Reciprocity, Discretization, and the Numerical Solution of Direct and Inverse Acoustic Radiation and Scattering Problems

Abstract

Except for the canonical problems in acoustics whose solution can be expressed in terms of analytic functions of a not too complicated nature, and for analytic approximation techniques (usually of an asymptotic nature) that can be applied to a wider variety of cases, wave propagation and scattering problems in acoustics have to be addressed with the aid of numerical techniques. Many of these methods can be envisaged as being discretized versions of appropriate ‘weak’ formulations of the pertinent operator (differential or integral) equations. For the relevant problems as formulated in the time Laplace-transform domain it is shown that the Rayleigh reciprocity theorem encompasses all known weak formulations, while its discretization leads to the discretized forms of the corresponding operator equations, in particular to their finite-element and integral-equation modeling schemes. Both direct (forward) and inverse problems are discussed.