Asymptotic approximations to angular-spectrum representations

Abstract

Under rather general conditiosn a time-harmonic wave field u (x,y,z) can be represented in a half-space z≳0 by a double integral known as the angular spectrum of plane waves. The representation divides naturally into the sum of two double integrals, one of which (uH) is a superposition of homogeneous plane waves and the other (ui) is a superposition of inhomogeneous plane waves. We obtain asymptotic approximations to u (x,y,z), uH, and UI valid when the point of observation of the field recedes towards infinity in a fixed direction through a fixed point. The results apply when the spectral amplitude of the plane waves belongs to a specific class which arises frequently in applications. Our approach is based on the method of stationary phase, which we extend in order to permit the presence of inhomogeneous waves in the integrand. Although the analysis of u requires that we distinguish the directions that are perpendicular to the z axis from the directions pointing into the half-space z≳0, the results for the former case are the same as would be obtained by taking appropriate limit in the results of the latter case. We obtain the general form of the asymptotic sequence appropriate for expanding u and present explicit expressions for the first two terms. Our derivation justifies the results of previous heuristic treatments. The analysis of uH and UI requires separate treatments for directions that are (i) perpendicular to the z axis, (ii) parallel to the z axis, and (iii) neither perpendicular nor parallel to the z axis. In contrast to the behavior of u, the asymptotic behavior of uH (and of uI) differs in the different cases. In each case, we obtain the general form of the appropriate sequence and present the first term explicitly.