Benchmark solutions for sound propagation in an ideal wedge

Sound propagation in a wedge-shaped waveguide with perfectly reflecting boundaries is one of the few range-dependent problems with an analytical solution, and hence provides an ideal benchmark for a full two-way solution to the wave equation. An analytical solution for the sound propagation in an ideal wedge with a pressure-release bottom was presented by Buckingham and Tolstoy [Buckingham and Tolstoy 1990 J. Acoust. Soc. Am. 87 1511]. The ideal wedge problem with a rigid bottom is also of great importance in underwater acoustics. We present an analytical solution to the ideal wedge problem with a perfectly reflecting bottom, either rigid or pressure-release, which may be used to provide a means for investigating the sound field in depth-varying channels, and to establish the accuracy of numerical propagation models. Closed-form expressions for coupling matrices are also provided for the ideal waveguides characterized by a homogeneous water column bounded by perfectly reflecting boundaries. A comparison between the analytical solution and the numerical solution recently proposed by Luo et al. [Luo W Y, Yang C M and Zhang R H 2012 Chin. Phys. Lett. 29 014302] is also presented, through which the accuracy of this numerical model is illustrated.     

水平变化波导中一种声场计算方法的数值实现

全局矩阵方法计算水平变化环境中的声场具有稳定性好、速度快、精度高等优点,在数值实现中如何快速、准确的求解大规模矩阵是该方法的一个关键问题。本文针对全局矩阵的特点,分别利用两种矩阵求解器,PARDISO和LAPACK,求解该问题。经过比较和讨论,得出结论:LAPACK对于全局矩阵的求解更有优势,求解速度快,而且可求解问题规模相对较大。使用不同的数值实现方法计算了楔形波导的传播损失(ASA标准问题),与解析解比较,证明全局矩阵方法计算精度很高。     

A three-dimensional coupled-mode model for the acoustic field in a two-dimensional waveguide with perfectly reflecting boundaries

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.     

楔形理想波导中球体散射的镜像方法

应用镜像原理和球波函数加法公式,把楔形理想波导中球体的散射声场等效为无界空间中多球体散射声场,镜像解析解与边界元数值结果相同,可以作为边界随水平距离变化波导中目标散射数值方法的标准解。比较无界空间和楔形理想波导中球体散射声场空间分布特征,结果表明:楔形理想波导中球体的散射声场在水平和倾斜边界反射;散射声场沿楔形理想波导下坡方向的衰减在相同的水平距离下小于无界空间的情况,散射声场在楔形理想波导上坡的空间分布受到限制。     

Underwater acoustic scattering from a radially layered cylindrical obstacle in a 3D ocean waveguide

A solution based on coupled mode expansions is presented for the 3D problem of acoustic scattering from a radially layered penetrable cylindrical obstacle in a shallow-water plane-horizontal waveguide. Each cylindrical ring is characterized by a general, vertical sound speed and density profile (ssdp), the ocean environment around the obstacle can be also considered horizontally stratified with a depth-arbitrary ssdp, and the bottom is assumed to be rigid. The total acoustic field generated by an harmonic point source is represented as a normal-mode series expansion. The expansion coefficients are calculated exploiting the matching conditions at the cylindrical interfaces, which results in an infinite linear system. The system is appropriately truncated and numerically solved by using a recursive relation, which involves the unknown coefficients of two successive rings. Results concerning the transmission loss outside and inside obstacles consisting of three cylindrical rings are given for a typical depth-dependent ocean sound-speed profile. The presented solution can serve as a benchmark solution to the general problem of 3D acoustic scattering from axisymmetric inhomogeneities in ocean waveguides at low frequencies.     

Simulation of the vector wave field of a low-frequency sound source in the ocean: Some important features

An approach to the simulation of low frequency vector wave fields in stratified media (mainly in the ocean) is considered. The approach is characterized by an improved stability with respect to dividing the medium into many layers of arbitrary thickness. The model for the sound field of a point source is based on an integral representation of two-dimensional, cylindrically symmetric vector wave fields in inhomogeneous media, so that the contributions of all types of waves are included automatically. The model medium is subdivided into N horizontally homogeneous layers for which 4(N?1) equations are formulated to satisfy the boundary conditions between adjacent layers. The method of the generalized Schmidt matrix is used to obtain the coefficients of the equations; these coefficients are substituted into the expressions (of the Fourier-Bessel integral type) for the local parameters of the field. The latter are calculated according to the numerical procedure, and the results are used to model the distributions of the acoustic pressure and the horizontal and vertical components of the particle velocity in liquid and elastic media. The instability of the calculation procedure may result in a disagreement between the model and the exact solution. However, the disagreement is shown to occur mainly in models containing excessively thick layers. A way for improving the stability of the numerical model is suggested. The simulation results are compared with the exact analytical solution for the simplest example and with the results obtained according to the commonly used generalized matrix procedure (the benchmark problem). The examples of the practical application of the model for investigating more complex seismoacoustic wave fields in the ocean are presented.     

浅海波导中目标散射的边界元方法

提出了一种计算三维散射体在声速剖面随深度变化、距离无关浅海波导中散射声场的数值方法波导边界元方法。当散射体不十分靠近波导界面因而边界多次散射可以忽略时,在边界元计算中可以用自由场格林函数近似波导格林函数。应用镜像法和球波函数加法定理推导了理想波导中球体散射声场的解析解,用来验证波导边界元方法的计算精度,证明该数值方法是准确的。对浅海波导中水下潜器散射声场数值模拟的结果表明,浅海波导海面、海底界面反射、声速剖面等对目标散射声场的幅值和方向性都有很大的影响。      

浅海波导中目标散射的简正波-Kirchhoff近似混合方法

提出了计算浅海波导中复杂目标散射的数值方法:简正波-Kirchhoff近似混合方法。通过把目标散射的Kirchhoff近似方法和简正波声传播模型相结合,可对大尺寸复杂目标在浅海波导中的散射声场进行计算。以浅海波导中刚性球体散射的解析解为标准解验证了本方法,说明简正波-Kirchhoff近似混合方法是有一定计算精度的工程预报方法。数值计算Pekeris浅海波导中球体目标散射声场在深度-频率平面和距离-频率平面上的二维干涉结构及其与自由空间中的差异。进而通过FFT获得目标时域回波随深度的分布图,分析浅海波导中目标姿态、声速剖面对Benchmark潜艇目标散射的影响。      

水平变化波导中的全局矩阵耦合简正波方法

提出了一种新的水平变化波导中声场的耦合简正波求解方法,该方法能够处理二维点源和线源问题,提供声场的双向解。该方法利用全局矩阵(DGM)一次性求解耦合模式的系数,消除了传播矩阵递推求解中存在的误差累积问题;此外,改善了现有模型中对距离函数的归一化方法,从而避免了泄露模式指数增长导致的数值溢出问题。本文还给出了绝对软海底理想波导中耦合矩阵的闭合表达式,并分析了单个阶梯下简正波耦合现象。此外,本文还计算了理想楔形波导中的声传播问题(ASA标准问题),并与解析解及COUPLE07计算结果进行了比较,结果表明该方法是一种稳定、精确的水平变化波导中的声场计算方法。      

Composition of propagated,reflected and transmitted waves in arbitrary and continuously stratified environments

First, a solution is presented for a canonical problem in wave propagation. Second, illustrations and applications of the results are carried out to study cases which are relevant to the propagation problem in the ocean and atmosphere.The canonical problem consists of a plane wave incident on an arbitrary and continuously stratified region with planar boundaries. The explicit composition of the reflected, transmitted and propagated waves are derived. The solution is systematic and allows for (i) discontinuities in the acoustic properties at boundaries and arbitrary variation within, (ii) attenuation, (iii) all angles of incidence. The general expressions are obtained by using an alternate procedure to one recently devised [1]. The present approach is straightforward and plainly amenable to physical interpretation of its auxiliary mathematical constants. The discontinuities at the boundaries are satisfied at the outset. The reflected and transmitted waves are directly and explicitly specified. Comparison to widely used techniques in both analytical and numerical works is made to demonstrate the viability of the present approach.A series of cases relevant to the problem at hand are considered. These cases illustrate the mechanics involved in use of the method, and expand its application to problems that appear to be at variance with the formulation of the canonical problem. The illustrations include attenuation in the medium, effect on the solution of different acoustic discontinuities at the boundaries, and use of an inhomogeneous background profile with known independent solutions. The expanded applications treat formally three types of problems: (i) the exact solution for plane waves in continuously stratified media where the well-used ray theory or W-K-B approximation serves only as a first approximation in a correct iterative solution; (ii) the scattering of a plane wave by non-planar boundaries, i.e., spherical or cylindrical acoustic lens with the stratification along the radial direction; (iii) the field due to a point source in a continuously stratified wave guide, like the ocean or atmosphere.     

On the use of higher-order azimuthal schemes in 3-D PE modeling

In this paper, the issue of using higher-order finite difference schemes to handle the azimuthal derivative term in a three-dimensional parabolic equation based model is addressed.The three-dimensional penetrable wedge benchmark problem is chosen to illustrate the accuracy and efficiency of the proposed schemes. Both point source and modal initializations of the pressure field are considered. For each higher-order finite difference scheme used in azimuth, the convergence of the numerical solution with respect to the azimuth is investigated and the CPU times are given. Some comparisons with solutions obtained from another 3-D model [J. A. Fawcett, J. Acoust. Soc. Am. 93, 2627-2632 (1993)] are presented. The numerical simulations show that the use of a higher-order scheme in azimuth allows one to reduce the required number of points in the azimuthal direction while still obtaining accurate solutions. The higher-order schemes have approximately the same efficiency as a FFT-based approach (in fact, may outperform it slightly); however, the finite difference approach has the advantage that it may be more flexible than the FFT approach for various PE approximations.     

Simulations of optoacoustic wave propagation in light-absorbing media using a finite-difference time-domain method

Optoacoustic (OA) imaging is an emerging technology that combines the high optical contrast of tissues with the high spatial resolution of ultrasound. Taking full advantage of OA imaging requires a better understanding of OA wave propagation in light-absorbing media. Current simulation methods are mainly based on simplified conditions such as thermal confinement, negligible viscosity, and homogeneous acoustic properties throughout the image object. In this study a new numerical approach is proposed based on a finite-difference time-domain (FDTD) method to solve the general OA equations, comprising the continuity, Navier-Stokes, and heat-conduction equations. The FDTD code was validated using a benchmark problem that has an approximate analytical solution. OA experiments were also conducted and data were in good agreement with those predicted by the FDTD method. Characteristics of simulated OA waveforms and OA images were discussed. The simulator was also employed to study wavefront distortion in OA breast imaging.     

A Chebyshev–Lagrangian method for acoustic analysis of a rectangular cavity with arbitrary impedance walls

A general Chebyshev–Lagrangian method is proposed to obtain the analytical solution for a rectangular acoustic cavity with arbitrary impedance boundary conditions. The originality of the present paper is the successful attempt of applying orthogonal polynomials, such as Chebyshev polynomials of the first kind, to the analysis of a rectangular sound field with general wall impedance. The sound pressure is uniformly expressed as triplicate Chebyshev polynomial series which is independent in each direction. The Chebyshev polynomial series solution is obtained using the Rayleigh–Ritz procedure after considering the influence of boundary impedance on the cavity as the work done by the impedance surfaces in the Lagrangian function. The accuracy and reliability of the proposed method are validated against the analytical solutions and some numerical results available in the literature. Excellent orthogonality and complete properties of the Chebyshev polynomials ensure the rapid convergence, numerical stability, high accuracy of the current solution. The simplicity and low computational cost of the present approach make it preferable to obtain the results of complex models even in the relative high frequency range by choosing enough truncated terms in the sound pressure expression. Numerous cases with various uniform or non-uniform impedance boundary conditions are analyzed numerically and some of the results can be used as benchmark. It is shown that the impedance boundary condition can effectively influence or modify the acoustic characteristics and response of a cavity.     

A high-order discontinuous Galerkin method for wave propagation through coupled elastic–acoustic media

We introduce a high-order discontinuous Galerkin (dG) scheme for the numerical solution of three-dimensional (3D) wave propagation problems in coupled elastic–acoustic media. A velocity–strain formulation is used, which allows for the solution of the acoustic and elastic wave equations within the same unified framework. Careful attention is directed at the derivation of a numerical flux that preserves high-order accuracy in the presence of material discontinuities, including elastic–acoustic interfaces. Explicit expressions for the 3D upwind numerical flux, derived as an exact solution for the relevant Riemann problem, are provided. The method supports h-non-conforming meshes, which are particularly effective at allowing local adaptation of the mesh size to resolve strong contrasts in the local wavelength, as well as dynamic adaptivity to track solution features. The use of high-order elements controls numerical dispersion, enabling propagation over many wave periods. We prove consistency and stability of the proposed dG scheme. To study the numerical accuracy and convergence of the proposed method, we compare against analytical solutions for wave propagation problems with interfaces, including Rayleigh, Lamb, Scholte, and Stoneley waves as well as plane waves impinging on an elastic–acoustic interface. Spectral rates of convergence are demonstrated for these problems, which include a non-conforming mesh case. Finally, we present scalability results for a parallel implementation of the proposed high-order dG scheme for large-scale seismic wave propagation in a simplified earth model, demonstrating high parallel efficiency for strong scaling to the full size of the Jaguar Cray XT5 supercomputer.     

Analytical and numerical prediction of harmonic sound power in the inlet of aero-engines with emphasis on transonic rotation speeds

Tone noise radiated through the inlet of a turbofan is mainly due to rotor-stator interactions at subsonic regimes (approach flight), and to the shock waves attached to each blade at supersonic helical tip speeds (takeoff). The axial compressor of a helicopter turboshaft engine is transonic as well and can be studied like turbofans at takeoff. The objective of the paper is to predict the sound power at the inlet radiating into the free field, with a focus on transonic conditions because sound levels are much higher. Direct numerical computation of tone acoustic power is based on a RANS (Reynolds averaged Navier–Stokes) solver followed by an integration of acoustic intensity over specified inlet cross-sections, derived from Cantrell and Hart equations (valid in irrotational flows). In transonic regimes, sound power decreases along the intake because of nonlinear propagation, which must be discriminated from numerical dissipation. This is one of the reasons why an analytical approach is also suggested. It is based on three steps: (i) appraisal of the initial pressure jump of the shock waves; (ii) 2D nonlinear propagation model of Morfey and Fisher; (iii) calculation of the sound power of the 3D ducted acoustic field. In this model, all the blades are assumed to be identical such that only the blade passing frequency and its harmonics are predicted (like in the present numerical simulations). However, transfer from blade passing frequency to multiple pure tones can be evaluated in a fourth step through a statistical analysis of irregularities between blades. Interest of the analytical method is to provide a good estimate of nonlinear acoustic propagation in the upstream duct while being easy and fast to compute. The various methods are applied to two turbofan models, respectively in approach (subsonic) and takeoff (transonic) conditions, and to a Turbomeca turboshaft engine (transonic case). The analytical method in transonic appears to be quite reliable by comparison with the numerical solution and with available experimental data.     

Least-squares Legendre spectral element solutions to sound propagation problems

This paper presents a novel algorithm and numerical results of sound wave propagation. The method is based on a least-squares Legendre spectral element approach for spatial discretization and the Crank-Nicolson [Proc. Cambridge Philos. Soc. 43, 50-67 (1947)] and Adams-Bashforth [D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications (CBMS-NSF Monograph, Siam 1977)] schemes for temporal discretization to solve the linearized acoustic field equations for sound propagation. Two types of NASA Computational Aeroacoustics (CAA) Workshop benchmark problems [ICASE/LaRC Workshop on Benchmark Problems in Computational Aeroacoustics, edited by J. C. Hardin, J. R. Ristorcelli, and C. K. W. Tam, NASA Conference Publication 3300, 1995a] are considered: a narrow Gaussian sound wave propagating in a one-dimensional space without flows, and the reflection of a two-dimensional acoustic pulse off a rigid wall in the presence of a uniform flow of Mach 0.5 in a semi-infinite space. The first problem was used to examine the numerical dispersion and dissipation characteristics of the proposed algorithm. The second problem was to demonstrate the capability of the algorithm in treating sound propagation in a flow. Comparisons were made of the computed results with analytical results and results obtained by other methods. It is shown that all results computed by the present method are in good agreement with the analytical solutions and results of the first problem agree very well with those predicted by other schemes.     

On the Method of Source Images for the Wedge Problem Solution in Ocean Acoustics: Some Corrections and Appendices

In this study the method of source images for the problem of sound propagation in a penetrable wedge [G. Deane and M. Buckingham, J. Acoust. Soc. Am. 93 (1993) 1319–1328] is revisited. This solution is very important three-dimensional (3D) benchmark in computational underwater acoustics, since a wedge bounded from above by the sea surface and overlying a sloping penetrable bottom is the simplest model of a shallow-sea waveguide near the coastline. The corrected formulae for the positions of the source images and bottom images are presented together with the explanation of their derivation. The problem of branch choice in the reflection coefficient is thoroughly discussed, and the corresponding explicit formulae are given. In addition, numerical validation of the proposed branch choice schemes and the resulting wedge problem solutions are presented. Finally, source images solution is computed for a series of examples with different ratios of shear and bulk moduli in the bottom. The interplay between the acoustic-elastic waves coupling and the horizontal refraction in the wedge is demonstrated.     

Burton–Miller-type singular boundary method for acoustic radiation and scattering

This paper proposes the singular boundary method (SBM) in conjunction with Burton and Miller?s formulation for acoustic radiation and scattering. The SBM is a strong-form collocation boundary discretization technique using the singular fundamental solutions, which is mathematically simple, easy-to-program, meshless and introduces the concept of source intensity factors (SIFs) to eliminate the singularities of the fundamental solutions. Therefore, it avoids singular numerical integrals in the boundary element method (BEM) and circumvents the troublesome placement of the fictitious boundary in the method of fundamental solutions (MFS). In the present method, we derive the SIFs of exterior Helmholtz equation by means of the SIFs of exterior Laplace equation owing to the same order of singularities between the Laplace and Helmholtz fundamental solutions. In conjunction with the Burton–Miller formulation, the SBM enhances the quality of the solution, particularly in the vicinity of the corresponding interior eigenfrequencies. Numerical illustrations demonstrate efficiency and accuracy of the present scheme on some benchmark examples under 2D and 3D unbounded domains in comparison with the analytical solutions, the boundary element solutions and Dirichlet-to-Neumann finite element solutions.     

Some illustrative examples of the use of a spectral-element method in ocean acoustics

Some numerical results in the time domain obtained with the spectral-element method are presented in order to illustrate the high potential of this technique for modeling the propagation of acoustic waves in the ocean in complex configurations. A validation for a simple configuration with a known solution is shown, followed by some simulations of the propagation of acoustic waves over different types of ocean bottoms (fluid, elastic, and porous) to emphasize the wide variety of media that can be considered within the framework of this method. Finally, a movie illustrating upslope propagation over a viscoelastic wedge is presented and discussed.     

Diffraction of a Plane Electromagnetic Wave by an Infinite Dihedral Wedge with a Magnetodielectric-Coated Cylinder at the Apex

We present a technique for obtaining a rigorous solution to the problem of diffraction of a plane electromagnetic wave by an infinite dihedral wedge with a magnetodielectric-coated cylinder at the apex. In the case of an E-polarized incident wave, we obtain a closed-form solution of the problem and present the results of exact calculations of the far-zone diffracted field for wide ranges of the structure parameters. In particular, it is shown that if the wedge apex is loaded with only a magnetodielectric cylinder, then the diffracted field for the very long-wavelength incident wave is not affected by the dielectric permittivity of the cylinder and the wave-incidence angle, but is affected only by the opening angle of the wedge, the diameter of the dielectric coating in terms of the free-space wavelength, the magnetic permeability of the coating, and the electric properties of the surrounding medium. If a half-plane or a wedge-shaped part are added to a single cylinder, then the backscattering coefficient of such a structure decreases, but the discovered resonance variation in the backscattering coefficient of the cylinder in the long-wavelength range becomes more pronounced.