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Band-Edge Homogenization and the Sound-Soft Limit of Finite Bubbly Crystals

by H. Ammari and Y. Du and X. Fu and W. Jing and M. Melcher

(Report number 2026-36)

Abstract
We establish a quantitative sound-soft scattering limit for a finite bubbly crystal near the upper edge of the first Bloch band of the corresponding infinite crystal. The inclusions form a dense periodic array of period \(\varepsilon\) in a bounded Lipschitz domain, and their density contrast is \(\delta=\varepsilon^2\), with fixed positive wave speeds. Under coordinate-reflection symmetry, we prove that the normalized capacitance symbol has a unique nondegenerate maximum on the Brillouin torus. Its Hessian defines a positive Dirichlet elliptic operator governing the limiting spectral detunings. A mean-constrained variational formulation allows us to compare the actual finite-array capacitance matrix with the truncated infinite-lattice operator and to prove exponential localization of their difference near the sample boundary. We obtain an \(O(\varepsilon)\) norm-resolvent approximation and connect it to the full acoustic scattering problem, retaining the second-order Bloch correction required by the frequency scaling. For rescaled detunings outside the effective Dirichlet spectrum, the exterior \(L^2\) and far-field discrepancies are \(O(\varepsilon)\), while the interior \(L^2\) field is \(O(\varepsilon^{1/2})\). Numerical experiments illustrate the far-field convergence and the effective spectral modes. Explicit one-dimensional calculations describe finer Fabry--Pérot transmission windows and show that a fixed frequency strictly inside the first band need not have a unique scattering limit.

Keywords: high-contrast homogenization; subwavelength resonances; Bloch band edges; variational capacitance operators; norm-resolvent convergence; sound-soft scattering limit

@Techreport{ADFJM26_1191,
  author = {H. Ammari and Y. Du and X. Fu and W. Jing and M. Melcher},
  title = {Band-Edge Homogenization and the Sound-Soft Limit of Finite Bubbly Crystals},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2026-36},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2026/2026-36.pdf },
  year = {2026}
}

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