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Abstract <p> In this paper we study a wave equation whose velocity has a localized perturbation at some point <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(x_0\)</EquationSource> </InlineEquation>. The initial condition has the form of a rapidly oscillating wave packet whose wavelength is not comparable with the scale of the inhomogeneity. In this case, the length of the initial wave is of the order of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\varepsilon\)</EquationSource> </InlineEquation>, and the width of the localized inhomogeneity is of the order of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\varepsilon^{1/m},\)</EquationSource> </InlineEquation> where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\varepsilon\)</EquationSource> </InlineEquation> is a small parameter that tends to 0, and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(m\)</EquationSource> </InlineEquation> is a positive integer greater than 2. </p> <p> <b> DOI</b> 10.1134/S1061920825600916 </p>

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Short-Wave Asymptotic Solutions of the Wave Equation with Localized Velocity Perturbations Whose Wavelength is not Comparable to the Scale of the Localized Inhomogeneity.

  • 期刊论文

  • 发表日期：

    2025年6月1日

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  • A.I. Allilueva,
  • A.I. Shafarevich

摘要

Abstract

In this paper we study a wave equation whose velocity has a localized perturbation at some point \(x_0\) . The initial condition has the form of a rapidly oscillating wave packet whose wavelength is not comparable with the scale of the inhomogeneity. In this case, the length of the initial wave is of the order of \(\varepsilon\) , and the width of the localized inhomogeneity is of the order of \(\varepsilon^{1/m},\) where \(\varepsilon\) is a small parameter that tends to 0, and \(m\) is a positive integer greater than 2.

DOI 10.1134/S1061920825600916