Interface-Shifted Minnaert Resonances

We develop a unified asymptotic theory for Minnaert resonances generated by a small bubble $D_{\epsilon }$ placed near an interface $\mathrm{\Gamma }$ of material discontinuity in an acoustic background. The bubble has characteristic size $\epsilon >0$, 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 $${\mathrm{\Theta }}_{\epsilon }:=\frac{\mathrm{dist}(D_{\epsilon },\mathrm{\Gamma })}{\epsilon },$$ and covers the three natural regimes $${\mathrm{\Theta }}_{\epsilon }\to +\mathrm{\infty },{\mathrm{\Theta }}_{\epsilon }\to {\mathrm{\Theta }}_{\ast }\in (0,\mathrm{\infty }),{\mathrm{\Theta }}_{\epsilon }\to 0:$$ 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 $\mathrm{dist}(D_{\epsilon },\mathrm{\Gamma })=o(\epsilon )$. 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.