Interface-Shifted Minnaert Resonances

Abstract

We develop a unified asymptotic theory for Minnaert resonances generated by a small bubble $D_\varepsilon$ placed near an interface $Γ$ of material discontinuity in an acoustic background. The bubble has characteristic size $\varepsilon>0$, with its density and bulk modulus depending on the same small parameter through the Minnaert high-contrast scaling. The analysis is organized by the scaled separation \[ Θ_\varepsilon := \frac{\operatorname{dist}(D_\varepsilon,Γ)}{\varepsilon}, \] and covers the three natural regimes \[ Θ_\varepsilon\to+\infty, \qquad Θ_\varepsilon\toΘ_*\in(0,\infty), \qquad Θ_\varepsilon\to0: \] 1) In the far-separation regime, the interface is invisible at leading order and the classical full-space Minnaert law is recovered. 2) In the critical regime, the rescaled geometry retains both the bubble and the interface at finite separation, and the Newtonian capacitance is replaced by a flat-interface capacitance determined by the limiting tangent transmission problem. 3) In the near-contact regime, this family converges to a contact-limiting capacitance, yielding the same leading-order Minnaert resonance for all gap scalings satisfying $\operatorname{dist}(D_\varepsilon,Γ) = o(\varepsilon)$. These three regimes provide a unified description of how the resonance frequency changes as the bubble approaches the interface, from the classical isolated-bubble limit to the contact limit. The analysis relies on a uniform reduction of the resonance problem to a scalar equation and on a variational characterization of the interface-dependent capacitances; in particular, Mosco convergence of the associated transmission energies yields convergence of the capacitary potentials and minimum energies up to the singular contact configuration.