{
  "abstract": "The Normalized Energy Density (NED) was introduced in Goto et al.(2011) as a quantity that, unlike the conventional energy, is conserved across material interfaces in a layered elastic medium. Its conservation was proved analytically only for a two-layer structure; for three or more layers it was supported by Monte Carlo simulation. I supply the missing proof for a general multi-layer (n-layer) structure in the 2D SH problem. The key step is to stop treating the angular frequency as a single variable and instead lift the layer phases to independent coordinates on the n-torus. The obstruction identified in the original paper---that the ratio of travel-time combinations is generically irrational, so that the periodicity argument used for two layers fails and disappears, and the proof reduces to Weyl's equidistribution theorem combined with n repetitions of the same elementary integral used in the two-layer case. I prove conservation in two independent settings that together cover all cases of practical interest: travel times that are linearly independent over Q, and commensurable travel times, including the equal-travel-time (Goupillaud) discretization. I also show that an arbitrary layered structure can be subdivided by reflectionless interfaces into one whose travel times take only d distinct, Q-independent values, where d is the dimension of the Q-span of the original travel times; this places the two proved cases at the endpoints d=1 and d=n and isolates the remaining open case 1<d<n in a canonical form.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.05924",
  "author_total": 1,
  "authors": [
    {
      "name": "Hiroyuki Goto"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cw84n10ck52sx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.05924",
  "fulltext_url": null,
  "id": "cw84n10ck52sx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.05924",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cond-mat.mtrl-sci"
    ],
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "physics.geo-ph"
    ],
    [
      "openalex",
      "1908|Seismic Waves and Analysis"
    ]
  ],
  "openalex": "W7212076025",
  "pdf_url": "https://arxiv.org/pdf/2609.05924",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cw84n10ck52sx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:48Z",
      "native_id": "2609.05924",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.05924"
    }
  ],
  "subjects": [
    {
      "field": "earth",
      "origin": "arxiv:physics.geo-ph",
      "subfield": "geophysics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.mtrl-sci",
      "subfield": "condensed-matter"
    }
  ],
  "title": "Conservation of Normalized Energy Density in multi-layered media: a proof for the general multi-layer 2D SH problem",
  "title_text": "Conservation of Normalized Energy Density in multi-layered media: a proof for the general multi-layer 2D SH problem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/cw84n10ck52sx",
  "venue": "arXiv",
  "versions": []
}