Acoustic scattering by a finite nonlinear elastic plate. I Primary, secondary and combination resonances

Abstract

A thin elastic plate of finite width is irradiated by time-harmonic acoustic waves. The fluid is assumed light compared with the plate mass, and the forcing term is of sufficient amplitude to necessitate the inclusion of a nonlinear term (due to mid-plane stretching) in the plate equation. The order-one scattered field is determined by the method of multiple scales when the forcing frequency approaches a free oscillation frequency (eigenfrequency) of the plate. This solution is shown to agree with previous work, for the linear problem, and can be multivalued for particular values of the plate-fluid parameters. The scattered wave may also exhibit jumps in its amplitude and phase angle as it varies with frequency, incident-wave angle or incident-wave amplitude. The non-linear term further allows the possibility of secondary and combination resonances. These are investigated and the scattered field is shown to contain terms of different frequencies to those of the incident waves. Multivalued solutions and the associated jump phenomenon are again found for these resonant cases.