On the source images method for sound propagation in ... - IEEE Xplore

Proceedings of the International Conference DAYS on DIFFRACTION 2017, pp. 304–309

c 2017 IEEE 978-1-5386-4796-7/17/$31.00 

On the source images method for sound propagation in a penetrable wedge: some corrections and appendices Jun Tang1,2 , Pavel S. Petrov3 , Sergey B. Kozitskiy3 , Shengchun Piao1,2 1

Acoustic Science and Technology Laboratory, Harbin Engineering University, Harbin 150001, China; College of Underwater Acoustic Engineering, Harbin Engineering University, Harbin 150001, China; 3 Il’ichev Pacific Oceanological Institute, 43 Baltiyskaya St., Vladiostok 690041, Russia; e-mail: [email protected],[email protected], [email protected],[email protected]

2

Some corrections and supplements for the source images method in the case of the penetrable wedge, proposed by Deane & Buckingham (1993), are introduced. The problem of sound propagation in a wedge-like waveguide is an important three-dimensional benchmark in underwater acoustics. Firstly, the formulae for positions of source images and bottom images are corrected. Secondly, a simple branch selection rule for the square root in the reflection coefficient is formulated and validated by several numerical tests. New examples, where the source images method is used to simulate time-harmonic acoustic fields and the pulse signal propagation, are presented.

1

Introduction

The source images method [1–3] is useful in providing benchmark solutions for the problem of threedimensional (3D) sound propagation in penetrable wedges [4–6]. The basic idea of this method is to decompose the total sound field into a sum of contributions from source images. Each source image corresponds to the waves that were emitted by the source and underwent a certain number of interactions with the wedge boundaries, i.e., the surface and the bottom. Some corrections and supplements of the source images method are introduced in this study. Firstly, the corrected formulae for angular coordinates of source images and inclination angles of bottom images are derived. Secondly, a branch selection rule for the square root in the reflection coefficient is presented (this issue was not touched upon in the original study [1]). This rule is validated in a series of numerical examples. We also present some interesting examples which were not given in the original study [1]. In the first example, the sound propagation in wedges with different ratios of shear and bulk moduli in the bottom is considered. In the second one, the propagation of

pulse signals in a wedge-like ocean is investigated. In principle, the results of our study can be used as benchmarks in validating sound propagation methods and codes. 2

Problem statement and general form of the solution

The geometry of a wedge-shaped ocean is shown in Fig. 1, where αw is the apex angle. Sound speed and density in the water column are c1 and ρ1 , respectively, while the corresponding parameters in the bottom are c2 and ρ2 , respectively. The attenuation in the bottom is β2 (in decibels per wavelength). In the case of elastic bottom, we also introduce the S-wave speed c2s and the S-wave attenuation β2s . A time-harmonic point source of frequency f is located in the water column (in Sec. 6 we consider the source emitting pulse signals). Accurate simulation of all propagation features in this waveguide requires extremely sophisticated mathematical technique [7]. Fortunately, contribution of the wave diffracted by the apex, recognized as the most difficult part of the solution, is negligible in typical shallow-water propagation sce-









 




  

Figure 1: Geometry of the wedge.



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S1,1


 







surface












α

S0,−1 αs







1,2

S0,2

w

  


S





bottom















S1,−2 S



1,−1





 

S

2,−2

Figure 2: The dashed and solid radial lines represent surface and bottom images, respectively. Local coordinate systems for either the upper or lower source images are exemplified. The physical ray of the wave emitted by S3,−2 is a polyline S0,−1 -3-2-1 and its equivalent ray is the straight line S3,−2 -3’-2’-1’ that directly connects S3,−2 and the receiver.

S2,−1

Figure 3: The source S0,−1 and its images. The arrows show how all images are obtained from the actual source S0,−1 and its reflection by the sea surface S0,2 .

By introducing the transformations kx = k sin θ cos ϕ, xr = R sin ζ cos ξ,

narios where αw is very small and the source is several kilometers away from the apex. According to Deane and Buckingham [1], the total sound field in this case can be represented as a sum over the waves emitted by the source images: p(x, y, z) =

N   nb =1

l

pnb ,l (x, y, z),

ky = k sin θ sin ϕ, yr = R sin ζ sin ξ, kz = k cos ϕ, zr = R cos ζ,

and using the expansion of resulting exponential in Bessel function series (see [1] for details), we can rewrite Eq. 2 in a form which is more convenient (1) for numerical implementation:

 where pnb ,l denotes the field contributed by the ik π/2−i∞ p = exp(iΩz ) sin θ n l b source image Snb ,l which is numbered by nb and 2π 0   l. Here nb is the number of bottom reflections un∞ a0 J0 (Ω)  ν dergone by the wave from Snb ,l on its way to the + i aν Jν (Ω) dθ, (3) × 2 receiver, and meanwhile l is used in addition to nb ν=1 to identify an image uniquely. With a plane-wave decomposition, pnb ,l can be where Ωz = kR cos θ cos ζ, Ω = kR sin θ sin ζ, and aν is given by expressed as    nb    π  nb i(−1)ns eik·R pnb ,l = V (ϕb ) dkx dky , (2) ns aν = (−1) 2 cos(νξ) V (ϕb ) cos(νφ) dφ. 2π kz R2 b=1

where ns is the number of surface reflections, which is determined by the values of nb and l; V (ϕb ) is the reflection coefficient for b-th bottom reflection; k = (kx , ky , kz ) is the wavenumber vector; R = (xr , yr , zr ) is the position of the receiver in a local coordinate system (see Fig. 2) which originates from Snb ,l . For the case zr ≥ 0, we require Im(kz ) ≥ 0 to ensure the infinity radiation condition [1].

0

3

b=1

Positions of source images and bottom images

Let us denote the angular position in a polar coordinate system centered at the apex and in the xz-plane by Φ, with the bottom along Φ = 0 (as shown in Fig. 3). Then, angular positions Φnb ,l of

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as

where αs is the angular position of the real source. For each nb , there are four images numbered by index l. The only exception is nb = 0, for which we have only l = −1, 2, representing the actual source S0,−1 and its first surface reflection S0,2 , respectively. In Fig. 3 we explain how all other images are obtained from S0,−1 and S0,2 (see also [1]) by reflecting them by turn from bottom and surface. Bottom images and surface images are simply distributed along lines that bisect the domains between adjacent source images as shown in Fig. 2. For source image Snb ,l , inclination angles Θb of the associated bottom images from the real bottom are

2(b − 1) αw , l < 0, Θb = 1 ≤ b ≤ nb . 2bαw , l > 0, In order to show how waves from a source image interact with the surface or bottom images, we have also displayed a physical ray and its equivalent ray in Fig. 2.

FEM Images

40

TL (dB)

l = −2, l = −1, l = 1, l = 2,

30

lossless fluid bottom

50 60 70 80

lossy fluid bottom

90 0

0.5

1

1.5

2

2.5

3

3.5

x (km) 30 FEM Images

40

TL (dB)

each source image can be expressed ⎧ −2(nb − 1) αw − αs , ⎪ ⎪ ⎪ ⎨−2n α + α , b w s Φnb ,l = ⎪ 2n α + α , b w s ⎪ ⎪ ⎩ 2(nb + 1) αw − αs ,

lossless elastic bottom

50 60 70 80

lossy elastic bottom

90 0

0.5

1

1.5

2

2.5

3

3.5

x (km)

Figure 4: TL curves along z = 30 m, y = 0 from the source images method and the FEM for both fluid (top plot) and elastic bottom (bottom plot) cases. The curves for lossy cases have been shifted downwards by 20 dB to avoid overlap.

A branch selection rule for the reflec- γ2 should be carefully chosen for every plane-wave component involved in (3). Basically, this issue can tion coefficient always be resolved by considering the problem of In the case of an elastic sea bottom, V (ϕb ) can be plane-wave reflection at the interface. In numeriexpressed as follows (see [8]): cal implementation, however, one needs a practical and simple rule for branch selection, which was not ρ2 cos2 2ϕs /γ2 + ρ2 sin2 2ϕs /γ2s − ρ1 /γ1 , (4) provided in the original paper [1]. By conducting V = ρ2 cos2 2ϕs /γ2 + ρ2 sin2 2ϕs /γ2s + ρ1 /γ1 some computational experiments to reproduce the results in Fig. 3 of [1], we find that the branch sewhere the subscript b has been omitted. Here γ1 , lection of γ should obey the following rule: 2 γ2 , and γ2s are the normal components of the wave vector for the incident wave, transmitted P-wave, (6) Re(γ1 ) · Re(γ2 ) > 0 for Re(γ2 ) = 0, and transmitted S-wave, respectively; ϕs is the reIm(γ2 ) > 0 for Re(γ2 ) = 0. (7) fraction angle of the transmitted S-wave. If c2s = 0, formula (4) is reduced to the case of a fluid bottom. Equation (6) means that the normal components For the sake of simplicity, we start the following dis- of incident and transmitted waves should have the cussion from the case of a fluid bottom. same direction. Equation (7) is enforced in the case By Snell’s law, γ2 can be expressed as where (6) is not applicable. Below we validate the branch selection rule by  γ2 = k 2 [(c1 /c2 )2 − 1] + γ12 . (5) solving the ASA (Acoustical Society of America) benchmark problem [11]. The geometry is shown Obviously, the square root makes γ2 a two-valued in Fig. 1. A point source of f = 25 Hz is located function. In order to compute pnb ,l , the branch of at x = 4000 m, y = 0, z = 100 m. In addition, 4

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Figure 5: The horizontal cut-planes of TLs at z = 30 m for the wedges with c2s = 0, 500, 1000 and 1700 m/s, respectively. c1 = 1500 m/s, ρ1 = 1 g/cm3 , c2 = 1700 m/s, and ρ2 = 1.5 g/cm3 . The cases of both lossless and lossy bottom (β2 = 0.5 dB/λ) are considered. Note that in the lossy case, (7) loses its efficacy since we always have Re(γ2 ) = 0. Solutions for both cases are shown in the upper plot of Fig. 4 and compared with those obtained by the finite element method (FEM) [9]. Excellent agreement between the two methods confirms the validity of our branch selection rule. In the case of an elastic bottom, we have square roots in the expressions for both γ2 and γ2s . Since the latter is the same as (5) but with c2 replaced by c2s , we conclude that the branch selection of γ2s should follow the same rule (6) and (7). Now

we validate our branch selection rule in the elastic ASA wedge. S-wave speed in the bottom is assumed to be c2s = 800 m/s. Results of both the lossless bottom case and the lossy bottom case (β2 = β2s = 0.5 dB/λ) are shown in the lower plot of Fig. 4. The good agreements between the two methods demonstrate the applicability of our branch selection rule in scenarios with elastic bottom. 5

Horizontal refraction in a wedge with elastic bottom

In this section, we present an example where the acoustic fields are computed in four wedges with

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2000 Pulse in Source 1500 1000

P(t)

500 0

−500 −1000 −1500 −2000 0.1

0.15

0.2

0.25

0.3 t, sec

0.35

0.4

0.45

0.5 y =6km

0.8

r

y =7.25km r

0.6

y =8.5km r

y =9.75km

0.4

r

P(t)

0.2 0

−0.2 −0.4 −0.6 −0.8 4

4.5

5

5.5

6

6.5

7

7.5

t, sec

Figure 6: Pulse signal emitted by the source (top plot) and signals computed at a series of receiver points at y1 = 6 km, y2 = 7.25 km, y3 = 8.5 km, y4 = 9.75 km (bottom plot). identical values of c2 and c1 but different values of c2s . The values of c2s in the four cases are 0, 500, 1000, 1700 m/s, respectively. Other parameters are c1 = 1500 m/s, ρ1 = 1 g/cm3 , c2 = 3400 m/s, ρ2 = 1.5 g/cm3 , β2 = β2s = 0. The contour plots of TL(x, y) on the plane of z = 30 m are presented in Fig. 5. In Case 1, six trapped modes are excited. The hyperbola-like caustics are formed by the turning points of horizontal rays [10]. In Case 2 and Case 3, where c2s < c1 < c2 , elasticity merely contributes to faster decay of the field magnitude due to transition of acoustic energy into S-wave energy. It is clear that a larger value of c2s results in larger transmission loss. Note that in Cases 2 and 3 the wavenumbers of all modes excited by the source have a nonzero imaginary part (by contrast to Case 1), although there is no bottom attenuation. In Case 4, we have c1 < c2s < c2 . The field magnitude in Case 4 is comparable with that in Case 1, while the interference pattern is absolutely different. The source excites four trapped modes, and three of them have wavenumbers which are nearly equal to wavenumbers of the 3 modes excited in a wedge-like waveguide with fluid bottom and c2 = 1700 m/s. The remaining one corresponds to the Scholte wave, though it is not efficiently excited in our case. Note that the caustics of horizontal rays have a more complicated pattern than that in Case 1.

6

Pulse signal propagation

In our final example, we consider the propagation of a pulse signal in the standard ASA wedge [11] with a fluid bottom, where c1 = 1500 m/s, ρ1 = 1 g/cm3 , c2 = 1700 m/s, ρ2 = 1.5 g/cm3 . The attenuation in the bottom is assumed to be β2 = 0.5 dB/λ for all frequencies. A series of receivers are located at the depth of zr = 30 m along the line x = 4 km at y1 = 6 km, y2 = 7.25 km, y3 = 8.5 km, y4 = 9.75 km, respectively. The source located at zs = 100 m, ys = 0 m, xs = 4 km emits a pulse signal with the waveform given by     t − 0.3 (t − 0.3)2 P (t) = 2000 βsHs , exp − 0.05 0.052 (8) where Hs (u) is s-th Hermite polynomial. The coHs (u) in such way efficient βs is used to normalize 2 that maxu∈R βs Hs (u)e−u  = 1. In our example, we use the signal of (8) with s = 25, and its time series are presented in Fig. 6 (top). The spectrum of the emitted signal is localized on the interval from 10 to 40 Hz. Formula (1) is used to solve the propagation problem for each tonal component of the pulse signal (i.e. for each frequency in its spectrum on a 1 Hz mesh), and the received signals are computed with the Fourier synthesis. The pulses observed at different receiver points are plotted in Fig. 6 (bottom subplot). These time-domain solutions were validated through di-

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rect comparisons against the pulse signals computed using the mode parabolic equation theory that accounts for modal interaction [6] (see also [12]). We observed excellent agreement of the nu[2] merical results from the two methods. 7

Conclusion

In this study, we revisited an important benchmark solution of the problem of sound propagation in a penetrable wedge. The wedge-shaped waveguide, which is a simplification of shallow-water environments near the coastline, features all the typical propagation effects of shallow-water acoustics, including mode interaction and horizontal refraction. Thus, it is an ideal benchmark example for various methods and tools for the simulation of acoustic fields. In the present work, we attempted to provide some useful corrections to the formulae from [1]. It was also shown that the source images technique can be successfully applied to the case of an elastic bottom and used for synthesizing time-domain solutions of a nonstationary wave equation. We have already published our codes for the computation of time-harmonic acoustic fields in both fluid and elastic bottom cases [2, 3]. The codes related to pulse signal propagation will be openly shared as well in the near future for the benefit of ocean acoustics research community. In our view, eventually a comprehensive open collection of analytical and numerical solutions of various kinds of propagation problems should be created to support new research efforts in the field of computational underwater acoustics (though this role is already played by OALIB [2, 3] partially). Acknowledgements

in a penetrable wedge with a stratified fluid or elastic basement, J. Acoust. Soc. Amer., Vol. 93, pp. 1319–1328. Petrov, P. S., 2015, The 3D penetrable wedge solution of G. Deane and M. Buckingham (MATLAB code), online at http://oalib.hlsresearch.com/ThreeD/ Petrov/PenetrableWedge/.

[3] Tang, J., 2017, Analytic solution for the elastic penetrable wedge shaped ocean (MATLAB code), online at http://oalib. hlsresearch.com/ThreeD/Penetrable% 20wedge%20(elastic%20bottom)/. [4] Petrov, P. S., Sturm, F., 2016, An explicit analytical solution for sound propagation in a three-dimensional penetrable wedge with small apex angle, J. Acoust. Soc. Amer., Vol. 139, pp. 1343–1352. [5] Petrov, P. S., Prants, S. V., Petrova, T. N., 2017, Analytical Lie-algebraic solution of a 3D sound propagation problem in the ocean, Phys. Lett. A, Vol. 381, pp. 1921–1925. [6] Trofimov, M. Y., Kozitskiy, S. B., Zakharenko, A. D., 2015, A mode parabolic equation method in the case of the resonant mode interaction, Wave Motion, Vol. 58, pp. 42–52. [7] Babich, V. M., Mokeeva N. V., Samokish B. A., 2012, The problem of scattering of a plane wave by a transparent wedge: a computational approach, J. Commun. Technol. Electron., Vol. 57, pp. 993–1000. [8] Brekhovskikh, L. M., Lysanov, Y. P., 2003, Fundamentals of Ocean Acoustics, Springer, New York.

[9] COMSOL AB, 2015, COMSOL Multiphysics The work of J. T. and S. P. was funded by the Reference Manual 5.2, Stockholm, Sweden. National Nature Science Foundation of China (No. 11234002). P. P. and S. K. were supported [10] Katsnelson, B. G., Petnikov, V. G., Lynch J., 2012, Fundamentals of Shallow Water Acousby POI FEBRAS Program “Nonlinear dynamtics, Springer, New York. ical processes in the ocean and atmosphere” (No. 0201363045). P. P. was also supported by the [11] Jensen, F. B., Porter, M. B., Kuperman, Russian Foundation for Basic Research (No. 16W. A., Schmidt, H., 2011, Computational Ocean 05-01074) and by the Russian President’s Council Acoustics, Springer, New York, et al. (No. MK-2262.2017.5). [12] Petrov, P. S., Trofimov, M. Yu., Zakharenko, A. D., 2012, Mode parabolic equations for the References modeling of sound propagation in 3D-varying shallow water waveguides, Proc. Intern. Conf. [1] Deane, G. B., Buckingham, M. J., 1993, An “Days on Diffraction 2012”, pp. 197–202. analysis of the three-dimensional sound field