Resonant Microstructures as Dirac-type Sources for Acoustic Wave Control

Abstract

We study finite-time acoustic wave tracking using effective interior Dirac-type actuators generated by clusters of subwavelength resonant bubbles driven from exterior transducers. Starting from a time-domain point-interaction approximation, the full scattering problem is reduced to a finite-dimensional delayed system of interacting retarded monopoles. After localization to a bounded observation domain, we formulate the tracking problem on a prescribed finite-dimensional Dirichlet spectral subspace. At the ideal level, a full-rank spatial coupling condition yields exact tracking of arbitrary sufficiently smooth compatible trajectories in this subspace by interior point sources. These ideal actuators are realized using small, high-contrast bubbles, whose Laplace-domain transfer operator encodes the Minnaert resonances and collective attenuation. The principal mode of each cluster is isolated by the Perron--Frobenius structure and the intra-cluster spectral gap. Clusters tuned to the same band are then treated as a collective resonant block. The selected Dirichlet bands are matched with the corresponding Minnaert bands. For prescribed slowly time-modulated resonant pressure profiles, with sufficiently flat initialization, we construct causal exterior controls and obtain a quantitative tracking estimate on every fixed time interval $[0,T]$. In contrast, on frequency ranges uniformly separated from the Minnaert resonances, the actuator gain is $O(\epsilon )$ and the corresponding realization cost is asymptotically unbounded as $\epsilon \ll 1$, $\epsilon$ being the maximum radius of the bubbles. Thus the Minnaert resonances convert accessible exterior excitations into robust effective interior controls.

The title of the manuscript has changed slightly. There are also updates to the proofs of some theorems/lemmas or propositions. Keywords. Wave equation; Effective Dirac actuators; Finite-time trajectory tracking; Causal control; Resonant microstructures; Minnaert resonance; Perron-Frobenius theory; Schur reduction; Matrix-block realization