Reflection coefficients for scattering from a pressure-release, sinusoidal surface

Abstract

This paper deals with the scattering of an incident plane wave from a pressure-release, sinusoidal rough surface. As was shown by Uretsky [Ann. Phys. 33, 400–427 (1965)] this problem is equivalent to solving an infinite set of linear equations (involving the matrix V0m,n) for the quantities ψm, the Fourier components (apart from a phase factor) of the normal derivative of the pressure on the surface. Holford showed that the ψm satisfied an infinite set of linear equations of the ‘‘second kind’’ (involving the matrix V1m,n) and deduced the existence and uniqueness of solutions. This paper shows that the matrix elements V0m,n and V1m,n can be expressed as a single sum over products of integer index Bessel functions, subject only to the slope restriction Kd<0.6627. These series representations of the matrices V0m,n and V1m,n are used to prove the result that Rayleigh’s equations are exact up to the slope limit Kd<0.6627. It is also shown that the reflection coefficients Rn satisfy an infinite set of linear equations of the second kind, valid provided the maximum slope Kd<0.6627. The matrix elements involved in the equations for the reflection coefficients Rn are proportional to a single integer Bessel function. Consequently, these equations may be easily solved by the method of reduction.