The Rayleigh equations for a multi-sinusoidal periodic surface

Abstract

It was shown by Purcell [J. Acoust. Soc. Am. 100, 2919–2936 (1996)] that the Rayleigh equations in the Fourier domain for the reflection coefficients for scattering of a plane wave from a pressure-release sinusoid are valid if the maximum slope of the sinusoid Kd<0.6627. This current work finds the corresponding constraint sufficient for the validity of Rayleigh’s equations for a more general periodic surface consisting of a finite sum of sinusoids. The mathematical basis of the derivation of the Rayleigh equations from the Helmholtz integral formula is a Fourier series given by Oberhettinger. Unlike the single sinusoid case developed by Purcell (referenced above), the analysis of the general periodic surface given here requires an analytic continuation argument. In addition, a set of (infinite linear) equations of the “second kind” is derived for the reflection coefficients for the general periodic surface. This guarantees that the truncation solutions for the reflection coefficients converge and are unique. The matrix elements involved in these equations of the second kind require the numerical evaluation of a finite integral (a generalization of Bessel’s integral for the integer index Bessel functions) and all calculations required can be performed by desktop computing.