A general analytical expression for the time-dependent mean-square incoherent field scattered from or through (penetrating) a 2-D fluid–fluid rough interface for a narrow-band incident plane-wave source is derived and expressed in terms of the second moment of the rough interface T-matrix. This analytical expression is independent of the scattering solution technique, and for distances greater than only a few wavelengths from the interface, is equivalently expressed in terms of the bistatic scattering cross section per unit area per unit solid angle (differential cross section) of the rough interface. Using this rigorously derived result, the scattered field for a narrow-band point source is heuristically derived. This derivation leads to the usual sonar equation in the limit as the narrow-band signal approaches the cw (continuous wave) case. First-order perturbation calculations for the case of a baseband Gaussian shaped source pulse illustrate narrow-band pulse dispersion effects of the incoherent field for forward scattering into a lossy sediment. For the case of incidence below the critical grazing angle, first-order perturbation computations also show that the incoherent field scattered through a rough interface can be much greater than the zeroth-order field (coherent) transmitted below the corresponding flat-surface depending on loss and receiver depth. These computations for the first-order mean square incoherent field penetrating the rough interface are compared to the results for the flat-surface case, for both plane-wave and point sources.

Seafloor roughness can cause acoustic energy to propagate into sediments for ‘‘subcritical’’ incident grazing angles, that is, grazing angles smaller than the critical angle determined by the compressional wave speed. Such effects must be considered in interpreting experimental data on sound penetration into sediments. The sediment is modeled as a fluid supporting only compressional waves and a twofold theoretical approach is used. First, an exact integral equation is solved numerically to obtain the penetrating field in two dimensions. Physical insights are abstracted from these results which are also used to show that perturbation theory is valid for problems of interest. Using perturbation theory, a numerically tractable three-dimensional model is compared to some of the experimental data of Chotiros and colleagues [N. P. Chotiros, J. Acoust. Soc. Am. 97, 199–214 (1995)]. The model and data match well subject to plausible assumptions about roughness statistics. [Work supported by ONR.]

A general analytical expression for the time-dependent field intensity scattered from or through (penetrating) a 2-D fluid-fluid rough surface due to a narrow-band incident plane wave and a narrow-band point source are derived and expressed in terms of the bistatic scattering cross section per unit area of the rough surface. Even though the scattering cross section is defined as a far-field quantity, this near-field result is general and exact for the special case of a continuous wave source and incident plane wave. Dispersion of a pulse is a function of medium parameters, the incident and scattered directions, as well the bandwidth and center frequency of the source signal. First-order perturbation calculations for the case of a Gaussian pulse illustrate intensity pulse dispersion effects due to forward scattering into a lossy sediment. [Work supported by ONR.]

Recent experimental results [F. E. Boyle and N. P. Chotiros, 2615–2619 (1992)] reveal acoustic penetration from water into sandy sediments at grazing angles below the compressional critical angle in relation to the mean surface. These authors interpret the results to indicate the excitation of a biot slow wave in the sediment. An additional mechanism for subcritical penetration will be discussed, based on assuming a small level of roughness at the water–sediment interface. Computer simulations of these experiments using theoretical calculations based on Rayleigh–Rice perturbation theory for 2-D surfaces reproduce experimental results, indicating that the acoustic penetration of the surface may be due to scattering (diffraction) from low levels of roughness. The accuracy of perturbation theory for the level of roughness being considered is verified using comparisons with exact calculations in the special case of 1-D surfaces. [Work supported by ONR.]

The first-order perturbation expression for the bistatic scattering cross section of a rough surface [Moe and Jackson, J. Acoust. Soc. Am. (to be published)] is used to study the effect of gradients on high-frequency bistatic bottom scattering. Strong gradients on scales of several centimeters are often observed in shallow-water environments. It is shown that upward refraction or reflection by strong gradients can lead to significant enhancement of high-frequency scattering. This enhancement occurs over a wide range of bistatic angles, but is strongest when the incident and scattered grazing angles are equal. These effects can be explained in terms of the reflection coefficient of the corresponding mean (flat) surface. [Work supported by ONR.]

The acoustic intensity penetrating a rough surface is analyzed using Rayleigh–Rice perturbation theory. When the grazing angle of the incident field is below the critical angle in relation to the mean surface, only the zero-order component of the transmitted field is evanescent; the higher-order components contain downward traveling waves. For an incident field below the critical angle, first-order computations using parameters appropriate to a sandy bottom show that the field below the rough surface can be much greater than the corresponding field below a flat surface. These computations are carried out using a low-frequency cutoff for the bottom relief spectrum. With regard to the accuracy of the calculation, both the short-wavelength portion of the relief profile that is retained as well as the long-wavelength portion that is discarded are considered. It is shown that the rms height of the portion retained and the rms slope of the portion discarded are sufficiently small to lend confidence in the perturbation approach. Further work is required, however, to unequivocally establish the accuracy of the method. [Work supported by ONR.]

The transition matrix relating the incident pressure field to the scattered field in a fluid above a rough surface is derived to first order using Rayleigh–Rice perturbation theory, resulting in a general expression for scattering strength. A small region bordered above by the rough surface and below by a horizontal plane intersecting the surface at its lowest point contains a lossy homogeneous fluid. Below this point, the sediment is allowed to be vertically stratified, viscoelastic, or porous, supporting shear or Biot slow waves. Gradients strongly affect the scattering cross section if they cause a substantial portion of the incident energy to be redirected toward the interface.