Band-Edge Homogenization and the Sound-Soft Limit of Finite Bubbly Crystals
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 ε in a bounded Lipschitz domain, and their density contrast is δ=ε2, 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(ε) 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 L2 and far-field discrepancies are O(ε), while the interior L2 field is O(ε1/2). 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.
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BibTeX
@misc{ammari2026bandedge,
title = {{Band-Edge Homogenization and the Sound-Soft Limit of Finite Bubbly Crystals}},
author = {Habib Ammari and Yuxin Du and Xin Fu and Wenjia Jing and Moritz Melcher},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27494},
eprint = {2609.27494},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27494},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Band-Edge Homogenization and the Sound-Soft Limit of Finite Bubbly Crystals
AU - Habib Ammari
AU - Yuxin Du
AU - Xin Fu
AU - Wenjia Jing
AU - Moritz Melcher
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.27494
UR - https://arxiv.org/abs/2609.27494
AB - 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 ε in a bounded Lipschitz domain, and their density contrast is δ=ε^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(ε) 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(ε), while the interior L^2 field is O(ε^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.
ER -