{
  "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 $δ=\\varepsilon^2$, 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27494",
  "author_total": 5,
  "authors": [
    {
      "name": "Habib Ammari"
    },
    {
      "name": "Yuxin Du"
    },
    {
      "name": "Xin Fu"
    },
    {
      "name": "Wenjia Jing"
    },
    {
      "name": "Moritz Melcher"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/76vbrk70bt4zt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27494",
  "fulltext_url": null,
  "id": "76vbrk70bt4zt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27494",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2204|Acoustic Wave Phenomena Research"
    ]
  ],
  "openalex": "W7214173220",
  "pdf_url": "https://arxiv.org/pdf/2609.27494",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/76vbrk70bt4zt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.27494",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27494"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Band-Edge Homogenization and the Sound-Soft Limit of Finite Bubbly Crystals",
  "title_text": "Band-Edge Homogenization and the Sound-Soft Limit of Finite Bubbly Crystals",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/76vbrk70bt4zt",
  "venue": "arXiv",
  "versions": []
}