Scattering by pressure-release sinusoidal surfaces in three dimensions is analyzed using the Fresnel phase approximation and realistic source and receiver directivity approximations. Geometrical shadowing and second-order scattering are explicitly included to explore the validity of the Kirchhoff approximation. No restrictions on the surface heights and slopes are made. The "goodness" of the resulting expressions is verified by requiring the pressure scattered by a sinusoidal surface to reduce to the image solution as the surface amplitude goes to zero. The first-order scattered pressure achieves a very good approximation to the image solution, and the second-order scattered pressure goes to zero, as expected, under this requirement. The theory is compared with available experimental scattering measurements, and the agreement is good. Because the slopes on the experimental surface are very steep, shadowing corrections are indispensible to achieving accurate first and second order scattering results. Shadowing has a greater impact on the scattering prediction than the second-order scattering contribution. This suggests that the Kirchhoff approximation may be more robust when incorporated into a theory including a detailed shadowing treatment as well as the Fresnel phase approximation and a good directivity approximation.

The potential method is extended to treat the reflection and scattering of acoustic waves by rough surfaces. The resulting expression for the scattered pressure contains terms which can be identified as the singly and multiply scattered field components. In addition, a boundary condition that approximately accounts for shadowing is used. Only the singly scattered field term is specialized to a form that is convenient for reflected or scattered pressure and intensity calculations. Finally, the assumption of a randomly rough surface is introduced and very general results concerning the coherent and total scattered fields are obtained.

Measured scattering coefficients are presented for four randomly rough, pressure-release surfaces which are statistically related. The scattering data were collected at frequencies of 100, 200, and 500 kHz for forward, specular, and backscattering geometries. The ratios of acoustic wavelength to surface rms height (λ/h) range from 0.33 to 6.5. The most obvious features of the data are the increasing diffuseness of the scattered energy with increasing roughness, and the substantially frequency independent behavior of the scattering coefficients.

Because of the complexity of the scattering integrals in three dimensions, numerous approximations are used to obtain closed-form solutions. By considering the scattering by an infinite, pressure-release plane surface, the effects of various phase approximations and source directivity approximations can be examined independently of the surface roughness. Calculations are carried out using the Fraunhofer and Fresnel phase approximations, and two directivity approximations. It has been shown experimentally that the image solution is valid for the reflection of an acoustic beam by an infinite, pressure-release plane surface if the plane is in the farfield of the source. Consequently, the image solution is used to compare analytical solutions obtained using various phase and directivity approximations, and it is found that both the Fresnel phase approximation and a realistic directivity approximation are required to achieve a good fit. The solution produced by the Fraunhofer phase approximation is obtained as an asymptotic limit of the modified Fresnel solution. Criteria for the validity of the Fraunhofer and Fresnel phase approximations are developed. The Fresnel phase approximation is valid under fairly broad conditions, but the Fraunhofer phase approximation is never valid for an infinite plane surface that must be in the farfield of the source.

Backscattering measurements from solid brass, aluminum, and lucite cylinders in water have been performed at a ka of approximately 240 using short acoustic pulses. Numerous discrete echoes are received due to multiple internal reflections of the acoustic pulse at the boundary of the cylinders. Excellent agreement between the measured echo arrival times and the echo arrival times predicted by the theory of Brill and Überall [J. Acoust. Soc. Am. 50, 921 (1979)] is obtained for all of the cylinders. The amplitudes of the backscattered echoes from the brass and aluminum cylinders are also compared with theory and fair agreement is obtained.