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Abstract <p> The paper studies a wave equation whose velocity has a perturbation localized at a 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 to the scale of the inhomogeneity. Specifically, 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">\(\varepsilon^{\alpha}\)</EquationSource> </InlineEquation>, where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\varepsilon\)</EquationSource> </InlineEquation> is a small parameter tending to <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(0\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\alpha\)</EquationSource> </InlineEquation> is any positive number. The cases <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\alpha&lt;1\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(\alpha&gt;1\)</EquationSource> </InlineEquation> are considered separately. </p>

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Multi-Scaled Short-Wave Asymptotic Solution of the Cauchy Problem to One-Dimensional Wave Equation with Smoothed Jump of the Velocity

  • 期刊论文

  • 发表日期：

    2025年12月1日

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

摘要

Abstract

The paper studies a wave equation whose velocity has a perturbation localized at a point \(x_0\) . The initial condition has the form of a rapidly oscillating wave packet whose wavelength is not comparable to the scale of the inhomogeneity. Specifically, the length of the initial wave is of the order of \(\varepsilon\) and the width of the localized inhomogeneity is of the order of \(\varepsilon^{\alpha}\) , where \(\varepsilon\) is a small parameter tending to \(0\) and \(\alpha\) is any positive number. The cases \(\alpha<1\) and \(\alpha>1\) are considered separately.