Mean-square responses in a plate with sprung masses, energy flow and diffusion

Abstract

Diagrammatic multiple-scattering theory is applied to the calculation of ensemble average square responses in an infinite homogeneous plate in flexure attached to a random distribution of undamped sprung masses. This system is a prototypical example of a wave-bearing master structure with a locally reacting “fuzzy” substructure. Results for mean fields were obtained in an earlier work. Here it is found that fluctuations away from the mean are weak if the spectral and areal number density of sprung masses is great. A radiative transfer equation is found to govern the flow of energy on time scales greater than the inverse of the frequency, and a diffusion equation is found to govern the flow of energy at times greater than the dwell time of energy in the substructure. The diffusion rate is very slow if the dwell time in the substructure is long. The effect of true damping on these results is discussed.