Conservation of energy in random media, with application to the theory of sound absorption by an inhomogeneous flexible plate

Abstract
This paper discusses the linear theory of wave propagation through conservative random media. By means of a simple illustrative example taken from acoustics it is verified that, at least uniformly to second order in an expansion parameter associated with the random fluctuations, the net flow of energy is from the mean, or coherent, wave field to the random component of the wave field (in accordance with the second law of thermodynamics). A general formula is derived for the distribution of the random modes of the system responsible for the power flux into the random field. These are demonstrated unambiguously (without recourse to the use of far field asymptotics) to be precisely those propagating modes which satisfy the homogeneous, non-random dispersion relation. The extension of the theory to a wider class of wave propagation problems is then outlined using an approach involving a Lagrangian density of wide generality. Finally the discussion is extended further to cover the case of coupled systems of wave-bearing media. An important analytical feature of such cases is the occurrence of ‘branch-cut’ integrals in the power flux formula. The situation is illustrated by an investigation of the power extracted from a plane sound wave incident on a flexible plate whose mass density is a random function of position. The division of the scattered power between the acoustic and plate bending modes is obtained, and comparison made with a heuristic argument leading to the same result.