Radiation and boundary-value problems in the frequency domain

Abstract

We return to the problem of computing the field u +( r , t ) radiated by a real-valued spaceand time-varying source q ( r , t ) embedded in an infinite homogeneous medium such as free space. As in Chapter 1 we will assume here that the time-dependent source q ( r , t ) is compactly supported in the space-time region { S 0| r ϵ τ 0 , ϵ t ϵ [0, T 0]}, where τ 0 is its spatial volume and [0, T 0 ] the interval of time over which the source is turned on. In the case in which the medium is non-dispersive the radiated wavefield satisfies the inhomogeneous scalar wave equation Eq. (1.1). More generally, if the background medium is dispersive it is necessary to replace the second time derivative in this equation by an integral (convolutional) operator, so that the wave equation is actually an integral-differential equation. In this chapter we will treat the radiation problem in the frequency domain so that this complication is avoided and our results apply both to dispersive and to non-dispersive backgrounds. In addition to treating the radiation problem we also treat the classical boundary value problem for the scalar wave Helmholtz equation in a (possibly dispersive) uniform background medium. Special attention is devoted to the famous Rayleigh–Sommerfeld boundary-value problem, which consists of computing a radiated field throughout a half-space that is exterior to the source region τ 0 from Dirichlet or Neumann conditions prescribed over an infinite bounding plane to the source.