Homogenization of resonant bubble screens: Influence of bubble shape and lattice arrangement

A time-domain effective model for acoustic wave propagation through a two-dimensional periodic array of gas bubbles embedded in a liquid is presented. The model is expressed as transmission conditions: pressure remains continuous, whereas the normal velocity exhibits a jump induced by the internal pressure of the bubbles. This internal pressure follows a damped mass-spring equation, with damping arising solely from radiative coupling to the surrounding liquid, which makes the resonance frequency and quality factor of the array emerge unambiguously. Aside from the bubble density in the lattice, these quantities are fully governed by two independent geometric parameters: a dimensionless capacitance, depending solely on bubble shape, and a lattice coefficient, depending solely on lattice geometry. For plane wave scattering, comparisons with direct numerical simulations demonstrate that the model accurately reproduces the resonant behavior of bubble screens across a range of configurations, including spherical, spheroidal, and cylindrical bubbles, as well as square and rectangular lattices. This generalizes the classical model of Leroy et al. [Eur. Phys. J. E 29(1), 123-130 (2009)] for spherical bubbles in square lattices. Notably, the model reveals-and simulations confirm-that the resonance frequency shift relative to an isolated bubble, usually positive (blue shift), can become negative (red shift) in rectangular lattices with aspect ratios exceeding seven.