Modal expansions for sound propagation in the nocturnal boundary layer Roger Waxlera) National Center for Physical Acoustics, University of Mississippi, University, Mississippi 38677 ͑Received 28 June 2003; revised 9 November 2003; accepted for publication 15 December 2003͒ A modal model is developed for the propagation of sound over an impedance ground plane in a stratified atmosphere which is downward refracting near the ground but upward refracting at high altitudes. The sound’s interaction with the ground is modeled by an impedance with both real and imaginary parts so that the ground is lossy as well as compliant. Such sound speed profiles are typical of the atmospheric boundary layer at night and, together with the ground impedance, have been used extensively to model ground to ground sound propagation in the nocturnal environment. Applications range from community noise and bioacoustics to meteorology. The downward refraction near the ground causes the propagation to be ducted, suggesting that the long range propagation is modal in nature. This duct is, however, leaky due to the upward refraction at high altitudes. The modal model presented here accounts for both the attenuation of sound by the ground as well as the leaky nature of the duct. © 2004 Acoustical Society of America. ͓DOI: 10.1121/1.1646137͔ PACS numbers: 43.20.Bi, 43.28.Js, 43.28.Fp ͓MSH͔ Pages: 1437–1448
I. INTRODUCTION Below the inversion the rate of increase of the sound speed can be large. Care will be taken that the techniques Sound propagation outdoors at night is characterized by developed here do not depend on any assumptions about the the duct that forms in the first few hundred meters of the magnitude of the sound speed gradient near the ground. At atmosphere. Typically, due to cooling of the atmosphere by high altitudes, however, the decrease of the sound speed is the earth, the temperature of the air increases as one moves gradual. It will be assumed here that at sufficiently high al- upward to an inversion point, generally at 100 to 300 m titudes the rate of decrease of the acoustic wavelength is elevation, above which it decreases slowly for the next few small. Some model sound speed profiles are presented in kilometers. This temperature inversion causes a part of the Fig. 1. sound field to propagate along the ground in a ducted fash- Propagation in such sound speed profiles, together with ion. a ground impedance condition6 with both imaginary ͑com- It should be noted that the sound speed depends on hu- pliant͒ and real ͑resistive͒ parts to model the sound’s inter- midity as well as temperature and that atmospheric winds action with the ground, has been widely studied with in- can have a profound effect on propagation.2,3 For low angle tended applications including physical meteorology,5 propagation, however, these effects can be modeled by using community noise modeling7,4 and bioacoustics.8 In modeling the effective sound speed3,4 the sound propagation the parabolic equation and horizontal wave number integration techniques have been preferred.4,9 ϭͱ␥ ͑ ϩ ͒ϩ " c RT 1 0.511q nˆ v. These are robust numerical schemes which work well regard- less of the sound speed profile, however, they are ‘‘black Here c is the sound speed, ␥ is the ratio of the constant box’’ routines which obscure the fact that the physics of pressure to constant volume specific heats for dry air, R is the ducted sound propagation is modal. gas constant for dry air, q is the ratio of the mass of water This paper continues the study begun in Ref. 10 of vapor to dry air, nˆ is the horizontal propagation direction, modal expansions for sound propagation in horizontally and v is the wind speed vector. stratified downward refracting atmospheres over lossy and Some nocturnal profiles for the effective sound speed, compliant ground planes. The main advantage of modal ex- determined by direct measurements of temperature, humid- pansions over other methods is that they give a transparent ity, and wind speed, can be found in Fig. 16 of Ref. 5. The decomposition of the sound field into physically different, distinguishing feature of such sound speed profiles is the independently propagating parts. In addition, once the modes presence of an inversion: the sound speed increases with in- have been computed, they are essentially analytical solutions creasing altitude up to the inversion height, typically found requiring negligible computation times. between 100 to 300 m elevation, above which it slowly de- In Ref. 10 sound speed profiles which become constant creases. The result is an atmosphere which is downward re- at sufficiently high altitudes were studied. A modal expan- fracting near the ground, causing a duct, but upward refract- sion similar to those commonly used in ocean acoustics9 was ing at high altitudes. developed and a technique for finding the corresponding horizontal wave numbers and attenuation coefficients was a͒Electronic mail: [email protected] presented. It was shown that for asymptotically constant
J. Acoust. Soc. Am. 115 (4), April 2004 0001-4966/2004/115(4)/1437/12/$20.00 © 2004 Acoustical Society of America 1437 modes if the bulk attenuation in the acoustic medium is sufficient.12 It will be seen that for most frequencies the modal sum is sufficient to describe the ducted part of the sound field. There are, however, certain narrow frequency bands in which the continuum integral also contributes to the ducted part of the sound field. These are the frequency bands in which the number of terms in the modal sum changes. At these transi- tion frequencies a mode emerges from the continuous spec- trum, accompanied by a sharp peak in the integrand in the continuum integral, reminiscent of the peaks one finds in the lossless case. It will be shown that at such frequencies the continuum integral has a significant contribution which can, at low altitudes, be estimated by a discrete sum over terms of the same form as the terms in the modal sum, obtained from the residue theorem of complex analysis. These additional terms will be referred to as quasi-modes. FIG. 1. Some model sound speed profiles. Profile 1 is used in the numerical The resulting sum of modes and possible quasi-modes simulations shown here. results in a model for sound propagation in the nocturnal boundary layer valid up to ranges of 10 km or more in a frequency range from a few Hz, the frequencies at which sound speed profiles there is a clean separation into an up- modes begin to appear, up to the kHz range, where the num- wardly propagating part and a ducted part, the ducted part of ber of modes begins to become prohibitively large, making the sound field completely described by a sum over a small the method unwieldy. The method is particularly powerful at number of vertical modes whose amplitudes decrease expo- low frequencies for which the number of modes is small. nentially with height. The upwardly propagating part is de- Throughout, the method is illustrated by explicit computa- scribed by an integral whose computation can be intensive, tions for a specific model sound speed profile. These compu- but which is negligible at low angles and long ranges and tations are restricted to frequencies below 100 Hz so that the thus need not be computed. The sum over modes will be number of modes is small and the required numerical tech- referred to as the modal sum while the remaining integral niques are straightforward. will be referred to as the continuum integral. In the asymptotically upward refracting case considered II. FORMULATION OF THE PROBLEM here the duct is imperfect in that sound trapped in the duct Consider the propagation of sound over a plane. Let x can leak upward to the region in which the sound speed H denote the horizontal coordinates ͑those parallel to the plane͒ profile is upward refracting and escape into the upper atmo- and let z denote the vertical coordinate ͑the height above the sphere. It follows that the separation into ducted and up- plane͒. Let t denote time and let ϭ2 f be the angular wardly propagating parts cannot hold with complete rigor. In frequency corresponding to f cycles per second. fact, in the lossless case, in which the real part of the ground 11 The sound speed, c(z), is assumed to depend only on z. impedance is zero, there is no modal sum at all. Rather the Further, it is assumed to be downward refracting near the entire sound field is contained in the continuum integral, the ground but to be upward refracting for large z. Explicitly, ducted behavior manifesting itself in sharp peaks in the inte- c(z) is assumed to have an absolute maximum at some grand. height. Above or below the absolute maximum there may be In the lossy case, in which the real part of the ground local maxima, but generally speaking the sound speed is in- impedance is not zero, it will be seen that there is a modal creasing at the ground and decreasing for sufficiently large z. sum. This is somewhat astonishing since there must be some Let H be the height above which c(z) is strictly decreasing. mechanism through which the ducted part of the sound field The rate of decrease of c(z) at high altitudes is assumed to leaks into the upper atmosphere. The vertical modes will be be slow in the sense that if zӷH, then seen to extend very high into the atmosphere, although typi- ϽϪ Ј͑ ͒Ӷ ͑ ͒ cally with negligible amplitude, before they begin to de- 0 c z . 1 crease exponentially with height. The physical picture is that This means that the relative rate of change of c(z), the power flux into the ground due to the attenuation of cЈ(z)/c(z), divided by 2, is small compared with an acous- sound by the ground is greater than the power flux upward tic wavelength f /c. into the atmosphere due to the asymptotic upward refraction. Recall the model sound speed profiles plotted in Fig. 1. Thus, although the ducted part of the field does leak into the In the numerical results presented in this paper sound speed upper atmosphere, the attenuation by interaction with the profile 1 is used. The analysis presented here applies equally ground is sufficient to prevent the field from extending up- well to sound speed profiles of the form of profiles 2 and 3, ward without bound. A similar phenomenon has been re- however, the numerical analysis depends on the results of cently found in an ocean acoustics propagation model in Ref. 10 which would require a slight generalization to handle which some of the classical leaky modes9 become trapped sound speed profiles with multiple maxima like profile 2.
1438 J. Acoust. Soc. Am., Vol. 115, No. 4, April 2004 Roger Waxler: Modal expansions in the nocturnal boundary layer Introduce the notation 2 Ϫ1/4 i͐z ͱk͑zЈ͒2Ϫ dzЈ Ͼ ͑k͑z͒ Ϫ͒ e z0 if Im 0, ͑z͒ϷAͭ Ϫ Ϫ ͐z ͱ ͑ Ј͒2Ϫ Ј ͑ ͑ ͒2Ϫ͒ 1/4 i z k z dz Ͻ k͑z͒ϭ ͑2͒ k z e 0 if Im 0, c͑z͒ for the height dependent wave number. Note that if zӷH, and those that grow exponentially as z decreases, which is to then 0ϽkЈ(z)Ӷk(z)2. In addition it is assumed kЈ(z)is say that they decrease exponentially as z increases. These bounded, which is to say that, as z→ϱ, k(z) grow no faster later solutions play a prominent role in the analysis presented here and will require their own notation: let, for Im Þ0, than linearly in z. Finally, the mean density, 0(z), is as- ϩ(,z) be the solution of ͑6͒ which has the large z sumed to vary so slowly with z that cЈ/ Ӷ1, so that the 0 0 asymptotic form relative change in density over an acoustic wavelength is negligible. Ϫ Ϫ ͐z ͱ ͑ Ј͒2Ϫ Ј ͑ ͑ ͒2Ϫ͒ 1/4 i z k z dz if Im Ͼ0, The acoustic pressure, P(x ,z,t), is assumed to be time k z e 0 H ϩ͑,z͒Ϸͭ 2 Ϫ1/4 i͐z ͱk͑zЈ͒2Ϫ dzЈ Ͻ harmonic ͑k͑z͒ Ϫ͒ e z0 if Im 0, ͑ ͒ ͒ϭ ˆ ͒ Ϫit ͑ ͒ 9 P͑xH ,z,t Re P͑xH ,z e , 3 for some z sufficiently large so that k(z)2ϾRe for z where the amplitude Pˆ (x ,z) satisfies the Helmholtz equa- 0 H уz . tion 0 The solutions ϩ(,z) do not generally satisfy the 2ϩk͑z͒2͒Pˆ ͑x ,z͒ϭ0. ͑4͒ boundary condition ͑7͒. However, there may be a discrete setٌ͑ H of values of , ͕ 1 , 2 ,..., N͖, for which the solution To model the effect of the ground an impedance condition is 13 ϩ(,z) given above does satisfy ͑7͒. The values assumed at zϭ0, ͕ 1 , 2 ,..., N͖ will be referred to as the point spectrum. .Here N may be any whole number from 0 to ϱ ͒ ˆ ץ P͑xH ,z ͯ ϭϪC͑͒Pˆ ͑x ,0͒, ͑5͒ It follows from the large z asymptotics ͑8͒ that the z H jץ zϭ0 must have nonzero imaginary parts. In particular, if C were where C͑͒ is related to the impedance Z() through real valued, then the boundary condition ͑7͒ would make the eigenvalue problem ͑6͒ self-adjoint forcing the point spec- ͑ ͒ i 0 0 13 C Z͑͒ϭ . trum to be real valued. It follows that if is real valued, C͑͒ then there is no point spectrum and Nϭ0. In general, both the real and imaginary parts of Z() are Given an j in the point spectrum, the mode corre- nonzero. In the explicit computations presented here C͑͒ is sponding to j is given by chosen as in Eq. ͑24͒ of Ref. 10. ϱ Ϫ1/2 2 Equation ͑3͒ will be solved by separating variables and ͑z͒ϭͩ ͵ ϩ͑ ,z͒ dzͪ ϩ͑ ,z͒. ͑10͒ j j j using an eigenfunction expansion10,13 in the vertical coordi- 0 nate. The vertical equation may be written This normalization is chosen so that d2 ͩ ϩk͑z͒2Ϫͪ ͑z͒ϭ0, ͑6͒ ϱ dz2 ͵ ͑ ͒2 ϭ ͑ ͒ j z dz 1. 11 0 where is the separation parameter. The impedance condi- ͑ ͒ ͑ ͒ 2 tion 5 leads to Note that in Eq. 11 the function squared, j(z) , rather than the complex modulus squared, ͉(z)͉2, is integrated. Ј͑0͒ϭϪC͑0͒. ͑7͒ This is the normalization appropriate to the eigenfunction Consider the large z behavior of the solutions of the expansion which arises from the non-self-adjoint boundary differential equation ͑6͒.Ifz is large enough so that ͑1͒ is condition ͑7͒.10,13 In practice it can be difficult to compute valid, then the WKB approximation9,14 may be used. For real the integral in ͑10͒. An alternative formula, ͑A8͒, is pre- valued one obtains the asymptotic form sented in the Appendix. →ϱ Ϫ ͐z ͱ ͑ Ј͒2Ϫ Ј The decrease in the magnitude of ϩ( j ,z)asz is ͑ ͒Ϸ͑ ͑ ͒2Ϫ͒ 1/4͑ i z k z dz z k z Ae 0 not quite exponential. From the asymptotic form ͑9͒ one ob- Ϫi͐z ͱk͑zЈ͒2Ϫ dzЈ tains ϩBe z0 ͒ ͑8͒
→ϱ 2Ͼ ϩ͑,z͒ as z ; here z0 is large enough so that k(z) for z у z0 insuring that no turning points are encountered in the Ϸ͑k͑z͒2Ϫ͒Ϫ1/4 integration and A and B are constants. Choose the square z ͱ 2 z ͱ 2 roots in ͑8͒ so that they have non-negative real parts. Note Ϫi͐ k͑zЈ͒ ϪRe dzЈϪ͑1/2͒Im ͐ ͓1/ k͑zЈ͒ ϪRe ͔ dzЈ ϫe z0 z0 . that for real values of the solutions are asymptotically os- cillatory and remain bounded as z→ϱ. Note that ͱk(z)2ϪRe is an increasing function of z for If Im Þ0, then the WKB approximation provides two large z. It follows that the first integral in the exponential types of solutions: those that grow exponentially as z in- function above grows more rapidly with z than a linear func- creases, tion of z,
J. Acoust. Soc. Am., Vol. 115, No. 4, April 2004 Roger Waxler: Modal expansions in the nocturnal boundary layer 1439 1 z N ϱ lim ͵ ͱk͑zЈ͒2ϪRe dzЈϭϱ, ˆ ͑ ͒ϭ ͑ ͒ ͑ ͒ϩ ͵ ͑⑀ ͒ ͑ ͒ ⑀ P xH ,z ͚ p j xH j z p ,xH ⑀ z d z→ϱ z z0 jϭ1 Ϫϱ ͑15͒ while the second integral grows more slowly than a linear function of z, with expansion coefficients ϱ ͑ ͒ϭ ͵ ͑ ͒ ˆ ͑ ͒ ͑ ͒ 1 z 1 p j xH j z P xH ,z dz 16 lim ͵ dzЈϭ0 0 →ϱ z z ͱk͑zЈ͒2ϪRe z 0 and ͓for example, if k(z)2 is linear for large z, then this last ϱ 1/2 ͑⑀ ͒ϭ ͵ ͑ ͒ ˆ ͑ ͒ ͑ ͒ integralϳconst z ]. Thus, ϩ(,z) asymptotically oscil- p ,xH ⑀ z P xH ,z dz. 17 • 0 lates more rapidly than a sinusoid with an amplitude that decreases more slowly that an exponential. In addition, to The expansion coefficients satisfy the two dimensional reach this asymptotic form, z must be large enough so that Helmholtz equations 2ӷ ͒ ͑ k(z) Re . Thus the asymptotic decrease in the magnitude ٌ͑2 ϩ ͒ ͑ ͒ϭ H j p j xH 0 18 of ϩ( j ,z) does not generally begin until quite high up in the atmosphere. and The set of values of for which the solutions of ͑6͒ ϩ⑀͒p͑⑀,x ͒ϭ0 ͑19͒ 2ٌ͑ subject to ͑7͒ are not exponentially decreasing but do remain H H bounded as z→ϱ is called the continuous spectrum. It fol- for xH in source free regions of space. Here ͑ ͒ ץ lows from 8 that the continuous spectrum is the entire real line. The associated solutions of ͑6͒ subject to ͑7͒ are called xץ continuum modes. Let ⑀(z) be the continuum mode associ- ٌ ϭ H ͩ ͪ . ץ ated to the real number ⑀. If appropriately normalized, then yץ one has biorthonormality in the sense that ϱ Given appropriate boundary conditions or sources, ͑18͒ and ͑ ͒ ͑ ͒ ϭ␦͑⑀Ϫ⑀Ј͒ ͵ ⑀ z ⑀Ј z dz , ͑19͒ are straightforward to solve. 0
ϱ ͵ ͑ ͒ ͑ ͒ ϭ␦ III. FINDING THE POINT SPECTRUM j z k z dz jk , 0 Once the spectrum is known, producing the associated and modes is a straightforward numerical exercise. In principle, given , one solves ͑6͒ subject to ͑7͒. For the continuum ϱ ͵ ͑ ͒ ͑ ͒ ϭ modes this procedure works well since the continuous spec- ⑀ z j z dz 0, 0 trum is known a priori and consists of real values of for ͑ ͒ 15,13 which the solutions of 6 are bounded. and completeness The chief difficulty is in finding the point spectrum. Fur- ͑ ͒ N ϱ ther, once the point spectrum is known, solving 6 subject to ␦͑ Ϫ Ј͒ϭ ͑ ͒ ͑ Ј͒ϩ ͵ ͑ ͒ ͑ Ј͒ ⑀ ͑7͒ is numerically unstable for complex valued because of z z ͚ j z j z ⑀ z ⑀ z d . jϭ1 Ϫϱ the exponential behavior of solutions to ͑6͒. It is preferable ͑12͒ to solve ͑6͒ subject to the condition that In the Appendix it is shown how one may normalize Ј͑z͒ Ϫiͱk͑z͒2Ϫ if Im Ͼ0, C ͑ ͒ Ϸͭ ͑ ͒ ⑀(z). Let Ϫ( , ,z) be the solution of 6 satisfying the 20 ͑z͒ iͱk͑z͒2Ϫ if Im Ͻ0, boundary condition ͑7͒ normalized by Ϫ(C,,0)ϭ1 and let ϩ(,z) be as above. Let ⑀ be a real number and let for large z. This forces the solution to have the asymptotic ͑ ͒ 9,14 ϩ ϩ form 9 and is numerically stable. FϮ͑C,⑀͒ϭϩЈ ͑⑀ϯi0 ,0͒ϩCϩ͑⑀ϯi0 ,0͒. ͑13͒ A. Solving the eigenvalue equation Here the prime indicates the derivative with respect to z and Consider the eigenvalue problem given by ͑6͒ and ͑7͒. ⑀Ϯi0ϩ indicates the limiting value as ϮIm ⑀ approaches Recall the solution, ϩ(,z), of ͑6͒ chosen to have the zero while remaining positive. One then has asymptotic form ͑9͒ for large z.Ifϩ(,z) satisfies ͑7͒, then is in the point spectrum and ϩ(,z) is, up to normaliza- 1 ϭ C ⑀ ͑ ͒ tion, the eigenfunction corresponding to . Dividing both ⑀͑z͒ ͱ Ϫ͑ , ,z͒. 14 FϪ͑C,⑀͒Fϩ͑C,⑀͒ sides of ͑7͒ by ͑0͒ it follows that the point spectrum is precisely the set of all with Im Þ0 for which ˆ One can now express the pressure amplitude P(xH ,z)in Ј ͑,0͒ an eigenfunction expansion with respect to the eigenfunc- ϩ ϩCϭ ͑ ͒ ͒ 0. 21 tions j and ⑀ . One obtains ϩ͑ ,0
1440 J. Acoust. Soc. Am., Vol. 115, No. 4, April 2004 Roger Waxler: Modal expansions in the nocturnal boundary layer FIG. 2. The decomposition of the wave number squared for soundspeed profile 1 into asymptotically constant and asymptotically upward refracting parts.
2 To specify ϩ( ,z) completely it is necessary to fix z0 d ͑ ͒ ͩ ϩv ͑z͒ϩv ͑z͒Ϫ͑Ϫk2͒ͪ ͑z͒ϭ0. in 9 . However, with the eigenvalue condition written in the dz2 1 2 0 ͑ ͒ form 21 it becomes clear that specifying z0 is not necessary since only the logarithmic derivative of ϩ(,z) is required. It will be shown that the point spectrum can be obtained by One need only find any solution of ͑6͒ which satisfies ͑20͒ treating v2(z) as a perturbation. for large z, track its logarithmic derivative back to zϭ0, and Let ͑ ͒ then substitute into 21 . In principle one could use this pro- CϭAϩiB cedure to search for the roots of ͑21͒. In practice such procedures are prone to failure.9 Not be the decomposition of C into its real and imaginary parts. only is it difficult to search blindly for roots in the complex Let L0 denote the differential operator obtained from plane, but such a search is further complicated here by the d2 ͑ ͒ ϭ ϩ exponential growth and decay of the solutions to 6 which L0 2 v1 can make numerical procedures unstable. To produce stable dz and robust algorithms for finding the point spectra one must by imposing ͑7͒ with Bϭ0, have some means to analytically estimate the roots with suf- Ј͑ ͒ϭϪA͑ ͒ ficient accuracy to allow standard numerical routines ͑the 0 0 , ͒ Newton–Rapheson method, for example to converge rap- let L1 denote the differential operator obtained from idly from the estimate to the exact value. d2 The technique used in Ref. 10 to find the point spectrum, L ϭ ϩv , that of treating Im C as a perturbation, cannot be applied here 1 dz2 1 Cϭ since if Im 0, then, as pointed out in the previous section, by imposing ͑7͒ with BÞ0, and let L denote the differential there is no point spectrum to perturb. Instead decompose operator obtained from k(z)2 as follows: d2 ϭ ϩ ϩ ͑ ͒2ϭ 2ϩ ͑ ͒ϩ ͑ ͒ ͑ ͒ L 2 v1 v2 k z k0 v1 z v2 z , 22 dz ͑ ͒ BÞ where by again imposing 7 with 0. The operator L0 is self-adjoint and numerous techniques 9,16,17 k ϭk͑H͒ exist for finding its spectrum. The point spectrum of L0 0 (0) (0) consists of M real, positive eigenvalues 1 ,..., M . The is the minimum value of k, continuous spectrum of L0 is the negative real half-line, ͑Ϫϱ,0͒. The techniques of Ref. 10 can then be applied to k͑z͒2Ϫk2 if zрH, produce the point spectrum of L1 from that of L0 ; the con- ͑ ͒ϭͭ 0 tinuous spectrum is unchanged. One finds that the point v1 z 0ifzуH, (1) (1) spectrum of L1 consists again of M values, 1 ,..., M , all (1) with positive real and imaginary parts. Let j (z) be the and ͓ ͑ ͔͒ eigenfunction normalized as in 11 of L1 corresponding to (1) . 0ifzрH, j ͑z͒ϭͭ It remains to be shown that the point spectrum of L v2 ͑ ͒2Ϫ 2 у ϭ ϩ k z k0 if z H. L1 v2 can be obtained from that of L1 . Note that v2(z)is a rather singular perturbation of L1 . In particular, the con- ͑ ͒ In Fig. 2 the decomposition 22 is depicted for the vertical tinuous spectra of L and L1 are drastically different. As dis- sound speed profile 1 from Fig. 1. cussed in the previous section the continuous spectrum of L ͑Ϫϱ ϱ͒ Since k(z) is slowly varying for large z it follows that is the entire real line, , , regardless of how slowly v2 ͑ ͒ B v2(z) is as well. Write Eq. 6 in the form increases with height. Further, in the self-adjoint case,
J. Acoust. Soc. Am., Vol. 115, No. 4, April 2004 Roger Waxler: Modal expansions in the nocturnal boundary layer 1441 FIG. 3. The modal dispersion rela- tions. Plotted are the horizontal wave
numbers relative to k0 , ͱ 2ϩ Re k0 j/k0 , and the attenuation ͱ 2ϩ constants, Im k0 j.
ϭ 0, it was seen that L1 has point spectrum while L does not. Note that for a given v2 the magnitude of the perturba- This self-adjoint case has been studied in detail in a classic tion is determined by the vertical extent of the unperturbed paper by Titchmarsh.11 What remains of the point spectrum mode. If the mode is concentrated near the ground and has in this case are poles, called resonances, in the analytic con- insignificant magnitude for zϾH, then the perturbation will tinuation in ⑀ of the continuum eigenfunctions. It is clear be insignificant as well. If the magnitude of the mode de- from ͑13͒ that the resonances are solutions of the eigenvalue creases only slowly with height and is still appreciable for equation ͑21͒ for which the corresponding solution of ͑6͒ is zϾH, then the perturbation can be significant, perhaps suf- not decreasing with height; it follows that the imaginary part ficiently so that j is not in the point spectrum. Since one has ͑ ͒ of the resonance must be negative. Titchmarsh shows that the ͑1͒ Ϫͱ 0 Ϫ 2 ͉ ͑ ͉͒ϳ j k0z real part of a resonance is well approximated by formal per- j z const e turbation theory18 while the imaginary part is smaller than (0) for large z the perturbation is significant only if j , the any power of v (z) ͑which is to say that it is exponentially 2 2 corresponding eigenvalue of L0 , is sufficiently close to k0. small in the size of the perturbation͒. An approach similar to that of Titchmarsh can be taken B. Numerical methods here in this non-self-adjoint context. Perturbation theory can be adapted in a straightforward manner to the bi- The form ͑21͒ for the eigenvalue equation is not the orthogonality relevant to the current situation. One finds that, most convenient for numerical analysis since it has poles at to leading order, the values of for which the Dirichlet problem ϩ(,0) ϭ ϱ 0 has a solution. An eigenvalue equation must be found ϭ͑1͒ϩ ͵ ͑1͒͑ ͒2 ͑ ͒ ϩ ͑ ͒ which has no poles in the region of interest. A convenient j j j z v2 z dz ¯ . 23 0 choice is ͑ ͒ In practice the approximation 23 has proven to be suffi- ϩЈ ͑,0͒ϩCϩ͑,0͒ ciently accurate to provide an initial point from which stan- ϭ0. ͑24͒ Ј ͑,0͒Ϫiϩ͑,0͒ dard numerical routines converge rapidly to the solution of ϩ the eigenvalue equation ͑21͒. This form still has poles, at the zeros of ϩЈ (,0) As discussed in the previous section, the mode j cor- Ϫiϩ(,0), but they are in the lower half-plane sufficiently responding to j extends much higher into the atmosphere distant from the real line. To compute ϩ(,0) fix some (1) than j , although in most cases with very small amplitude. value of z0 which is large enough for the asymptotic form ͑ ͒ ͑ ͒ In particular j has slightly less contact with the ground than 20 to be accurate and then solve 6 , using any convenient (1) ͑ ͒ j and thus is attenuated slightly less by the ground. One numerical solver, subject to 20 imposed as a boundary con- thus expects that dition at z0 . In practice the estimate ͑23͒ is found to be quite accu- Ͻ ͑1͒ Im j Im j , (1) rate, except for those occasions in which j is too close to 2 ͑ ͒ just as in the self-adjoint case. This expectation is borne out k0. Even in these cases, however, it has been found that 23 by the numerical results presented below. Indeed, it is pos- is sufficiently accurate to provide a starting point from which ͑ ͒ sible that the perturbation causes the imaginary part of j to the Newton–Rapheson method applied to 24 converges be negative; as pointed out above, in the self-adjoint case all rapidly, even when Im j turns out to be negative. In the of the eigenvalues aquire a negative imaginary part. If the examples worked in this paper the bisection method was imaginary part becomes negative, then the corresponding so- employed to find the point spectrum of L0 . The perturbative lution of ͑6͒ and ͑7͒ grows exponentially as z→ϱ and the estimate of Ref. 10 was then used as a starting point for a eigenvalue becomes a resonance and ceases to be in the point Newton–Rapheson solver applied to the eigenvalue equation spectrum. It follows that NрM: the number of eigenvalues for L1 . in the point spectrum of L is less than or equal to the number The dispersion obtained from applying these methods to of eigenvalues in the point spectrum of L0 , the operator sound speed profile 1 from Fig. 1 is plotted in Fig. 3. Plotted ϭͱ 2ϩ relevant to a strictly downward refracting atmosphere with are the real and imaginary parts of kHj k0 j as a func- no attenuation at the ground. tion of frequency. The real part is normalized by k0 . The real
1442 J. Acoust. Soc. Am., Vol. 115, No. 4, April 2004 Roger Waxler: Modal expansions in the nocturnal boundary layer FIG. 4. The modal dispersion relations for the fifth mode in the asymptoti- cally constant case and the asymptotically upward refracting case are com- pared near a transition frequency. Plotted are the attenuation constants, ͱ 2ϩ ͱ 2ϩ Ϫ Im k0 j, versus the horizontal wave numbers, Re k0 j k0 , and then the attenuation constants versus frequency.
FIG. 5. The modes at 50 Hz for sound speed profile 1. part is referred to as the horizontal wave number and the imaginary part as the attenuation constant. In Fig. 4 the dis- persion of mode 5 for the asymptotically constant approxi- at 50 Hz is plotted at high altitudes. Physically this long tail mation and the full asymptotically upward refracting profile at high amplitudes represents the leaking of acoustic energy ͱ 2ϩ Ϫ are compared. In the upper plot k0 5 k0 and from the duct up into the atmosphere. The eventual decay of ͱ 2ϩ(1)Ϫ k0 5 k0 are tracked in the complex plane. In the the mode with height reflects the fact that the rate at which ͱ 2ϭ lower plot the attenuation coefficients Im k0 5 and energy is lost into the ground is much greater than the rate at ͱ 2ϩ(1) Im k0 5 are plotted as functions of frequency. Note that which it can leak into the upper atmosphere. the asymptotically constant approximation can have more modes than in the asymptotically upward refracting case. In both cases the mode vanishes at the frequency at which the attenuation constant becomes zero; however, in the asymp- totically constant approximation the horizontal wave number approaches k0 while in the asymptotically upward refracting case the horizontal wave number remains slightly larger than k0 . Once the eigenvalues are found, the corresponding modes are obtained by computing ϩ( j ,z) and then nor- malizing according to ͑11͒. Computing the normalization from ͑10͒ can be difficult, however, since ϩ(,z) can de- crease very slowly as z→ϱ. Instead ͑A8͒ is used. In Fig. 5 the modes at 50 Hz for sound speed profile 1 are plotted. They are labeled by their corresponding horizon- tal wave number and attenuation coefficient. In Fig. 6 the same plots are found, but for 41.6 Hz. This is a transition frequency at which the number of modes increases from three to four ͑see Fig. 3͒. Accordingly, the fourth mode has horizontal wave number very close to k0 . Note that the fourth mode at 41.6 Hz extends high up into the atmosphere oscillating before it eventually slowly decays in accordance with ͑9͒. In fact, all of the modes have this behavior, although except in the exceptional case of a transition frequency at which the number of modes changes, the oscillations at high altitude have negligably small ampli- FIG. 6. The modes at the transition frequency 41.6 Hz for sound speed tude. This is demonstrated in Fig. 7 in which the fourth mode profile 1.
J. Acoust. Soc. Am., Vol. 115, No. 4, April 2004 Roger Waxler: Modal expansions in the nocturnal boundary layer 1443 are oscillatory with behavior given, for sufficiently large z, ͑ ͒ ⑀ Ͻ by 8 . Between the two turning points, for 0( ) z Ͻ ⑀ 14 1( ), the modes grow exponentially with height. It fol- lows that in this case the modes have negligible amplitude near the ground. Their amplitude increases rapidly with ⑀ height up to the vicinity of the turning point 1( ), above which they oscillate with gradually decreasing amplitude. These modes contain that part of the sound field radiated from a source above the sound speed inversion which is re- fracted upwards away from the ground, not penetrating be- low the inversion. ⑀ 2 When is comparable to k0 two factors can complicate the behavior of the continuum modes: there is the coales- cence of the two turning points discussed in the last para- graph and there is the proximity of the point spectrum. At high altitudes the continuum modes will be oscillatory. At low altitudes the coalescence of the turning points means that no simple and general behavior can be expected. The behav- FIG. 7. The high altitude behavior of mode 4 at 50 Hz for sound speed ior of the modes reflects the details of the low altitude form profile 1. of the sound speed profile. The proximity of the point spectrum produces sharp peaks in the low altitude behavior of those modes for which ⑀ is in the vicinity of the real part of an eigenvalue in the IV. THE MODES OF THE CONTINUOUS SPECTRUM point spectrum. This is seen clearly in ͑14͒: the eigenvalue ͑ ͒ ϩ C ϭ Consider the modes of the continuous spectrum ⑀(z) condition 21 is precisely the condition that F ( , ) 0so given by ͑14͒. The dependence of ⑀(z)on⑀ may be divided that ⑀ has poles where the eigenvalue condition is satisfied. ⑀ into three general types according to whether ⑀ is somewhat The effect of these poles is tempered by the fact that if is 2 2 2 sufficiently greater than k2, then the corresponding mode is less than k0, comparable to k0, or somewhat greater that k0. 0 In Fig. 8 continuum modes at 50 Hz for sound speed profile still exponentially decreasing with decreasing z below the 1 displaying the three types of behavior are plotted. inversion. ⑀ 2 This behavior is demonstrated in Figs. 9͑a͒ and 9͑b͒ in When is somewhat less than k0 the behavior of the modes is oscillatory over the entire domain. These are the which the continuum mode amplitude on the ground, modes which contain that part of the sound field which ͉⑀(0)͉, for sound speed profile 1 is plotted as a function of propagates from low altitudes up into the atmosphere or the ⑀ for 50, 52, 54, 56, and 58 Hz. Both ͉⑀(0)͉ and log ͉⑀(0)͉ reverse. Note that in this case the magnitude of a mode does are plotted in Figs. 9͑a͒ and 9͑b͒, respectively. Note that the not change dramatically with increasing height. peaks in ͉⑀(0)͉ are sharpest for transition frequencies near ⑀ 2 When is somewhat greater than k0 there are one or which the number of modes changes. Indeed these are the more turning points at height z˙ for which k(z)2ϭ⑀. For frequencies for which one of the eigenvalues in the point sound speed profile 1 there is always one turning point, to be spectrum can be arbitrarily close to the continuous spectrum. ⑀ ͉ ͉ denoted 1( ) for purposes of discussion, above the sound The plot of log ⑀(0) shows that all of the eigenvalues in the speed inversion. If ⑀Ͻk(0)2, there is a second, to be denoted point spectrum show up as peaks in the continuous spectrum; ⑀ ⑀у 2 2 0( ), below the sound speed inversion. If k(0) , let however, unless their real parts are sufficiently close to k0, ⑀ ϭ Ͼ ⑀ 0( ) 0. Above the turning points, for z 1( ), the modes the exponential behavior of the mode overwhelms the peak.
FIG. 8. Continuum modes at 50 Hz for sound speed
profile 1. The three types of modal behavior, for kH ϭͱ⑀Ͻ ϭ Ͼ k0 , k0 , k0 , are displayed: below k0 the modes are oscillatory with little variation in amplitude,
near k0 the modes are strongly influenced by the duct, above k0 the modes have negligible amplitude near the ground.
1444 J. Acoust. Soc. Am., Vol. 115, No. 4, April 2004 Roger Waxler: Modal expansions in the nocturnal boundary layer N ͒ϭ ͒ ͒ ͒ G͑ ,xH ,z0 ,z ͚ GH͑ j ,xH j͑z0 j͑z jϭ1
ϱ ϩ ͵ ͑⑀ ͒ ͑ ͒ ͑ ͒ ⑀ ͑ ͒ GH ,xH ⑀ z0 ⑀ z d , 27 Ϫϱ ⑀ where GH( ,xH) is the free space Green’s function satisfying ͒ ͑␦ϩ⑀͒ ͑⑀ ͒ϭ 2ٌ͑ H GH ,xH xH . In polar coordinates r, , GH is independent of . One finds9,19,20 i G ͑⑀,x ͒ϭϪ H͑1͒͑ͱ⑀r͒, ͑28͒ H H 4 0 14 where H0 is the Hankel function. Once the modes j have been computed, computing the contribution to ͑27͒ from the modal sum is straightforward. ͑ ͒ ͉ ͉ ͑ ͒ FIG. 9. Continuum mode amplitudes on the ground, a ⑀(0) and b The contribution from the integral over the continuum modes ͉ ͉ log ⑀(0) , for sound speed profile 1 for several frequencies. must be computed numerically. This computation can be in- tensive. However, if one is interested in the sound field near the ground and far from the source, then it can be expected To extract the contribution to integrals such as those that that the integral over the continuous spectrum is appear in ͑15͒ from these peaks it is useful to know the insignificant10,9 so that one need only compute the modal residues of the poles that cause them. Consider sum. For the sound speed profiles considered here it will be 1 shown that it is indeed the case for most frequencies that the ͒ ͒ϭϪ C ⑀ ͒ C ⑀ ͒ ⑀͑z1 ⑀͑z2 Ϫ͑ , ,z1 Ϫ͑ , ,z2 . continuous spectrum is insignificant for low altitudes at long Fϩ͑⑀͒FϪ͑⑀͒ ͑25͒ ranges. The exceptions are those narrow bands of transition frequencies in which the number of modes changes. As dis- cussed in the previous section, for these frequencies the Let ,..., be the solutions of the eigenvalue condition 1 M emerging mode manifests itself as a sharp peak in the con- ͑7͒ which were obtained in Sec. III by perturbing off of the tinuum mode amplitudes. asymptotically constant case. Recall that, depending on the The long range approximation to the Green’s function G signs of their imaginary parts, not all of the are necessar- j is obtained by replacing the Hankel function in ͑28͒ by its ily in the point spectrum and thus not all represent true large argument asymptotic form,9,14,20 ducted modes. FϮ(⑀) are defined for ϮIm ⑀у0, but have straightforward analytic continuations into regions with i ͱ Ϯ ⑀Ͻ ϭ G ͑⑀,r͒ϷϪͱ ei ⑀r, Im 0. One then has Fϩ( j) 0, regardless of the sign H ͱ ⑀ 8 ⑀r of Im j . Thus, ⑀(z0) ⑀(z) has poles, as a function of ,at ⑀ Ͼ the j . When Im j 0, so that j is in fact in the point and dropping the part of the integral in ͑27͒ which does not spectrum, then Ϫ2i times the residues at these poles gives propagate horizontally, that is, that part of the integral with the modes, j , corresponding to j : ⑀р0. One obtains N i Ϫ ͱ 2i ͑ ͒ϷϪͱ ͫ 1/4 i jr ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ϭ ͑C ͒ ͑C ͒ G ,r,z0 ,z ͚ j e j z0 j z j z1 j z2 ͒ Ј ͒ Ϫ , j ,z1 Ϫ , j ,z2 8 r jϭ1 FϪ͑ j Fϩ͑ j ͑26͒ ϱ ϩ ͵ ⑀Ϫ1/4 iͱ⑀r ͑ ͒ ͑ ͒ ⑀ͬ ͑ ͒ e ⑀ z0 ⑀ z d . 29 ͓ ⑀ϩ ϩ C 0 this follows from the fact that ϩ( i0 ,0) Ϫ( , j ,z) ϩ ϭϩ(⑀ϩi0 ,z) and from ͑A4͒ and ͑A5͒; see the Appen- Note that for fixed z the continuum modes ⑀(z)goto ͔ ⑀ Ͻ dix . When Im j 0, introduce the notion of the quasi-mode zero exponentially in ⑀ as ⑀→ϱ so that the improper integral ͑ ͒ j defined by 26 . in ͑29͒ converges rapidly. As mentioned above, performing these integrals is computationally intensive. However, it is generally expected to be unnecessary to compute the con- iͱ⑀r V. LONG RANGE PROPAGATION FROM A POINT tinuum integral since at long ranges the term e oscillates ⑀Ϫ1/4 SOURCE rapidly compared to the variations of ⑀(z0) ⑀(z), causing the integral to be negligible.9 Consider the sound field produced by a point source at An exception to this rule is the case in which ϭ ϭ ͑ ͒ xH 0, z z0 . The solution to 4 for a point source at xH ⑀(z0) ⑀(z) is sharply peaked. As discussed in the previous ϭ ϭ 0, z z0 is proportional to the Green’s function section this occurs near those frequencies at which the num-
J. Acoust. Soc. Am., Vol. 115, No. 4, April 2004 Roger Waxler: Modal expansions in the nocturnal boundary layer 1445 ͉ ͉ tude of the Green’s function times range, xH G( ,xH,0,z), is displayed. This quantity is the magnitude in dB of the sound field produced by a point source relative to the field that would be produced by a point source of equal strength at ͉ ͉ a distance xH from the source in an empty homogeneous FIG. 10. The integration contour to extract the effect of poles passing near space. The full field is compared to the individual contribu- the continuous spectrum. tions from the modal sum and the continuum integral. Fre- quencies 50 Hz, which is not a transition frequency, and 54 Hz, which is a transition frequency, are shown in Figs. 11͑a͒ ber of modes changes. The contribution from these peaks can and 11͑b͒, respectively. The small fluctuations in the compu- ͑ ͒ be extracted by deforming the integration contour in 29 tation of the continuum integral at short ranges is the result ⌫ below the real line to the contour depicted in Fig. 10. The of numerical noise associated with the vertical discretization iͱ⑀r ⑀Ͻ term e grows exponentially with range for Im 0, how- of the continuum modes as ⑀↓0. The sum of modes and ⌫ ever, the contour need only descend below the real line a quasi-modes alone, ͑30͒, is seen to be an excellent approxi- ͉ ͉ distance large compared to the Im j and thus may be cho- mation to the full field. sen so that this exponential growth is insignificant for ranges Ӷ ͉ ͉ ͑ r 1/ Im N here N is used here since the imaginary parts ͒ of the j typically decrease with increasing j . In the ex- VI. CONCLUSIONS amples considered here ͉Im ͉Ӷ10Ϫ4 per meter so that for j A vertical modal expansion has been produced for the rр10 km this deformation can be made. In addition, the ana- model commonly used to describe outdoor sound propaga- lytic continuations of the continuum modes to complex ⑀ will tion on calm nights over flat ground. In this model the sound grow exponentially as z→ϱ. However, for low altitudes, for field is refracted downwards near the ground, upwards at ranges up to 10 km, and for ⌫ sufficiently far from the poles high altitudes and is attenuated by its interaction with the the contribution from the resulting integral is negligible. ground. The full sound field is given by a sum over ducted If any poles are encountered in making this deformation, modes plus an integral over continuum modes as given in their residues ͑26͒ must be subtracted from the integral. The ͑15͒. The ducted modes decrease exponentially at high alti- long range contribution from the continuum integral is con- tudes while the continuum modes oscillate with very slowly tained entirely in these residues. Ignoring the continuum in- decreasing amplitude at high altitudes. tegral over the deformed contour ⌫ one obtains as an ap- Except for a discrete set of narrow frequency bands in proximation to ͑29͒ which the number of ducted modes changes, at long ranges N the sound field near the ground is well approximated by the i Ϫ ͱ ͑ ͒ϷϪͱ ͫ 1/4 i jr ͑ ͒ ͑ ͒ G ,r,z0 ,z ͚ j e j z0 j z modal sum alone. For these exceptional frequencies the con- 8r jϭ1 tinuum integral has a long range contribution to the sound M field. At low altitudes, however, the contribution from the Ϫ ͱ ϩ 1/4 i jr ͑ ͒ ͑ ͒ͬ ͑ ͒ ͚ j e j z0 j z , 30 continuum integral is well approximated by a sum over ϭ ϩ j N 1 quasi-modes. The quasi-modal sum has the same form as the where, for jϾN, the quasi-modes defined by ͑26͒ are used. modal sum except that the quasi-modes increase exponen- To check the accurracy of the approximation ͑30͒ the tially at high altitudes. continuum integral has been computed using a trapezoidal The modes and quasi-modes, and the associated wave rule approximation with variable step size ͑a finer step size is numbers and attenuation constants, can be obtained by per- ⑀ϳ 2 used in the region near Re k0 where the integrand is more turbing about a sound speed profile which is constant at high rapidly varying͒. The integral is performed along the de- altitudes, reducing the problem to the type solved in Ref. 10. formed contour ⌫. In Fig. 11, ground to ground propagation, Given the modes and quasi-modes, the result, presented in ϭ ϭ ͑ ͒ z0 z 0, is studied. Ten times the logarithm of the magni- 30 for propagation from a point source, is an easily evalu-
FIG. 11. A comparison of the full sound field with the individual contri- butions from the modal sum and the continuum integral for ground to ground propagation. The pressure field produced by a unit point source on the ground is shown as a function of range and height in dB relative to 1/range: ͑a͒ 50 Hz, no quasi-modes, ͑b͒ 54 Hz, one quasi-mode.
1446 J. Acoust. Soc. Am., Vol. 115, No. 4, April 2004 Roger Waxler: Modal expansions in the nocturnal boundary layer ated analytical expression valid at low altitudes for frequen- Ϫ͑C,,0͒ϭ1, ͑A3͒ cies greater than a few Hz. ϩ(,z), defined only for R, is the exponentially de- W ACKNOWLEDGMENTS creasing solution, and denotes the Wronskian. Note that it follows from ͑A2͒ and ͑A3͒ that It is a pleasure to thank Ken Gilbert with whom every aspect of this work has been discussed. His insight and guid- W͑ ͑C ͒ ͑ ͒͒ϭϪ͑Ј ͑ ͒ϩC ͑ ͒͒ Ϫ , ,• , ϩ ,• ϩ ,0 ϩ ,0 . ance have been invaluable. In addition I’d like to thank Car- ͑A4͒ rick Talmadge for many helpful discussions. This work was supported by ARDEC Contract No. DAAE30-02-D-1017. Further, using the asymptotic form ͑9͒, one finds that APPENDIX: NORMALIZING THE MODES W͑ ͑⑀ϩ ϩ ͒ ͑⑀Ϫ ϩ ͒͒ϭϪ ͑ ͒ ϩ i0 ,• , ϩ i0 ,• 2i. A5 In this appendix it is shown how, under rather general In addition to ͑A1͒ one has the eigenfunction expansion, conditions, the ducted and continuum modes can be normal- N 1 ized. Normalizing the ducted modes by computing the nor- ͑C ͒ϭ ͑ ͒ ͑ ͒ g , ,z1 ,z2 ͚ Ϫ j z1 j z2 malization integral ͑10͒ can be difficult since the modes can jϭ1 j extend very high up into the atmosphere. Instead, a form ϱ 1 similar to that of Appendix B5 of Ref. 9 will be used. In Ref. ϩ ͵ ͑ ͒ ͑ ͒ ⑀ ͑ ͒ ⑀ z1 ⑀ z2 d , A6 10 the continuum modes were normalized using the fact that Ϫϱ⑀Ϫ for asymptotically constant sound speed profiles the con- which follows directly from the definitions of j and ⑀ and tinuum modes become trigonometric functions for large z so from the completeness relation ͑12͒. Using the fact that when that a comparison could be made to the well known sine and ϭ cosine transforms. For the sound speed profiles considered in j ͑ ͒ ͑C ͒ϭ ͑ ͒ ͑ ͒ this paper this is no longer the case. Instead, formulas are ϩ j,0 Ϫ , j ,z ϩ j ,z A7 C developed which do not depend on the specific large z and comparing the residues of g( , ,z1 ,z2) at the j com- asymptotic form of the modes. The approach used is a gen- puted from ͑A1͒ and ͑A6͒, one finds that eralization of a technique developed to normalize continuum ϩ͑ ,0͒ modes in atomic physics.21 ͒ ͒ϭ j j͑z1 j͑z2 ͒W͑Ϫ͑C,, ͒,ϩ͑, ͉͒͒ϭץ/ץ͑ Let R and consider the Green’s function, • • j g(C,,z ,z ), given by 1 2 ϫ C ͒ C ͒ Ϫ͑ , j ,z1 Ϫ͑ , j ,z2 . d2 ͩ ϩv͑z ͒Ϫͪ g͑C,,z ,z ͒ϭ␦͑z Ϫz ͒, Using ͑A7͒ the equality dz2 1 1 2 1 2 1 ϱ C ϭ C ͵ ͑ ͒2 ϭϪ ͑ ͒ g( , ,z1 ,z2) g( , ,z2 ,z1) and ϩ j ,z dz ϩ j,0 0 ץ ץ .͒ g͑C,,z ,z ͒ͯ ϭϪCg͑C,,0,z ͯ͒͒ z 1 2 2 ϫ ͑Ј ͑ ͒ϩC ͑ץ ϩ ,0 ϩ ,0ץ z ϭ0 1 1 ϭ j Here v(z)ϭk(z)2 is positive. One has ͑A8͒ 1 ͑C ͒ϭ for the normalization of the eigenfunctions in the point spec- g , ,z1 ,z2 W͑ ͑C ͒ ͑ ͒͒ Ϫ , ,• , ϩ ,• trum follows. As regards the normalization of the eigenfunctions in the ϫϪ͑C,,zϽ͒ϩ͑,zϾ͒, ͑A1͒ continuous spectrum, note that where zϽ Ͼ is the smaller ͑larger͒ of z and z , Ϫ(C,,z) ( ) 1 2 g͑C,⑀ ϩi0ϩ,z ,z ͒Ϫg͑C,⑀ Ϫi0ϩ,z ,z ͒ and ϩ(,z) are solutions of the differential equation 0 1 2 0 1 2 2 ϱ 2i␦ d ϭ ͵ ͑ ͒ ͑ ͒ ⑀ ͩ ϩ ͑ ͒Ϫͪ ͑ ͒ϭ lim 2 2 ⑀ z1 ⑀ z2 d 2 v z z 0, ͑⑀Ϫ⑀ ͒ ϩ␦ dz ␦↓0 Ϫϱ 0 C Ϫ( , ,z) satisfies the initial conditions ϭ2i⑀ ͑z ͒⑀ ͑z ͒. 0 1 0 2 Ј C ͒ϭϪC C ͒ ͑ ͒ Ϫ͑ , ,0 Ϫ͑ , ,0 , A2 Let ⑀R and note that
J. Acoust. Soc. Am., Vol. 115, No. 4, April 2004 Roger Waxler: Modal expansions in the nocturnal boundary layer 1447 ϱ d2 ͑ ͒ ͑ ͒ϭ ͵ ͫ ͑C ⑀ϩ ϩ ͒ͩ ϩ ͑ ͒Ϫ⑀ͪ ͑C ⑀Ϫ ϩ ͒ 2 i ⑀ z1 ⑀ z2 g , i0 ,z1 ,w 2 v w g , i0 ,w,z2 0 dw d2 Ϫg͑C,⑀Ϫi0ϩ,w,z ͒ͩ ϩv͑w͒Ϫ⑀ͪ g͑C,⑀ϩi0ϩ,z ,w͒ͬ dw 2 dw2 1 d d ϱ ϭͫg͑C,⑀ϩi0ϩ,z ,w͒ g͑C,⑀Ϫi0ϩ,w,z ͒Ϫg͑C,⑀Ϫi0ϩ,w,z ͒ g͑C,⑀ϩi0ϩ,z ,w͒ͬͯ 1 dw 2 2 dw 1 wϭ0 ϩ ϩ W͑ϩ͑⑀ϩi0 , ͒,ϩ͑⑀Ϫi0 , ͒͒ ϭ • • ͑C ⑀ ͒ ͑C ⑀ ͒ ͑ ͒ W͑ ͑C ⑀ ͒ ͑⑀ϩ ϩ ͒͒W͑ ͑C ⑀ ͒ ͑⑀Ϫ ϩ ͒͒ Ϫ , ,z1 Ϫ , ,z2 . A9 Ϫ , ,• , ϩ i0 ,• Ϫ , ,• , ϩ i0 ,• Here the fact that Ϫ(C,,z) is continuous in as crosses the real line while ϩ(,z) is not has been used. Equations ͑14͒ and ͑13͒ now follow from ͑A4͒ and ͑A5͒.
1 R. B. Stull, An Introduction to Boundary Layer Meteorology ͑Kluwer 13 N. Dunford and J. T. Schwartz, Linear Operators, Part III ͑Wiley, New Academic, Dordrecht, 2001͒. York, 1971͒. 2 A. D. Pierce, ‘‘Propagation of acoustic-gravity waves in a temperature and 14 F. W. J. Olver, Asymptotics and Special Functions ͑Academic, New York, wind stratified atmosphere,’’ J. Acoust. Soc. Am. 37, 218–227 ͑1965͒. 1974͒. 3 V. E. Ostashev, Acoustics in Moving Inhomogeneous Media ͑E&FN 15 In the mathematical literature such eigenfunction expansions have been Spon, London, 1997͒. shown to exist for sound speed profiles which are asymptotically constant 4 E. Salomons, Computational Atmospheric Acoustics ͑Kluwer, Dordrecht, ͑for example in Ref. 13͒. To this author’s knowledge the extension to 2001͒. sound speed profiles which are asymptotically decreasing ͑indefinitely͒ 5 D. Keith Wilson, J. M. Noble, and M. A. Coleman, ‘‘Sound propagation in has not been considered. A proof of the existence of such an extension will the nocturnal boundary layer,’’ J. Atmos. Sci. 60͑20͒, 2473–2486 ͑2003͒. not be attempted here. One may view the expansion used here either as a 6 K. Attenborough, ‘‘Acoustical impedance models for outdoor ground sur- hypothetical extension of the asymptotically constant case or as an ap- faces,’’ J. Sound Vib. 99, 521–544 ͑1985͒. proximate form for the case in which the sound speed profile eventually 7 T. F. W. Embleton, ‘‘Tutorial on sound propagation outdoors,’’ J. Acoust. becomes constant, but at a height well above the domain of interest, and at Soc. Am. 100,31–48͑1996͒. a value well below the sound speed on the ground. 8 M. Garstang, D. Larom, R. Raspet, and M. Lindeque, ‘‘Atmospheric con- 16 W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Nu- trols on elephant communication,’’ J. Exp. Biol. 198, 939–951 ͑1995͒. merical Recipies in C, 2nd ed. ͑Cambridge U.P., New York, 1992͒. 9 F. B. Jensen, W. A. Kuperman, M. B. Porter, and H. Schmidt, Computa- 17 J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, 2nd ed. tional Ocean Acoustics ͑American Institute of Physics, New York, 1994͒. ͑Springer, New York, 1991͒. 10 R. Waxler, ‘‘A vertical eigenfunction expansion for the propagation of 18 L. D. Landau and E. M. Lifshitz, Quantum Mechanics, 3rd ed. ͑Pergamon, sound in a downward-refracting atmosphere over a complex impedance New York, 1977͒, Sec. 38. plane,’’ J. Acoust. Soc. Am. 112, 2540–2552 ͑2002͒. 19 A. D. Pierce, Acoustics ͑Acoustical Society of America, Woodbury, NY, 11 E. C. Titchmarsh, ‘‘Some theorems on perturbation theory,’’ Proc. R. Soc. 1989͒. London, Ser. A 210,30–44͑1951͒. 20 P. M. Morse and K. U. Ingaard, Theoretical Acoustics ͑Princeton U.P., 12 F. S. Henyey, D. Tang, and S. A. Reynolds, ‘‘Vertical modes with sediment Princeton, NJ, 1986͒. absorption in generalized Pekeris waveguides,’’ J. Acoust. Soc. Am. 113, 21 R. Froese and R. Waxler, ‘‘The spectrum of a hydrogen atom in an intense 2187 ͑2003͒. magnetic field,’’ Rev. Math. Phys. 6, 699–832 ͑1995͒, Appendix B.
1448 J. Acoust. Soc. Am., Vol. 115, No. 4, April 2004 Roger Waxler: Modal expansions in the nocturnal boundary layer