David H. Berman

Simulations of rough interface scattering

This paper describes numerical simulations of rough interface scattering. Both Dirichlet and fluid–solid boundary conditions are treated. The Rayleigh–Fourier method is used to compute exact plane‐wave scattering amplitudes and results are compared to various approximations. The small‐slope approximation of Voronovich performs remarkably well, while the tangent‐plane approximation is shown to miss some essential physics of scattering. It is observed that at the Rayleigh angle there is a peak in the backscattering amplitude, even for plane‐wave incidence. It is argued that the statistics of plane‐wave scattering amplitudes are Gaussian.

Rayleigh method for scattering from random and deterministic interfaces

This paper describes Rayleigh methods for computing plane‐wave scattering amplitudes for rough interface scattering. For Dirichlet, fluid–fluid, and fluid–solid interfaces, the Rayleigh–Fourier (RF) method is shown to give good results for interfaces with slopes exceeding the limit of the Rayleigh hypothesis. In the case of sinusoidal Dirichlet surfaces, described by z=h cos(2πx/L), the method breaks down for 2πh/L∼2. For gentler surfaces the RF method gives high‐quality results more cheaply than boundary integral equation or least‐squares methods. In the case of fluid–solid interfaces, results of the RF method, which are apparently presented here for the first time, compare favorably with published results obtained by extinction theorem methods. It is concluded that the RF method is a suitable candidate for studying rough interface scattering by Monte Carlo simulations, even for fluid–solid interfaces.

Manifestly reciprocal scattering amplitudes for rough interface scattering

A new exact expression for scattering amplitudes for rough interface scattering is presented. This expression is explicitly reciprocal and it is shown to hold for a variety of boundary conditions: Dirichlet (pressure release), Neumann (rigid surface), impedance, fluid‐fluid, and fluid‐solid interfaces. This expression is shown to be a convenient starting point for deriving approximations that also respect reciprocity. Examples include a small‐slope approximation, a reciprocal phase perturbation approximation, and a reciprocal smoothing approximation. [Work supported by ONR.]

An optimal PE‐type wave equation

A one‐way wave equation is presented with the following properties. (1) For low angles and small sound‐speed variations, it reduces to the standard parabolic approximation. (2) It allows a split‐step solution. (3) The rays associated with this equation are exactly the rays of the Helmholtz equation in a range‐independent environment. It is in the last sense an optimal one‐way wave equation. Results of the split‐step solution of this equation are presented and compared to normal‐mode calculations and results of another modification of the standard parabolic equation, which was given by Thomson and Chapman [D. J. Thomson and N. R. Chapman, J. Acoust. Soc. Am. 74, 1848–1854 (1983)].

Reverberation in waveguides with rough surfaces

An impedance formalism for treating scattering at multiple rough boundaries in waveguides with depth-varying sound speeds is developed. The formalism is used to give a compact result for the time history of multiply scattered reverberant sound. In general, the multiple scattering contribution to reverbation decays more slowly than the single scattering contribution. When attenuation mechanisms other than surface scattering are small, the reverberant field has a diffusive character, and this leads to algebraic (∼1/t) decay of the reverberation of originally pulsed signals. As a by-product of the formalism, conventional reverberation models are derived from “first principles.” The derivation provides the precise form of the scattering strength used in such models. In addition, it shows that the mean field should be used for propagation to and from the scattering surface. Finally, the computations present the possibility of treating the scattering by two rough boundaries simultaneously.

Effective reflection coefficients for the mean acoustic field between two rough interfaces

A formalism is presented that demonstrates that the mean Green’s function for the acoustic field between two rough interfaces can be expressed as a Green’s function associated with two flat interfaces with effective reflection coefficients. This result incorporates all orders of the fluctuations in the half‐space scattering amplitudes associated with each interface considered separately. From the mean Green’s function modal attenuations can be found. To lowest order in the surface height fluctuations it is shown that it is not sufficient to use mean half‐space scattering amplitudes as effective reflection coefficients. The formalism is designed to provide approximations for the Green’s function in layered media which are based on previously developed approximations for half‐space scattering amplitudes.

Computing effective reflection coefficients in layered media

Computations are presented which show that the effective reflection and transmission coefficients for a rough interface embedded in a layered medium can differ significantly from the mean reflection and transmission coefficients computed for the same rough interface when it separates two homogeneous half-spaces. These differences are large when the correlation length of the roughness is long compared to the skip distance of rays associated with normal modes in the layered medium. Otherwise, these differences may be generally neglected. However, increasing the rms roughness decreases the ratio of correlation length to skip distance at which the effect of the layering is important. The case of a Pekeris waveguide with a rough fluid–fluid interface and the case of a rough Dirichlet surface bounding an upwardly refracting medium are considered.

Renormalization of propagation in a waveguide with rough boundaries

A scheme is presented for computing the mean field in a waveguide with rough boundaries using nonperturbative half‐space scattering amplitudes. It is shown that the presence of a second scattering surface renormalizes the mean half‐space amplitudes even when surface roughness on the two bounding surfaces is uncorrelated. The method of ‘‘smoothing’’ is used to show how recent work on half‐space scattering amplitudes might be incorporated into solutions of the wave equation in more realistic settings.

The mean acoustic field in layered media with rough interfaces

An algorithm is presented for determining the mean acoustic field in a layered medium containing rough interfaces. It is assumed that scattering by the rough interfaces when considered separately and in the absence of sound speed and density variation can be well‐approximated. It is also assumed that propagation in layered media with flat interfaces can be well approximated. The present work shows how these results can be combined to yield the mean field in a stack of layers with variable sound speeds and densities which are separated by rough interfaces.

Approximating moments of acoustic fields in random media

Abstract Starting from a path integral representation of the fourth moment of an acoustic field, two types of approximations are considered in detail: a ‘zero-kink’ approximation and a ‘single-kink’ approximation. The expressions ‘zero-kink’ and ‘single-kink’ describe families of paths which are used to approximate path integrals over all continuous paths. The single-kink approximation reduces the propagation problem to a phase screen problem with fluctuations on the phase screen containing information about fluctuations along entire paths from source to receiver.