A time-domain energy theorem for scattering of plane acoustic waves in fluids

Abstract

A time-domain energy theorem for the scattering of plane acoustic waves in fluids by an obstacle of bounded extent is derived. It is the counterpart in the time domain of the ‘‘optical theorem’’ or the ‘‘extinction cross section theorem’’ in the frequency domain. No assumptions as to the acoustic behavior of the obstacle need to be made; so, the obstacle may be fluid or solid, acoustically nonlinear, and/or time variant (a kind of behavior that is excluded in the frequency-domain result). As to the wave motion, three different kinds of time behavior are distinguished: (a) transient, (b) periodic, and (c) perpetuating, but with finite mean power flow density. For all three cases the total energy [case (a)] or the time-averaged power [cases (b) and (c)] that is both absorbed and scattered by the obstacle is related to a certain time interaction integral of the incident plane-wave and the spherical-wave amplitude of the scattered wave in the farfield region, when observed in the direction of propagation of the incident wave.