Scattering by a three-part impedance plane: a new spectral approach

Abstract

The problem of diffraction by an arbitrary three-part impedance plane is delicate. Generally the Green's functions involved in this problem are very complicated and the Wiener-Hopf method leads to coupled integral equations not easy to use when the central strip is small. A new method based on the Sommerfeld-Maliuzhinets integral representation is presented. We develop an expression of the spectral function, where we can simply isolate the contribution of any element of the surface. For the three-part impedance plane with arbitrary passive impedances, it then becomes possible to derive simple functional equations on spectral functions of Maliuzhinets type, and, after some calculus, to obtain Fredholm integral equations that are uncoupled. We can then use a semi-inversion that modifies the kernels so that they are convenient when the central strip of the three-part plane is small. An asymptotic analysis of the integral equation, for large imaginary arguments, then permits us to derive analytical expressions of the field at both discontinuities. An explicit approximate expression is then possible for the scattering diagram. For the particular case when the impedances of the semi-infinite planes are equal, we detail the expression and give an analytical approximation of the scattering diagram exhibiting some angular behaviour and resonances that we then validate.