Abstract

This study concerns the propagation of acoustic waves through a thin screen made of a periodic arrangement of air bubbles in water. The bubbles are oscillators of the Minnaert type whose dynamics is modified by the containment. This nonlinear dynamics is obtained in the time domain using asymptotic analysis and a homogenization technique involving three scales, those being the scale of a bubble, that of the array and eventually that of the wavelength. The resulting effective model is set in the water (the screen has disappeared) and it encapsulates the effect of the screen in a jump of the normal acoustic velocity. The jump is linked to the continuous version of the bubble radius which satisfies a non-linear equation of the Rayleigh–Plesset type. Our findings are exemplified by numerical experiments of the scattering of a short pulse in the linear and nonlinear regimes.