Leaky Rayleigh and Scholte waves at the fluid–solid interface subjected to transient point loading Jinying Zhu and John S. Popovicsa) Department of Civil and Environmental Engineering, University of Illinois at Urbana, Illinois 61801 Frank Schubert Lab EADQ, Fraunhofer Institute for Nondestructive Testing, D-01326 Dresden, Germany ͑Received 6 February 2004; revised 9 July 2004; accepted 18 July 2004͒ The analysis of acoustic waves generated by a transient normal point load applied on a fluid–solid interface is presented. The closed-form exact solution of the wave motion is obtained by using integral transform techniques. The obtained analytical solution provides necessary theoretical background for optimization of fluid-coupled ultrasonic and acoustic wave detection in experiments. Numerical simulation ͑elastodynamic finite integration technique͒ is performed to verify the obtained analytical solution. Detailed descriptions of leaky Rayleigh and Scholte wave solutions are presented. A simplified solution to isolate the contributions of leaky Rayleigh and Scholte waves generated by a transient point load is proposed, and closed-form formulations for displacement and stress components are then presented. The simplified solution is compared to the exact solution for two configurations: water/concrete and air/concrete. The excitation effectiveness of leaky Rayleigh waves for the air/concrete configuration is studied, which has practical significance to air-coupled sensing in civil engineering structures. © 2004 Acoustical Society of America. ͓DOI: 10.1121/1.1791718͔ PACS numbers: 43.35.Pt, 43.20.El, 43.35.Zc ͓YHB͔ Pages: 2101–2110
I. INTRODUCTION tally by Glorieux et al. ͑2001͒. The propagation of leaky Rayleigh waves under the influences of viscous damping and The propagating leaky Rayleigh wave that emanates heat conduction at the fluid–solid interface was studied by from a fluid–solid interface has been used as an effective Qi ͑1994͒, who concluded that the effect of viscosity can be means for surface and subsurface defect detection. With re- neglected for fluids with Reynolds number larger than 2500. cent improvements in instrumentation, air-coupled transduc- For common fluids such as water and air at normal driving ers have been used for detection of leaky surface ͑Zhu et al., frequencies ͑Ͻ10 MHz͒, the viscous effect of the fluid can 2001͒ and leaky guided waves ͑Castaings et al., 2001͒. With be neglected because the Reynolds numbers are far above the the advantage of noncontact sensing, air-coupled transducers critical value of 2500. provide an opportunity for quick scanning and imaging of The response of leaky waves owing to transient point large civil engineering structures by detecting the leaky Ray- loading is of great practical interest to nondestructive evalu- leigh wave. Therefore, detailed study of leaky surface waves ation ͑NDE͒ researchers, especially in civil engineering, for this case is needed. where an impulse hammer or a point impactor applied to the Extensive studies and applications of leaky surface surface of the solid is often used as a transient wave source. waves have been reported during the past 40 years. A com- Gusev et al. ͑1996͒ provided detailed theoretical analyses of prehensive study of Rayleigh waves and leaky Rayleigh laser-induced wave motions at the fluid–solid interface, waves has been given by Viktorov ͑1967͒, where leaky Ray- which include Scholte, leaky Rayleigh, and lateral waves. leigh waves at the interfaces of a solid half-space with both a General solutions for interface wave motion were given us- fluid layer and a fluid half-space were investigated in great ing a 2-D formulation. 2-D and 3-D analytical solutions for a detail. Leaky Rayleigh waves propagate with a velocity fluid–solid configuration subjected to implosive line and slightly higher than the ordinary Rayleigh wave, and attenu- point sources in the fluid have been given by de Hoop et al. ate more intensively with distance due to continuous energy ͑1983, 1984͒. However, for the case of the ‘‘Lamb’’ problem radiation into the fluid. It was initially believed that the leaky in fluid–solid configuration, where a normal transient point Rayleigh wave exists when the fluid wave velocity cF is load is applied at the fluid–solid interface, no closed-form smaller than the leaky Rayleigh wave velocity cLR . How- exact solution has been reported so far. ever, Mozhaev and Weihnacht ͑2002͒ showed the actual In this paper, Laplace and Hankel integral transforms are threshold phase velocity for leaky Rayleigh wave existence employed to derive the full analytical solution to the was 1.45% lower than the acoustic wave velocity of the ‘‘Lamb’’ problem in a fluid–solid half-space system, where a fluid. The character and existence conditions of leaky Ray- point load, varying with time as a step function, is applied at leigh and Scholte waves were also investigated experimen- the interface. The obtained step response solution is shown in integral format, which can be calculated numerically. Im- a͒Author to whom correspondence should be addressed. Electronic mail: pulse responses are then obtained by differentiating the step [email protected] responses. Therefore, for any transient impact loading that
J. Acoust. Soc. Am. 116 (4), Pt. 1, October 2004 0001-4966/2004/116(4)/2101/10/$20.00 © 2004 Acoustical Society of America 2101 ͒ ץ ץ ץ ץ 2 2 1 ͑r u ϭ Ϫ , w ϭ ϩ , ͑2͒ rץ z rץ z 2ץ rץ 2 where the subscripts 1 and 2 represent quantities related to the fluid and solid, respectively.
B. Continuity and initial condition A point load is applied at the origin as a Dirac delta function in space and varies as a Heaviside step function in FIG. 1. A transient point load applied at the interface of a fluid–solid half- time, which can be expressed as QH(t)␦(r)/2r. Because space system. ideal fluid and shear-free interface conditions are assumed, only the normal stress and vertical component of the dis- has arbitrary temporal variation and spatial distribution, the placement are continuous at the interface. The continuity ϭ responses can be obtained by convolving the impulse re- conditions at z 0 are sponse in time and space domains. The obtained analytical ϭ w1 w2 , solutions are verified by EFIT ͑elastodynamic finite integra- tion technique͒ numerical simulation, which is a powerful ␦͑r͒ ␦͑r͒ ϭ ϩ͑ϪQ͒H͑t͒ ϭϪPϪQH͑t͒ , ͑3͒ tool for elastodynamic wave field analysis. Then a simplified zz2 zz1 2r 2r analytical formulation for the pressure field in the fluid is ϭ ϭ derived and illustrated. zr2 zr1 0, where zz and zr are the normal and shear components of stress, and P the pressure in the fluid. II. COMPLETE FORMULATION Assuming the system is at rest prior to tϭ0, we have Consider a fluid–solid half-space system as shown in ͑r,z,0͒ϭ¨ ͑r,z,0͒ Fig. 1. The solid half-space is given by zϾ0, and the fluid by 1 1 Ͻ ´ ϭ ͒ϭ ͒ z 0. The properties of the fluid are given by the Lame con- 2͑r,z,0 ¨ 2͑r,z,0 stant 1 and mass density 1 , and those of the solid by the ϭ ͑r,z,0͒ϭ¨ ͑r,z,0͒ϭ0. ͑4͒ Lame´ constants 2 and and density 2 . The interface be- 2 2 tween the fluid and solid half-space is subjected to a normal point load of magnitude QH(t), where H(t) is the Heaviside C. Integral transform step function. Because the wave motion in the fluid and solid One-sided Laplace and Hankel transforms are used to generated by the point load is axially symmetric about the z obtain solutions to the equations. The Laplace and nth-order axis, cylindrical coordinates are employed, where the origin Hankel transforms are defined respectively as is located at the load point on the interface. ϱ ϱ Ϫ A. Fluid–solid half-space ¯͑ ͒ϭ ͵ ͑ ͒ pt Hn͑͒ϭ ͵ ͑ ͒ ͑ ͒ ͑ ͒ f p f t e dt, f f r Jn r rdr, 5 0 0 Introducing displacement potential functions 1 in the fluid and 2 and in the solid, the governing equations for where p and are variables of the Laplace and Hankel trans- the fluid and the solid half-spaces are forms, respectively. Applying the Laplace transform to Eq. ͑1͒ with respect 2ץ ץ 2ץ 1 1 1 1 1 ϩ ϩ ϭ ¨ , to time t and the zeroth- and first-order Hankel transform 1 2 2 ץ rץ r 2 ץ r z cF with respect to the radial distance r yields
2 2H0ץ ץ 2ץ 2 1 2 2 1 d ¯ ϩ ϩ ϭ ͑ ͒ 1 2 H0 ¨ 2 , 1 Ϫ␣ ¯ ϭ0, 1 1 2 2 2 ץ rץ r 2 ץ r z c P dz ͑6͒ 2 2 Hץ ץ 2ץ 1 1 d ¯ 0 2¯H1 ϩ ϩ Ϫ ϭ ¨ 2 H d , Ϫ␣2¯ 0ϭ Ϫ2¯H1ϭ ,0 ,0 2 2 2 2 2 ץ rץ r 2 ץ r z r cS dz2 dz2 2 ϭ where cF 1 / 1 is the acoustic wave velocity in the fluid, where 2 ϭ ϩ 2ϭ and c P ( 2 2 )/ 2 and cS / 2 are the P- and S-wave ␣2ϭ2ϩ 2 2 ␣2ϭ2ϩ 2 2 2ϭ2ϩ 2 2 ͑ ͒ velocities in the solid. The double dots represent a double 1 sFp , 2 s Pp , sSp , 7 derivative with respect to time. The displacements are related to potential functions by and sF , s P , sS are P- and S-wave slowness in the fluid and the solid. Only choosing the terms which lead to finite re- ץ ץ 1 1 sponses for large values of ͉z͉, we obtain the solutions to Eq. u ϭ , w ϭ , ,z ͑6͒ץ r 1ץ 1
2102 J. Acoust. Soc. Am., Vol. 116, No. 4, Pt. 1, October 2004 Zhu et al.: Leaky Rayleigh and Scholte waves H0 ␣ z ¯ ϭ⌽ ͑,p͒e 1 , H Q 1 Ϫ␣ 1 1 ¯ 0 ϭϪ ͓͑s2p2ϩ22͒2e 2z zz2 S H Ϫ␣ Ϫ 2 p ¯ 0ϭ⌽ ͑ ͒ 2z ¯H1ϭ⌿͑ ͒ z ͑ ͒ 2 2 ,p e , ,p e , 8 1 ⌽ ⌽ ⌿ Ϫ4␣ 2eϪz͔ , where 1( ,p), 2( ,p), and ( ,p) are functions of p 2 D ͑,p͒ and that need to be determined. From Eq. ͑2͒ and the H displacement–stress relationships, the transformed displace- ␣ ͑ 2 2ϩ 2͒ H Q 1 Ϫ␣ Ϫ 2 2 sSp 2 ͑ ͒ ¯ 1 ϭ ͓Ϫe 2zϩe z͔ . ments and stresses pressure for the fluid in the fluid are zr2 ͒ 2 p DH͑ ,p d¯H0 H 1 H H H w¯ 0ϭ , ¯u 1ϭϪ¯ 0, ¯PH0ϭϪ p2¯ 0, ͑9͒ 1 dz 1 1 1 1 III. CHARACTERISTIC EQUATION FOR LEAKY RAYLEIGH AND SCHOLTE WAVES and in the solid are D (,p)ϭ0 is the characteristic equation for leaky H H d¯ 0 ¯H1 H 2 H d H Rayleigh waves at the interface of fluid–solid half-spaces. w¯ 0ϭ ϩ¯H1, ¯u 1ϭϪ Ϫ¯ 0, ␥ϭ 2 dz 2 dz 2 Introducing the substitution i /psS and notations q ϭ ϭ ϭ sL /sS and u sF /sS , the equation DH 0 can be changed d¯H1 to the following in terms of ␥: ¯H0 ϭͫ͑ 2 2ϩ 2͒¯H0ϩ ͬ ͑ ͒ zz2 sSp 2 2 2 , 10 dz ͱ␥2Ϫ 2 1 q ͑1Ϫ2␥2͒2Ϫ4␥2ͱ␥2Ϫ1ͱ␥2Ϫq2ϩ ϭ0. d¯H0 ͱ␥2Ϫ 2 H 2 2 u ¯ 1 ϭϪͫ 2 ϩ͑s2p2ϩ22͒¯H1ͬ. ͑ ͒ zr2 dz S 15 ͑ ͒ ͑ ͒ Applying the same integral transforms to the continuity Equation 15 is the same as that given by Viktorov 1967 , conditions Eq. ͑3͒, then substituting Eqs. ͑9͒ and ͑10͒ into it, which differs from the regular Rayleigh equation in the third ⌽ term due to the pressure of the fluid. Equation ͑15͒ produces generates a group of linear equations in terms of 1( ,p), ⌽ ⌿ eight Riemann sheets owing to the square roots. The roots 2( ,p), and ( ,p). Solving the equations yields have the physical meaning of normalized slowness with re- 2 2 Q ␣ s p2 Q s p2ϩ22 spect to s . When leaky Rayleigh waves exist, we can obtain ⌽ ϭ 2 S ⌽ ϭϪ S S 1 ␣ ͒ , 2 ͒ , 2 p 1 DH͑ ,p 2 p DH͑ ,p six roots, which include two pairs of complex conjugate ͓ ␥2 pairs corresponding to the leaky Rayleigh wave Re( R) Q 2␣ Ͻ 2 ⌿ϭϪ 2 ͑ ͒ u ͔, and two opposite real roots corresponding to the ͑ ͒ , 11 ␥2 Ͼ 2 2 p DH ,p Scholte wave ( sch u ). The two pairs of complex conjugate roots corresponding where to the leaky Rayleigh wave take the form of Ϯ͓Re(␥) ␣ Ϯi Im(␥)͔, in which only two roots are acceptable. Because p ͑ ͒ϭ͑ 2 2ϩ 2͒2Ϫ 2␣ ϩ 1 4 4 2 ͑ ͒ DH ,p sSp 2 4 2 p sS ␣ . 12 2 1 is the Laplace transform variable, Re(p) must be negative to have a finite time domain response. Therefore, only the roots Substituting Eq. ͑11͒ into Eqs. ͑9͒ and ͑10͒, we obtain the Ͻ ϭ ␥ which lead to Re(p) 0 are acceptable, where p i / sS .A displacements and pressure in the fluid, quick analysis shows that the two acceptable roots take the 2 ␥ ϭϮ ␥ Ϫ ␥ ␣ form of R1,R2 Re( ) i Im( ). Here, we denote them as H Q 1 2sSp ␣ w¯ 0ϭ e 1z, ␥ , ␥ , and note ␥ ϭϪ¯␥ , where the top bar indicates 1 2 D ͑,p͒ R1 R2 R2 R1 H complex conjugate. ␣ 2 Analysis also shows that ͉Re(␥ )͉Ͻs /s , where s H Q 1 2 sSp ␣ R1,R2 R S R ¯u 1ϭϪ e 1z, ͑13͒ 1 2 ␣ D ͑,p͒ is the slowness of Rayleigh wave in free surface solid half- 1 H space. This result confirms that the slowness of the leaky ␣ 4 3 Q 1 2 sSp ␣ Rayleigh wave is smaller than sR , i.e., in a fluid–solid half- ¯ H0ϭϪ 1z P ␣ ͒ e , space system, the leaky Rayleigh wave will travel faster than 2 2 1 DH͑ ,p the pure Rayleigh wave in the same free surface solid. This and the displacements and stresses in the solid conclusion is reasonable considering the influence of fluid ␥ pressure applied on solid surface. The imaginary part of R H Q 1 Ϫ␣ ¯u 1ϭ ͓͑s2p2ϩ22͒e 2z accounts for the energy leaked into fluid. The larger the value 2 S 2 p ͉ ␥ ͉ of Im( R1) , the more energy leaks into the fluid. The Scholte wave velocity can be obtained from the real Ϫ2␣ eϪz͔ , Ϯ␥ ␥ Ͼ ϭ␥ 2 D ͑,p͒ roots sch , where sch u and ssch schsS . Scholte H waves always exist for any fluid–solid combination case. For
H Q 1 Ϫ␣ the lighter fluids lying on stiffer solids case, ssch is only w¯ 0ϭ ͓͑s2p2ϩ22͒e 2z 2 2 p S slightly smaller than the acoustic wave velocity of the fluid, and the Scholte wave contribution is relatively small. The ␣ Ϫ 2 Ϫz͔ 2 ͑ ͒ property of Scholte waves will be discussed in more detail in 2 e ͒ , 14 DH͑ ,p Sec. VII.
J. Acoust. Soc. Am., Vol. 116, No. 4, Pt. 1, October 2004 Zhu et al.: Leaky Rayleigh and Scholte waves 2103 IV. SOLUTION OF PRESSURE IN THE FLUID
The solution of fluid pressure in the space–time domain is derived because pressure in the fluid is the most often measured quantity in practical application. By taking the in- verse zeroth-order Hankel transform to Eq. ͑13͒ and intro- ducing the substitution ϭp, we have
4 ϱ ͱ2ϩ 2 QsS 1 s P p ¯PϭϪ ͵ 2 ͱ2ϩ 2 D͑͒ 2 0 sF
ͱ2ϩ 2 ϫ pz sF ͒ ͑ ͒ e J0͑p r d , 16 where D() is defined as FIG. 2. Complex v plane and the modified integral path. The indices denote: D͑͒ϭ͑22ϩs2͒2Ϫ42ͱ2ϩs2 ͱ2ϩs2 P→leaky P wave, S→leaky S wave, LR→leaky R wave, f→fluid acoustic S P S wave, SCH→Scholte wave. ͱ2ϩ 2 1 s P ϩ s4. ͑17͒ ͱ2ϩ 2 S P 2 sF where G (t) is defined as the Green’s function for pressure. ␥ ͑ ͒ ϭϪ␥ The variable v is related to in Eq. 15 by v sS . The inverse Laplace transform is evaluated by the According to the analysis in the previous section, there are Cagniard–de Hoop method, as described by Achenbach two poles in the right half complex v plane, which corre- ͑ ͒ 1973 . Introducing the following representation of J0(x), spond to leaky Rayleigh and Scholte wave arrivals. The in- 2 ϱ eixs tegral path along the negative imaginary axis can be de- J ͑x͒ϭ Im ͵ ds, ͑18͒ formed to path ⌫ as shown in Fig. 2. The new integration 0 ͱ 2 1 s Ϫ1 ⌫ ϭ Ϫ ͱ 2 Ϫ 2 path is defined by the equation vr z sF v , which and substituting it into Eq. ͑16͒ yields can be solved for v to yield
4 ϱ ͱ2ϩ 2 QsS 1 s P ϩ ͱ 2 2Ϫ2 ¯PϭϪ Im ͵ d r z sFR 2 ͱ2ϩ 2 D͑͒ v͑͒ϭ , is real and 0ррϱ, 2 0 sF R2 2 ͑23͒ iprsϩpzͱ2ϩs ϱ e F ϫ ͑ ͒ ͵ p ds. 19 ϭͱ 2ϩ 2 1 ͱs2Ϫ1 where R r z . The benefit of deforming the original integration path to ⌫ is obvious, because there is no pole ϭϪ Ϫ ͱ2ϩ 2 Introducing the substitution t i rs z sF and along the new path. The Green’s function integration along ϭiv yields the new path is
4 Ϫiϱ ͱ 2 Ϫ 2 QsS 1 s P v v ¯Pϭ Im ͵ dv t ͱs2 Ϫv͑͒2 v͑͒ 2 2 2 ͑ ͒ P P 2 0 ͱs Ϫv D iv G ͑t͒ϭͩ Im ͵ F ͱ 2 Ϫ ͑͒2 D͓iv͔͑͒ tp sF v ϱ eϪpt ϫ ͵ ͑ ͒ Ϫ ͱ 2 2Ϫ2 p ds, 20 r z/ sFR 1 ͱs2Ϫ1 ϫ d ͪ , ͑24͒ R2ͱ͓tϪϩv͑͒r͔2Ϫv͑͒2r2 where 2 2 Ϫ 2 where t ϭrs ϩ͉z͉ͱs Ϫs is the leaky P wave arrival time. ϱ e pt d H͑tϩzͱs Ϫv2Ϫvr͒ p P F P Ϫ1 F P ϭ Ͻ L ͩ ͵ p dsͪ ϭ ͩ ͪ G (t) 0 when t t p . 1 ͱ 2Ϫ dt ͱ͑ ϩ ͱ 2 Ϫ 2͒2Ϫ 2 2 s 1 t z sF v v r However, when the receiver is located on the interface, ͑21͒ i.e., zϭ0, the integral path ⌫ is along the real axis, and and L represents the Laplace transform. Therefore, the in- passes by the Scholte pole. In this case, the principal value of verse Laplace transform of Eq. ͑20͒ is Eq. ͑24͒ must be taken. The integrand in Eq. ͑24͒ has a square root singularity at 4 Ϫiϱ ͱ 2 Ϫ 2 QsS 1 d s P v v end point ϭt, which increases the difficulty of numerical P͑t͒ϭ ͩ Im ͵ 2 2 2 ͑ ͒ integration. This problem can be solved by further introduc- 2 dt 0 ͱs Ϫv D iv F ing the following transformation, ͑ ϩ ͱ 2 Ϫ 2Ϫ ͒ H t z sF v vr ϫ dv ͪ ϭt cos2͑͒ϩt sin2͑͒, ͑25͒ ͱ͑ ϩ ͱ 2 Ϫ 2͒2Ϫ 2 2 1 2 t z sF v v r where t , t are the lower and upper limits of the integration. 4 1 2 QsS 1 d рр ϭ G P͑t͒, ͑22͒ Then the integration interval t1 t2 is mapped to the 2 2 dt fixed interval 0рр/2, while
2104 J. Acoust. Soc. Am., Vol. 116, No. 4, Pt. 1, October 2004 Zhu et al.: Leaky Rayleigh and Scholte waves FIG. 3. Green’s function G P(t) for pressure in the fluid at position r ϭ1.5 m, zϭϪ0.05 m. Material parameters for the fluid ϭ1000 kg/m3, 1 FIG. 4. Absolute value of pressure ͑in fluid͒ and stress ͑in solid͒ field c ϭ1500 m/s; for the solid, ϭ2400 kg/m3, c ϭ4000 m/s, vϭ0.25. zz F 2 P snapshot at tϭ0.72 ms given by EFIT analysis. The interface is subject to a transient point load that varies with time as f (t)ϭsin2 (t/T) with T ϭ200 s. ϭ Ϫ ͒ ͒ ͒ d 2͑t2 t1 sin͑ cos͑ , V. VERIFICATION BY NUMERICAL SIMULATION ͱ͓ Ϫϩ ͑͒ ͔2Ϫ ͑͒2 2 t2 v r v r To verify the obtained analytical solutions, numerical ϭͱ Ϫ ͒ ͒ͱ Ϫ ͒ 2 ͒ϩ ͒ ͑ ͒ analyses were performed to simulate the response for fluid– ͑t2 t1 cos͑ ͑t2 t1 cos ͑ 2v͑ r. 26 solid half-space cases. The elastodynamic finite integration technique ͑EFIT͒ is a numerical time-domain scheme to ϭ ϭ Another square root singularity at point t f sFR can be model elastic wave propagation in homogeneous and hetero- processed similarly. A similar technique was also used by de geneous, dissipative and nondissipative, as well as isotropic ͑ ͒ Hoop 1984 . and anisotropic elastic media ͑Fellinger et al., 1995͒. EFIT The vertical component of displacement in the fluid is uses a velocity-stress formalism on a staggered spatial and also obtained in the same way: temporal grid complex. The starting point of EFIT is the integral form of the linear governing equations, i.e., the 4 t ͱ 2 Ϫ ͑͒2 ͑͒ Q sS s P v v Cauchy equation of motion, and the equation of deformation w͑t͒ϭϪ Im ͵ 2 2 rate. EFIT performs integrations over certain control vol- 2 t D͓iv͔͑͒R p umes V, and the surfaces of these cells S, assuming constant Ϫ ͱ 2 2Ϫ2 velocity and stress within V and on each S. This method r z/ sFR ϫ d. ͑27͒ requires staggered grids and leads to a very stable and effi- ͱ͓ Ϫϩ ͑͒ ͔2Ϫ ͑͒2 2 t v r v r cient numerical code, which also allows easy and flexible treatment of various boundary conditions. In recent years, Figure 3 shows the Green’s function G P(t) at a near- EFIT has been successfully used for a wide variety of appli- surface position rϭ1.5 m, zϭϪ0.05 m in the fluid. The ma- cations, especially in the field of nondestructive testing terial configuration simulates a concrete/water system. The ͑Schubert and Marklein, 2003͒. arrival times of all wave types are marked. It is noticed that In the present case of a transient point load at a fluid– ϭ the slope is discontinuous at positions t t p , ts , and t f , solid interface, we used a special axisymmetric EFIT code in which correspond to the leaky P-, leaky S-, and fluid acoustic cylindrical coordinates ͑Schubert et al., 1998͒. The water/ waves in water. From the mathematics viewpoint, these dis- concrete case shown in Fig. 3 was studied. The material pa- ϭ 3 ϭ continuities come from the branch points along the integral rameters are for water, 1 1000 kg/m , cF 1500 m/s, and ⌫ ϭ 3 path . The poles corresponding to the leaky Rayleigh and for concrete 2 2400 kg/m , P wave velocity c P Scholte wave arrivals contribute to the large smooth peaks. ϭ4000 m/s, and Poisson’s ratio ϭ0.25. The vertical tran- When the receiving position is very close to the interface, the sient point load varies with time as function f (t) integral path ⌫ in Fig. 2 will bend to the real axis, and a ϭsin2(t/T), where the force duration is Tϭ200 s. A grid sharp peak will appear nearby the Scholte wave arrival in spacing of ⌬rϭ⌬zϭ2.5 mm and a time step of 0.44 s are Fig. 3. used in order to guarantee stability as well as sufficient dis- The impulse response to a point loading on the interface cretization of the shortest wavelengths. The dimensions of between a fluid and a solid half-space can be obtained from the model are2minradial and3minaxial direction, re- the corresponding step response solution by taking differen- sulting in 800ϫ1200 grid cells. At the outer boundaries of tiation with respect to time. Then for any transient loading the model, highly effective absorbing boundary conditions that has arbitrary temporal variation and spatial distribution, based on the perfectly matched layer ͑PML͒ are used in order the response can be obtained by convolving the impulse re- to suppress interfering reflections ͑Liu, 1999͒. sponse in both time and space domains. Figure 4 shows the cross-sectional snapshot image of
J. Acoust. Soc. Am., Vol. 116, No. 4, Pt. 1, October 2004 Zhu et al.: Leaky Rayleigh and Scholte waves 2105 leigh wave effects in a free surface half-space by Chao et al. ͑1961͒. Achenbach ͑1973͒ reproduced Chao’s results in de- tail. The simplified solution provides an easy and quick way to estimate leaky Rayleigh and Scholte wave effects.
A. Displacements and stresses in the fluid Applying the inverse Hankel and Laplace transforms to the transformed solution in Eq. ͑13͒, the pressure in fluid can be expressed as Q s4 ϱ ͑ ͒ϭϪ S 1 ͵ ͑ ͒ P r,z,t J0 r 2 2 i 2 0
⑀ϩiϱ ␣ 3 2 p ␣ ϩ ϫ ͵ 1z pt ͑ ͒ ␣ ͒ e dp d . 28 FIG. 5. Comparison of the analytical and the numerical solutions for pres- ⑀Ϫiϱ 1 DH͑ ,p sure in the fluid at rϭ1.5 m, zϭϪ0.05 m. The interface is subject to a point load that varies with time as f (t)ϭsin2 (t/T) with Tϭ200 s. Material Considering the integrand term, the contributions from the ϭ 3 ϭ parameters for the fluid 1 1000 kg/m , cF 1500 m/s; for the solid 2 leaky Rayleigh poles are ϭ 3 ϭ ϭ 2400 kg/m , c P 4000 m/s, v 0.25. ϱ ⑀ϩiϱ ␣ 3 1 2 p ␣ ϩ ϭ ͵ ͑ ͒ ͵ 1z pt I1 J0 r ␣ ͑ ͒ e dp d pressure field ͑absolute value͒ as generated by EFIT, at time 2 i 0 ⑀Ϫiϱ 1 DH ,p ϭ Ͼ t 0.72 ms. Only the region with r 0 was calculated, but ϱ for better illustration the reversed region with rϽ0 is also ϭ ͵ ͑ ͒ J0 r shown here. The half-circle in the upper half-plane repre- 0 sents the acoustic wave front in water, and the inclined lines ␣ 3 2 p ␣ ϩ represent the leaky Rayleigh wave fronts, which are tangent ϫͫ 1z ptͬ , e d ץ ͒ ͑ ץ ␣ 1 DH ,p / p ϭ ␥ ␥ to the half circle at the leaky angle direction. In this case, the p i /sS R1 ,i /sS R2 leaky Rayleigh angle determined by Snell’s law is ϭ43.8°, ͑29͒ measured from the vertical axis. The leaky Rayleigh wave ͓ ␣ ␣ front is separable from the subsequent Scholte and fluid where the expression ( 2 / 1) 3 p͔͔ ϭ ␥ ␥ represents a pair ofץ/(D (,pץ/ ϫ͓p acoustic wave fronts at larger values of radial distance r. The H p i /sS R1 ,i /sS R2 ¯ 3-D shape of the combined leaky Rayleigh and fluid acoustic complex conjugate constants, denoted as A1 and A1 . wave fronts looks like a domed cone. The leaky Rayleigh Introducing wave in concrete is also seen, which behaves similarly to the i z t ordinary Rayleigh wave, and attenuates exponentially with ϭϪ ͫ ͱ 2Ϫ␥2 ϩ ͬ ͑ ͒ m1,2 ␥ u R1,R2 30 increasing depth in the solid. In the near-interface region, the R1,R2 r sSr Scholte wave effect is strong in both the fluid and the solid. generates Figure 5 shows the analytical and numerical time do- ͓␣ zϩpt͔ ϭ ␥ ϭϪrm , ͑31͒ main near-surface response of the pressure in water, at posi- 1 p i /sS R1,R2 1,2 tion rϭ1.5 m, zϭϪ0.05 m. Very good agreement between ϭ ͑ ͒ where m1 m¯ 2 . Thus, Eq. 29 can be expressed as analytical and numerical responses is observed. ϱ ϱ Ϫ Ϫ ϭ ͵ ͑ ͒ m1r ϩ¯ ͵ ͑ ͒ m2r I1 A1 J0 r e d A1 J0 r e d VI. SIMPLIFIED LEAKY RAYLEIGH WAVE RESPONSE 0 0 ϱ In Sec. IV, the complete analytical solution of pressure Ϫ ϭ ͵ m1r ͑ ͒ 2Re A1e J0 r d and displacement in the fluid are obtained. However, the in- 0 tegral form solution is not always convenient to use. In this
and the next sections, the authors provide a simplified 2 A1m1 ϭ Reͫ ͬ,Re͑m ͒Ͼ0, ͑32͒ closed-form solution to the same problem by considering 1 r2 ͑1ϩm2͒3/2 only Rayleigh and Scholte pole contributions. This simplifi- 1 cation is acceptable when measurements are taken at a large where we use the zeroth-order Hankel transform formula distance from the source. ϱ a According to the previous analysis, the wave field ex- ͵ ͑ ͒ Ϫa ϭ ͑ ͒Ͼ ͑ ͒ J0 r e d ,Rea 0. 33 cited by a normal point load at the interface includes contri- 0 ͑r2ϩa2͒3/2 butions from leaky P, S, Rayleigh, fluid acoustic, and Scholte From Eq. ͑28͒, the pressure P is expressed as waves. Analysis shows that, at large horizontal distances from the source, disturbances near the interface are domi- Q 1 1 A1m1 PϭϪ s4 Reͫ ͬ. ͑34͒ nated by leaky Rayleigh and Scholte wave contributions, 2 S 2 3/2 2 r ͑1ϩm ͒ which can be obtained from the residues at corresponding 1 poles. The similar idea was already used to investigate Ray- Similarly, by introducing
2106 J. Acoust. Soc. Am., Vol. 116, No. 4, Pt. 1, October 2004 Zhu et al.: Leaky Rayleigh and Scholte waves ␣ p ϭͫ 2 ͬ Q 1 B5n p1 B6ns1 A2 , ϭϪ Reͫ Ϫ ͬ, zz2 ץ ͒ ͑ ץ DH ,p / p ϭ ␥ 2 ͑ ϩ 2 ͒3/2 ͑ ϩ 2 ͒3/2 p i /sS R1 r 1 n p1 1 ns1 ͑35͒ ␣ p2 Ϫ ϭͫ 2 ͬ Q 1 B7 B7 A3 , ϭ Reͫ ϩ ͬ, zr2 ץ ͒ ͑ ץ ␣ 1 DH ,p / p ϭ ␥ 2 ͑ ϩ 2 ͒3/2 ͑ ϩ 2 ͒3/2 p i /sS R1 r 1 n p1 1 ns1 the vertical and radial components of the displacement in the where the coefficients are fluid are obtained: s2p2ϩ22 ϭͫ S ␣ ͬ , 2 ץ ͒ ͑ ץ Q s4 1 A B1 S 2 p DH ,p / p ϭ ␥ w ϭ Reͫ ͬ, p i /sS R1 1 r ͑ ϩ 2͒1/2 2 1 m1 23␣ ͑36͒ ϭͫ 2 ͬ , ץ ͒ ͑ ץ Q s2 1 m B2 S 1 p DH ,p / p ϭ ␥ u ϭϪ Reͩͫ 1Ϫ ͬA ͪ , p i /sS R1 1 3 r ͑1ϩm2͒1/2 1 ͑ 2 2ϩ 2͒ sSp 2 ͑ ͒Ͼ B ϭͫ 2ͬ , ͑40͒ pץ/D ͑,p͒ץRe m1 0. 3 p H pϭi/s ␥ Care should be taken with the square root when calcu- S R1 ͑ ͒ 22␣  lating m1,2 from Eq. 30 . To have bounded results, only the ϭͫ 2 ͬ , ץ ͒ ͑ ץ branch that gives Re(m )Ͼ0 should be selected. B4 1 p DH ,p / p ϭ ␥ p i /sS R1 B. Displacement and stress in the solid half-space ͑s2p2ϩ22͒2 ϭͫ S ͬ , ץ ͒ ͑ ץ B5 The response of the leaky Rayleigh wave in the solid p DH ,p / p ϭ ␥ p i /sS R1 can be obtained in a similar way. Introducing B ϭ2B , B ϭ2B . 1 z 6 4 7 1 n ϭ ͫϮ ͱ␥2 Ϫq2Ϫiͬ, ͑ ͒ p1,2 ␥ r R1,R2 The expressions in Eq. 39 are only valid in the region R1,R2 where Re(n )Ͼ0 and Re(n )Ͼ0. Other stress compo- ͑37͒ p1,p2 s1,s2 1 z nents in the solid can be derived from the following stress– ϭ ͫϮ ͱ␥2 Ϫ Ϫ ͬ ns1,2 ␥ R1,R2 1 i R1,R2 r displacement relations: wץ u uץ yields ϭ͑ ϩ ͒ 2 ϩ ͩ 2 ϩ 2 ͪ rr2 2 2 2 , zץ r rץ Ϫ␣ ϩ ͓ 2z pt͔pϭi/s ␥ S R1,R2 ͑41͒ wץ uץ u r z ϭ͑ ϩ ͒ 2 ϩ ͩ 2 ϩ 2 ͪ 2 . zץ rץ ϭϪ ͫϮͱ␥2 Ϫ 2 Ϫ ͬϭϪ 2 2 r 2 ␥ R1,R2 q i rnp1,2 , R1,R2 r ͑ ͒ Ϫ ϩ 38 ͓ z pt͔pϭi/s ␥ C. Attenuation and dispersion of leaky Rayleigh S R1,R2 waves r z ϭϪ ͫϮͱ␥2 Ϫ Ϫ ͬϭϪ In addition to the geometric decay due to the effect of ␥ R1,R2 1 i rns1,2 . R1,R2 r point loading, which varies as 1/ͱr along the interface for The real part of Eq. ͑38͒ must be negative to have bounded the Rayleigh wave, there is another type of attenuation caused by continuous radiation ͑leakage͒ of energy into the responses, therefore the real parts of n p1 , n p2 , ns1 , ns2 must be positive. Using the similar argument for dealing with fluid. In frequency domain, the solutions are exponential functions of (Ϫr), where has the physical meaning of m1,2 , only those results of n p1 , n p2 , ns1 , ns2 that have wavenumber. According to Eq. ͑32͒, higher frequency ͑larger positive real parts are acceptable. In addition, n p1 , n p2 and ͒ contents give more attenuation during propagation. There- ns1 , ns2 should be complex conjugate pairs. Applying the inverse Hankel and Laplace transforms to fore the waveform generated by a transient loading becomes Eq. ͑14͒ and calculating the residues at Rayleigh poles yields wider with increasing distance, i.e., it shows dispersion prop- the displacements and stresses in the solid, erty due to leakage-induced attenuation, although the phase velocity of leaky Rayleigh waves does not vary with fre- 2 Q sS 1 B1 B2 quency. w ϭ Reͫ Ϫ ͬ, 2 r ͱ ϩ 2 ͱ ϩ 2 2 1 n p1 1 ns1
2 Q sS 1 n p1 VII. SCHOLTE WAVE RESPONSE u ϭ Reͫ B ͩ 1Ϫ ͪ 2 3 2 2 r ͱ1ϩn p1 The real roots of the Scholte equation correspond to Scholte wave propagation along the interface. The Scholte ns1 ϪB ͩ 1Ϫ ͪͬ, wave solutions can be obtained by calculating the residues at 4 ͱ1ϩn2 ␥ϭϮ␥ s1 the poles sch . For common cases of light fluids lying ͑39͒ J. Acoust. Soc. Am., Vol. 116, No. 4, Pt. 1, October 2004 Zhu et al.: Leaky Rayleigh and Scholte waves 2107 ͉␥ ͉ϭ on stiff solids, the root sch ssch /sS is slightly larger than ϭ ͑ ͒ ͑ ͒ u sF /sS . Therefore, Eqs. 31 and 38 are changed to sch ͓␣ zϩpt͔ ϭ Ϯ ␥ ϭϪrm , 1 p i / sS sch 1,2 sch ͓Ϫ␣ zϩpt͔ ϭ Ϯ ␥ ϭϪrn , ͑42͒ 2 p i / sS sch p1,2 sch ͓Ϫzϩpt͔ ϭ Ϯ ␥ ϭϪrn , p i / sS sch s1,2 where 1 z t schϭϪ ͩͱ␥2 Ϫ 2 Ϯ ͪ m1,2 ␥ sch u i , sch r sSr 1 z t sch ϭ ͩͱ␥2 Ϫ 2 Ϯ ͪ ͑ ͒ n p1,2 ␥ sch q i , 43 sch r sSr 1 z t sch ϭ ͩͱ␥2 Ϫ Ϯ ͪ ns1,2 ␥ sch 1 i . sch r sSr The solutions for the leaky Rayleigh wave are also valid for sch sch sch the Scholte wave by substituting m , n p , and ns for m, ͑ ͒ n p , and ns . The first terms in the expressions of Eq. 43 represent the real parts, which are positive and result in the decay in z direction. It can be seen that the Scholte wave decays exponentially in z direction in both the fluid and the solid. For lighter fluid cases, i.e., where the acoustic imped- ͱ␥2 Ϫ 2 ance of the fluid is less than that of the solid, sc u is ͱ␥2 Ϫ 2 ͱ␥2 Ϫ much smaller than sc q and sc 1. This indicates that the Scholte wave attenuates much more quickly in the solid than in the fluid. Therefore, in contrast to the leaky Rayleigh wave, most of Scholte wave energy is localized in the fluid ͑Gusev et al., 1996͒. The Scholte wave generation efficiency increases with increasing acoustic impedance of the fluid. For example, it is much easier to generate Scholte waves in the water/concrete configuration than the air/ FIG. 6. Comparison of the exact and simplified solutions for water/concrete concrete configuration. In fact, almost no Scholte wave effect case. Pressure in the fluid at position ͑a͒ rϭ1.5 m, zϭϪ0.05 m, and ͑b͒ r ϭ ϭϪ can be observed in the air/concrete case. With increasing 1.5 m, z 0.5 m. The interface is subject to a point load that varies with time as f (t)ϭsin2(t/T) with Tϭ200 s. Material parameters for the fluid impedance of the fluid, Scholte waves have deeper penetra- ϭ 3 ϭ ϭ 3 1 1000 kg/m , cF 1500 m/s; and for the solid 2 2400 kg/m , c P tion depth in the solid. This property provides the possibility ϭ4000 m/s, vϭ0.25. for NDT application of Scholte waves, which was studied experimentally by Glorieux et al. ͑2001͒. Because there is no leakage during Scholte wave propagation along the radial simplified and exact analytical solutions decreases. Generally direction, the decay in the radial direction is only attributed speaking, the simplified solution provides better estimation ͉ ͉ to the geometrical effect. In 2-D cases, the Scholte wave of pressure for larger r and smaller z cases, where the con- travels without attenuation along the propagation direction tribution of body waves is negligible. ͑ ͒ ͑ ͒ ͑Glorieux et al., 2001͒. Figures 7 a and b show the comparison of the exact and simplified solutions for the air/concrete case. The re- ceiver positions are rϭ1.0 m, zϭϪ0.05 m and rϭ1.0 m, z VIII. COMPARISON OF THE EXACT AND SIMPLIFIED ϭϪ0.5 m, respectively. Good agreement is observed near SOLUTIONS the leaky Rayleigh wave arrival time for both positions, Figure 6 shows the comparison of the exact and simpli- while obvious differences can be seen near the Scholte and fied analytical solutions for fluid pressure for the water/ fluid acoustic wave arrival times. The reason is that the leaky concrete case. In Fig. 6͑a͒, when the receiver position is Rayleigh wave is well separated from the fluid acoustic close to the interface (rϭ1.5 m, zϭϪ0.05 m), good agree- wave, and Scholte wave contribution is very small compared ment is observed around the leaky Rayleigh and Scholte to the acoustic wave contribution in the fluid for air/concrete wave arrival times. The small yet noticeable differences be- case, even in the near-interface region. Therefore, for the fore the leaky Rayleigh and Scholte wave arrivals are due to air/concrete configuration, the pressure field in the fluid is the absence of leaky body waves and fluid acoustic waves in dominated by leaky Rayleigh and fluid acoustic waves. In the simplified solution. When the receiver is away from the air-coupled sensing, the leaky Rayleigh wave is usually the interface, as shown in Fig. 6͑b͒ for receiver position r component in which we are interested. The acoustic wave ϭ1.5 m, zϭϪ0.5 m, the degree of agreement between the contribution in the fluid can be separated by increasing mea-
2108 J. Acoust. Soc. Am., Vol. 116, No. 4, Pt. 1, October 2004 Zhu et al.: Leaky Rayleigh and Scholte waves Hoop method. Simplified formulations are also derived, which provide an easy and quick way to estimate leaky Ray- leigh and Scholte wave contributions. The following conclu- sions can be drawn based on the analysis: ͑1͒ A transient point load applied to the interface is an ef- fective way to generate leaky Rayleigh waves in the fluid. For air-coupled wave detection in concrete, the ex- citation effectiveness of leaky Rayleigh waves is around 0.1–1.0 Pa/kN, depending on the impact force duration. ͑2͒ For the light fluid/heavy solid case, the leaky Rayleigh wave is separable from Scholte and acoustic waves in the fluid when distance r is large enough, where r de- pends on velocities of leaky Rayleigh, Scholte, and acoustic waves, vertical distance ͉z͉, and force duration. For the air–concrete configuration shown in Fig. 7͑a͒, where ͉z͉ϭ0.05 m and rϾ0.2 m, the difference in arrival time between leaky Rayleigh and acoustic waves is Ͼ362 s. Therefore the received signals will be domi- nated by leaky Rayleigh waves, which provide important material information of the underlying solid. ͑3͒ Simplified solutions are obtained when contributions from leaky Rayleigh waves and Scholte waves poles only are considered. Equations ͑34͒–͑36͒ and ͑39͒ give the solution to responses in the fluid and solid, respec- tively. The simplified solution accurately simulates the transient pressure field response for the air/concrete case when the fluid acoustic wave contribution is removed or separated. ͑4͒ The Scholte wave contribution is prominent in the near- interface region for the water/concrete case. Because most of the energy of Scholte waves is localized in the FIG. 7. Comparison of the exact and simplified solutions for air/concrete fluid, however, Scholte wave properties are not very sen- ͑ ͒ ϭ ϭϪ ͑ ͒ case. Pressure in the fluid at position a r 1.0 m, z 0.05 m, and b r sitive to the variation of the underlying solid materials, ϭ1.0 m, zϭϪ0.5 m. The interface is subject to a point load that varies with time as f (t)ϭsin2(t/T) with Tϭ200 s. Material parameters for the fluid which limits the NDE application of Scholte waves for ϭ 3 ϭ ϭ 3 1 1.21 kg/m , cF 343 m/s; for the solid 2 2400 kg/m , c P the common light fluid/heavy solid cases. ϭ4000 m/s, ϭ0.25. suring distance r between receivers and the source, or elimi- ACKNOWLEDGMENTS nated by employing the directional property of the air- This work was carried out in the course of research ͑ ͒ coupled sensor Zhu et al., 2001 . sponsored by the National Science Foundation under Grant The excitation effectiveness of leaky Rayleigh waves No. 0223819. The authors also wish to thank Dr. Nelson Hsu induced by an impact point load can be inferred from Fig. 7. from National Institute of Standards and Technology for his ϭ 2 For a typical impulse force f (t) sin ( t/T) with a modest helpful suggestions. peak value of 1 kN and duration Tϭ200 s, the output pres- sure of the leaky Rayleigh wave is 0.1–0.15 Pa, which is Achenbach, J. D. ͑1973͒. Wave Propagation in Elastic Solids ͑North- approximately equivalent to a sound pressure level of 75 dB. Holland, Amsterdam, The Netherlands͒, Chap. 7, pp. 262–321. Such a pressure is large enough to be detected readily by an Castaings, M., and Hosten, B. ͑2001͒. ‘‘Lamb and SH waves generated and air-coupled sensor, even when material attenuation effects detected by air-coupled ultrasonic transducers in composite material are considered. The excitation effectiveness of leaky Ray- plates,’’ NDT & E Int. 34, 249–258. Chao, C. C. ͑1961͒. ‘‘Surface waves in an elastic half space,’’ J. Appl. Mech. leigh waves is dependent on the impact force duration. 28, 300–301. Shorter force durations give higher output pressure of leaky de Hoop, A. T., and van der Hijden, J. ͑1983͒. ‘‘Generation of acoustic Rayleigh waves. For example, when the duration is de- waves by an impulsive line source in a fluid/solid configuration with a ϭ plane boundary,’’ J. Acoust. Soc. Am. 74, 333–342. creased to T 50 s, the output peak pressure of the leaky de Hoop, A. T., and van der Hijden, J. ͑1984͒. ‘‘Generation of acoustic Rayleigh wave will increase to 1.2 Pa ͑95 dB͒. Most impac- waves by an impulsive point source in a fluid/solid configuration with a tors used for concrete testing will generate transient forces plane boundary,’’ J. Acoust. Soc. Am. 75, 1709–1715. with duration between 50 and 200 s. Fellinger, P., Marklein, R., Langenberg, K. J., and Klaholz, S. ͑1995͒. ‘‘Nu- merical modelling of elastic wave propagation and scattering with EFIT— IX. CONCLUSIONS Elastodynamic finite integration technique,’’ Wave Motion 21,47–66. Glorieux, C., and Van de Rostyne, K. ͑2001͒. ‘‘On the character of acoustic The exact analytical solution to the ‘‘Lamb’’ problem in waves at the interface between hard and soft solids and liquids,’’ J. Acoust. a fluid/solid half-space system is derived by the Cagniard–de Soc. Am. 110, 1299–1306.
J. Acoust. Soc. Am., Vol. 116, No. 4, Pt. 1, October 2004 Zhu et al.: Leaky Rayleigh and Scholte waves 2109 Gusev, V., Desmet, C., Lauriks, W., Glorieux, C., and Thoen, J. ͑1996͒. gration Technique ͑EFIT͒,’’ in Proc. IEEE Ultrasonics Symp., Munich, ‘‘Theory of Scholte, leaky Rayleigh, and lateral wave excitation via the Germany, 8–11 October, 2002, Article 5G-5, 778–783 ͑on CD͒. laser-induced thermoelastic effect,’’ J. Acoust. Soc. Am. 100, 1514–1528. Schubert, F., Peiffer, A., Koehler, B., and Sanderson, T. ͑1998͒. ‘‘The elas- ͑ ͒ Liu, Q. H. 1999 . ‘‘Perfectly matched layers for elastic waves in cylindrical todynamic finite integration technique for waves in cylindrical geom- and spherical coordinates,’’ J. Acoust. Soc. Am. 105, 2075–2084. etries,’’ J. Acoust. Soc. Am. 104, 2604–2614. Mozhaev, V. G., and Weihnacht, M. ͑2002͒. ‘‘Subsonic leaky Rayleigh Viktorov, I. A. ͑1967͒. Rayleigh and Lamb Waves ͑Plenum, New York͒. waves at liquid-solid interfaces,’’ Ultrasonics 40, 927–933. ͑ ͒ Qi, Q. ͑1994͒. ‘‘Attenuated leaky Rayleigh waves,’’ J. Acoust. Soc. Am. 95, Zhu, J., and Popovics, J. S. 2001 . ‘‘Non-contact detection of surface waves 3222–3230. in concrete using an air-coupled sensor,’’ in Review of Progress in Quan- Schubert, F., and Marklein, R. ͑2003͒. ‘‘Numerical Computation of Ultra- titative Nondestructive Evaluation ͑American Institute of Physics, sonic Wave Propagation in Concrete using the Elastodynamic Finite Inte- Melville, NY͒, Vol. 20B, 1261–1268.
2110 J. Acoust. Soc. Am., Vol. 116, No. 4, Pt. 1, October 2004 Zhu et al.: Leaky Rayleigh and Scholte waves