikonal, perturbation and the heat release from the nonequilib- ζ rium degrees of freedom and, hence, to a sound ampli- and = x – U∞t is the “traveling” wave coordinate, and fication. substitute it into Eq. (4). Ignoring the diffraction term, we finally obtain It should be noted that, in stationary media with ξ < 0, the perturbations of the velocity, pressure, and ∂I density in an acoustic wave increase with the same ------+++θ∇⊥ II∇⊥θ 2gI = 0, (7) α ∂ζ increment `, whereas in quasi-stationary media, the increments of these perturbations have different forms ∂θ ∇ ()0 [15Ð17]. For example, the pressure perturbation θ∇ θ 2 ∇ ()0 ⊥T increases with the increment 2-----∂ζ-2+ ⊥ +0,------⊥U + ------= (8) U∞ T 0 ∂T ()2 + γ ∞ 1 0 α 2 θ ∇ ψ gP ==gV Ð ------∂ - ∞ Ð ------τ . where I = A and = ⊥ is the inclination angle of a 1T 0U∞ t 4CP∞U∞ v ray with respect to the x axis. In the case of a beam char- The second equation of the set describes the devel- acterized by a radius a0, an initial parabolic profile of opment of a longitudinal acoustic streaming: intensity, and a plane wave front, the solutions to Eqs. (7) and (8) can be tried in the form ∂ ()0 η ∂ ()0 U ∆ ()0 1 P ()1 2 ------∂ - Ð ----ρ- ⊥U + ----ρ------∂ - = gU . (5) ζ t 0 0 x 2 2 Ð2 ∫ gdζ a0 2r θβζ(), ----- 0 In this equation, we ignore the diffraction changes ==t r, II01 Ð ------2 e , of the amplitude U(0). Moreover, in the following calcu- a a lations, we ignore in Eq. (5) the value of the longitudinal pressure gradient, which is insignificant when where I = A2 , A is the initial amplitude of the beam at (0) Ӷ 0 0 0 U U` [3, 4]. ζ The right-hand side of Eq. (5) represents the driving ζ βζ force caused by the acoustic radiation pressure and its axis, and a( , t) = a0exp∫ d is the beam width. determined by the change of the amplitude of an acous- 0 tic wave in a dissipative medium. Equation (5) coin- Then, in the case of a dimensionless beam width cides with the equation describing, e.g., an Eckart flow f = a/a , Eq. (8) can be reduced to the form correct to the form of the absorption coefficient [20]. 0 However, one should note that, for g > 0, this flow is 2 ()0 ∂ f 1 2 ()0 ∇⊥T always directed in the direction of the sound propaga------∇⊥ ------2 = Ð U +.------(9) tion, whereas in an amplifying medium with g < 0, the ∂ζ 2r U∞ T 0 sign of the driving force changes and the acoustic streaming becomes opposite. We assume the dependences of T(0) and U(0) on r to The third equation of the set describes the tempera- also be parabolic, which is a good approximation of an τ ρ 2 η ture change in a medium under the effect of intense exact solution to Eqs. (5) and (6) for ti < T, 0a / [6]: sound: ()0 2 ()0 T = F + r F , ∂T ()0 U∞gm ()1 2 1 2 ------Ð χ ∆⊥T = ------U , (6) (10) ∂t 0 0 ()0 2 CP U = B1 + r B2.

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 212 MOLEVICH

Substituting Eqs. (10) into Eqs. (5), (6), and (9), we absorption coefficient is δ ≈ 10–3 cm–1 and the amplifi- ≈ –1 obtain cation coefficient is gV 0.5 cm . We obtain the value 2 of the critical energy Ecr necessary for the observation 1∂ f 2 F2 ------2 = Ð ------B2 +,----- (11) of the self-focusing from the condition LF = Ld, where f ∂ζ U∞ T 0 2 Ld = ka0 /2 is the characteristic diffraction length: ∂ Ð2∫ gdζ F2 2U∞gI0me 18.8 ------= Ð,------(12) E ≈ ------= 12 mJ. ∂t 0 2 4 cr 2 4α CPa0 f k a0 ζ ∂ Ð2∫gd The condition LF < Ld must be satisfied for the B2 2gI0e ------= Ð------. (13) observation of self-focusing. For example, at E = 9Ecr ∂t 2 4 գ a0 f and ti = 1 s, we have LF = Ld/3 2 cm. In this case, the distance at which a shock is formed, is equal to [20] If the background values of temperature change 2 ωγ()≈ ӷ insignificantly within the characteristic time of the L p = 2U∞/ ∞ + 1 A0 22 cm LF. sound beam focusing τ (τ ∂T /∂t Ӷ T ), the set of F F 0 0 τ ≈ equations (11)Ð(13) can be reduced to a single equation Such a pulse also satisfies the condition ti < T 1.5 s. after differentiating Eq. (11) with respect to time: Thus, this paper demonstrates the fundamental possibility to observe the self-focusing of sound in a quasi- ∂ ∂2 α ζ 1 f N Ð2∫gd stationary, vibrationally excited gas. Two mechanisms -----------= Ð------e , (14) ∂t f ∂ζ2 f 4 of self-action cause the self-focusing of sound in an acoustically active medium: the cooling of gas by π 2ρ where N = a0 0U∞ I0/2 is the total power of the sound sound and the excitation of acoustic streamings in beam and opposite directions.

4g γ ∞ α ------= Ð 4 2 2 + ------0 . REFERENCES πa U∞ρ C 0 0 P 1. G. A. Askar’yan, Pis’ma Zh. Éksp. Teor. Fiz. 4, 144 The exponential factor on the right-hand side of (1966) [JETP Lett. 4, 99 (1966)]. Eq. (14) enhances the nonlinear refraction in an ampli- 2. E. A. Zabolotskaya and R. V. Khokhlov, Akust. Zh. 22, fying medium (g < 0) and reduces it in an absorbing 28 (1976) [Sov. Phys. Acoust. 22, 15 (1976)]. medium (g > 0). We can ignore this factor in the 3. E. A. Zabolotskaya, Akust. Zh. 22, 222 (1976) [Sov. | | approximation of a thick lens ( g LF < 1). In this case, Phys. Acoust. 22, 124 (1976)]. Eq. (14) coincides with the corresponding equation 4. F. V. Bunkin, K. I. Volyak, and G. A. Lyakhov, Zh. Éksp. from [6], correct to the form of the quantity α. Teor. Fiz. 83, 575 (1982) [Sov. Phys. JETP 56, 316 According to [6], when α > 0, one can observe the (1982)]. self-focusing of sound with the characteristic focusing 5. N. S. Bakhvalov, Ya. M. Zhileœkin, and E. A. Zabo- length L ≈ (4.7/αE)1/2, where E = t Ntd is the energy lotskaya, Nonlinear Theory of Sound Beams (Nauka, F ∫0 Moscow, 1982; AIP, New York, 1987). of the sound beam. 6. M. Yu. Romanovskiœ, Akust. Zh. 33, 331 (1987) [Sov. The quantity α is always positive, e.g., in liquids Phys. Acoust. 33, 192 (1987)]. (apart from water), when thermal self-action prevails 7. O. V. Rudenko and O. A. Sapozhnikov, Zh. Éksp. Teor. over the streaming one. In equilibrium gaseous media, Fiz. 106, 395 (1994) [JETP 79, 220 (1994)]. both mechanisms of the self-action always lead to a α 8. O. V. Rudenko and A. A. Sukhorukov, Akust. Zh. 41, 822 defocusing of sound ( < 0). On the contrary, in an (1995) [Acoust. Phys. 41, 725 (1995)]. acoustically active nonequilibrium medium, both 9. O. V. Rudenko, A. Sarvazyan, and S. G. Emilianov, mechanisms make a positive contribution to the coefﬁ- J. Acoust. Soc. Am. 99, 2791 (1996). cient α. Exceptions are strongly nonequilibrium media, 0 10. Tjotta Sigve, Rept/Dep. Appl. Math. Univ. Bergen, in which CP < 0. No. 119, 1 (1998). Now, we present the estimates of the characteristic 11. A. V. Krasnoslobodtsev, G. A. Lyakhov, and K. F. Shipi- lov, Akust. Zh. 45, 832 (1999) [Acoust. Phys. 45, 750 quantities for a typical laser medium ëé2 : N2 : He = τ ≈ –5 (1999)]. 1 : 2 : 3 with P0 = 1 atm, T0 = 300 K, v 10 s, and τ ≈ 12. E. Ya. Kogan and N. E. Molevich, Pis’ma Zh. Tekh. Fiz. vT –3.4 [14]. 12, 96 (1986) [Sov. Tech. Phys. Lett. 12, 40 (1986)]. In the case of a speciﬁc energy contribution to the 13. E. Ya. Kogan and N. E. Molevich, Zh. Tekh. Fiz. 56, 941 vibrational degrees of freedom W = 50 mJ/cm3, the (1986) [Sov. Phys. Tech. Phys. 31, 573 (1986)]. ≈ value of S is S 0.5. Then, for a sound beam with the 14. N. E. Molevich and A. N. Oraevskiœ, Tr. Fiz. Inst. im. ω × 5 radius a0 = 1 cm and the frequency = 5 10 Hz, the P. N. Lebedeva, Ross. Akad. Nauk 222, 45 (1992).

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15. T.-Y. Toong, P. Arbeau, C. A. Garris, et al., in Proceed- 19. O. V. Rudenko, M. M. Sagatov, and O. A. Sapozhnikov, ings of 15th Symposium (International) on Combustion Zh. Éksp. Teor. Fiz. 98, 808 (1990) [Sov. Phys. JETP 71 (The Combustion Inst., Pittsburgh, USA, 1975), p. 87. (3), 449 (1990)]. 16. G. E. Abouseif, T.-Y. Toong, and J. Converti, in Proceed- 20. O. V. Rudenko and S. I. Soluyan, Theoretical Founda- ings of 17th Symposium (International) on Combustion tions of Nonlinear Acoustics (Nauka, Moscow, 1975; (Univ. of Leads, England, 1978), p. 1341. Consultants Bureau, New York, 1977). 17. N. E. Molevich and A. N. Oraevskiœ, Akust. Zh. 35, 482 21. S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, (1989) [Sov. Phys. Acoust. 35, 282 (1989)]. Usp. Fiz. Nauk 93, 19 (1967) [Sov. Phys. Usp. 10, 609 (1967)]. 18. A. A. Karabutov, O. V. Rudenko, and O. A. Sapozhnikov, Akust. Zh. 34, 644 (1988) [Sov. Phys. Acoust. 34, 371 (1988)]. Translated by M. Lyamshev

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Acoustical Physics, Vol. 48, No. 2, 2002, pp. 214Ð219. Translated from Akusticheskiœ Zhurnal, Vol. 48, No. 2, 2002, pp. 253Ð259. Original Russian Text Copyright © 2002 by Pishchalnikov, Sapozhnikov, Sinilo.

Increase in the Efficiency of the Shear Wave Generation in Gelatin Due to the Nonlinear Absorption of a Focused Ultrasonic Beam Yu. A. Pishchalnikov, O. A. Sapozhnikov, and T. V. Sinilo Moscow State University, Vorob’evy gory, Moscow, 119899 Russia e-mail: [email protected] Received December 18, 2000

Abstract—Experimental results and theoretical estimates are presented to demonstrate the prospects of using the acoustic nonlinearity of a gel-like medium for increasing the efﬁciency of the shear wave generation in it by a pulsed ultrasonic beam. The experiment is based on the propagation of a focused beam of longitudinal acoustic waves at a frequency of 1.1 MHz in a gelatin sample and on the detection of shear waves by the optical method [1]. It is demonstrated that the amplitude of the shear wave excited by a nonlinear acoustic pulse can be increased by an order of magnitude owing to the formation of shock fronts in the proﬁle of this pulse. © 2002 MAIK “Nauka/Interperiodica”.

An ultrasonic wave that experiences both absorption wave signals. Experiments on samples made of gelatin and scattering in the course of its propagation transfers with different concentration (i.e., with different values part of its momentum to the medium. As a conse- of the shear modulus) are described. It is demonstrated quence, an amplitude-modulated wave produces low- that, in the case of a constant total energy of the acous- frequency (medium) elastic stresses in the medium, i.e., tic pulse, it is possible to obtain an amplification of the the radiation pressure [2]. In the case of the ultrasound shear wave by reducing the duration of the excitation localization in the form of a beam, the corresponding pulse and by simultaneously increasing its amplitude to shear stresses produce a transverse wave propagating in the values at which shock fronts are formed in the wave the direction perpendicular to the beam axis (Fig. 1). In profile. A theoretical calculation is conducted to com- common solids, this effect is minor; however, it can pare the efficiencies of the shear wave excitation with become noticeable in gel-like media, where the shear and without allowance for the medium nonlinearity and modulus is small and the corresponding shear strain is to evaluate the gain. relatively large. Such media are rubbers, gels, and soft biological tissues. The detection of the amplitude or the propagation velocity of the transverse disturbances pro- Shear vides an opportunity to measure the shear modulus of wave the medium [1]. This method can be promising for Pulsed medical applications, e.g., for an early detection of can- ultrasonic cer, since the values of the shear moduli for healthy and beam cancer-affected tissues differ by orders of magnitude [3]. If a focused acoustic beam is used, an efficient generation of shear waves occurs only in the focal region. Thus, it is possible to obtain a local excitation of shear waves in a medium at a large distance from the source of radiation. The main difficulty in the utilization of this effect is connected with the fact that the excited shear waves are Fig. 1. Excitation of a shear wave due to the absorption of a usually very weak, and, therefore, difficult to detect. focused ultrasonic pulse. The dashed lines indicate the Hence, it is important to find ways to increase the effi- boundaries of the ultrasonic beam. The solid lines illustrate ciency of the generation of shear stresses. Here we sug- the deformation of the medium: in the initial unperturbed gest one such method namely, to use large-amplitude state, they formed a family of equidistant vertical straight lines. The absorbed ultrasonic pulse exerts a radiation pres- focused ultrasonic pulses whose profiles are nonlin- sure on the medium, and the shear stress arising in this case early distorted in the course of their propagation for produces a quasi-cylindrical shear wave traveling away increasing the efficiency of the generation of shear- from the beam axis.

INCREASE IN THE EFFICIENCY OF THE SHEAR WAVE GENERATION 215

The setup used in the experiments is schematically represented in Fig. 2. An ultrasonic beam was produced by a focusing piezoceramic transducer 10 cm in diam- 34 10 eter with a 20-cm curvature radius. The transducer operated at a resonance frequency of 1.1 MHz [4]. The longitudinal and transverse dimensions of the focal region, which were measured according to the zeros of the amplitude distribution of the pressure field, were 87 6 and 7 mm, respectively. The transducer was excited by 9 an electric signal from an HP33120A generator using an ENI AP400B power amplifier. We used a pulsed 7 regime of excitation with rectangular envelopes of pulses. The pulse duration varied from 40 to 700 µs. 2 The source was submerged into a water basin with the 5 dimensions 20 × 20 × 60 cm3 and could be moved with the help of a positioning system (Velmex VP9000, 1 USA) in three mutually perpendicular directions. A gel- 8 atin sample shaped as a cylinder with a diameter of 80 mm and a generatrix of 65 mm was placed into the focal region of the ultrasonic beam. Figure 3 shows a Fig. 2. Experimental setup: (1) a water basin, (2) an ultra- photograph of the ultrasonic transducer (on the left) and sonic transducer, (3) an electric power amplifier, (4) a generator, (5) a gelatin sample, (6) a micropositioning sys- one of the samples (on the right). The sample was posi- tem, (7) a shutter particle, (8) a beam of a He-Ne laser, tioned in such a way that the cylinder axis coincided (9) a photodiode, and (10) a digital oscilloscope. with the acoustic axis and the central section of the cylinder lay in the focal plane of the source. An optical system described in [1] was selected to detect the shear waves. The beam of a helium-neon laser was focused at the edge of an opaque particle 60Ð300 µm in size, which was placed in the medium. On the shear wave arrival, the particle moved and modulated the transmitted energy of the laser beam. The light signal detected further by a photodiode was proportional to the shear displacement. To incorporate such modulator particles into the medium, gelatin samples were manufactured in two stages, which provided an opportunity to put the particles into the central section perpendicular to the cylinder axis. The purpose of our measurements was to demonstrate that the use of an acoustic wave with shocks in its profile makes the excitation of shear-wave signals Fig. 3. Photograph of the piezoceramic transducer (on the much more efficient. Ultrasonic pulses with different left) and the gelatin sample (on the right). amplitudes, but with the same energy (which was attained by the corresponding choice of the pulse dura- of pulses on a wide-aperture target absorber [5]. The tion) were used. If the ultrasonic propagation in the medium were linear, the amplitude of the shear waves target was shaped as a cylinder 12 cm in diameter and excited by acoustic pulses with the same energy would 5 cm in height and was made of rubber of the type of an be the same [1]. However, in the presence of nonlinear- RTV-2 two-component silicon elastomer, which had a ity, the profile of an ultrasonic wave in the focal region large absorption coefficient and an acoustic impedance of the beam becomes distorted, and in the case of a close to the impedance of water. To measure the radia- rather large amplitude, shock fronts arise, i.e., a saw- tion force, an acoustic beam was directed to the tooth profile is formed. As a consequence, the wave is absorber from below and the absorber was weighted absorbed more efficiently and transfers a greater part of both under the ultrasonic irradiation and immediately its momentum to the medium, as compared to the case after switching off the source of ultrasound [6]. In such of linear propagation. Thus in the nonlinear regime, one measurements, the change in the absorber weight ∆P can expect a considerable increase in the amplitude of and the average ultrasonic power W are related as the shear disturbance generated by the ultrasonic wave. ∆P/W = 67 mg/W [5]. On the basis of these measure- The regime calibration according to the total energy ments, several regimes of operation with different of acoustic pulses was performed by measuring the lengths and amplitudes of pulses were selected for a average radiation force exerted by a periodic sequence preset acoustic energy.

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216 PISHCHALNIKOV et al.

p, atm p, atm

(‡) (c) 100

–50 0 200 400 74 75 76 50 (b) (d)

–50 0 200 400 t, µs 316 317 318 t, µs

Fig. 4. Shape of an ultrasonic pulse at the focal point in the (a, b) broad and (c, d) narrow time windows: (a, c) a nonlinear regime and (b, d) a quasi-linear regime.

u/umax To verify that, in the selected regimes, the acoustic r = 5 mm (a) signals were really subjected to nonlinear distortions, the measurements of the wave form in the focal region 0.2 of the beam were also conducted using a self-made broadband PVDF membrane hydrophone with a sensitive area 0.5 mm in diameter. The rear side of the PVDF 6 mm film was acoustically loaded by a thick layer of trans- 0.1 former oil, which provided an opportunity to eliminate 7 mm the stray capacitance and enhance the locality of reception [7]. Figure 4 shows the wave profiles of the shortest and longest pulses in the focal region of the source. Fig- 0 ures 4a and 4b show the whole pulses, and Figs. 4c and r = 5 mm 4d, three periods from their central parts. It should be (b) noted that the electric signal at the piezoelectric transducer and, therefore, the ultrasonic wave emitted by it 0.02 6 mm were sinusoidal in both cases. As one can see from Fig. 4, in the focus, the wave is strongly distorted, especially, in the case of a large amplitude: shock fronts are 0.01 7 mm clearly visible in the profile of the first pulse, whereas the second profile is distorted to a lesser extent. Thus, the effect of the acoustic nonlinearity of the medium must manifest itself in the comparison of the efficien- 0 cies of the shear wave generation in the two indicated 357 9 regimes. t, ms Figure 5 demonstrates the experimentally measured profiles of a shear pulse in the focal plane at different distances from the acoustic axis of the beam (5, 6, and Fig. 5. Time profiles of a shear pulse (the photodiode sig- 7 mm, respectively). Figure 5a corresponds to the shear nal u) measured in the focal plane at three different dis- wave excitation by a short high-amplitude pulse (a non- tances from the acoustic axis r: (a) a nonlinear regime and (b) a quasi-linear regime. The values of the displacement u linear regime) and Fig. 5b corresponds to the excitation (for all curves) are normalized to its maximal value in the by a long pulse with a much smaller amplitude (a quasi- nonlinear regime umax. linear regime) and with the same total energy E = 4.2 mJ.

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INCREASE IN THE EFFICIENCY OF THE SHEAR WAVE GENERATION 217

u/u max t, ms

1.0 15

10 0.5 5

0 –15 –10 –5 0 5 10 r, mm 0 –15 –10 –5 0510r, mm

Fig. 6. Transverse distribution of the shear wave amplitude measured in the focal plane of the source in the nonlinear Fig. 7. Delay time of the peak of the shear wave versus the regime (the solid line) and the transverse proﬁle of intensity transverse coordinate r. The measurements were performed (the dashed line). The curves are normalized to the corre- in the nonlinear regime for two gelatin concentrations: 4.5% sponding maximal values. (the dashed line) and 6.7% (the solid line).

All curves are normalized to the maximal displacement Figure 7 presents the detection time of the peak of a at the beam axis in the nonlinear excitation regime. It is shear pulse versus the distance r between the detection necessary to note that the vertical scales in Figs. 5a and point and the axis of the ultrasonic beam. The two 5b differ by a factor of ten. Thus, although the signal curves correspond to two different concentrations of profiles in the nonlinear and quasi-linear regimes look gelatin (4.5 and 6.7%) and are obtained using nonlinear almost the same at the corresponding distances, the acoustic pulses. The slope of these curves is inversely amplitudes of shear waves differ essentially. In the proportional to the propagation velocity of shear waves quasi-linear regime, the signals were weak and their in the corresponding material. The experimental results amplitude was comparable to the noise amplitude. shown in Fig. 7 for the gelatin concentrations 4.5 and 6.7% give the values of the velocity c = 0.8 and 1.2 m/s Therefore, the curves were obtained by averaging over µ t 300Ð400 realizations. Thus, the assumption that the and the shear modulus = 690 and 1450 Pa, respec- efficiency of the shear wave generation grows when the tively. These values coincide with the shear modulus measured by the indentation method using a rigid acoustic nonlinearity of the medium comes into play sphere [8, 9]. Thus, the remote excitation and detection was confirmed. Similar results were obtained for other of shear waves provide an opportunity to determine the regimes of excitation with different pulse energies and shear modulus. for samples with different gelatin concentrations. Let us proceed to the theoretical description of the It is of interest to know how far from the point of observed effects. The shear disturbances can be excitation it is possible to detect the shear waves. Fig- described in the framework of the linear theory because ure 6 demonstrates the transverse distribution of the of their smallness. The displacement u of the particles peak displacement in the shear wave excited in the non- of the elastic medium in the force field F can be linear regime (with the ultrasonic pulse duration equal described by the equation to 55 µs and the total energy of a single pulse equal to ∂2 µ 4.2 mJ) in a sample with a gelatin concentration of ρ u  µ∆ ρ ------= K + --- graddivu ++u F, (1) 4.5%. The dashed line in the same figure shows the ∂2t 3 experimentally measured transverse distribution of the intensity of an acoustic wave in the focal plane, which where ρ is the density of the medium; K and µ are the corresponds to the distribution of the sources of shear bulk modulus and the shear modulus, respectively; F is waves. Both curves are normalized to the correspond- the volume density of forces acting from the side of the ing maximal values. One can see clearly that, although ultrasonic beam; and t is time. Let a weakly divergent a shear wave decays rapidly, it still propagates to a con- beam of acoustic waves, which has a circular cross sec- siderable distance from the acoustic beam. This pro- tion, propagate in the medium. We denote the longitu- vides an opportunity to measure the velocity of shear dinal and the transverse coordinates by z and r, respec- waves c with a sufficiently high accuracy and, hence, to tively. Then, we can ignore the transverse component Fr t of the force F because of its smallness in comparison µ ρ 2 determine the shear modulus = ct . with the longitudinal component Fz.

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218 PISHCHALNIKOV et al. We note that, if the transverse dimension of the The force of radiation pressure in a viscous heat-con- beam is much smaller than the absorption length, the ducting medium can be written in the form [10] force Fz varies mainly in the transverse direction by decreasing with the distance from the axis, whereas its b ∂p 2 Fz = ------------, (5) dependence on the longitudinal coordinate is weak. In c5ρ3 ∂t the indicated conditions, we can ignore the dependence l of the deformation of the medium on the longitudinal where cl is the velocity of longitudinal waves; p is the coordinate and assume that the displacement of parti- 4 acoustic pressure in the ultrasonic beam; b = ζ + ---η + cles occurs only along the z axis. In this case, Eq. (1) 3 takes the form of an inhomogeneous wave equation for 1 1 a shear cylindrical wave: κ ------Ð ------is the dissipation coefficient; ζ and η are C C ∂2 v p uz 2 κ ∆⊥ (), the volume and shear viscosities; is the coefficient of ------2- Ð ct uz = Fz rt, (2) ∂t heat conductivity; Cp and Cv are the specific heats at constant pressure and at constant volume, respectively; µ ρ where ct = / is the velocity of shear waves and and the overbar means averaging over the period of the 1 ∂ ∂2 ultrasonic wave. When small-amplitude ultrasound is ∆⊥ = ------+ ------is the transverse Laplacian. The form used, the wave form within a pulse is sinusoidal: p = r ∂r ∂ 2 ω r p0sin t. In this case, we obtain from Eq. (5): of the solution to the wave equation (2) is determined lin α 2 ρ2 2 by the dependence of the volume force Fz on the coor- F0 = p0/ cl , (6) dinate and time. If the dependence of the force on time α ω2 ρ 3 at all space points is described by the same function where = b /2 cl is the ultrasonic absorption coef- ϕ Φ ϕ Φ (t), the expression Fz = (r) (t), where (r) describes ficient and the superscript “lin” indicates the linear the transverse profile of the acoustic beam, is true. As case. we demonstrated in our previous paper [1], in the case 2 ρ of a short pulse ϕ = δ(t) with the transverse profile of Taking into account the fact that I = p0 /2 cl is the Φ 2 2 –3/2 wave intensity, from Eqs. (4) and (6) we obtain the beam in the form (r) = F0(1 + r /a ) , the solution to Eq. (2), i.e., the pulsed response, is expressed lin αa analytically as umax = ρ------t0 I. (7) clct 2 2 2 aF0 ()AB+ Ð 41()+ r /a As one can see from this expression, the medium dis- hrt(), = ------θ()t ------, (3) placement under the effect of the radiation force is pro- 2ct AB portional to the quantity t0I, i.e., it is determined by the 2 2 2 2 where A = 1 + (r + ctt) /a , B = 1 + (r – ctt) /a , a is the energy of the ultrasonic pulse rather than by its inten- radius of the sound beam, and θ(t) is the Heaviside step sity. Therefore, in the linear regime, ultrasonic pulses function. According to Eq. (3), the characteristic dura- with different amplitudes p0 but with the same total tion of the pulsed response coincides with the traveling energy produce identical shear-wave signals. time of the shear wave through the excitation region Now let us consider the case of the shear wave exci- ϕ ta = a/ct. In the case of an arbitrary function (t), the tation by an acoustic pulse of the same duration t but ϕ 0 solution has the form of a convolution uz = h ∗ . If the with a saw-tooth carrier instead of the sinusoidal one. Ӷ action duration is t0 ta, the profile of the shear wave An ultrasonic wave acquires such a shape as a result of coincides with the profile of the pulsed response: uz = its nonlinear evolution (see Fig. 4c). The wave propaga- t0h(r, t). As one can see from Eq. (3), the displacement tion in a nonlinear dissipative medium is described by of the medium manifests itself most strongly at the the Burgers equation beam axis (r = 0), where it depends on time as uz = F t/(1+(c t/a)2). Hence, the maximal displacement of ∂p ε ∂p b ∂2 p 0 t ------Ð ------p------= ------, (8) the medium under the effect of a pulsed radiation ∂z ρc3 ∂τ 2ρc3 ∂τ2 force is l l τ where = t – z/cl is time in the moving coordinate sys- at0F0 tem and ε is the acoustic nonlinearity parameter of the umax = ------. (4) 2ct medium. The profile of a saw-tooth wave within a single period is described by the Khokhlov solution [11] The result given by Eq. (4) corresponds to the following time dependence of the volume force at the beam axis: p ωτ p τ ≤ ≤ p = -----s Ð------+,tanh ------s - Ðπωτπ≤ ≤ , (9) Fz(0, t) = F0 in the interval 0 t t0 and Fz = 0 outside 2 π 2b this interval. To compare the linear and nonlinear regimes of where ps is the value of the pressure jump at the shock π σ excitation, it is necessary to calculate the volume force. front. It depends on the distance as ps = 2 p0/(1 + ),

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 INCREASE IN THE EFFICIENCY OF THE SHEAR WAVE GENERATION 219 σ εω ρ 3 where = p0z/ cl is the distance from the source times greater than in the case of linear propagation. This estimate is confirmed by the experiment (see Fig. 5). divided by the length of the shock formation and p0 is a certain characteristic value of the wave amplitude. Therefore, the efficiency of the shear wave genera- However, the specific form of the function ps(z) is tion can be considerably increased by using pulses unimportant for our calculations. We are interested in whose wave profiles contain shock fronts. Certainly, ω ӷ the case of waves with large amplitudes, ps/b 1. when using saw-tooth signals for medical purposes, it Formally, the consideration of large-amplitude waves is necessary to be sure that they cause no tissue damage, corresponds to the transition b 0 in Eq. (9). In this i.e., there is no cavitation and no overheating of the case, a shock is formed in the profile. Substituting Eq. (9) medium. Such conditions are obtained by using rarely into expression (5) for the force of radiation pressure, repeated short pulses of the megahertz frequency range. performing the averaging, and passing to the limit For example, saw-tooth profiles are observed in the b 0, we obtain course of the operation of some diagnostic ultrasonic devices intended for cardiological applications [12]. nl ε f 3 F = ------p . (10) 0 ρ3 5 s 6 cl ACKNOWLEDGMENTS We are grateful to V.V. Yakushev for manufacturing Here, f = ω/2π is the wave frequency. The superscript “nl” indicates that the estimate given by Eq. (10) the PVDF films and for discussing the characteristic belongs to the case of the nonlinear ultrasonic propaga- features of the calibration of our acoustic sensors. We tion. Let us indicate two important features of the non- are also grateful to D. Cathignol for providing us with linear case (Eq. (10)) that make it different from the lin- the components of the device for measuring the total ear case (Eq. (6)). First, the radiation force does not power of the beam by the radiation force. depend on the linear absorption coefficient of the The work was supported by the CRDF, project medium. Second, the radiation force in the case of a no. RP2-2099, the FIRCA-NIH, project no. DK43881, and saw-tooth profile is proportional to the third power of the program “Universities of Russia,” project no. 1-5286. the wave amplitude rather than to the second power. The latter fact is fundamental and testifies to the possi- REFERENCES bility of increasing the efficiency of the shear wave gen- 1. V. G. Andreev, V. N. Dmitriev, Yu. A. Pishchal’nikov, et al., eration in the nonlinear regime. We note that the inten- Akust. Zh. 43, 149 (1997) [Acoust. Phys. 43, 123 (1997)]. 2 ρ sity of a saw-tooth wave is expressed as I = ps /12 cl. 2. L. K. Zarembo and V. A. Krasil’nikov, Introduction to From Eqs. (4) and (10), we obtain Nonlinear Acoustics (Nauka, Moscow, 1966), pp. 178Ð 205. nl ε fa 3. A. P. Sarvazyan, O. V. Rudenko, S. D. Swanson, et al., u = ------t Ip . (11) max ρ2 4 0 s Ultrasound Med. Biol. 24 (9), 1419 (1998). cl ct 4. R. O. Cleveland, O. A. Sapozhnikov, V. R. Bailey, and L. A. Crum, J. Acoust. Soc. Am. 107, 1745 (2000). As one can see, in the case of a fixed wave energy (t0I = const), the generation efficiency is directly proportional 5. IEEE Std 790-1989, IEEE Guide for Medical Ultra- sound Field Parameter Measurements (ANSI) (IEEE, to the value of the pressure jump ps at the shock fronts of New York, 1990). the saw-tooth wave. Remember that there is no depen- 6. A. E. Ponomarev, Yu. A. Pishchal’nikov, T. V. Sinilo, and dence on the amplitude in the linear case (see Eq. (7)). O. A. Sapozhnikov, in Proceedings of the VII All-Rus- Comparing Eqs. (7) and (11) in the case of the same sian SchoolÐSeminar on Wave Phenomena in Inhomoge- intensities of the sinusoidal and saw-tooth waves, we neous Media, Krasnovidovo, Moscow region, May 2000 obtain the following amplitude ratio of the shear-wave (Moscow State Univ., Moscow, 2000), Vol. 1, p. 34. signals: 7. J. Tavakkoli, A. Birer, and D. Cathignol, Shock Waves 5, 369 (1996). unl εfp K ==------max ------s . (12) 8. N. E. Waters, Br. J. Appl. Phys. 16, 557 (1965). ulin αρc3 9. V. G. Andreev and A. V. Vedernikov, Vestn. Mosk. Univ., max l Ser. 3: Fiz., Astron., No. 1, 34 (2001). Thus, we obtain that the factor K is proportional to 10. N. S. Bakhvalov, Ya. M. Zhileœkin, and E. A. Zabo- lotskaya, Nonlinear Theory of Sound Beams (Nauka, the amplitude of the acoustic wave ps, i.e., it can be much greater than unity for a high-intensity ultrasound. Moscow, 1982; AIP, New York, 1987). The parameters in gelatin have the following character- 11. O. V. Rudenko and S. I. Soluyan, Theoretical Founda- α –1 ε ρ 3 3 tions of Nonlinear Acoustics (Nauka, Moscow, 1975; istic values: = 1 m , = 4, = 10 kg/m , and cl = 1.5 × 103 m/s. For ultrasonic waves with the amplitude Consultants Bureau, New York, 1977). 7 6 12. Output Measurements for Medical Ultrasound, Ed. by ps = 10 Pa and the frequency f = 10 Hz used in the experiment, we obtain that, at a given pulse energy, the R. C. Preston (Springer, Berlin, 1991). maximal displacement value observed in a medium with a pronounced nonlinearity is approximately ten Translated by M. Lyamshev

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Acoustical Physics, Vol. 48, No. 2, 2002, pp. 220Ð224. Translated from Akusticheskiœ Zhurnal, Vol. 48, No. 2, 2002, pp. 260Ð265. Original Russian Text Copyright © 2002 by A. Pushkarev, M. Pushkarev, Remnev.

Sound Waves Generated Due to the Absorption of a Pulsed Electron Beam in Gas A. I. Pushkarev, M. A. Pushkarev, and G. E. Remnev Research Institute of High Voltages, Tomsk Polytechnical University, pr. Lenina 2, Tomsk, 634050 Russia e-mail: [email protected] Received June 29, 2001

Abstract—The results of an experimental investigation of acoustic vibrations (their frequency, amplitude, and attenuation coefficient) generated in a gas mixture as a result of the injection of a high-current pulsed electron beam into a closed reactor are presented. It is shown that the change in the phase composition of the initial mixture under the action of the electron beam leads to a change in the frequency of the sound waves and to an increase in the attenuation coefficient. By measuring the change in frequency, it is possible to evaluate with sufficient accuracy (about 2%) the degree of conversion of the initial products in the plasmochemical process. Relations describing the dependence of the sound energy attenuation coefficient on the size of the reactor and on the thermal and physical properties of the gases under study are derived. It is shown that a simple experimental setup measuring the parameters of acoustic waves can be used for monitoring the plasmochemical processes initiated by a pulsed excitation of a gas mixture. © 2002 MAIK “Nauka/Interperiodica”.

INTRODUCTION EXPERIMENTAL SETUP The plasmochemical processes that accompany the This paper presents the results of investigation of the injection of an electron beam in a gas or the volume sound waves generated in one-, two-, and three-compo- gaseous discharges are currently considered as an alter- nent gas mixtures as a result of the dissipation of the native to thermodynamical-equilibrium chemical pro- energy of a pulsed high-current electron beam in a cesses. Unlike these, in plasmochemical reactions the closed plasmochemical reactor (PCR). A Temp acceler- main part of energy of the source of excitation (up to ator [12] operating in an electron mode was used as a 80Ð90%) is delivered to the vibrational degrees of free- source of a high-current electron beam (HCEB) with dom of molecules, which provides a high efficiency of the following parameters: the maximal electron energy chemical processes [1, 2]. The plasmochemical meth- 300 keV, the current of the beam at the maximum 12 kA, ods of film deposition from ionized vapor make it possi- and the pulse width at the half-amplitude level 60 ns. ble to fabricate high-quality multilayer structures [3, 4]. The PCR had the form of a tube. Electrons were In volume gaseous discharges and under an electron injected in the tube at its end through a titanium foil. beam injection, an efficient dissociation of NOx and Two reactors with different dimensions were used in SO2 molecules is observed [5Ð7]. Plasmochemical pro- the experiments: one with a diameter of 6 cm and a cesses find applications in dry etching of Ni, Fe, and length of 11.5 cm and the other with a diameter of 9 cm other thin films [8] and in sterilization [9]. For monitor- and a length of 30 cm. The sound waves were detected ing the plasmochemical processes, optical methods by a differential pressure transducer capable of measur- (emission and absorption spectroscopy, Rayleigh scat- ing pressure variations in the reactor with frequencies tering, and so on) and mass spectrometry methods are up to 3.5 kHz. The gases studied in the experiment were used [10]. These methods require complicated equip- as follows: argon, nitrogen, oxygen, methane, silicon ment and optical access to the reaction zone. tetrachloride, tungsten hexafluoride, and their mixtures. When the energy of a pulsed source of excitation (a pulsed microwave discharge, a pulsed high-current electron beam, a gaseous discharge, and so on) dissi- INVESTIGATION OF THE SOUND WAVE pates in a closed plasmochemical reactor, the radiation- FREQUENCY acoustic effect [11] leads to the generation of acoustic vibrations, which are determined by the nonuniformity In a closed reactor with rigid walls, after the dissipa- of excitation (and, accordingly, of heating) of the tion of a pulsed electron beam, standing waves are gen- reagent gases. The measurement of the parameters of erated whose frequency in an ideal gas is [13] sound waves does not require complicated equipment n γRT but provides ample data on the processes that occur in f = ------, (1) the plasmochemical reactor. n 2l µ

SOUND WAVES GENERATED DUE TO THE ABSORPTION OF A PULSED ELECTRON BEAM 221 where n is the serial number of harmonic (n = 1, 2, …), fsw, Hz l is the length of the reactor, γ is the adiabatic exponent, R is the universal gas constant, and T and µ are the temperature and molar mass of gas in the reactor. 1500 In the experiments, we recorded the sound vibrations corresponding to the generation of standing waves 1 along and across the reactor. For our investigations, we 1000 chose the lowest component of the sound waves with the fundamental frequency of waves propagating along the reactor (n = 1 in Eq. (1)). 500 2 The dependences of the frequency of sound vibrations in the PCR on the parameter (γ/µ)0.5 for the one- γ µ 0.5 component gases in the reactors 11.5 and 30 cm long 0.05 0.10 0.15 0.20 ( / ) are shown in Fig. 1. The dots correspond to experimen- Fig. 1. Frequency of sound vibrations in the PCRs (1) 11.5 tal data and the lines correspond to calculations by and (2) 30 cm long as a function of the ratio of the adiabatic Eq. (1) for l = (1) 11.5 and (2) 30 cm. As seen from this exponent to the molar mass for single-component gases. figure, in the investigated frequency range, the sound vibrations are adequately described by the relation for ideal gases. written in the form that is more convenient for data processing The measurement of the frequency of sound vibra- γ tions generated due to the dissipation of energy of the RT ∑ iPi pulsed source of excitation makes it possible to monitor i f sw = ------. (3) the plasmochemical reactions resulting in the formation µ 2l ∑ iPi of solid products, e.g., CO2 C + O2, WF6 W + i 3F2, etc. The minimal degree of conversion of the initial gas that can be detected by the change in frequency The measurements of the frequency of sound vibra- does not exceed 2% for the attained accuracy of the fre- tions excited in the PCR due to the injection of an HCEB in two- and three-component mixtures showed quency measurement. The low attenuation of sound that the discrepancy between the values calculated by waves makes it possible to measure the vibration fre- Eq. (3) and the experimental values did not exceed quency accurate to 0.5%. 10%, and at the frequencies lower than 400 Hz the dis- In actual plasmochemical reactions, multicompo- crepancy was less than 5%. The dependence of the fre- nent gas mixtures are used, and the products of reac- quency of sound vibrations generated in the PCR due to the injection of an electron beam in two- and three- tions are also gas mixtures. In calculating the frequency component mixtures on the parameter ϕ, of sound vibrations, one has to take into account the weight coefficient of every component of the gas mix- ϕγ= ∑ P ∑µ P , ture and perform the calculation by the formula [14] i i i i i i is shown in Fig. 2. The dots correspond to experimental γ RT imi data and the line corresponds to the calculation by Eq. (3). f sw = ------∑------µ -, (2) 2lm0 i By simple calculations, it is possible to show that, i for a chemical reaction in which the initial mixture of reagents and the final mixture obtained in the reaction, where m0 is the total mass of all components of the gas γ µ are gases, the frequency of sound vibrations measured mixture and mi, i, and i are the mass, the adiabatic exponent, and the molar mass of the ith component, after the reaction is equal to the vibration frequency in respectively. Taking into account that the mass of the ith the initial mixture. However, if solid or liquid products are obtained in the reaction, the frequency of the sound component is waves will change. P V The proposed technique for monitoring the plasmo- × –27µ µ i mi = 1.66 10 iNi = K i ------, chemical reaction by the change in frequency of sound P0 vibrations was used in studying the direct reduction of tungsten from tungsten hexafluoride under the action of where Ni is the number of molecules of the ith compo- an HCEB [15]. The results were in good agreement nent, Pi is its partial pressure, V is the volume of the with the data obtained by weighing the substrate placed PCR, P0 = 760 torr, and K is a constant, Eq. (2) can be in the reactor.

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222 PUSHKAREV et al.

fsw, Hz The dependence of the energy of sound vibrations in the reactor on the energy of the electron beam absorbed in gas is shown in Fig. 3. The energy of the HCEB absorbed by the gas was measured by the change in pressure in the reactor after the injection of the HCEB 400 (as in [16]). The pressure was measured by the same differential pressure transducer that was used for detecting the sound waves. 300 For nitrogen and argon, a good correlation between the energy of sound vibrations and the energy contribution of the electron beam to gas was observed in a wide 200 range of pressures (and, accordingly, in a wide range of energy contributions of the beam to gas), which makes 0 0.10 0.15 ϕ it possible to evaluate the energy contribution of the electron beam to gas by the sound wave amplitude. The Fig. 2. Frequency of sound vibrations in the 30-cm-long energy of sound waves was about 0.2% of the energy of PCR as a function of the parameter ϕ for a gas mixture. the electron beam absorbed in the gas.

E, J INVESTIGATION OF THE SOUND WAVE ATTENUATION 0.15 Since the form of sound vibrations generated in the reactor due to the injection of the HCEB is nearly sinusoidal, the change in the energy of sound waves due to 0.10 the absorption is described by the relation [13] () Ðαt Et = E0e , 0.05 where α is the absorption coefficient. When a sound wave propagates in a tube closed at both ends, the absorption coefficient equals ααα α α 0255075Q, J = 1 +++2 3 4, where α is the sound absorption coefficient for the Fig. 3. Energy of sound vibrations in the PCR as a function 1 α of the energy of the electron beam absorbed in a gas. propagation in an unbounded gas, 2 is the absorption coefficient due to the reflection from the side walls of α the tube for the propagation along the tube, 3 is the INVESTIGATION OF THE SOUND WAVE absorption coefficient due to the reflection from the α ENERGY tube ends, and 4 is the absorption coefficient due to the friction at the tube walls. In a closed volume of the PCR, with the injection of an electron beam, standing waves are generated, the The absorption coefficient of the sound wave energy form of which is in our case nearly sinusoidal. Then, in gas due to the heat conduction and the shear viscosity the energy of the sound vibrations is described by the of gas can be determined by the StokesÐKirchhoff for- expression [13] mula [13] 2 2 ()2π f 4  E = 0.25β∆P V, α sw ηχ1 1 sw 1 = ------2 --- +,------Ð ------(4) 2ρc 3 Cv C p β ∆ sw where is the compressibility of the medium, Psw is the amplitude of the sound waves, and V is the volume where η is the coefficient of shear viscosity of gas of the reactor. When the degree of compression is low (g/cm s), χ is the coefficient of heat conduction ∆ Ӷ (cal/cm s deg), and C and C are the heat capacities of ( Psw 1) and the law of momentum conservation is v p valid (a low attenuation), the compressibility of the gas at constant volume and at constant pressure, respec- medium can be calculated by the formula [13] tively. For a low-frequency sound wave propagating along βρ()2 Ð1 λ λ = csw , a circular tube, when the condition > 1.7d (where is the wavelength and d is the tube diameter) is satisfied, ρ where is the gas density and csw is the sound velocity the wave front is plane and the sound energy attenua- in gas. tion coefficient for the propagation along a tube with

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 SOUND WAVES GENERATED DUE TO THE ABSORPTION OF A PULSED ELECTRON BEAM 223 ideally heat-conducting walls can be calculated by the where Kirchhoff formula [17] χ η ()γ π K2 = csw 0.39 Ð 1 ------+.0.37 --- (10) 1 f sw χ γ γ α = ------()γ Ð 1 ------+,η (5) C p 2 r ρ γ C 0 p The inclusion of the sound wave energy loss due to where r0 is the tube radius. the friction at the walls is important when the diameter ρ ρ Taking into account that = 0P/P0, where P is the of the tube is comparable with the mean free path of gas pressure of gas in the reactor, ρ is the gas density at molecules, i.e., for capillary tubes. In our case, we can 0 α ≈ normal conditions, and P0 = 760 torr, we obtain from assume that 4 0. Eq. (5): Numerical estimates of the contributions of various mechanisms of sound absorption in the reactor show α K1 f sw 2 = ------, that the influence of volume absorption (due to the heat r0 P conduction and gas viscosity) is insignificant. For where example, for sound waves generated in a 30-cm PCR filled with nitrogen at a pressure of 500 torr, we have α × –3 –1 α –1 α –1 πP χ 1 = 1.8 10 s , 2 = 5.9 s , and 3 = 7.7 s . K = ------0 ()γ Ð 1 ------+.η (6) 1 ρ γ C The summary absorption coefficient taking into 0 p account only the normal incidence of sound waves (i.e., α α –1 It can be shown that the temporal absorption coeffi- 2 = 0, 3 = 1.2 s ) is much smaller than the experi- cient for the sound wave energy in a closed reactor with mentally measured coefficient of sound absorption for multiple reflection from its ends is equal to these conditions (14.7 s–1). Then, the expression for the c summary absorption coefficient of the sound wave α = ------sw- ln()1 Ð δ , (7) energy in a closed reactor can be written as 3 l where l is the length of the reactor and δ is the energy α K1 K2 f sw = ------+ ------, (11) absorption coefficient of a sound wave at a single r0 l P reflection. For normal incidence of a plane sound wave on a where K1 and K2 are calculated by Eqs. (6) and (10), r0 metal wall, which is a good heat conductor, the energy and l are the radius and length of the reactor in centime- absorption coefficient equals [17] ters, fsw is measured in hertz, and P is in torrs. The values of the coefficients K and K for the π χ 1 2 δ ()γ f sw investigated gases are given in the table. = 4 Ð 1 ------γ -. (8) C pP It is important to note that, for the propagation of However, if we consider only normal incidence of a sound waves in a closed reactor, the main contribution sound wave on the ends of the reactor, we should (60Ð80%) to the absorption is made by the gas viscos- neglect the absorption of the sound wave energy at ity. The magnitude of the second term in Eqs. (5) and α (9) is 3Ð9 times greater (for various gases) than the reflection from the side walls of the reactor (i.e., 2 = 0). In this case, the sound absorption will be deter- magnitude of the first term. The contributions of the mined only by the heat conduction and the gas viscos- side walls and ends of the reactor to the absorption of ity (Eq. (4)) and by the absorption at reflection from the the sound wave energy are approximately equal for a reactor ends (Eqs. (7) and (8)). As will be shown below, large reactor. the experimentally measured sound energy absorption The dependence of the sound energy absorption coefficients in the reactor are several times greater than coefficient in the reactor on the pressure for various the values calculated by formulas (4), (7), and (8). Con- gases is shown in Fig. 4. For the comparison of the sequently, in the reflection from the ends of the reactor, attenuation coefficients observed in different plasmo- it is necessary to take into account the dependence of chemical reactors (with the lengths 11.5 and 30 cm) and the absorption coefficient on the angle of incidence and in various gases, the value of the attenuation coefficient perform calculations by the formula [17] f χ Table δ = ------sw 0.39()γ Ð 1 ------+. 0.37 η (9) γP C p Gas K1 K2 δ Ӷ δ ≈ δ Taking into account that, for 1, ln(1 Ð ) , from N 25 218 Eqs. (9) and (7) we derive 2 O2 27 189 α K2 f sw Ar 32 254 3 = ------, l P WF6 6.5 52

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 224 PUSHKAREV et al.

α/K it possible to monitor with a high accuracy the plasmochemical process that is accompanied by a change in 0.7 2 the phase composition of the initial reagent mixture. The formation of clusters in the volume of the reactor 0.6 leads to a change in the frequency of sound waves and to a considerable increase in the attenuation of the 0.5 vibration amplitude. Thus, the sound-wave diagnostics 0.4 can be used for monitoring plasmochemical processes. 0.3 REFERENCES 0.2 1. V. D. Rusanov and A. A. Fridman, Physics of a Chemi- cally Active Plasma (Nauka, Moscow, 1984). 0.1 1 2. A. A. Ivanov and T. K. Soboleva, Nonequilibrium 0 Plasma Chemistry (Atomizdat, Moscow, 1978). 200 300 400 500 600 700 , torr P 3. Ionized Physical Vapor Deposition, Ed. by Jeffrey A. Hop- wood (San Diego, CA, Academic, 2000). Fig. 4. Normalized coefficient of sound absorption in the reactor as a function of pressure. 4. P. Czuprynski, O. Joubert, L. Vallier, and N. Sadeghi, J. Vac. Sci. Technol. A 17 (5), 2572 (1999). 5. Mak Yong Sun, IEEE Trans. Plasma Sci. 27 (4), 1188 was normalized to the coefficient K calculated by the (1999). formula 6. G. V. Denisov, Yu. N. Novoselov, and R. M. Tkachenko, Pis’ma Zh. Tekh. Fiz. 26 (16), 30 (2000) [Tech. Phys. K K Lett. 26, 710 (2000)]. Kf= ------1 + ------2 . swr l 7. T. Ikegaki, S. Seino, Y. Oda, et al., in Proceedings of 13th 0 International Conference BEAMS 2000 (Nagaoka, Japan, 2000), p. 934. The dots in Fig. 4 correspond to experimental mea- 8. H. Cho, K. B. Jang, and D. C. Hays, Appl. Surf. Sci. 140 surements, and curve 1 is calculated by Eq. (11). For (1Ð2), 215 (1999). nitrogen, argon, and oxygen, the discrepancy between the calculated and experimental values of the absorp- 9. S. Yu. Socovnin, Yu. A. Kotov, and M. E. Balesin, in Pro- ceedings of 12th Symposium on High Current Electron- tion coefficient does not exceed 30%. ics (Tomsk, 2000), Vol. 2, p. 512. For sound waves generated due to the dissipation of 10. V. K. Zhivotov, V. D. Rusanov, and A. A. Fridman, Diag- a pulsed electron beam in the WF6 vapor (curve 2 in nostics of Nonequilibrium Chemically Active Plasma Fig. 4), the experimentally obtained values of the (Energoatomizdat, Moscow, 1985). absorption coefficient far exceed (by a factor of 14Ð15) 11. L. M. Lyamshev, Radiation Acoustics (Nauka, Moscow, the values calculated by Eq. (11). This may be caused 1996). by the formation of clusters in the reactor with the 12. G. E. Remnev, I. F. Isakov, M. S. Opekunov, et al., Izv. injection of the electron beam. The presence of large Vyssh. Uchebn. Zaved., Fiz., No. 4, 92 (1998). particles in the volume of gas leads to an increase in the 13. M. A. Isakovich, General Acoustics (Nauka, Moscow, absorption of sound waves. At the injection of the 1973). HCEB in WF , the direct reduction of tungsten in the 6 14. B. M. Yavorskiœ and A. A. Detlaf, A Handbook on Phys- form of nano-sized particles take place, which results ics (Nauka, Moscow, 1968). not only in an increase in the frequency of sound waves [15], but also in a considerable growth of the sound 15. G. E. Remnev, A. I. Pushkarev, M. A. Pushkarev, et al., Izv. Vyssh. Uchebn. Zaved., Fiz., No. 5, 33 (2001). energy absorption. 16. Yu. F. Bondar’, S. I. Zavorotnyœ, A. L. Ipatov, et al., Fiz. Plazmy (Moscow) 8, 1192 (1982). CONCLUSION 17. B. P. Konstantinov, Hydrodynamic Sound Formation and Propagation in a Bounded Medium (Nauka, Mos- The described investigations of the sound waves cow, 1974). generated in a closed reactor due to the dissipation of the energy of a pulsed electron beam show that a simple experimental setup detecting acoustic vibrations makes Translated by A. Svechnikov

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002

Diffraction of Sound by an Elastic Cylinder Near the Surface of an Elastic Halfspace E. L. Shenderov† Morﬁzpribor Central Research Institute, Chkalovskiœ pr. 46, St. Petersburg, 197376 Russia Received September 25, 2000

Abstract—Diffraction of an acoustic wave by an elastic cylinder near the surface of an elastic halfspace is considered. The solution relies on a Helmholtz-type integral equation and uses the Green function of an elastic halfspace. The latter function is represented in the form of an integral over the Sommerfeld contour on the plane of a complex variable that has the meaning of the angle of the wave incidence on the halfspace boundary. An integral equation for the sound pressure distribution over the cylinder surface is derived. This equation is reduced to an inﬁnite system of equations for the Fourier-series expansion coefﬁcients of this distribution. The results obtained are valid for the diffraction of a cylindrical wave and a plane wave. They also describe the diffraction of a spherical wave when the transmitter and receiver are far from the cylinder and lie in one plane that is orthogonal to the cylinder axis. © 2002 MAIK “Nauka/Interperiodica”.

† If a body is located near a sound-reflecting bound- The solution presented below is a rigorous solution ary, an emitted or diffracted sound experiences multiple to the two-dimensional problem, i.e., to the problem of reflections between the body and the boundary, which diffraction of a cylindrical acoustic wave by an elastic can significantly change the pattern of the acoustic cylinder when the cylinder axis and the axes of the field. The sound diffraction by bodies located near a cylindrical source and the receiver are parallel to each planar boundary has been extensively studied. The other and to the surface of the halfspace. However, as T-matrix method combined with the summation of shown in the appendix, the results obtained are valid for multiply reflected waves was used to study the scatter- the diffraction of the cylindrical wave and for the dif- ing from particular bodies located near a liquid halfs- fraction of the spherical wave when the receive and pace [1, 2]. A cylinder with rounded ends and an elastic transmit points lie in one plane that is orthogonal to the spherical shell were considered. In [3Ð5], the problem cylinder axis and the distance from one of these points was solved by replacing the reflecting boundary with to the cylinder is large in terms of the wavelength. In the object image symmetric about this boundary. Such this case, the only difference in the solutions for the a replacement leads to a problem of diffraction by two incident cylindrical and spherical waves is that, for the bodies, which is solved using the summation theorem cylindrical wave, the amplitude of the wave scattered for the special functions involved in the sound field from the cylinder (and, therefore, the radius of the equiv- expansions and is reduced to an infinite system of equa- alent sphere) is by a factor of 2 greater than that for the tions. The methods for solving such problems can be found in [6Ð8]. However, the replacement of the bound- spherical wave. This fact was noted earlier in [10]. ary with a symmetric object image is only valid for a Let us derive a system of equations for the coeffi- perfect (acoustically hard or soft) boundary. In the case cients of the Fourier series expansion of the total acous- of an elastic or impedance boundary, this method is tic field on the cylinder surface. inapplicable. The situation is the same as in the classical problem of the spherical wave reflection from an The coordinate system is illustrated in Fig. 1. Let a impedance plane, whose solution is described in [9]. line source M0 oriented normally to the plane of the drawing emit the cylindrical wave This paper addresses the diffraction by an elastic () ρ (), cylinder located near an elastic or impedance halfspace. pi r1 = Ðik cQG0 r0 r1 , (1) The halfspace may be stratified or covered with an elastic plate. The dependence of the reflection coefficient or where k = ω/c is the wave number; ρ and c are the density the input impedance of the halfspace surface on the of the upper halfspace and the acoustic velocity in it, incidence angle is assumed to be known. respectively; Q is the source strength; and G0(r0, r1) = () † 1 | | Deceased. iH0 (k r1 – r0 )/4 is the Green function for a free space.

226 SHENDEROV

y incidence angle, whereas, for an elastic surface, the impedance depends on this angle. The following M0 r0 ϕ1 expression for this impedance is presented in [9]: ϕ 2 2 0 cos 2θ sin 2θ w = w ------t + w ------t . (5) r1 h l θ t θ ϕ cos l cos t Scyl 2 ρ ρ ρ ρ O Here, wl = ( 1cl)/( c) and wt = ( 1ct)/( c) are the wave x impedances of the lower halfspace normalized by ρc. The subscripts l and t refer to longitudinal and shear waves, respectively; cl and ct are the velocities of these r2 waves; and ρ is the density of the medium. The angles Sh = – 1 y b of refraction θ and θ satisfy the Snell law: sinθ = θ l θ t θ l sin (cl /c) and sin t = sin (ct/c). Fig. 1. Coordinate system. Let us write the distributions of the sound pressure and of the normal component of the particle velocity over the surface of the halfspace as the Fourier integrals In the upper halfspace, the total acoustic field satis- representing the expansions in the wave numbers: fies the Helmholtz-type integral equation ∞ () ρ (), px()= Pu()exp()ikux du, p r1 = Ðik cQG r1 r0 2 ∫ 2 (6) ∂G()r , r ∂p()r (2) Ð∞ () 1 2 2 (), ∞ + ∫ p r2 ------∂ - Ð ------∂ -G r1 r2 dS. n2 n2 1 ∂p SS= + S v ()x ==------Vu()exp()ikux du. h cyl n 2 ωρ∂ ∫ 2 (7) i n2 Here, the surface of integration, S, consists of the sur- Ð∞ face of the cylinder and the surface of the halfspace. In these expressions, x2 denotes a point on the surface The derivation of Eq. (2) can be found in [11] (appen- of the halfspace. For each plane wave, the complex dix). The subscript 2 indicates that the derivative is amplitudes of the sound pressure and particle velocity taken along the normal at the point r2, which lies at the are related as P(u) = –Zh(u)V(u). These expressions surface of the cylinder in the integral over the surface yield the formula for the normal derivative of the sound Scyl and at the surface of the halfspace in the integral pressure at the surface of the halfspace, which enters over the halfspace surface Sh. As the Green function into Eq. (2): G(r , r ), any Green function of the Helmholtz differen- ∞ 1 2 ∂ () () tial equation can be used. To eliminate the integration p r2 Pu () over the infinite surface, we choose the Green function ------∂ - = Ðik ∫ ------()- exp ikux2 du. (8) n2 wh u for the upper halfspace so as to satisfy the boundary Ð∞ conditions at the surface (see [11]): Substituting Eqs. (3), (6), and (8) into Eq. (2), we obtain ∞ that the integral over the halfspace surface Sh is zero i G()r , r = ------exp()iku() x Ð x and only the integral over the cylinder surface Scyl is 1 2 4π ∫ 1 2 left. Thus, the use of the Green function given by Ð∞ (3) expression (3) eliminates the integration over the infi- × []()γ ()γ ()du nite surface. exp ik y1 Ð y2 + Ah' exp ik y1 + y2 ------, γ Expression (3) can be represented as G = G0 + G1, where G0 is the Green function for a free space and the γ 2 γ γ term G determines the field reflected from the bound- where = 1 Ð u , Ah' = exp(i2kb )Ah, Ah(u) = (wh – 1 γ ρ ary of the halfspace, i.e., 1)/(wh + 1), wh = Zh/( c), and Zh is the input impedance of the elastic halfspace. The change of variables (), i ()1 () θ G0 r1 r2 = ---H0 k r1 Ð r2 u = sin transforms the expression for Ah(u) into the 4 formula ∞ () (9) J ()kr H 1 ()kr r > r , θ i n 1 n 2 2 1 wh cosÐ 1 = --- ∑ exp()in()ϕ Ð ϕ , A ()θ = ------, (4) 4 ()1 1 2 > h θ J ()kr H ()kr r1 r2, wh cos+ 1 n = Ð∞ n 2 n 1 (), which gives the reflection coefficient for the plane wave G1 r1 r2 incident on the boundary of the elastic halfspace at the ∞ angle θ. Note that, for a locally responding surface (i.e., i du (10) = ------A' exp[]ik() u()γ x Ð x + ()y + y ------. a surface whose properties are described by the normal 4π ∫ h 1 2 1 2 γ impedance), the impedance wh is independent of the Ð∞

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 DIFFRACTION OF SOUND BY AN ELASTIC CYLINDER 227

Expression (9) represents the summation theorem for of the particle velocity of the cylinder and can be repre- the cylindrical functions. Using the change of variables sented as the expansion u = sinθ and γ = cosθ and switching to the cylindrical ∞ ϕ ϕ ϕ ∂ coordinates x1 = r1 sin 1, y1 = r1cos 1, x2 = r2sin 2, ()ϕ ()ϕ 1 p ϕ v ==∑ v q exp iq ------. and y2 = r2cos 2, we obtain iωρ∂n q = Ð∞ ra= (), i ()θ The coefficients p and v are related as v = –p /Z , G1 r1 r2 = ------∫ Ah' q q q q q 4π where Z are the mode impedances of the elastic cylin- Γ (11) q der oscillations, which are given by Eq. (A.3) of the × []()ϕ θ []()ϕ θ θ exp ikr1 cos 1 Ð exp ikr2 cos 2 + d . appendix. Therefore, we obtain ∞ Here, Γ is the Sommerfeld contour: –π/2 + i∞, π/2 – i∞. ∂ p pq ()ϕ Using the expansion of a plane wave in cylindrical ∂ = Ðik ∑ ------exp iq . (16) n wq functions ra= q = Ð∞ ρ ∞ Here, wq = Zq/( c) are the mode impedances normal- m ized by the wave impedance of the medium. exp()ikrcosα = ∑ i J ()kr exp()imα , (12) m Let us place the observation point at the surface of m = Ð∞ the cylinder, substitute the above expressions into inte- we obtain gral relationship (2), and take into account that, with the Green function chosen above, the integration can be ∞ performed only over the cylinder surface. The follow- i n G ()r , r = --- ∑ ()Ði J ()kr exp()inϕ ing transformations are simple, but cumbersome. 1 1 2 4 n 1 1 Therefore, we only describe their main stages: n = Ð∞ ϕ ϕ ∞ (13) (a) Substitute expansion (15) at = 1 in Eq. (2) on its left-hand side. × ()Ði m J ()kr exp()Ðimϕ f , ∑ m 2 2 mn+ (b) Substitute the Green function with Eqs. (9) and m = Ð∞ (13) it involves into the first term on the right-hand side ϕ ϕ where f are the coefficients defined as by replacing r2 and 2 with ‡ and 0. m + n (c) Substitute expansions (15) and (16) of the distri- ()s butions of the sound pressure and its normal derivative Ði ()θ []θ θ θ f s = ------π -∫ Ah cos exp i2kbcos + is d . (14) over the surface of the cylinder into the integrand. Γ (d) Substitute the Green function into the integrand by performing the differentiation along the normal with For an acoustically hard or acoustically soft surface of respect to the variable r . After completing the differen- the halfspace, A = 1 or Ð1, respectively. Then, expres- 2 h tiation, set r1 = r2 = a. When calculating the derivative sion (14) takes the form of an integral representation of in Eq. (9), one should first let the point r2 tend to the ()1 the Hankel function, which yields f = ±H (2kb), surface, i.e., assume that r1 > r2 and use the lower line s s of formula (9); take the derivative with respect to r ; where the plus and minus signs refer to the acoustically 2 and then set r1 = r2 = a. hard and acoustically soft surfaces, respectively. ϕ Applying the summation theorem twice, one can show (e) Integrate with respect to 2 with allowance for the orthogonality relationships for the exponential fac- that, in these particular cases, the term G1 is expressed () tors. As a result, the sums over m and n involved in the by the formula G iH 1 kR , which describes the ϕ 1 = 0 ( ')/4 integrand and containing 2 vanish except for the terms field of the image line source; in this case, R' = with n = q or m = q. ()x Ð x 2 + ()y + y 2 . (f) Change the order of the summation in the double 1 2 1 2 sums, i.e., use the change of variables n q, to obtain ϕ Let us represent the unknown distribution of the the factors exp(iq 2) in all their terms. Since all the sums total sound pressure on the surface of the cylinder as the over q are Fourier series, the equation obtained must be expansion satisfied for each term of the sum over q. ()1 ∞ (g) Using the well-known identity Jn' (ka)(Hn ka) = (), ϕ ()ϕ ()1 ' π pr ra= = ∑ pq exp iq , (15) Jn(ka)(Hn ka) – 2i/( ka), obtain the system of equa- ∞ q = Ð tions for the coefficients pq: ∞ where p are the unknown coefficients. The normal q ∞ … ∞ component of the particle velocity in the medium at the pq + ∑ pnzqn ==bq, q Ð , (17) surface of the cylinder is equal to the radial component n = Ð∞

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 228 SHENDEROV where problem of the spherical wave reflection from an elastic halfspace. w J ()ka Ð iw J' ()ka z = ------q()Ð1 qn+ ------n n n f , (18) qn ()1 ()1 qn+ To find the scattered field, substitute expansion (15) wn H ()ka Ð iw H '()ka q q q and the Green function, written as the sum of expres- ρcQ w sions (9) and (13), into formula (23). Assume that the b = ------q q π ()1 ()1 point r2 resides on the cylinder surface, i.e., r2 = a. 2 a H ()ka Ð iw H '()ka q q q Since, in this case, r1 > r2, we should use the lower line in formula (9). Then, we obtain  () × H 1 ()kr exp()Ðiqϕ q 0 0 ∫ (19) πka  p = Ð------s 2 ∞  ∞ ∞ mq+ (25) () () ()ϕ  () +1∑ Ð Jm kr0 exp Ðim 0 f mq+ . × 1 () ()ϕ  ∑ pqsq Hq kr1 exp iq 1 +,∑ pqsquq m = Ð∞ q = Ð∞ q = Ð∞ Note that the coefficients zqn of the matrix depend on neither the source position nor the position of the obser- where vation point. The free terms bq depend on the source position but are independent of the position of the ∞ observation point. ()nq+ () ()ϕ uq = ∑ Ð1 Jn kr1 exp in 1 f nq+ , (26) To find the sound pressure p at the observation point n = Ð∞ M1, we again resort to Eq. (2) and represent it, using ()() () Eqs. (9) and (13), as a sum of the incident cylindrical sq = Jq ka Ð iwq Jq' ka /wq. (27) wave pi, the wave pr reflected from the halfspace boundary, and the wave ps scattered by the cylinder System of equations (17) and expression (25) deter- (with allowance for the multiple scattering between the mine the scattering field in terms of the coefficients of cylinder and the halfspace): the Fourier series expansion of the total field on the cyl-

pp= i ++pr ps, (20) inder surface. Therefore, when the radius of the cylinder decreases, these coefficients tend to the expansion where coefficients of the sum of the incident and scattered () p = kρcQH 1 ()k r Ð r /4, (21) fields rather than to zero. Hence, it is reasonable to i 0 1 0 transform these coefficients so as to extract the coeffi- ∞ cients of expansion of the scattered field, which tend to kρcQ n p = ------∑ ()Ði J ()kr exp()inϕ zero as the radius of the cylinder decreases. To this end, r 4 n 1 1 ∞ we introduce new expansion coefficients aq, which are n = Ð (22) ∞ related to the coefficients pq as × ()m () ()ϕ ∑ Ði Jm kr0 exp Ðim 0 f mn+ , 2 wq m = Ð∞ p = Ða ------. (28) q qπkaJ ()ka Ð iw J' ()ka ∂ (), ∂ () q q q ()G r1 r2 p r2 (), ps = ∫ p r2 ------∂ - Ð ------∂ -G r1 r2 dS. (23) n2 n2 Then, the system of equations (17)Ð(19) will have the Scyl form First, consider Eq. (22). Substitute representation (14) into it and, in each sum, expand the plane wave in ∞ ∞ … ∞ cylindrical functions. As a result, we obtain the integral aq + ∑ an Dnq ==Eq, q Ð , (29) representation for the reflected wave: n = Ð∞ kρcQ ()θ [ ( θ () pr = ------∫ Ah exp ik 2bcos 0 ()qn+ 4π Dnq = Ðaq Ð1 f qn+ , (30) Γ (24) ()ϕ θ ()ϕ θ ] θ + r1 cos 1 + + r0 cos 0 Ð d . ()0 kρcQ ()1 E = a ------H ()kr exp()Ðiqϕ This expression differs from Eq. (11) in the constant q q 4 q 0 0 ∫ factor and in the subscripts that indicate the positions of the source and the observation point with the coordi- (31) ϕ ϕ ∞ nates (r0, 0) and (r1, 1), respectively. Expression (24) ()mq+ () ()ϕ is an expansion of the reflected cylindrical wave similar +1∑ Ð Jm kr0 exp Ðim 0 f mq+ , to the WeylÐBrekhovskikh integral [9] in the classical m = Ð∞

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 DIFFRACTION OF SOUND BY AN ELASTIC CYLINDER 229 and the scattered field (25) will take the form ϕ M1 M0 ∞ 1 r0 ()1 () ()ϕ ϕ ps = ∑ aq Hq kr1 exp iq 1 ∫ 0 θ ∞ 0 h q = Ð O h 0 (32) r 1 ϕ ∞ 1 θ 0 b 0 ()nq+ () ()ϕ +1∑ Ð Jn kr1 exp in 1 f nq+ . ∞ n = Ð S () B In Eqs. (30) and (31), the coefficients a 0 are deter- b q h mined by formulas (A.2). O' 1 π ϕ θ A /2 – ( 1 + 0) To obtain an asymptotic expression for the case ϕ when the receiver and transmitter are far from the cyl- 1 inder, we first consider the field reflected from the r1 M1' plane. If at least one of the wave distances, kr1 or kr0, is much greater than unity, the integral in Eq. (24) can be calculated by the saddle-point method. Poles of the Fig. 2. To the derivation of Eqs. (33) and (34). θ function Ah( ) that may be crossed in the course of the deformation of the integration path can be neglected, If the source is at a long distance from the cylinder, because the residues at these poles produce surface ӷ waves, which decay exponentially with distance from i.e., kr0 1, Eq. (31) can be simplified. To this end, we the surface. In the exponent, we can single out the term represent the sum that appears in this expression as ik(h + h )cosθ, where h = b + r cosθ and h = b + ∞ 0 θ 1 0 0 1 q r1cos are the distances from the transmitter and the i ()θ ∑ = --- ∫ Ah observation point to the surface, respectively. There- π (35) fore, the calculation of the residues gives rise to the fac- m = Ð∞ Γ θ × ()()θ ()θϕ θ θ tor exp(–k(h0 + h1)Re(cos )). The amplitude of the exp ik 2bcos + r0 cos Ð 0 + iq d . surface wave thus depends on the sum h0 + h1 rather ӷ than on each distance individually. Hence, if the dis- When kr0 1, this integral can be calculated by the tance from at least one of these points to the surface is saddle-point method. If r ӷ b and kr ӷ q, the saddle θ ≈ ϕ 0 0 longer than the wavelength, the contribution of the point is 0 0. When transforming the integration path residue can be neglected. This fact has been noted in into the steepest descent path, it can cross the poles of θ relation to the problem of the spherical wave reflec- the function Ah( ). The residues at these poles deter- tion from a halfspace [9]. mine the surface waves near the halfspace boundary. The value of the variable of integration correspond- However, since the contributions of these poles ing to the saddle point is determined as decrease exponentially with the distance from the boundary, they can be ignored when calculating the r sinϕ Ð r sinϕ field produced by a distant source. Note that all contri- tanθ = ------0 0 1 1 -. (33) 0 2br++cosϕ r cosϕ butions of the poles and the corresponding surface 0 0 1 1 waves produced by the interaction between the closely θ As follows from Fig. 2, 0 is equal to the angle of inci- spaced cylinder and boundary are taken into account dence of sound at the point corresponding to the specu- rigorously, because they are described by the coeffi- lar reflection. After calculating the integral, we obtain cients fq + m, which enter into the matrix coefficients zqm. ρ After calculating the integral, we obtain the system ≈ ()θ k cQ 2 ()π pr Ah 0 ------exp iklÐ i /4 , of equations 4 πkl (34) ӷ () ӷ ∞ kr1 1, kh0 + h1 1, a + ∑ a D ==F , q Ð∞ … ∞, (36) where l = r cos(ϕ + ϕ ) + 2bcosθ + r cos(ϕ – θ ) is q n qn q 1 1 0 0 0 0 0 n = Ð∞ the total distance M0SM1 (Fig. 2). The first, second, and third terms of the expression for l are the lengths of the where the coefficients of the matrix coincide with the segments M 1'A, AB, and BM0, respectively. Thus, in the coefficients given by Eq. (30) and the right-hand sides far-field region, the reflected field is expectedly formed are written as as a result of the emission of a cylindrical image source ≈ ()0 ()0 ()q with the amplitude that is proportional to the reflection Fq pi aq Ði coefficient of sound at the angle of incidence corre- (37) × []()ϕ ()q ()ϕ ()ϕ sponding to the geometrical optics reflection. expÐiq 0 + Ð 1 Ah' 0 exp iq 0 .

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 230 SHENDEROV

()0 Here, pi represents the sound pressure in a free space Consider the calculation of coefficients fn + q defined at a point lying on the cylinder axis, i.e., by integral (14). Let us split this integral into integrals over the segments (–π/2 + i∞, –π/2), (–π/2, π/2), and ()0 kρcQ 2 (π π i∞). In the first and third of these integrals, p ≈ ------exp()ikr Ð iπ/4 . /2, /2 – i 4 πkr 0 we change the variables θ = –π/2 + iα and π/2 – iα, 0 θ θ ± π α respectively. Denoting Ah( ) at = ( /2 – i ) as For the diffraction of a unit-amplitude plane wave, () 1 α s π ()0 Ah ( ), we obtain fs = (–i) (I1 – iI2)/ , where one should set pi = 1 and replace the sign of approximate equality with the sign of exact equality. π/2 ()θ ()θ ()θ θ Let us calculate the sound pressure in the scattered I1 = ∫ Ah exp i2kbcos cos n d , (41) wave at the observation point located far from the cyl- ӷ 0 inder when kr1 1. The sum over n in Eq. (32) can be ∞ represented as ()1 ()α ()α ∞ I2 = ∫ Ah exp Ð2kbsinh iq (42) ()θ 0 ∑ = --- ∫ Ah π (38) n = Ð∞ Γ × []exp()iπn/2 + nα + exp()Ð()iπn/2 + nα dα, × exp[]ik()2bcosθ + r cos()ϕ + θ + iqθ dθ. α 1 1 ()1 ()α iwh sinhÐ 1 Ah = ------α , (43) In the far-field region, i.e., under the constraints iwh sinh+ 1 imposed when calculating integral (35), integral (38) θ θ can also be calculated by the saddle point method. The and the refraction angles l and t, which enter into θ ϕ Eq. (5), are written as saddle point is determined as 0 = – 1. Then, we obtain the scattered field as θ () α θ () α sin l ==cl/c cosh , sin t ct/c cosh . (44) θ 2 ()Φπ ()ϕ The function Ah( ) does not have any poles on the ps = π------exp ikr1 Ð i /4 s 1 , (39) θ kr1 real axis of the complex plane of . Therefore, the integral I is calculated in a straightforward manner. When where 1 ()1 α ∞ the medium is lossless, the function Ah ( ) can Φ ()ϕ ≈ ()q increase without limit at certain points of the integra- s 1 ∑ aq Ði (40) tion path, which corresponds to the condition of the q = Ð∞ generation of surface waves. In this case, a residue at × []exp()iqϕ + ()Ð 1 q exp()Ðiqϕ A' ()ϕ . the pole should be added. The pole is however found as 1 1 h 1 a solution to an intricate transcendental equation. It is The system of equations (36) and expressions (39) therefore simpler to assume that the reflecting halfs- and (40) completely determine the scattered field pace is always lossy and that the velocities of the longi- formed in the far zone as a result of the diffraction of a tudinal and transverse waves are complex-valued. plane wave and also of a cylindrical wave by an elastic Integral (42) converges fast, the integrand starting to cylinder located near a boundary of an elastic halfs- α α pace. decrease rapidly when sinh > n (2kb). Note that, due to the sharp maxima in the integrand, in order to In system of equations (36), the indices assume pos- decrease the computation time, it is reasonable to split itive and negative values, whereas the existing pro- the integration interval into a number of shorter inter- grams for solving systems of linear equations handle vals and calculate each of these integrals using a only positive indices. If the variables are renumbered quadrature formula rather than integrate over the entire through shifting their indices by a constant value, the interval at once. In this study, we used the Gaussian most significant terms (for example, a0 and a1) will quadrature formula with the automatic selection of the occur in the middle of the system, which is inconve- number of nodes. nient when an infinite system of equations is solved by the reduction method. It is therefore reasonable to rear- The system of equations was solved by reduction. range the system so as to make all indices positive and To obtain the result with at least three true decimal dig- α place the most significant terms at the beginning of the its, it was sufficient to retain 1.2k + 3 terms in the system. It is also useful to separate the solution into the series and, accordingly, the same number of equations ϕ in the system. The convergence of the system depends symmetric and antisymmetric parts about 1 = 0. The system of equations can thus be split into two systems, on the distance between the cylinder and the surface of each of them being twice as small as the original sys- the halfspace and persists until the cylinder touches the tem, which reduces the computation time when the boundary. dimension of the system is large. The manipulations When calculating the bistatic scattering patterns and final formulas are omitted here for brevity. produced by the elastic cylinder (Fig. 3), we took into

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 DIFFRACTION OF SOUND BY AN ELASTIC CYLINDER 231

90° 7 7 90° (‡)120° 60° (b) 120° 60° 6 6 α1 α1 5 5 4 150° 30° 4 150° 30° 3 3

2 α0 2 α0 1 1 0 180° 0° 0 180° 0° 1 1 2 2 3 3 4 210° 330° 4 210° 330° 5 5 6 6 240° 7 300° 7 240° 300° 270° 270°

Fig. 3. Bistatic scattering patterns for an elastic cylinder located near the surface of a liquid halfspace (the solid lines) and in a free ρ 3 ()cyl ()cyl ε()cyl space (the dotted lines): kb = (a) 7.0 and (b) 3.0, m = 2000 kg/m , ka = 3.0, cl = 2000 m/s, ct = 300 m/s, l = 0.1, ε()cyl ρ 3 ε ϕ α t = 0.2, 1 = 1800 kg/m , cl = 2000 m/s, l = 0.01, and 0 = 80° ( 0 = 10°).

3.0

2.5 b/a = 1.0 b/a = 4.0

2.0

1.5

1.0

0.5

0 2468100246810

Fig. 4. Relative amplitude of the backscattered field for a cylinder located near the boundary of a halfspace (the solid lines) and in ρ 3 a free space (the dotted line) versus the wave radius for different distances between the cylinder and the surface: m = 2000 kg/m , ()cyl ()cyl ε()cyl ε()cyl ρ 3 ε ϕ ka = 3.0, cl = 2000 m/s, ct = 300 m/s, l = 0.1, t = 0.2, 1 = 1800 kg/m , cl = 2000 m/s, l = 0.01, and 0 = 45°. account the active loss in the cylinder material. The boundary is, the smaller the angular spacing between coefficients that characterize the sound attenuation for the extrema of the plots. At certain directions, the the transverse waves were assumed to be twice as large amplitude of the wave scattered by the cylinder located as those for the longitudinal waves, which is typical of near the halfspace boundary was found to be 2Ð3 times elastic media (rubber-like materials). In Fig. 3 and the greater than the amplitude of the wave scattered by the subsequent figures, the plots are constructed versus the cylinder located in a free space. α ϕ angular coordinate 1 = 90° – 1. The dependence of the scattering amplitude on the As compared to the scattering patterns in a free wave radius of the cylinder exhibits an oscillatory behav- space, the patterns produced by a cylinder located near ior (Fig. 4). Oscillations of two types are observed here. a boundary are more nonuniform. The additional max- One of them is associated with the resonance oscilla- ima and minima in the patterns occur due to the multi- tions of the cylinder, and the other is produced by the ply reflected waves and the interference between them. interference of the waves multiply scattered between The longer the distance between the cylinder and the the cylinder and the plane. The frequency of the oscil-

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 232 SHENDEROV

90° 90° 3 (a)120° 60° 4 (b) 120° 60°

2 3 150° 30° 150° 30° 2 1 1

0 180° 0° 0 180° 0° 1 1 2 210° 330° 210° 330° 2 3

300° 4 3 240° 240° 300° 270° 270°

Fig. 5. Backscattering patterns for a cylinder located near the surface of an elastic halfspace (the solid lines) and in a free space (the ρ 3 ()cyl ()cyl ε()cyl ε()cyl dotted lines): kb = (a) 7.0 and (b) 3.0, m = 2000 kg/m , ka = 3.0, cl = 1800 m/s, ct = 300 m/s, l = 0.1, t = 0.2, ρ 3 ε ε 1 = 2700 kg/m , cl = 6100 m/s, ct = 3050 m/s, l = 0.001, and t = 0.002. lations increases with the distance between the cylinder dimensional problem, while a solution for the spherical and the halfspace surface. incident wave is obtained for the asymptotic case when The above results refer to a cylinder located near a the transmitter and receiver are far from the cylinder. liquid halfspace. Figure 5 shows the backscatter Let a cylinder of radius ‡ be centered at the origin of (monostatic) pattern for the elastic halfspace. The back- ϕ ϕ coordinates and r1, 1 and r0, 0 be the coordinates of scatter patterns exhibit sharp maxima due to the excita- the observation point and of a line source of a cylindri- tion of the longitudinal and transverse waves. The cal wave, respectively. The diffracted ﬁeld is given by angles at which these maxima occur satisfy the condi- the well-known expansion (see, e.g., [10]) tions sinϕ ≈ c/c and sinϕ ≈ c/c . In Fig. 5, these angles 1 αl π 1 ϕ t correspond to 1 = /2 – 1 of 57° and 123° for the ()cyl kρcQ p = ------shear waves and 76° and 104° for the longitudinal s 4 waves. ∞ (A.1) () () () × ∑ a 0 H 1 ()kr H 1 ()kr exp()in()ϕ Ð ϕ , ACKNOWLEDGMENTS n n 0 n 1 1 0 n = Ð∞ This work was supported by the Russian Foundation for Basic Research, project no. 00-02-17840. where

() J ()ka Ð iw J' ()ka 0 () ------n n n APPENDIX an ka = Ð () ()1 . (A.2) H 1 ()ka Ð iw H '()ka RELATION BETWEEN THE SOLUTIONS n n n TO THE PROBLEMS OF THE DIFFRACTION ρ Here, wn = Zn/ c are the mode impedances of the elastic OF CYLINDRICAL AND SPHERICAL WAVES cylinder normalized by ρc. Expressions for the imped- BY A CYLINDER ances Zn can be found in [7]: Paper [12] was the first to consider the diffraction of () a plane wave by an elastic cylinder. Later, similar prob- iρ c cyl Z = ------m t lems in various formulations were solved in a number n k a of works (see, e.g., [13, 14]). The diffraction of a spher- t (A.3) ical acoustic wave by an elastic cylinder was first con- F ()k a Fk()a Ð 2F ()k a F ()k a × ------1 l t 3 l 2 t , sidered in [15]. However, as it is indicated above, the nJ ()k a F ()k a Ð k J' ()k a F ()k a solution is expressed in terms of rather intricate com- n t 1 l l n l 2 t plex integrals of cylindrical functions of a complex ω ()cyl ω ()cyl argument, which are very difficult to calculate. There- where kl = /cl and kt = /ct are the wave num- fore, in this paper, we use an exact solution to the two- bers of the longitudinal and shear waves in the material

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 DIFFRACTION OF SOUND BY AN ELASTIC CYLINDER 233 ρ of the cylinder, m is the density of the material of the cylinder, and r0 y z0 F ()x = 2nxJ()' ()x Ð J ()x ; 1 n n ϕ 1 z1 r1 F ()x = x2 J''()x Ð xJ' ()x + n2 J ()x ; 2 n n n (A.4) ϕ0 () 2[]() ()()()cyl ()cyl 2 () F3 x = x Jn'' x + 1 Ð cl /ct /2 Jn x .

In expression (A.3), a misprint present in [7] is cor- S x rected. cyl ӷ ӷ z When kr0 1 and kr1 1, we use asymptotic expressions for the Hankel functions and normalize the y = –b scattered ﬁeld by the sound pressure of the incident Sh wave at the cylinder axis, i.e., by

()cyl kρcQ ()1 p = ------H ()kr 0 4 0 0 Fig. A. Coordinate system for the three-dimensional problem. ρ ≈ k cQ 2 ()π ------π - exp ikr0 Ð i /4 , 4 kr0 ӷ ӷ Since we consider the case when kr0 1 and kr1 1, to obtain the ratio of the sound pressure amplitude of the integral can be calculated by the stationary phase the scattered wave to the amplitude of the incident wave method. For simplicity, assume that the source and the at the origin receiver are located in one plane that is orthogonal to the axis, i.e., z = z . Then, we obtain ()cyl 1 0 p 2 ------s = ------exp()Φikr Ð iπ/4 ()ka , (A.5) ()cyl π 1 s ()sph ρcQ k p kr1 p = Ð------0 s 3/2 () ()2π r0r1 r1 + r0 (A.9) where ×Φ() ()() s ka exp ik r1 + r0 . n = ∞ () Φ ()ka = ()Ð1 na 0 ()ka exp()in()ϕ Ð ϕ . (A.6) We normalize this quantity by the sound pressure at the s ∑ n 0 1 axis in a free space (i.e., in the absence of the cylinder), n = Ð∞ which is given by Eq. (A.7). As a result, we arrive at the Assume that the sound is radiated by a spherical expression source and the distance between the source and the cyl- () p sph 2r inder axis is r0. Then, the sound pressure in a free space ------s = ------0 exp()Φikr ()ka . (A.10) at a point located at this distance will be ()sph πkr ()r + r 0 s p0 1 1 0 () ()sph ρ exp ikr0 The comparison of expressions (A.5) and (A.10) yields p0 = Ðik cQ------π -. (A.7) Φ 4 r0 that the function s(ka), which depends on the radius of the cylinder and its elastic characteristics and on the To derive an expression for the scattered ﬁeld in the scattering angles, is the same for the cylindrical and three-dimensional case, i.e., for a spherical wave, we use spherical sources, whereas the distance dependence of the following familiar technique [10, 16]: we replace k with the scattering amplitudes are different. In a particular 2 ξ2 2 ξ2 case when the source and the receiver are at the same k Ð everywhere; multiply by exp[i k Ð (z1 – π point, i.e., r1 = r0, we have z0)]/(2 ), where z1 and z0 are the coordinates of the observation point and the source along the cylinder axis ()sph p 1 (Fig. A); and integrate with respect to ξ between the ------s = ------exp()Φikr ()ka . (A.11) ()sph πkr 1 s inﬁnite limits. As a result, we obtain p0 1

()sph ÐiρcQ Contrasting this expression with formula (A.6), we p = ------s π2 obtain that, for the spherical source, the scattering 4 r0r1 ∞ amplitude is 2 times smaller than that for the line × []2 ξ2()ξ()(A.8) source. As was noted in [10], this fact can be explained ∫ exp ikÐ r1 + r0 + i z1 Ð z0 as follows. When the spherical wave is diffracted by a Ð∞ cylinder, the Fresnel zones, i.e., antiphase regions whose contributions partly compensate for each other, ×Φ()2 ξ2 ξ s k Ð a d . occur on the surface of the cylinder along its axis. The

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 234 SHENDEROV radius of the equivalent sphere for the diffraction of the Since the radius of the equivalent sphere increases with spherical wave by an infinite cylinder can be obtained the distance between the cylinder and the source and through the comparison of expression (A.11) with a receiver position as r1/2 , the amplitude of the scattered similar expression for the acoustically hard or acousti- 1 Ð1/2 cally soft sphere of radius aequiv that is large in terms of field decreases by the cylindrical law r1 rather than the wavelength. In this case, the following well-known Ð1 expression is valid: by the spherical law r1 . ()sph a ps = ------equiv. (A.12) ------() REFERENCES p sph 2r1 0 1. G. C. Bishop and J. Smith, J. Acoust. Soc. Am. 101, 767 Equating expressions (A.11) and (A.12), we obtain (1997). the radius of the equivalent sphere for the spherical 2. G. C. Bishop and J. Smith, J. Acoust. Soc. Am. 105, 130 wave incident on the cylinder: (1999). 3. J. C. Bertrand and J. W. Young, J. Acoust. Soc. Am. 60, r1 1265 (1976). a = 2 ------Φ ()ka . (A.13) equiv πk s 4. G. C. Gaunaurd and H. Huang, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 43, 690 (1996). To obtain the radius of the equivalent sphere for the ϕ ϕ 5. G. C. Gaunaurd and H. Huang, J. Acoust. Soc. Am. 96, backscattered field, we should set 1 = 0 in the expres- 2526 (1994). sions for Φ . Note that, in [7], the positive angular s 6. V. Tversky, J. Appl. Phys. 23 (3), 407 (1952). direction is opposite to that accepted in this paper; 7. E. L. Shenderov, Wave Problems in Hydroacoustics therefore, in the tables given there in the appendix, the (Sudostroenie, Leningrad, 1972). values associated with the backscatter are denoted as Φ (π). 8. E. A. Ivanov, Electromagnetic Wave Diffraction by Two s Bodies (Nauka i Tekhnika, Minsk, 1968). The radius of the equivalent sphere for the incidence 9. L. M. Brekhovskikh, Waves in Layered Media, 1st ed. of a cylindrical or plane wave on a cylinder can be (Nauka, Moscow, 1957; Academic, New York, 1960). found by equating expressions (A.5) and (A.12), which 10. E. L. Shenderov, Radiation and Scattering of Sound yields (Sudostroenie, Leningrad, 1989). 2r 11. E. L. Shenderov, Akust. Zh. 46, 816 (2000) [Acoust. 1 Φ () Phys. 46, 716 (2000)]. aequiv = 2 ------π s ka . (A.14) k 12. J. J. Faran, J. Acoust. Soc. Am. 23, 405 (1951). Thus, the radius of the equivalent sphere for a plane or 13. L. M. Lyamshev, Akust. Zh. 2, 358 (1956) [Sov. Phys. cylindrical wave incident on an infinite cylinder is Acoust. 2, 382 (1956)]. found to be 2 times greater than that for a spherical 14. W. H. Lin and A. C. Raptis, J. Acoust. Soc. Am. 73, 736 incident wave. (1983). 15. Tao-bao Li and Mitsuhiro Ueda, J. Acoust. Soc. Am. 87, As follows from Eqs. (A.13) and (A.14), the radius 1871 (1990). of the equivalent sphere for an infinite cylinder depends 16. L. B. Felsen and N. Marcuvitz, Radiation and Scattering on the distance between the source and the cylinder. of Waves (Prentice-Hall, Englewood Cliffs, NJ, 1973; This property is also associated with the Fresnel zones Mir, Moscow, 1978), Vols. 1, 2. that occur along the cylinder axis (see, e.g., [10]). The size of the first zone, which makes the greatest contribution to the scattered field, increases with the distance. Translated by A. Khzmalyan

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002

SHORT COMMUNICATIONS

Resonance Absorber for Flexural Waves in Beams and Plates A. D. Lapin Andreev Acoustics Institute, Russian Academy of Sciences, ul. Shvernika 4, Moscow, 117036 Russia e-mail: [email protected] Received March 15, 2001

In practice, a vibration insulation for flexural waves effect of the field w(0)(x, t), the resonator is excited and in beams and plates is obtained with the use of resona- generates a field w(1)(x, t). The total field w in the beam tors [1Ð7]. The simplest resonator is a springÐmass sys- with the resonator is equal to the sum w(0) + w(1). Let us tem [8Ð10]. When such a resonator is positioned nor- show that, at the resonance frequency, at a certain value mally to a plate and attached to it by a spring, it pro- of the dissipation factor ε, the travelling flexural wave vides effective scattering of flexural waves propagating exp[i(kx – ωt)] propagating in the total field w(x, t) in in the plate. The vibration insulation occurs because of the region x > H is absent. the scattering (reflection) of flexural waves from the resonators [3, 4, 6]. A dissipative loss in a resonator Denote the displacement of the resonator mass by reduces the efficiency of its operation as a scatterer for w'(t). The equation of motion of this mass has the form flexural waves. Earlier [11], it was shown that, at a cer- d2w' tain value of the dissipation factor, the power absorbed m------= ÐFt(), (2) by a resonator reaches its maximum and becomes equal dt2 to the power scattered by this resonator. The latter property of resonators with dissipation can be used for where the force F is determined by the formula designing efficient absorbers of flexural waves propagating in beams and plates. Wide-band absorbers consisting Ft()= κ()1 Ð iε []w'()t Ð wHt(), . (3) of several resonant elements were studied in [12]. The equation of motion of the beam connected with Consider a thin semi-infinite cantilever beam coin- the resonator can be written as cident with the semiaxis x > 0 so that its fixed end is at x = 0. A harmonic flexural wave propagating along the 2 4 beam is characterized by the displacement ρd w d w ()δ() ------2 + G------4 = Ft xHÐ , (4) () dt dx w i ()xt, = exp[]Ðikx()+ ωt , ρ ω where and G are the linear density and flexural rigid- where k and are the wave number and the circular fre- ity of the beam, respectively, and δ(x) is the delta func- quency, respectively. The reflection from the fixed end tion. The wave number of the flexural wave is equal to gives rise to two waves one of which is homogeneous 1 --- and the other inhomogeneous. The reflected field has 4 ρ 2 the form ----ω . Since w(i) and w(r) are free waves, we can G () ω w r ()xt, = {}ieikx Ð ()1 + i eÐkx eÐi t. replace w by w(1) on the left-hand side of Eq. (4). The displacement w(1) satisfies the boundary condition The total field in the beam is expressed as ()1 ()0 ()i ()r ()1 dw w ()xt, = w ()xt, + w ()xt, w ==------0 at x =0. (5) ω (1) dx = ()1+ i {}cos()kx Ð sin()kx Ð eÐkx eÐi t. The scattered field in the beam can be derived as fol- In this standing field, the antinodes occur at the points lows. We calculate the displacement produced in a ()π4n Ð 1 semi-infinite beam by a harmonic point force F(t) = x ≈ ------, ω n 4k F0exp(–i t), where F0 is a complex amplitude. In an infinite beam, this point force causes the displacement where n represents any arbitrary positive integer. At (0) some of the antinodes of the field w (e.g., at x = x ≡ ()2 iF0 q w ()xt, = ------{exp[]ik xÐ H H), we attach a resonator to the beam and assume that 4Gk3 the resonator has a mass m and an elastic coefficient κ(1 – iε), where ε is the dissipation factor. Under the + ikxHexp[]Ð Ð }exp()Ðiωt .

1063-7710/02/4802-0235 $22.00 © 2002 MAIK “Nauka/Interperiodica” 236 LAPIN Taking into account the waves reflected from the fixed At the resonance frequency, the force amplitude is end x = 0, we obtain the following expressions for w(1): εω Ð1 ω ()0 () F0 = i w0 H ReY + ------2 << ()1 (), iF0 { []() κ()1 + ε 0 xH, w xt = ------3 exp ik HÐ x 4Gk ε 3 Ð1 ≈ 3 ÐikH 2 Gk ikH ÐkH ikx (6) i4Gk e 1 + ------. + ikHxexp[]Ð ()Ð + []ie Ð ()1 + i e e κ()1 + ε2 ÐkH ikH Ðkx + []e Ð ()1 + i e e }exp()Ðiωt , Let us separate the wave travelling along the x axis from the total field w = w(0) + w(1). According to Eqs. (1) () xH> , w 1 ()xt, and (7), the amplitude of this wave is expressed as

iF ÐikH ikH ÐkH ikx iF = ------0 {[]e + ie Ð ()1 + i e e (7) Ai= + ------0 []eÐikH + ieikH Ð ()1 + i eÐikH . 4Gk3 4Gk3 kH ÐkH ikH Ðkx ω ω + []ie + e Ð ()1 + i e e }exp()Ðiωt . At the frequency = 0, we approximately obtain

3 Ð1 We select the force amplitude F0 so as to satisfy 2εGk  Eq. (3). According to Eq. (2), the displacement of the Ai= 121Ð + ------2 . mass will have the form κ()1 + ε F When the dissipation factor ε is approximately equal to () 0 ()ω κ 3 w' t = ------2 exp Ði t . (8) /(2Gk ), the amplitude A becomes zero. This means mω that the resonator completely absorbs the incident wave. Substituting Eqs. (1), (6), and (8) in Eq. (3), we obtain the desired force amplitude A similar consideration is possible for a plate. Let a semibounded plate lie in the upper xy half-plane and be  εω rigidly clamped along the boundary y = 0. A harmonic ω ()0 () flexural wave incident on this boundary is characterized F0 = i w0 H  ReY + ------ κ()1 + ε2 by the displacement (9) ()i 0 0 Ð1 w ()xyt,, = exp[]ik()xkÐ y Ð ωt , 1 ω  x y + i ImY + ------Ð ------ , mω κ()ε2  0 0 1 + where kx and Ðky are the projections of the wave vector of the incident wave on the x and y axes, respec- where tively. The total field w(0) in the plate is equal to the sum () () of the incident and reflected waves: w 0 ()H = w 0 ()Ht, exp()iωt , 0 ()0 (),, { ()0 ϕ0 () w xyt = Ði2 sin ky y Ð Ðiωw 1 ()Ht, (11) Y = ------ϕ0 ()α0 } []()0 ϕ0 ω Ft() + sin exp Ð y expikx x Ð Ð t , is the compliance of the semi-infinite beam with respect where to the point force. Neglecting small quantities of the 0 2 k π α0 ==k2 Ð()k0 and sinϕ0 ------y -. order of exp(–kH) ≡ exp- –(4q – 1) --- , we obtain the x 4 2k approximate expressions ≈ In this field, the antinodes occur on the lines yn 0 () ω π()2n Ð 1 + 2ϕ 0 () () () ------, where n is any integer. We attach w0 H ==2exp ÐikH , Y ------3 2 + i . 0 4Gk 2ky identical resonators with masses m and elastic coeffi- (1) The scattered field w is calculated by substituting cients κ(1 – iε) to the plate along one of these lines F in Eqs. (6) and (7). ≡ ≡ 0 (e.g., y = yq H) at the points x = xs sL, where s = 0, ω ± ± (0) The resonance scattering occurs at the frequency 0 1, 2, …. The resonators are excited by the field w determined from the equation and generate the field w(1). The total field in the plate with the resonators is w = w(0) + w(1). 1 ω ImY +0.------Ð ------= (10) Let us denote the displacement of the mass of the ω 2 m κ()1 + ε sth resonator [which is attached to the plate at the

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 RESONANCE ABSORBER FOR FLEXURAL WAVES IN BEAMS AND PLATES 237 point (x , H)] by w' (t). The equation of motion of this 1 2 n n n s s + i ----- Ð ---- sinϕ exp()ik HiÐ ϕ Ð α H mass has the form αn n y ky 2 n n  d ws' × []()αω () ------() --- exp ikx x Ð t Ð yHÐ , m 2 = ÐFs t , (12)  dt where the force F is determined by the formula kn s where sinϕn = ------y -. In Eq. (14), the first term in the () κ()ε []() (),, Fs t = 1 Ð i ws' t Ð wxs Ht . (13) 2k n The equation describing the motion of the plate con- braces represents a homogeneous plane wave for kx < nected with the resonators can be written in the form n k and an inhomogeneous plane wave for kx > k, and ∞ 2 s = the second term in the braces always represents an ρdw ∆2 ()δ()δ() + G w = ∑ Fs t xxÐ s yHÐ , inhomogeneous wave. dt2 s = Ð∞ We select the force amplitude F so as to satisfy where ρ and G are the surface density and the flexural Eq. (13). The displacement of the plate at the point of rigidity of the plate, respectively, and ∆ is the Lapla- the resonator attachment is understood as the displace- cian. On the left-hand side of this equation, the quantity ment averaged over the contact area between the plate w can be replaced by w(1). and the resonator. According to Eq. (12), the displace- () The structure of the scattered field is determined by Fs t (0) (1) ment ws' (t) is equal to ------. Substituting w , w , and the period of the scattering array (chain) of resonators, mω2 and the quantity w(1)(x, y, t)exp(–ixk0 ) is a periodic x ws' in Eq. (13), we derive the desired force amplitude function of x with a period L. Then, in the presence of Ð1 the exciting field given by Eq. (11), the force Fs(t) can ω ω ()0 (), 1 be represented in the form Fi= w0 0 H Yi+ ------Ð ------, ωm κ()1 Ð iε () []()0 ω Fs t = Fkexp x xs Ð t , where where F is the force amplitude at s = 0. The scattered () field in the plate is obtained in the same way as the scat- w 0 ()0, H = w0()0,,Htexp()iωt ≈ 2exp()Ðik0 H , tered field in a beam. In an unbounded plate, the chain 0 y of point forces Fs(t) generates the field ω ()1 (),, Ði w xs Ht n = ∞  Y = ------() ()2 iF 1 n Fs t (),, ---------- () n = ∞ n w xyt = ∑ 2 n exp iky HyÐ ωε  4LGk k n n n ky n = Ð∞ y ≈ ∑ ------1 Ð exp[]i2()k H Ð ϕ + i----- , 2 n y αn ∞ 4LGk ky   n = Ð i ()αn []()n ω + ----- exp Ð HyÐ exp ikx x Ð t , n n αn  sin()k a exp()ik a Ð 1 ε = ------x ------y . n ()n ()n where kxa ikya

n 0 2π Averaging is performed over a square area whose side 2a k = k + ------n, ε ≈ x x L is small compared to the flexural wavelength; n 1 when kn a Ӷ 1. n 2 ()n 2 αn 2 ()n 2 x ky ==k Ðkx , and k +.kx The scattered field is obtained from Eq. (14) by sub- Taking into account the waves reflected from the stituting F into it. The resonance scattering occurs at clamped boundary y = 0, we obtain the following the frequencies determined from the equation expression for w(1) at y ≥ H: ()1 1 ω w ()xyt,, ImY +0.------Ð ------= mω 2 n = ∞ κ()1 + ε iF  1 n n = ---------- []1 Ð exp()i2k HiÐ 2ϕ ∑ 2 n y This dispersion equation is identical in form to Eq. (10). 4LGk k n = Ð∞ y However, because of the complex dependence of Y on frequency, it may have several solutions. At the reso- × []()n ω n() exp ikx x Ð t + iky yHÐ (14) nance frequency, the amplitude of the nth scattered

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 238 LAPIN homogeneous spectrum (i.e., scattered homogeneous When the dissipation factor ε is approximately equal to plane wave) has the form κ Ð1 ------2 -0 , the amplitude A becomes zero. This means ω εω 2LGk ky An = Ð------Y1 + ------2LGk2knκ()1 + ε2 that the chain of resonators completely absorbs the inci- y (15) dent wave. × []()0 n {}[]()n ϕn exp Ðiky + ky H 1 Ð exp i2 ky H Ð , where REFERENCES ω ' {}[]()s ϕs 1. I. I. Klyukin, Akust. Zh. 6, 213 (1960) [Sov. Phys. Y1 = ReY = ∑------2 s 12Ð cos ky H Ð . Acoust. 6, 209 (1960)]. 4LGk k (16) s y 2. I. I. Klyukin and Yu. D. Sergeev, Akust. Zh. 10, 60 Here, the prime denotes the summation over all s at (1964) [Sov. Phys. Acoust. 10, 49 (1964)]. s which ky is real. 3. M. A. Isakovich, V. I. Kashina, and V. V. Tyutekin, in λ Marine Instrument Making, Ser. Acoust. (Inst. of Acous- When the period of the chain is smaller than (1 + tics, USSR Acad. Sci., Moscow, 1972), Issue 1, pp. 117Ð k0 125. sinθ)–1, where θ = arcsin----x is the angle of the wave k 4. M. A. Isakovich, V. I. Kashina, and V. V. Tyutekin, USSR Inventor’s Certificate No. 440509, Byull. Izobret., 2π incidence and λ = ------, only the “zeroth” spectrum is No. 31 (1974). k 5. M. A. Isakovich, V. I. Kashina, and V. V. Tyutekin, Akust. homogeneous in the scattered field given by Eq. (14). Zh. 23, 384 (1977) [Sov. Phys. Acoust. 23, 214 (1977)]. Then, from Eqs. (15) and (16), we derive the expres- 6. L. S. Tsil’ker, Akust. Zh. 26, 606 (1980) [Sov. Phys. sions Acoust. 26, 336 (1980)]. ω 7. R. J. Nagem, I. Velikovik, and G. Sandri, J. Sound Vibr. Y1 = ------2 -0, 2LGk ky 207, 429 (1997). Ð1 8. Acoustics Handbook, Ed. by Malcolm J. Crocker (Wiley, ε 2 0 ()ϕ0 2 LGk ky New York, 1997). A0 = 2exp Ði2 1 + ------. κ()1 + ε2 9. M. Gurgoze and H. Batan, J. Sound Vibr. 195, 163 (1996). Combining the homogeneous reflected wave with the zeroth scattered spectrum, we obtain a travelling homo- 10. M. Gurgoze, J. Sound Vibr. 223, 667 (1999). geneous wave with the amplitude A equal to 11. A. D. Lapin, Akust. Zh. 33, 278 (1987) [Sov. Phys. Acoust. 33, 164 (1987)]. []()ϕ0 Ðexp Ði2 + A0 12. V. V. Tyutekin and A. P. Shkvarnikov, Akust. Zh. 18, 441 Ð1 (1972) [Sov. Phys. Acoust. 18, 369 (1972)]. 2εLGk2k0 ()ϕ0 y = Ðexp Ði2 121Ð + ------2 . κ()1 + ε Translated by E. Golyamina

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002

Acoustical Physics, Vol. 48, No. 2, 2002, pp. 239Ð242. Translated from Akusticheskiœ Zhurnal, Vol. 48, No. 2, 2002, pp. 281Ð284. Original Russian Text Copyright © 2002 by Ostrovskiœ. SHORT COMMUNICATIONS

Statistical Properties of the Sidelobe Level of an Acoustic Parametric Antenna D. B. Ostrovskiœ Morﬁzpribor Central Research Institute, Chkalovskiœ pr. 46, St. Petersburg, 197376 Russia e-mail: [email protected] Received May 14, 2001

An advantage of a parametric transmitting antenna where (PTA) is that it has lower sidelobes than traditional ()θβ, (), β antennas [1], which provides a low level of interfer- 1/AR= f 1 R f 2 q . (4) ences created by the surface and volume reverberation and also enhances hiding of the transmission. The For simplicity sake, we denote |R (θ, β)| = R , i = 1, 2. θ β fi i angular pressure dependence P( , ) at the difference The pumping signal can be applied to the antenna in frequency can be written as a two-channel or one-channel excitation mode [1]. In the ﬁrst mode, the source array is divided into two ()θβ, ()θβ, ()Ψθβ, P = P f 1 P f 2 , (1) equal, though statistically independent, overlapping subarrays. In the one-channel excitation mode, the beat θ β θ β where ( , ) are the angular coordinates; Pf 1, 2( , ) is signal between the pumping frequencies is applied to the pressure produced by the source at the partial pump- each element of the source. Ψ ing frequencies f1 and f2, respectively; and is a func- First, consider the two-channel excitation mode. tion independent of angles (θ, β), which is different for Since the subarrays that constitute the PTA pumping different PTA models [1Ð5]. source are statistically independent, the mean value and Since the directional pattern determines only the variance of the pattern at the difference frequency are angular antenna characteristics, the PTA factor [6], or expressed in terms of the mean values and variances of the nonnormalized directional pattern, can be written as the partial directional patterns: 2 2 ()θβ, ()θβ, ()θβ, RR==1 R2, DR DR1 DR2 ++R1 DR2 R2 DR1. R = R f 1 R f 2 , (2) As shown in [8], the functions R1 and R2 are θ β described by the Rician distribution: where Rf 1, 2( , ) are the partial patterns at the frequencies f1 and f2. 2 2 R R + Q R Q As for real sources, the parameters of their elements fR()= -----i expÐ------i i I ------i -i , (5) θ β θ β i σ2 σ2 0σ2 are spread; therefore, Rf 1( , ), Rf 2( , ), and, hence, i 2 i i R(θ, β) are random variables. The statistical character- θ β istics of R( , ) are determined by the statistical char- where Q and σ (i = 1, 2) are the distribution parame- θ β θ β i i acteristics of Rf 1( , ) and Rf 2( , ). The only known ters. work on the statistical characteristics of PTA [7] studies For a traditional antenna representing the source, only the pressure at the difference frequency at the i.e., when Q/σ > 3Ð4, the Rician distribution (5) goes maximum of the directional pattern versus the spread in over into the Gaussian distribution; when Q/σ < 0.5, it the partial pumping frequencies. Therefore, it is of changes to the Rayleigh distribution [9] interest to study the statistic of PTA’s sidelobes. Consider the statistical characteristics of a random R R2 fR()= ----- exp Ð------, (6) PTA’s pattern in the general case, as this is customary σ2 σ2 in the statistical theory of antennas [6, 8]; i.e., with 2 allowance for errors in the excitation amplitude and whose mean value and variance are phase, for errors in element settings, and for element failures. We will normalize function (2) by its mean [] σπ [] σ2()π value; then the magnitude of the normalized pattern is MR ==/2, DR 2 Ð /2 . (7) Since our primary interest is in low sidelobe levels, ()θβ, ()θβ, ()θβ, R = ARf 1 R f 2 , (3) we assume that the magnitude of the directional pattern

1063-7710/02/4802-0239 $22.00 © 2002 MAIK “Nauka/Interperiodica” 240 OSTROVSKIŒ is characterized by distribution (6). In this case, the par- tion [6]); M is the number of elements (for the two- tial patterns are independent and, therefore, the magni- channel mode, M = N /2); N is the nominal number of 0 0 π tude of the normalized pattern at the difference fre- elements in the source; and ki = 2 fi /c. quency is characterized by the two-dimensional Ray- As follows from Eq. (11), the quantities dependent leigh distribution on the frequency fi are the element pattern Rel and the ≈ 2 2 wave number k. Since the condition fi/F 10 (F is the xy x + y ≈ fxy(), = ------exp Ð------, (8) difference frequency) is usually met and, therefore, f1 σ2σ2 2σ σ ≈ σ σ σ 1 2 1 2 f2 f0 = ( f1 + f2)/2, we can assume that 1 = 2 = ( f0). Replacing σ in Eq. (8) with σ, we obtain the distribu- σ 1, 2 where the parameters 1, 2 of the distribution corre- tion density of PTA’s pattern in the form of the product spond to the random variables X and Y, i.e., to R1 and R2. of independent random variables [11]: Let us express σ in terms of the statistical characteristics of ﬂuctuations of the antenna elements. The array fR()= RK ()R/σ2 /σ4, (12) factor R(θ, β) is 0

N0 where K0(z) is the Macdonald function, whose mean ()θβ, ∆ []() value and variance are R = ∑ Am0 m exp i kkÐ 0 rm0 , (9) m = 1 MR[]== πσ2/2, DR[] σ4()4 Ð π2/4 . (13) where Am0 is the nominal excitation coefficient of the mth element, N0 is the nominal number of elements, k Figure 1a shows the histogram and distribution (12), is the wave vector, and rm0 is the nominal position vec- which approximates it. We performed simulations for tor of the array element. an antenna with the average pumping frequency f0 = Summarizing the data presented in [6, 8, 10], we 50 kHz, difference frequency F = 5 kHz, and total num- ∆ ber of elements N × . The levels of partial direc- represent the scatter coefficient m for the parameters of 0 = 18 18 the mth element as tional patterns were 0.0266 and 0.0264. The numerical characteristics of the fluctuations obtained from n = ∆ ()δ ()ϕ []()δ σ m = am 1 + m exp i m exp i kkÐ 0 rm , (10) 1000 realizations were as follows: p = 0.9, ∆ = 0.7, σϕ = 0.3, and σ = 2 mm. The solid curve refers to the σ d where am is the failure parameter, which equals unity parameter obtained from formula (13); the dashed with a probability p when the element is operable and curve, from formula (11). The two curves virtually zero with probability 1–p when it fails; δ is the relative coincide and approximate with statistical confidence δ m amplitude error; and rm is the fluctuation in the ele- the directional pattern distribution of the parametric ment position. radiator excited in the two-channel mode. Let us assume for definiteness sake that the fluctua- Consider the one-channel excitation mode. As fol- tions possess the following properties: lows from the expressions that describe the models of (i) elements and fluctuations of different types are the parametric radiator, in the one-channel excitation statistically independent; mode, the pressures Pf 1 and Pf 2 are created by the (ii) fluctuations are uniformly distributed over the whole antenna aperture. Accordingly, the partial direc- antenna aperture; tional patterns R1 and R2 are formed by all elements of (iii) elements are statistically indistinguishable; the source. The PTA pattern is calculated by formula (3), but, in this case, the partial patterns R1 and R2 are not (iv) phase errors and adjustment errors have the independent. Gaussian distribution with zero mean values; standard deviations of the amplitude, phase, and adjustment Each of the partial patterns is a random (in general, errors are σ∆, σϕ, and σ , respectively. nonstationary) process in frequency. At the same time, d ӷ As follows from [8], the parameter σ of distribution at f/F 1, in the theory of the parametric radiator [4, 5], (6) subject to these assumptions has the form the product of pressures at the frequencies f1 and f2 in formula (1) is replaced by the squared pressure at the 2 2 2 2 2 1 + σ∆ Ð pexp()Ðσϕ Ð k σ frequency f : P P ≅ P . Then, PTA pattern (3) goes σ2 = ------i d - 0 f 1 f 2 f 0 i ()σ2 2σ2 over into 2 pexp Ð ϕ Ð k d (11) 2 2 M M 2 ()θβ, ()θβ, ()θβ,  R ==ARf 0 ,1/ARf 0 , × R2 ()θβ,,f ∑ A2 /∑ A , el i m0 m0 i.e., becomes equal to a squared directional pattern of m = 1 m = 1 the source at the frequency f0. where Rel is the directional pattern of one element (in The original directional pattern of the source has the case of a planar antenna and complete compensa- Rician distribution (5). The square of this random vari-

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 STATISTICAL PROPERTIES OF THE SIDELOBE LEVEL 241

Table Mean value Variance Mode ζ ζ formula M formula D

Linearσπ/2 1 σ2(2 Ð π/2) 1

Two-channel πσ2 σ 2π 4σ4(4 Ð π2/4) 2σ2(4 + π)

One-channel 2σ2 2σ 2π 4σ4 8σ2/(4 Ð π) X(2)/X(1) Ð π/2 > 1 Ð 4 Ð π2/4 > 1 able, i.e., the directional pattern of the parametric tom row of the table compares the two excitation modes antenna, will have the distribution of the parametric radiator, X(2) and X(1). σ 2 The parameter is on the order of 1/N0; therefore, () 1 RQ+ RQ the sidelobe level of the PIA directional pattern is more fR = ------expÐ------I0------, (14) 2σ2 2σ2 σ2 stable to random ﬂuctuations both in mean value and in variance, i.e., it has a lower average level and a smaller | | where the parameter Q = R0( f0) is the nominal pattern of the source and σ is determined by formula (11) at n 300 M = N0. σ If Q/ > 3, the directional pattern of the PTA is dis- (a) tributed as a squared Gaussian variable 250

2 200 1 ()RQÐ fR()= ------exp Ð.------σ π σ2 2 2 R 2 150 When Q/σ < 0.5, the directional pattern of the PTA operating in the one-channel mode has the distribution 100

() 1 R 50 fR = ------expÐ------. 2σ2 2σ2 0 The results of simulations shown in Fig. 1b refer to 0123456789 the same source antenna as in the two-channel mode. In R, 10–3 this case, we consider the direction, in which Q/σ ≈ 1; 200 therefore, the approximating curve is calculated by for- 180 mula (14). 160 (b) Let us compare the parametric radiator operating in 140 different excitation modes and the traditional (linear acoustic) antenna in terms of their stability to ﬂuctua- 120 tions of the statistical characteristics at the same num- 100 ber of elements N0 and at the same ﬂuctuation parame- 80 σ σ σ ters ∆, ϕ, d, and p. We consider small directional pat- 60 tern levels satisfying the condition Q/σ < 0.5. The table summarizes the formulas for the mean value and vari- 40 ance and for the ratios of the PTA’s statistical character- 20 istics to similar characteristics of the linear antenna. In ζ 0 the table, M is the ratio of the mean value of the PTA 123456789 pattern in the two-channel (one-channel) excitation R, 10–2 mode to that of the linear antenna, and ζ is the variance σ D ratio. As the parameter , we use the parameter of the Fig. 1. Distribution histograms and the approximating func- linear antenna, which is taken into account in the tions: (a) the two-channel mode and (b) the one-channel respective formulas for the two-channel mode. The bot- mode.

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002 242 OSTROVSKIŒ spread. The one-channel excitation mode is more stable 6. M. D. Smaryshev and Yu. Yu. Dobrovol’skiœ, Hydroa- to element parameter ﬂuctuations than the two-channel coustic Antennas (Sudostroenie, Leningrad, 1984). mode. 7. V. N. Zuev and N. I. Kalyagin, Prikl. Akust. (Taganrog), No. 9, 47 (1983). 8. V. B. Zhukov and D. B. Ostrovskiœ, Parametric Reliabil- REFERENCES ity of Hydroacoustic Antennas (Sudostroenie, Lenin- 1. B. K. Novikov and V. I. Timoshenko, Parametric Anten- grad, 1980). nas in Underwater Detection and Ranging (Sudostroe- 9. P. Beckmann, J. Res. Natl. Bur. Stand., Sect. D 66 (3), nie, Leningrad, 1989). 231 (1962). 2. J. N. Tjøtta, S. Tjøtta, and E. H. Vefring, J. Acoust. Soc. 10. V. M. Khrustalev, Tr. Taganrog. Radiotekh. Inst., Issue 22, Am. 88, 2859 (1990). 107 (1971). 3. V. B. Zheleznyœ, Prikl. Akust. (Taganrog), No. 11, 23 11. D. B. Ostrovskiœ, Sudostroit. Prom., Ser. Obshchetekh., (1985). No. 37, 49 (1992). 4. M. B. Moffett and R. H. Mellen, J. Acoust. Soc. Am. 61, 325 (1977). 5. R. L. Rolleigh, J. Acoust. Soc. Am. 58, 964 (1975). Translated by A. Khzmalyan

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002

CHRONICLE

Yuriœ Mikhaœlovich Sukharevskiœ (On His 95th Birthday)

ment for reception and high-fidelity reproduction of speech and music produced in broadcasting studios and concert halls. The change in the scientific interests of Sukharevskiœ was explained by his natural desire to combine his two professions—engineering and music. In 1931Ð1935, he published a number of papers on electroacoustics and architectural acoustics and, in 1936, he published a monograph entitled Modern Elec- troacoustics and Wire Broadcasting. Another book written in collaboration with A.V. Rabinovich entitled Broadcasting Studios and Microphones appeared in 1938. Sukharevskiœ was also involved in the problems of electroacoustic metrology. In 1934, at the Central Research Institute of the People’s Commissariat of Communication, he developed and designed Russia’s first test bench for the absolute calibration of microphones and for testing loudspeaker characteristics, including nonlinearity. In 1936, he completed Russia’s first acoustic test site for powerful sound sources, where he performed extensive studies of the directional characteristics of acoustic horns. The results of these studies were published in 1938 in Elektrosvyaz’. At that time, Sukharevskiœ was working on the development of acoustical systems for the All-Union agricultural exhi- bition in Izmailovo Park, Moscow. There, he supervised the installation of the first outdoor anechoic distributed system of loudspeakers, which simulated the effect of large concert hall reverberation. Professor Yuriœ Mikhaœlovich Sukharevskiœ—a doc- In 1938, Sukharevskiœ became a senior researcher at tor of engineering, a laureate of the USSR State Award, the Physical Institute of the Academy of Sciences of the and a prominent scientist whose activity has been USSR. In 1939Ð1940, he performed theoretical and closely connected with the development of Russian experimental studies of acoustic feedback that hydroacoustics—turned ninety-five. restricted the possibilities of sound amplification in Sukharevskiœ was born in Moscow on September 8, sound-amplifying systems. The results of these studies 1906. In 1930, he graduated from the Moscow Power were published in Doklady Akademii Nauk SSSR. Institute, where he specialized in electrical engineering. In 1939, Sukharevskiœ received his candidate Simultaneously, he studied music at the Piano Faculty degree, and, in 1940, he became a doctor of engineer- of the Moscow Conservatory. After receiving his mas- ing. His doctoral dissertation was entitled Methods for ter’s degree in music in 1931, he became a post-gradu- Calculating Sound-Amplification Systems. One of his ate student and continued his studies at the conserva- official reviewers was N.N. Andreev. tory until 1935. Upon the German invasion of the USSR in World Sukharevskiœ’s first research project dates back to War II, Sukharevskiœ began working on military acous- 1929, when he was still a student. His first publication tical problems. The research and development projects appeared in 1930. At that time, he was already working carried out by Sukharevskiœ and his colleagues during at the Moscow Electrical Plant. Later that same year, he the war were described in his paper in Acoustical Phys- became a researcher at the Acoustic Laboratory of the ics in 1996. Specifically, Sukharevskiœ described his Central Research Institute of the People’s Commissar- work in collaboration with D.I. Blokhintsev on the iat of Communication. There, his attention was directed improvement of sound-detecting horns used in anti-air- toward the problems of using electroacoustic equip- craft artillery in 1942 and the full-scale experiments

244 YURIŒ MIKHAŒLOVICH SUKHAREVSKIŒ (ON HIS 95th BIRTHDAY) performed in collaboration with V.S. Grigor’ev in the long-range underwater detection and ranging in the Pacific Ocean in 1944. These experiments dealt with audio frequency range. He proposed the 3/2 power law the characteristics of sonars used on Russian and for- to describe the frequency dependence of sound absorp- eign naval vessels and submarines. Simultaneously, tion in the sea. Sukharevskiœ studied sound reflection from the hulls of In 1954, the Acoustical Laboratory of the Physical ships and the acoustic parameters of the ocean that Institute of the Academy of Sciences of the USSR was determine the operating range of sonars. reorganized into the Acoustics Institute of the Academy For his contribution to the defense potential of the of Sciences of the USSR. Simultaneously, the Sukhumi USSR, Sukharevskiœ was awarded an Order of the Red expedition received the title of the Sukhumi Marine Banner of Labor. Research Station of the Acoustics Institute. At that The experience gained by Sukharevskiœ during his time, the capital construction of the marine research work in the Pacific determined his continued career in station was in progress and new equipment was hydroacoustics. The program proposed by Sukha- installed. The program of studying the far zones of revskiœ for extensive studies in this area was the impetus insonification was extended, and other hydroacoustic for organizing an experimental hydroacoustic base for investigations were carried out. They included the the Physical Institute of the Academy of Sciences of the sound reflection from wakes of ships, the sound fluctu- USSR, namely, a marine research station with a coastal ations caused by the inhomogeneity and dynamics of laboratory, stationary transmittingÐreceiving antennas, the marine environment (in relation to underwater com- and research ships. The idea of organizing such a sta- munication and target indication), and the static charac- tion was approved by S.I. Vavilov, director of the Phys- teristics of sea reverberation. ical Institute. During the next fifteen years, Sukha- In 1959, Sukharevskiœ proposed a method to revskiœ worked on the realization of this idea. increase the operating range of existing hydroacoustic In 1945Ð1948, using the ships of the Black Sea equipment by more than an order of magnitude using Fleet, Sukharevskiœ continued his studies (including the effect of far zones of insonification. He calculated those of sea reverberation) that were started during his the parameters of the corresponding shipborne hydroa- work in the Pacific. The results of these studies were coustic systems that allowed such long-range opera- published in Doklady Akademii Nauk SSSR (1948). tion. He initiated and supervised the development of the Sukharevskiœ was authorized to select the optimal loca- first Russian long-range hydroacoustic system for mass tion for the marine station. The main criterion was that production. For this work, Sukharevskiœ received a the station should be near a deep-water region. The USSR State Award. Caucasian coastal region of the Black Sea satisfied this Sukharevskiœ combined his scientific work with the requirement. In addition, the warm climate allowed full- education of two research groups in Moscow and scale investigations year-round in any weather condition. Sukhumi. His friend and colleague, theoretical physi- The region could also be considered as a 1 : 2Ð1 : 3 scale cist G.D. Malyuzhinets, assisted him in his work. model of the northwestern Pacific. The final choice was Together, they contributed largely to the development Cape Sukhumi, with a bottom slope of 35°. This place of the theoretical studies of sound scattering from thin- was most convenient because of the nearby port facili- walled elastic shells, hydrodynamic cavitation, and ties for research ships. sound scattering from the sea surface. In 1948, Sukharevskiœ organized an expedition to In 1959, Sukharevskiœ supervised the SovietÐChina the Black Sea. With a small group of researchers and marine expedition on hydroacoustics. the support of the Black Sea Fleet, he established a tem- In 1961Ð1966, Sukharevskiœ held the office of Dep- porary marine station at Cape Sukhumi equipped with uty Director of the Acoustics Institute. While perform- hydroacoustic antennas, which he designed, and mock- ing his administrative duties, he continued his collabo- ups of the electronic and recording systems installed at ration with industrial institutes, design offices, and the Sukhumi lighthouse. Using this equipment, he con- naval institutions in developing new hydroacoustic ducted investigations of sound reflection from ships equipment. Using his experience in studying different and submarines. The newly developed stationary equip- aspects of hydroacoustic problems, Sukharevskiœ ment was also used for studying the acoustic parame- undertook the complex investigation of the triad repre- ters of the marine environment: the sound absorption in sented by a hydroacoustic system, the environment, and sea water and the sound scattering from the sea bottom a ship with the aim to optimize the operating frequency. and surface. In contrast to the conventional deterministic represen- In 1953, studies of the sound propagation through tation of marine environment parameters, he consid- the sea began and, specifically, the studies of sound ered the statistics of the main parameters for a global fields in the regions of geometric shadow. In 1954, set of various acoustic conditions in the ocean and stud- these studies resulted in the discovery of the far zones ied the operating range of a hydroacoustic system as a of underwater insonification, or the so-called conver- probabilistic quantity. gence zones. Describing this effect in 1956, Sukha- The probabilistic approach developed by Sukha- revskiœ suggested that it could open up possibilities for revskiœ for describing the operating range of hydroa-

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002

YURIŒ MIKHAŒLOVICH SUKHAREVSKIŒ (ON HIS 95th BIRTHDAY) 245 coustic systems and the new technique he proposed for educated more than 30 doctors and candidates of sci- range estimation offered the possibility of making the ence, including some members and corresponding operation of hydroacoustic systems more reliable. This members of the Academy of Sciences, honored scien- possibility is based on the fact that the parameter indi- tists and engineers, and honored inventors. Some of his cating the efficiency of a hydroacoustic system, i.e., the students hold leading posts at various research insti- operating range, is guaranteed with a given integrated tutes and design offices. probability. The approach proved to be of dramatic sig- Sukharevskiœ carries authority with broad circles of nificance for this area of research. The main results of acoustical physicists, designers of hydroacoustic equip- these studies were published by Sukharevskiœ in Acous- ment, and Navy specialists. For his services to the tical Physics in 1995. Throughout the years, Sukha- country, Sukharevskiœ was awarded two Orders of the revskiœ investigated the prospects of hydroacoustic sci- Red Banner of Labor, an Order of the October Revolu- ence and engineering. During a ten-year period, he gave tion, a Badge of Honor, a Valiant Labor during the lectures on hydroacoustics at the Institute of the Patriotic War Medal, and other medals. Improvement of Professional Skills for the leading spe- Sukharevskiœ is a man of duty, responsibility, and cialists of the shipbuilding industry. At the present time, exceptional, enduring capabilities. In 1996, he became Sukharevskiœ is a principle researcher at the Acoustics a laureate of the competition among the best publica- Institute. tions of 1995 appearing in the journals of the Russian Sukharevskiœ’s achievements in various fields of Academy of Sciences. Sukharevskiœ was awarded a acoustics, as well as his entire career in science, testify special grant from the President of the Russian Federa- to his outstanding abilities. Sukharevskiœ is a prominent tion as a prominent scientist of Russia. and far-seeing scientist. He made a significant contribu- The interests of Sukharevskiœ reach far beyond his tion to the introduction of scientific results into modern professional occupation in science. His friends and col- engineering. leagues enjoy his musical performances at parties held Sukharevskiœ is the author of more than 150 scien- at the Acoustics Institute. tific publications. He is not only a talented researcher We wish Yuriœ Mikhaœlovich Sukharevskiœ good but also an excellent teacher of young scientists. His health and many happy years to come. scientific school is well known as the one of top-level acousticians. The ability to select and educate students is one of the most remarkable skills of Sukharevskiœ. He Translated by E. Golyamina

ACOUSTICAL PHYSICS Vol. 48 No. 2 2002

CHRONICLE

In Memory of Evgeniœ L’vovich Shenderov (June 1, 1935ÐSeptember 1, 2001)

scientist and outstanding engineer. He founded a scientific school, the work of which made a critical contribution to the development of underwater acoustics. The scope of Shenderov’s scientific interests was rather wide. He published a number of fundamental works devoted to the diffraction of sound waves by elastic plates and shells, the optical visualization of sound fields diffracted by complex-shaped bodies, and the theoretical methods of analyzing the sound fields and sound wave diffraction by ship hulls. Shenderov is the author of more than 100 scientific publications and 50 inventions. Many of his papers were published in Acoustical Physics and in the Journal of the Acoustical Society of America (JASA). Shenderov wrote two fundamental monographs on theoretical acoustics: Wave Problems of Underwater Acoustics (1972) and Radia- tion and Scattering of Sound (1989). He was also involved in tutorial activities and was known as an excellent lecturer. In different higher educational institutes of St. Petersburg, he gave lectures on sound radiation, wave propagation, and oscillation of mechanical systems. Shenderov presented his papers at many scientific conferences. The last paper he prepared was presented by his colleagues at the 17th International Con- gress on Acoustics in Rome, two days after his death. Shenderov was a member of the Scientific Council on Acoustics of the Russian Academy of Sciences and a member of the Editorial Council of Acoustical Phys- ics. For years, he was a co-chairman of the Leningrad Evgeniœ L’vovich Shenderov—Honored Scientist (now, St. Petersburg) seminar of the Scientific Council and Engineer of the Russian Federation, Doctor of on Acoustics of the Russian Academy of Sciences. Engineering, Professor, and Head of the Research Sec- Shenderov was a charming and benevolent person. tor of the Morfizpribor Central Research Institute— Being a true intellectual, he was fairly democratic and passed away after a brief illness. friendly toward other people. He carried indisputable authority with all who knew him not only in his profes- Shenderov started working at the Morfizpribor Cen- sion but also in other areas. Shenderov had a wide vari- tral Research Institute immediately after his graduation ety of hobbies: he was a yachtsman, tourist, downhill from the Leningrad Elecrotechnical Institute in 1957. skier, free diver, underwater photographer, cabinet- For more than 44 years, his work was related to the for- maker, gardener, and oven maker. mation and development of one of the most important This outstanding scientist, talented engineer, and areas of modern underwater acoustics: the study of the good friend has passed away. The shining memory of effect of sonar domes on the parameters of hydroacous- Evgeniœ L’vovich Shenderov will forever remain in the tic arrays. He developed techniques for calculating the hearts of those who were lucky enough to have known acoustic permeability of sonar domes and proposed this wonderful person. new designs of sound-transparent parts of domes, which have found practical application in the design of naval ships and submarines. Shenderov was a talented Translated by E. Golyamina