Scattering of sound by a finite non-linear elastic plate bounding a nearly resonant cavity

Abstract

A thin elastic plate of finite width is set in an infinite rigid baffle. A rectangular cavity with pressure release walls is appended to the underside of the plate, and a compressible inviscid fluid occupies the region above the baffle and inside the cavity. The fluid is assumed to be light compared to the plate mass. Time-harmonic plane acoustic waves are incident onto the plate from above, and are of sufficient amplitude to necessitate the inclusion of a non-linear term (due to mid-plane stretching) in the plate equation. The plate deflection and scattered sound field are obtained for non-resonant frequencies, and are shown to increase in magnitude as a cavity resonance frequency is approached. The method of multiple scales, involving two slow-time variables, is employed to obtain the leading-order asymptotic solution, and the orders of magnitude of the potentials and plate deflection are shown to agree with previous results for the fully linear problem. The plate non-linearity is found to introduce jump discontinuities in the scattered wave amplitude as it varies with frequency, incident-wave angle or incident-wave amplitude. Secondary and combination resonances are possible, but coupled primary and secondary resonances are shown to be impossible for the particular configuration chosen.