Scattering of acoustic waves by a nonlinear resonant bubbly screen

Our 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 an equation of the Rayleigh-Plesset type. This allows us to highlight two important effects. Firstly, a bubble within the array has a much larger radiative damping than an isolated bubble. Secondly it perceives a pressure which differs from the acoustic pressure imposed by the source due to bubble-bubble interactions; it results in a term of mass correction deduced from the Green's function for a Laplace problem which accounts for the bubble arrangement. Our findings are exemplified by numerical experiments of the scattering of a short pulse in the linear and nonlinear regimes.