{
  "id": "1402.1081",
  "version": "v1",
  "published": "2014-02-05T16:42:20.000Z",
  "updated": "2014-02-05T16:42:20.000Z",
  "title": "On the nonlocality of the state and wave equation of Treeby and Cox",
  "authors": [
    "Richard Kowar"
  ],
  "categories": [
    "math-ph",
    "math.AP",
    "math.MP"
  ],
  "abstract": "In this paper it is shown that the state equation of Treeby and Cox [B. E. Treeby and B. T. Cox, J. Acoust. Soc. Am. \\textbf{127} 5, (2010)] is \\emph{nonlocal}, more precisely, a \\emph{local} density variation causes an \\emph{instant global} pressure variation and a \\emph{local} pressure variation can only be caused by an \\emph{instant global} density variation. This is in contrast to all frequency dependent dissipative state equations known to the author. Moreover, it is shown that the Green function $G$ of the wave equation of Treeby and Cox cannot have a \\emph{finite} wave front speed, i.e. there exists no finite $c_F>0$ such that $$ G(\\mathbf{x},t) = 0 \\qquad\\mbox{for}\\qquad |\\mathbf{x}|/c_F > t $$ holds, where $|\\mathbf{x}|/c_F$ corresponds to the \\emph{travel time} of a wave propagating with speed $c_F$ from point $\\mathbf{0}$ to point $\\mathbf{x}$. As a consequence, the density and pressure waves satisfying (i) the state equation of Treeby and Cox, (ii) the equation of motion and (iii) the equation of continuity do not have a \\emph{finite} wave front speed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-02-05T16:42:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35L05",
      "35A08",
      "35Q70",
      "74J05",
      "97M10"
    ],
    "keywords": [
      "wave equation",
      "nonlocality",
      "pressure variation",
      "wave front",
      "density variation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1402.1081K"
    }
  }
}