REFLECTION COEFFICIENTS FOR SCATTERING FROM A PRESSURE-RELEASE, SINUSOIDAL SURFACE

This paper deals with the scattering of an incident plane wave from a pressure-release, sinusoidal rough surface. As was shown by Uretsky [A nn. Phys. 33, 400-427 (1965)] this problem is equivalent to solving an infinite set of linear equations (involving the matrix V-m,n(0)) for the quantities psi(m), the Fourier components (apart from a phase fact or) of the normal derivative of the pressure on the surface. Holford s howed that the psi(m) satisfied an infinite set of linear equations of the ''second kind'' (involving the matrix V-m,n(1)) and deduced the e xistence and uniqueness of solutions. This paper shows that the matrix elements V-m,n(0) and V-m,n(1) can be expressed as a single sum over products can be expressed as a single sum over products of integer ind ex Bessel functions, subject only to the slope restriction Kd<0.6627. These series representations of the matrices V-m,n(0) and V-m,n(1) are used to prove the result that Rayleigh's equations 1 are exact up to the slope limit Kd<0.6627. It is also shown that the reflection coeffi cients R(n) 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 R, are propo rtional to a single integer Bessel function. Consequently, these equat ions may be easily solved by the method of reduction. (C) 1996 Acousti cal Society of America.