Sound propagation and scattering in media with random inhomogeneities of sound speed,density and medium velocity

Abstract

This review presents both classical and new results of the theory of sound propagation in media with random inhomogeneities of sound speed, density and medium velocity (mainly in the atmosphere and ocean). An equation for a sound wave in a moving inhomogeneous medium is presented, which has a wider range of applicability than those used before. Starting from this equation, the statistical characteristics of the sound field in a moving random medium are calculated using Born-approximation, ray, Rytov and parabolic-equation methods, and the theory of multiple scattering. The results obtained show, in particular, that certain equations previously widely used in the theory of sound propagation in moving random media must now be revised. The theory presented can be used not only to calculate the statistical characteristics of sound waves in the turbulent atmosphere or ocean but also to solve inverse problems and develop new remote-sensing methods. A number of practical problems of sound propagation in moving random media are listed and the further development of this field of acoustics is considered.     

An effective quiescent medium for sound propagating through an inhomogeneous,moving fluid

The idea of similarity between acoustic fields in a moving fluid and in a certain "effective" quiescent medium, first put forward by Lord Rayleigh, proved very helpful in understanding and modeling sound propagation in an atmosphere with winds and in an ocean with currents, as well as in other applications involving flows with small velocity compared to sound speed. Known as effective sound speed approximation, the idea is routinely utilized in the contexts of the ray theory, normal mode representation of the sound field, and the parabolic approximation. Despite the wide use of the concept of effective sound speed in acoustics of moving media, no theoretical justification of Rayleigh's idea was published that would be independent of the chosen representation of the sound field and uniformly apply to distinct propagation regimes. In this paper, we present such a justification by reducing boundary conditions and a wave equation governing sound fields in the inhomogeneous moving fluid with a slow flow to boundary conditions and a wave equation in a quiescent fluid with effective sound speed and density. The derivation provides insight into validity conditions of the concept of effective quiescent fluid. Introduction of effective density in conjunction with effective sound speed is essential to ensure accurate reproduction of acoustic pressure amplitude in the effective medium. Effective parameters depend on sound speed, flow velocity, and density of the moving fluid as well as on sound propagation direction. Conditions are discussed under which the dependence on the propagation direction can be avoided or relaxed.     

Mean field of a low-frequency sound wave propagating in a fluctuating ocean

Statistical characteristics of low-frequency sound waves propagating over long distances in a fluctuating ocean are important for many practical problems. In this paper, using the theory of multiple scattering, the mean field of a low-frequency sound wave was analytically calculated. In these calculations, the ratio of the sound wavelength and the scale of random inhomogeneities can be arbitrary. Furthermore, the correlation function of inhomogeneities is expressed in terms of a modal spectrum (e.g., internal waves modes). The obtained mean sound field is expressed as a sum of normal modes that attenuate exponentially. It is shown that the extinction coefficients of the modes are linearly related to the spectrum of random inhomogeneities in the ocean. Measurements of the extinction coefficients can therefore be used for retrieving this spectrum. The mean sound field is calculated for both 3D and 2D geometries of sound propagation. The results obtained can be used to study the range of applicability of the 2D propagation model.     

Parabolic equation for nonlinear acoustic wave propagation in inhomogeneous moving media

A new parabolic equation is derived to describe the propagation of nonlinear sound waves in inhomogeneous moving media. The equation accounts for diffraction, nonlinearity, absorption, scalar inhomogeneities (density and sound speed), and vectorial inhomogeneities (flow). A numerical algorithm employed earlier to solve the KZK equation is adapted to this more general case. A two-dimensional version of the algorithm is used to investigate the propagation of nonlinear periodic waves in media with random inhomogeneities. For the case of scalar inhomogeneities, including the case of a flow parallel to the wave propagation direction, a complex acoustic field structure with multiple caustics is obtained. Inclusion of the transverse component of vectorial random inhomogeneities has little effect on the acoustic field. However, when a uniform transverse flow is present, the field structure is shifted without changing its morphology. The impact of nonlinearity is twofold: it produces strong shock waves in focal regions, while, outside the caustics, it produces higher harmonics without any shocks. When the intensity is averaged across the beam propagating through a random medium, it evolves similarly to the intensity of a plane nonlinear wave, indicating that the transverse redistribution of acoustic energy gives no considerable contribution to nonlinear absorption. Published in Russian in Akusticheskiĭ Zhurnal, 2006, Vol. 52, No. 6, pp. 725–735. This article was translated by the authors.     

Moment-screen method for wave propagation in a refractive medium with random scattering

Direct numerical solution of a parabolic equation (PE) for the second moment of the sound field in a refracting medium with random scattering is described. The method determines the mean-square sound pressure without requiring generation of random realizations of the propagation medium. The second-moment matrix is factored into components that are independently propagated with a conventional PE algorithm. A moment screen is periodically applied to attenuate the coherence of the wavefield, much as phase screens are often applied in the method of random realizations. An example involving upwind and downwind propagation in the near-ground atmosphere shows that the new direct method converges to an accurate solution faster than the method of random realizations and is particularly well suited to calculations at low frequencies.     

海洋声学中三维抛物方程非均匀网格模型

在海洋声学中,三维抛物方程模型可以有效考虑三维空间的声传播效应。然而,采用三维抛物方程模型分析三维空间内的声传播问题时,计算时间较长,并且需要消耗较大的计算机内存,因此给远距离声场的快速精确计算带来了很大困难。为此,将非均匀网格Galerkin离散化方法用于三维直角坐标系下的水声抛物方程模型中,深度算子和水平算子Galerkin离散方式由均匀网格变为非均匀网格。仿真结果表明,三维直角坐标系下非均匀网格离散的抛物方程模型,在保持计算精度、提高计算速度的同时,可以实现远距离声场的快速预报。另外,针对远距离局部海底地形与距离有关的三维声传播问题,给出了声场快速计算方法;在海底保持水平的区域,采用经典Kraken模型,重构抛物方程算法的初始场,随后依次递推求解地形与距离有关海底下的三维声场。采用改进模型,证明了远距离楔形波导声强增强效应。      

聚焦超声波在层状生物媒质中的二次谐波声场的理论与实验研究

基于Khokhlov-Zabolotkaya-Kuznetsov(KZK)方程，在频域提出了聚焦超声波在层状生物媒质中传播的理论模型，该模型计及生物媒质的吸收、非线性和边界，同时考虑声源的衍射对声传播的影响.数值研究了聚焦超声波的基波和二次谐波在层状生物媒质中的声传播，并与实验结果相比较.研究结果表明，此方法可以有效地描述聚焦超声波在层状生物媒质中的二次谐波声场. 关键词： 聚焦超声波 层状生物媒质 二次谐波     

Stochastic differential equation analysis for sound scattering by random internal waves in the ocean

The scattering of a weakly divergent narrow sound beam by random inhomogeneities of a fluctuating ocean is considered in the coupled-mode approximation. The random index of sound refraction is described using the Garrett-Munk internal wave spectrum. The problem is solved using the stochastic differential equations for the first-and second-order statistical moments of the acoustic field. The equations are formulated according to the cumulant expansion method. The existence of weakly divergent narrow sound beams in long-range sound propagation was one of the last discoveries of L.M. Brekhovskikh, to which he attached much importance. The concentration of sound into narrow beams away from the axis of the underwater sound channel was first observed experimentally and then explained by Brekhovskikh and his former students Goncharov, Kurtepov, and Petukhov. In the present paper, the scattered field intensity of a sound beam is calculated for different frequencies and source depths. Analytical expressions are obtained for the coefficients of the differential equation. The intermode energy transfer that accompanies the long-range propagation of a weakly divergent sound beam is analyzed. A comparison with the conventionally used Monte Carlo simulation in the parabolic equation approximation is performed.     

Equations for finite-difference, time-domain simulation of sound propagation in moving inhomogeneous media and numerical implementation

Finite-difference, time-domain (FDTD) calculations are typically performed with partial differential equations that are first order in time. Equation sets appropriate for FDTD calculations in a moving inhomogeneous medium (with an emphasis on the atmosphere) are derived and discussed in this paper. Two candidate equation sets, both derived from linearized equations of fluid dynamics, are proposed. The first, which contains three coupled equations for the sound pressure, vector acoustic velocity, and acoustic density, is obtained without any approximations. The second, which contains two coupled equations for the sound pressure and vector acoustic velocity, is derived by ignoring terms proportional to the divergence of the medium velocity and the gradient of the ambient pressure. It is shown that the second set has the same or a wider range of applicability than equations for the sound pressure that have been previously used for analytical and numerical studies of sound propagation in a moving atmosphere. Practical FDTD implementation of the second set of equations is discussed. Results show good agreement with theoretical predictions of the sound pressure due to a point monochromatic source in a uniform, high Mach number flow and with Fast Field Program calculations of sound propagation in a stratified moving atmosphere.     

Spectral-energy characteristics of sound fields scattered by fine-structured inhomogeneities of sound velocity into the geometrical shadow zone

In analyzing experimental results obtained with explosive sources in the tropical zone of the Indian Ocean, a good agreement was obtained for spectral-energy characteristics of signals observed in the first geometrical shadow zone with computer calculations of the sound field scattered by fine structure inhomogeneities of the fractal type. From the comparison of the results of calculating the frequency characteristics of sound fields in the shadow zone by the wave code and by the method combining ray acoustics with the wave theory of sound scattering, it was found that both methods are appropriate for describing the real processes of scattering and propagation of sound in the ocean with fine-structured stratification and that these methods can be used for solving inverse problems.     

Scattering of electromagnetic waves by ensembles of particles and discrete random media

Current problems of the theory of multiple scattering of electromagnetic waves by discrete random media are reviewed, with an emphasis on densely packed media. All equations presented are based on the rigorous theory of electromagnetic scattering by an arbitrary system of non-spherical particles. The main relations are derived in the circular-polarization basis. By applying methods of statistical electromagnetics to a discrete random medium in the form of a plane-parallel layer, we transform these relations into equations describing the average (coherent) field and equations for the sums of ladder and cyclical diagrams in the framework of the quasi-crystalline approximation. The equation for the average field yields analytical expressions for the generalized Lorentz-Lorenz law and the generalized Ewald-Oseen extinction theorem, which are traditionally used for the calculation of the effective refractive index. By assuming that the particles are in the far-field zones of each other, we transform all equations asymptotically into the well-known equations for sparse media. Specifically, the equation for the sum of the ladder diagrams is reduced to the classical vector radiative transfer equation. We present a simple approximate solution of the equation describing the weak localization (WL) effect (i.e., the sum of cyclical diagrams) and validate it by using experimental and numerically exact theoretical data. Examples of the characteristics of WL as functions of the physical properties of a particulate medium are given. The applicability of the interference concept of WL to densely packed media is discussed using results of numerically exact computer solutions of the macroscopic Maxwell equations for large ensembles of spherical particles. These results show that theoretical predictions for spare media composed of non-absorbing or weakly absorbing particles are reasonably accurate if the particle packing density is less than ∼30%. However, a further increase of the packing density and/or absorption may cause optical effects not predicted by the low-density theory and caused by near-field effects. The origin of the near-filed effects is discussed in detail.     

Reference-wave solutions for the high-frequency fields in inhomogeneous-background random media

Ray theory plays an important role in determining the propagation properties of high-frequency fields and their statistical measures in complicated random environments. For computations of the statistical measures it is therefore desirable to have a solution for the high-frequency field propagating along an isolated ray trajectory. A new reference wave is applied to obtain an analytic solution of the parabolic wave equation that describes propagation along the ray trajectory of the deterministic-background medium. The methodology is based on defining a paired-field measure as a product of an unknown field propagating in a disturbed medium and the complex-conjugate component propagating in a medium without random fluctuations. When a solution of the equation for the paired-field measure is obtained, the solution of the deterministic component can be extracted from the paired solution to determine the solution of the unknown field in an explicit form.     

On the possibility of using acoustic reverberation for remote sensing of ocean dynamics

The two-point correlation function of diffuse noise fields produced by distributed random sound sources carries useful information on the medium of sound propagation. Such information can be used for performing passive acoustic tomography of the ocean. In a number of cases that are important for practice, the noise field in the ocean is predominated by contributions of individual point sources. Here, a theoretical study is presented on the possibility of determining the sound speed and current velocity in the water column by the correlation processing of reverberation signals measured by two vertical receiving arrays. In other words, we study the possibility of replacing the diffuse noise produced by a great number of delta-correlated sources by waves generated by a localized source and scattered at the rough surface and bottom of the ocean for sensing the medium. The correlation function of scattered waves is calculated by using the method of small perturbations. It is shown that the correlation processing of the scattered waves offers an opportunity of measuring the acoustic nonreciprocity and reconstructing the field of sound speed in the fluid, without using any acoustiLc transceivers.     

A High-Efficiency Spectral Method for Two-Dimensional Ocean Acoustic Propagation Calculations

The accuracy and efficiency of sound field calculations highly concern issues of hydroacoustics. Recently, one-dimensional spectral methods have shown high-precision characteristics when solving the sound field but can solve only simplified models of underwater acoustic propagation, thus their application range is small. Therefore, it is necessary to directly calculate the two-dimensional Helmholtz equation of ocean acoustic propagation. Here, we use the Chebyshev–Galerkin and Chebyshev collocation methods to solve the two-dimensional Helmholtz model equation. Then, the Chebyshev collocation method is used to model ocean acoustic propagation because, unlike the Galerkin method, the collocation method does not need stringent boundary conditions. Compared with the mature Kraken program, the Chebyshev collocation method exhibits a higher numerical accuracy. However, the shortcoming of the collocation method is that the computational efficiency cannot satisfy the requirements of real-time applications due to the large number of calculations. Then, we implemented the parallel code of the collocation method, which could effectively improve calculation effectiveness.     

A numerical technique for solving the coupled radiative transfer and heat equations in a random medium with plane geometry

A Monte Carlo based numerical method is presented which generates statistical information equivalent to the unknown flux density solution of the radiative transfer equation within a plane geometry, for a useful class of problems. This method scores all vectorial intersections of randomly generated light paths with a set of internal data surfaces spanning the random medium. This approach has not been previously considered, but can be implemented on a simple and compact computer code that is straightforward to utilize and adapt. The method is presented as an alternative or complement to standard numerical methods and is tested with diffusion theory. Many useful computations can be performed within several minutes. It is particularly well suited to modelling laser-generated heat sources within finite, artificial or biological random media. A numerical solution of the heat equation is then possible on the same set of data surfaces used in the optical model. Illustrative calculations are presented.     

Wave mode computing method using the step-split Padé parabolic equation

Models based on a parabolic equation (PE) can accurately predict sound propagation problems in range-dependent ocean waveguides. Consequently, this method has developed rapidly in recent years. Compared with normal mode theory, PE focuses on numerical calculation, which is difficult to use in the mode domain analysis of sound propagation, such as the calculation of mode phase velocity and group velocity. To broaden the capability of PE models in analyzing the underwater sound field, a wave mode calculation method based on PE is proposed in this study. Step-split Padé PE recursive matrix equations are combined to obtain a propagation matrix. Then, the eigenvalue decomposition technique is applied to the matrix to extract sound mode eigenvalues and eigenfunctions. Numerical experiments on some typical waveguides are performed to test the accuracy and flexibility of the new method. Discussions on different orders of Padé approximant demonstrate angle limitations in PE and the missing root problem is also discussed to prove the advantage of the new method. The PE mode method can be expanded in the future to solve smooth wave modes in ocean waveguides, including fluctuating boundaries and sound speed profiles.     

Computer modeling of sound propagation in the ocean with fractal inhomogeneities

The wave-field computer code based on the wide-angle parabolic equation is modified and adapted to the problems of sound scattering in a medium with anisotropic inhomogeneities of fractal type. To verify the computer code, a model numerical experiment on determining the angular dependence of the scattered sound field is performed for different anisotropy coefficients of the sound speed inhomogeneities. The comparison of the computed data with the theoretical dependences shows their rather good agreement and indicates that the computer code can be applied to calculations of sound propagation in the ocean with fine-structure inhomogeneities possessing fractal properties.     

Proper orthogonal decomposition and cluster weighted modeling for sensitivity analysis of sound propagation in the atmospheric surface layer

Outdoor sound propagation predictions are compromised by uncertainty and error in the atmosphere and terrain representations, and sometimes also by simplified or incorrect physics. A model's predictive power, i.e., its accurate representation of the sound propagation, cannot be assessed without first quantifying the ensemble sound pressure variability and sensitivity to uncertainties in the model's governing parameters. This paper describes fundamental steps toward this goal for a single-frequency point source. The atmospheric surface layer is represented through Monin-Obukhov similarity theory and the acoustic ground properties with a relaxation model. Sound propagation is predicted with the parabolic equation method. Governing parameters are modeled as independent random variables across physically reasonable ranges. Latin hypercube sampling and proper orthogonal decomposition (POD) are employed in conjunction with cluster-weighted models to develop compact representations of the sound pressure random field. Full-field sensitivity of the sound pressure field is computed via the sensitivities of the POD mode coefficients to the system parameters. Ensemble statistics of the full-field sensitivities are computed to illustrate their relative importance at every down range location. The central role of sensitivity analysis in uncertainty quantification of outdoor sound propagation is discussed and pitfalls of sampling-based sensitivity analysis for outdoor sound propagation are described.     

Ray dynamics of the propagation of sound in nonuniform moving media

This paper presents a new ray theory for the propagation of sound waves in nonuniformly moving media. It is found that the ray equations in weakly inhomogeneous and slowly moving media are analogous to the equations of motion of charged particles in nonuniform electric and magnetic fields. The adiabatic approximation is used to study the problem of the propagation of sound rays in a model of near-ocean-bottom waveguide with horizontal flow and slowly varying parameters along the direction of propagation of the wave. A general formula is derived that describes the transverse displacement of the trajectory of the ray relative to the direction of propagation of the wave.     

A review of research progress in air-to-water sound transmission

International and domestic research progress in theory and experiment and applications of the air-to-water sound transmission are presented in this paper. Four classical numerical methods of calculating the underwater sound field generated by an airborne source, i.e., the ray theory, the wave solution, the normal-mode theory and the wavenumber integration approach, are introduced. Effects of two special conditions, i.e., the moving airborne source or medium and the rough air-water interface, on the air-to-water sound transmission are reviewed. In experimental studies, the depth and range distributions of the underwater sound field created by different kinds of airborne sources in near-field and far-field, the longitudinal horizontal correlation of underwater sound field and application methods for inverse problems are reviewed.