Cite this Article
Luo Wen-Yu, Yu Xiao-Lin, Yang Xue-Feng, Zhang Ze-Zhong, Zhang Ren-He. A three-dimensional coupled-mode model for the acoustic field in a two-dimensional waveguide with perfectly reflecting boundaries. Chinese Physics B, 2016, 25(12): 124309  
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A three-dimensional coupled-mode model for the acoustic field in a two-dimensional waveguide with perfectly reflecting boundaries
Luo Wen-Yu1, †, , Yu Xiao-Lin1, 2, Yang Xue-Feng2, 3, Zhang Ze-Zhong1, 2, Zhang Ren-He1
State Key Laboratory of Acoustics, Institute of Acoustics, Chinese Academy of Sciences, Beijing 100190, China
University of Chinese Academy of Sciences, Beijing 100049, China
Shanghai Acoustic Laboratory, Chinese Academy of Sciences, Shanghai 200032, China

 

† Corresponding author. E-mail: lwy@mail.ioa.ac.cn

Project supported by the National Natural Science Foundation of China (Grant Nos. 11125420, 11434012, and 41561144006) and the Knowledge Innovation Program of the Chinese Academy of Sciences.

Abstract
Abstract

This paper presents a three-dimensional (3D) coupled-mode model using the direct-global-matrix technique as well as Fourier synthesis. This model is a full wave, two-way three-dimensional model, and is therefore capable of providing accurate acoustic field solutions. Because the problem of sound propagation excited by a point source in an ideal wedge with perfectly reflecting boundaries is one of a few three-dimensional problems with analytical solutions, the ideal wedge problem is chosen in this work to validate the presented three-dimensional model. Numerical results show that the field results by analytical solutions and those by the presented model are in excellent agreement, indicating that the presented model can serve as a benchmark model for three-dimensional sound propagation problems involving a planar two-dimensional geometry as well as a point source.

PACS: 43.30.Bp;43.30.Gv;43.20.Fn;43.20.El
Keyword:coupled-mode method;wedge-shaped waveguide;direct-global-matrix approach;3D acoustic field
1. Introduction

Modeling three-dimensional (3D) sound propagation is of great importance in underwater acoustics. A variety of approaches have been applied to this problem. Finite difference and finite element methods seem to be the most suitable methods for this problem. However, the extraordinarily large computational size might make it unrealistic to apply these methods to large-scale sound propagation. A significant effort has gone into modeling 3D sound propagation based on the parabolic equation method.[1–3] The normal mode method is another choice for this problem.[4] Some hybrid approaches, such as the adiabatic mode-parabolic equation method,[5] have also been presented in recent years.

Generally it is very intensive computationally to obtain the full solution of the acoustic field in a 3D waveguide. The problem becomes less complex if the waveguide is two-dimensional. Specifically, in this case the 3D field can be obtained by combining a two-dimensional (2D) model and Fourier synthesis.[6]

A 2D coupled-mode model was developed by Luo et al.,[7] which is referred to as DGMCM2D hereafter in this paper. Unconditional stability and great accuracy are key features of this model. The unconditional stability is achieved by applying the direct-global-matrix approach in combination with an appropriate normalization scheme. This model is capable of handling two-dimensional problems with either a point source in cylindrical geometry or a line source in plane geometry. Especially, appropriate source conditions are introduced in the formulation for the line-source problem in plane geometry.[7] As a result, this model is capable of addressing the scenario where a line source is located in a range-dependent region, for example, on top of a sloping bottom.

An exact solution in the form of uncoupled normal modes was proposed by Buckingham for the acoustic field excited by a point source in an ideal wedge-shaped ocean with perfectly reflecting boundaries.[8] In this solution, each mode coefficient is the sum of two integrals, representing the image field from the source plus images, and the diffraction field arising from diffraction by the apex.

Another analytical solution for sound propagation in an ideal wedge-shaped ocean was derived by Luo et al.[9] This solution is achieved by combining the analytical solution to the 2D ideal wedge problem with a line source[10] and Fourier synthesis.

This paper presents a 3D full wave coupled-mode model, hereafter referred to as DGMCM3D, for sound propagation in a waveguide with a planar 2D geometry. First, we give the formulation of this model, followed by details on the inverse Fourier transform and integration contour chosen in this model. Then we give a brief review of the 2D coupled-mode model DGMCM2D, which applies the direct-global-matrix technique, for sound propagation in a 2D planar waveguide with a line source. We also describe briefly the analytical solutions for the acoustic field excited by a point source in a wedge-shaped ocean with perfectly reflecting boundaries. Section 3 is devoted to validation of the proposed 3D coupled-mode model, DGMCM3D, through two numerical examples. Considering that exact solutions are present for 3D sound propagation excited by a point source in an ideal wedgelike ocean with perfectly reflecting boundaries, we choose this problem to validate the presented model. In numerical examples, both the pressure-release bottom case and the rigid bottom case are considered.

2. Theory

Consider the problem illustrated in Fig. 1, which involves a waveguide with a planar 2D geometry and a point source. First, we introduce the 3D coupled-mode model DGMCM3D, followed by a brief review of the analytical solutions adopted in this work to validate the numerical model DGMCM3D. In this work, the boundaries are assumed to be perfectly reflecting, that is, the surface is pressure-release whereas the bottom is either pressure-release or rigid.

Fig. 1. Schematic diagram of the 3D problem involving a waveguide with a planar 2D geometry and a point source. The surface is pressure-release and the bottom is pressure-release or rigid.
2.1. Three-dimensional direct-global-matrix coupled-mode method

In Cartesian coordinates, with the assumption that the point source is located in the vertical plane y = 0, that is, the source position is (xs, 0, zs), the 3D Helmholtz equation is[11]

In this method, the following Fourier transform pairs are adopted:

Applying

to Eq. (1), we obtain

By comparing Eq. (4) with the 2D Helmholtz equation with a line source,[7,12] viz

it can be observed that equation (4) is exactly the governing equation for the 2D line-source problem, except that k2 is replaced by Hence, for a specific ky, we may solve Eq. (4) by the 2D direct-global-matrix coupled mode model DGMCM2D[7,12] to obtain p̃(x, ky, z). Once a sequence of p̃(x, ky, z) is obtained, we may apply the inverse Fourier transform in Eq. (2b) to compute the 3D field solution p(x, y, z).

2.1.1. Inverse Fourier transform and complex integration contour

Once a sequence of the spectrum p̃(x, ky, z) is obtained, we can compute p(x, y, z) with the inverse Fourier transform. However, because p̃(x, ky, z) has singularities on the real ky axis, this model evaluates the integral in Eq. (2b) by moving the integration contour out into the complex plane, instead of evaluating that integral directly. Note that unlike Ref. [6], in this model we use the complex integration contour in the following form, which is also illustrated in Fig. 2,

Fig. 2. Illustration of the complex integration contour for the inverse Fourier transform. The dots indicate horizontal wavenumbers of the normal modes, where the spectrum p̃(x, ky, z) is singular. For the ideal waveguides with perfectly reflecting boundaries considered in this work, the horizontal wavenumbers are purely real or imaginary. However, they are in the first quadrant for general problems with attenuation in the bottom.

In addition, with symmetry, it can be shown that the inverse Fourier transform in Eq. (2b) can be rewritten as (see Appendix A for details)

The inverse Fourier transform in Eq. (7) is used in the model DGMCM3D.

In the model DGMCM3D, we choose δ = 1/(6Δs) with Δs = (kmax − kmin)/(Ns − 1), and the contour offset ε = 3Δs/(2π log e), with Ns denoting the total number of sampling points in the wavenumber space.

2.1.2. Two-dimensional direct-global-matrix coupled-mode solution for line-source problem in plane geometry

Below we give a brief review of the 2D direct-global-matrix coupled-mode model DGMCM2D for the line-source problem in plane geometry. See Refs. [7] and [12] for details.

In Cartesian coordinates, the Helmholtz equation with a line source located at (xs, zs) is[7]

where p = p(x, z) is the complex pressure.

As illustrated in Fig. 3, we divide a range-dependent waveguide into a number of range-independent segments. Note that a virtual interface is introduced at the source range x = xs. The solution in segment j is represented by

where N denotes the total number of normal modes involved in mode coupling, and are modal amplitudes corresponding to the forward and backward propagating waves, respectively, are the local eigenfunctions, and and are normalized exponential functions defined respectively as

with denoting the horizontal wavenumber of mode n in segment j.

Fig. 3. Schematic diagram of a 2D range-dependent problem with a line source in plane geometry. The line source can be located at an arbitrary range, for instance, on top of a sloping bottom.

Modal amplitudes aj and bj can be obtained by solving the following linear system of equations

which is derived by imposing the boundary conditions at each interface, together with the radiation conditions a1 = 0 and bJ = 0. Here, C is the global coefficient matrix,

x is the column vector

and υ is the column vector

where the superscript T stands for transpose and s is the column vector with entries

The coupling matrices in segment j, and are respectively defined as

In addition, perfectly reflecting boundaries are considered in this work, and hence we have analytical expressions for the modal solutions and the coupling matrices. Thus, the eigenfunctions are in the form

where ρ is density in water and D is water depth. For the pressure-release bottom case, we have

whereas for the rigid bottom case we have

For the pressure-release bottom case, we have the following closed-form expressions for and

where

The expressions for and are different for the rigid bottom case, where for upslope propagation we have

and for downslope propagation we have

2.2. Analytical solution for three-dimensional ideal wedge problem

The problem of 3D sound propagation in an ideal wedge-shaped ocean is illustrated in Fig. 4, which involves a wedge of infinite extent in the y direction and a homogeneous water column with sound speed and density denoted by c and ρ, respectively. Note that unlike DGMCM3D, a cylindrical coordinate system is used in deriving the analytical solutions, with the y axis along the shore line and the vertical plane y = 0 passing through the point source. Thus, the location of the source is denoted by (r, θ, y) = (rs, θs,0). The boundaries are perfectly reflecting, where the surface is pressure-release and the bottom is either pressure-release or rigid.

Fig. 4. Schematic diagram of the 3D wedge problem, where the wedge angle is denoted by θ0.

The 3D Helmholtz equation in cylindrical coordinates is[9,11]

For the 3D ideal wedge problem, Buckingham presented an analytical solution,[8] viz,

where

As pointed out in Ref. [8], the solution in Eq. (28) involves two components contributing to the field in a corner: the image field, consisting of the direct radiation from the source plus specular reflections from the boundaries, and the diffraction field due to diffraction at the apex. For the pressure-release bottom case, the diffraction field is identically zero as the wedge angle is a submultiple of π. However, for the rigid bottom case, the diffraction field is identically zero as the wedge angle is a submultiple of π/2. As the diffraction field is identically zero, the total field consists solely of the image field, and the integral in Eq. (28) can be rewritten as

where

It is clear that a takes its maximum value of 1/2 when r = rs and y = 0.

The authors also derived another analytical solution for the 3D ideal wedge problem using Fourier synthesis in combination with the well-established analytical solution for the 2D line-source ideal wedge problem.[9,10] This analytical 3D solution is of the same form as that in Eq. (28), except that in this solution the integral is defined as follows:

where r< = min(r, rs) and r> = max(r, rs).

As y ≠ 0, both of the integrands in Eqs. (29b) and (32) decay rapidly for kr > k for the former integral and ky > k for the latter one. However, as y = 0, the integrand decays rather slowly for kr > k for the integral in Eq. (29b). As a result, in this case, an integration interval much longer than [0,k] is required to achieve convergent results, which is undesirable for computational reasons. On the contrary, for the integral in Eq. (32), as y = 0, the integrand still decays rapidly for ky > k, such that the integration interval [0,k] is almost sufficient to reach convergence.

2.3. Transmission loss

In this work, sound propagation is expressed in transmission loss (TL) in dB re 1 m defined as

where r is the location of a field point, and the reference pressure p0(1) is the pressure at 1-m distance from the point source,

with k0 denoting the wavenumber at the source location.

3. Numerical examples

In this section we consider the ideal wedge problem with two different wedge angles: one is 3° and the other is approximately 2.86°. For both wedges the bottom is either pressure-release or rigid. From Subsection 2.2, for the 3° wedge problem, because the wedge angle is a submultiple of π/2, and of course also a submultiple of π, for both pressure-release bottom and rigid bottom cases, the diffraction field is identically zero and hence the image solution is sufficient to provide benchmark solutions. However, for the 2.86° wedge problem, since the wedge angle is neither a submultiple of π nor π/2, the diffraction field is not identically zero for the pressure-release bottom case or the rigid bottom case. Thus, for the 2.86° wedge problem, we need to apply the analytical total solution with the integral given in Eq. (32) to provide benchmark solutions.

Note that the waveguides in the numerical examples considered in this paper are ideal waveguides involving a homogeneous water column, a pressure-release sea surface, and a pressure-release or rigid bottom. Closed-form expressions are present for the local modal solutions as well as coupling matrices, and these closed-form expressions are used in DGMCM3D in this work to avoid numerical errors in that regard. For general realistic problems, a numerical method should be adopted to compute the local modal solutions and coupling matrices.

It should be pointed out that although a wedge-shaped ocean is considered in the numerical examples in this paper, the model DGMCM3D is not limited to this scenario. As described in Section 2, DGMCM3D is capable of solving sound propagation in a general waveguide with a planar 2D geometry, and the 2D planar wedge is just a special case of such waveguides.

3.1. 3° ideal wedge problem

This problem is illustrated in Fig. 5, which involves a wedge-shaped ocean with the wedge angle of 3°, a homogeneous water column, a free surface and a perfectly reflecting bottom, and a point source of frequency 25 Hz. The point source is located in the vertical plane y = 0 km at depth 100 m and at the range where the water depth is 200 m.

Fig. 5. Schematic diagram of the 3-degree ideal wedge problem.

First, we consider the pressure-release bottom case. For this ideal wedge problem, because the bottom is pressure-release and the wedge angle is a submultiple of π, the diffraction field is identically zero. Transmission loss along the range with a fixed cross-range and depth computed by the analytical total solution, the analytical image solution, and DGMCM3D is shown in Fig. 6. Note that in this work the analytical total solution with the integral given in Eq. (32) is applied to provide benchmark total field solutions. Figure 6 shows that the results by the analytical image solution, the analytical total solution, and DGMCM3D are in excellent agreement, indicating that the diffraction field is identically zero for this specific problem and that the 3D model DGMCM3D is accurate.

Fig. 6. TL along the range computed by the analytical image solution (blue solid curves), the analytical total solution (red dashed curves), and DGMCM3D (green dashed curves) for the 3° wedge problem with a pressure-release bottom. (a) At depth 30 m, y = 0 km; (b) at depth 150 m, y = 0 km; (c) at depth 30 m, y = 10 km; (d) at depth 150 m, y = 10 km.

Transmission loss in a vertical plane computed by the analytical image solution (equivalent to the analytical total solution for this specific problem) and by DGMCM3D is shown in Fig. 7, which, once again, shows that the results by the analytical image solution and by DGMCM3D are in excellent agreement.

Fig. 7. TL in a vertical plane for the 3° wedge problem with a pressure-release bottom. (a) y = 0 km, by the analytical image solution; (b) y = 0 km, by DGMCM3D; (c) y = 10 km, by the analytical image solution; (d) y = 10 km, by DGMCM3D.

In addition, transmission loss in a horizontal plane is shown in Fig. 8, which shows that the results by the analytical solution and by DGMCM3D are in excellent agreement.

Fig. 8. TL in the horizontal plane z = 30 m computed by (a) the analytical image solution and (b) DGMCM3D.

Then we consider the rigid bottom case. For this ideal wedge problem, since the bottom is rigid and the wedge angle is a submultiple of π/2, the diffraction field is identically zero. Transmission loss along the range with a fixed cross-range and depth computed by the analytical image solution, the analytical total solution, and DGMCM3D is shown in Fig. 9, where excellent agreement between these three solutions is observed.

Fig. 9. TL along the range computed by the analytical image solution (blue solid curves), the analytical total solution (red dashed curves), and DGMCM3D (green dashed curves) for the 3° wedge problem with a rigid bottom. (a) At depth 30 m, y = 0 km; (b) at depth 150 m, y = 0 km; (c) at depth 30 m, y = 10 km; (d) at depth 150 m, y = 10 km.

Transmission loss in a vertical plane computed by the analytical image solution (equivalent to the analytical total solution in this specific problem) and by DGMCM3D for this 3° wedge problem with a rigid bottom is shown in Fig. 10, which, once again, indicates that the results by the analytical image solution and by DGMCM3D are in excellent agreement.

Fig. 10. TL in a vertical plane for the 3° wedge problem with a rigid bottom. (a) y = 0 km, by the analytical image solution; (b) y = 0 km, by DGMCM3D; (c) y = 10 km, by the analytical image solution; (d) y = 10 km, by DGMCM3D.

In addition, transmission loss in a horizontal plane is shown in Fig. 11, which shows that the results by the analytical image solution and by DGMCM3D are in excellent agreement.

Fig. 11. TL in the horizontal plane z = 30 m for the 3° wedge problem with a rigid bottom computed by (a) the analytical image solution and (b) DGMCM3D.
3.2. 2.86° ideal wedge problem

The original Acoustical Society of America (ASA) benchmark wedge[13] was extended to a 3D version by Deane and Buckingham.[14] Note that in Ref. [14] a penetrable wedge was considered, whereas in this paper the boundaries of the wedge are perfectly reflecting. The 2.86° ideal wedge problem is illustrated in Fig. 12, which involves a wedge-shaped ocean, a homogeneous water medium, a free surface and a perfectly reflecting bottom, and a point source of frequency 25 Hz. The point source is located at depth 100 m and range 4 km, where the water depth is 200 m. Thus, the wedge angle is approximately 2.86° in this problem.

Fig. 12. Schematic diagram of the 2.86-degree ideal wedge problem.

For this ideal wedge problem, the wedge angle is neither a submultiple of π nor a submultiple of π/2. As a result, the diffraction field is not identically zero for either the pressure-release bottom case or the rigid bottom case. Hence, the analytical total solution, rather than the analytical image solution, should be used to provide benchmark field solutions.

For this 2.86° wedge problem, first we consider the pressure-release bottom case. Transmission loss in vertical planes y = 0 km and y = 10 km by the analytical solutions and DGMCM3D are shown in Fig. 13, from which we observe significant difference between the first solution and the other two in the vicinity of the corner. This difference is due to the fact that the diffraction field is not identically zero for this problem with a pressure-release bottom since the wedge angle is not a submultiple of π. Transmission loss along the range with a fixed cross-range and depth computed by the analytical image solution, the analytical total solution, and DGMCM3D is shown in Fig. 14, which depicts that the results by the analytical total solution and by DGMCM3D are in excellent agreement, both of which are different from that by the analytical image solution. Besides, comparison of transmission loss in the horizontal plane z = 30 m between the analytical image solution, the analytical total solution, and DGMCM3D is shown in Fig. 15, where excellent agreement is observed between the analytical total solution and DGMCM3D. The result by the analytical image solution shows a wider shadow zone compared to the other two results.

Fig. 13. TL in vertical planes for the 2.86° wedge problem with a soft bottom. (a) y = 0, by the analytical image solution; (b) y = 0, by the analytical total solution; (c) y = 0 km, by DGMCM3D; (d) y = 10 km, by the analytical image solution; (e) y = 10 km, by the analytical total solution; (f) y = 10 km, by DGMCM3D.
Fig. 14. TL along the range computed by the analytical image solution (blue solid curves), the analytical total solution (red dashed curves), and DGMCM3D (green dashed curves) for the 2.86° wedge problem with a pressure-release bottom. (a) At depth 30 m, y = 0 km; (b) at depth 150 m, y = 0 km; (c) at depth 30 m, y = 10 km; (d) at depth 150 m, y = 10 km.
Fig. 15. TL in the horizontal plane z = 30 m by (a) the analytical image solution, (b) the analytical total solution, and (c) DGMCM3D for the 2.86° wedge problem with a pressure-release bottom.

Then we consider the rigid bottom case. Transmission loss in vertical planes y = 0 km and y = 10 km by the analytical image solution, the analytical total solution, and DGMCM3D are shown in Fig. 16, from which we observe significant difference between the first solution and the other two in the vicinity of the corner. Like the pressure-release bottom case, this difference is also due to the fact that the diffraction field is not identically zero for this problem with a rigid bottom since the wedge angle is not a submultiple of π/2. Transmission loss along the range with a fixed cross-range and depth computed by the analytical image solution, the analytical total solution, and DGMCM3D is shown in Fig. 17, which depicts that the results by the analytical total solution and by DGMCM3D are in excellent agreement, both of which are different from that by the analytical image solution. Besides, comparison of transmission loss in the horizontal plane z = 30 m between the analytical image solution, the analytical total solution, and DGMCM is shown in Fig. 18, where excellent agreement is observed between the analytical total solution and DGMCM3D. The result by the analytical image solution shows a wider shadow zone compared to the other two results.

Fig. 16. TL in vertical planes for the 2.86° wedge problem with a rigid bottom. (a) y = 0, by the analytical image solution; (b) y = 0, by the analytical total solution; (c) y = 0 km, by DGMCM3D; (d) y = 10 km, by the analytical image solution; (e) y = 10 km, by the analytical total solution; (f) y = 10 km, by DGMCM3D.
Fig. 17. TL along the range computed by the analytical image solution (blue solid curves), the analytical total solution (red dashed curves), and DGMCM3D (green dashed curves) for the 2.86° wedge problem with a rigid bottom. (a) At depth 30 m, y = 0 km; (b) at depth 150 m, y = 0 km; (c) at depth 30 m, y = 10 km; (d) at depth 150 m, y = 10 km.
Fig. 18. TL in the horizontal plane z = 30 m by (a) the analytical image solution, (b) the analytical total solution, and (c) DGMCM3D for the 2.86° wedge problem with a rigid bottom.
4. Conclusions

A full wave 3D coupled-mode model is presented in this paper, which is capable of computing acoustic fields excited by a point source in a waveguide with a planar 2D geometry. This 3D model is based on Fourier synthesis in combination with the 2D coupled-mode model DGMCM2D. When dealing with problems with an ideal waveguide involving a homogeneous water column as well as perfectly reflecting boundaries, closed-form expressions for the local modal solutions as well as coupling matrices are used in this model to avoid numerical errors in that regard. However, for realistic waveguide problems, a numerical method should be adopted to compute the local modal solutions and coupling matrices instead.

The presented model DGMCM3D is validated through two numerical examples, both of which involve a point source and an ideal wedge-shaped ocean consisting of a homogeneous water column and perfectly reflecting boundaries. Analytical solutions are present for such 3D ideal wedge problems, and they are used in this work to provide benchmark solutions. The wedge angle of the first problem is 3°, which is a submultiple of π and also π/2, and hence the diffraction field is identically zero for both the pressure-release bottom case and the rigid bottom case. The second problem is actually an extension of the 2D ASA benchmark wedge problem, where the wedge angle is approximately 2.86°. In this problem, since the wedge angle is neither a submultiple of π nor a submultiple of π/2, the diffraction field cannot be neglected when applying the analytical solutions. For both numerical examples, excellent agreement is observed between the results by the analytical total solution and those by DGMCM3D, indicating that the model DGMCM3D is very accurate, regardless of whether the diffraction field is identically zero or not.

Note that although we concentrate on the wedge-shaped ocean in this paper, the model DGMCM3D is not limited to this scenario. As a matter of fact, DGMCM3D is suitable for treating sound propagation in a general waveguide with a planar 2D geometry.

In conclusion, an accurate 3D coupled-mode model is developed. This model can serve as a benchmark model for 3D sound propagation excited by a point source in an ideal waveguide with a planar 2D geometry.

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