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<p>We&#xa0;consider the nonlinear Westervelt equation.In&#xa0;this equation, the term describing sound diffusion is dropped.Then the Westervelt equation becomes a&#xa0;quasilinear hyperbolic equation.Next, this equation is linearized, and for it we pose and study an&#xa0;inverse problem of determining the coefficient of the nonlinearity depending on the spatial variables.We&#xa0;write down the structure of a&#xa0;solution to the Cauchy problem in a&#xa0;neighborhood of the wave front.On&#xa0;this basis, the inverse problem is reduced to a&#xa0;problem of integral geometry on a&#xa0;family of straight lines with a&#xa0;given weight function.The&#xa0;weight function, as well as the family of straight lines, is invariant under all rotations around a&#xa0;fixed point.This property allows us to establish a&#xa0;uniqueness theorem for a&#xa0;solution to the inverse problem and to propose a&#xa0;convenient and effective method for constructing its solution.</p>

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An Inverse Problem for the Westervelt Equation

  • 期刊论文

  • 发表日期：

    2026年7月30日

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  • V. G. Romanov

摘要

We consider the nonlinear Westervelt equation.In this equation, the term describing sound diffusion is dropped.Then the Westervelt equation becomes a quasilinear hyperbolic equation.Next, this equation is linearized, and for it we pose and study an inverse problem of determining the coefficient of the nonlinearity depending on the spatial variables.We write down the structure of a solution to the Cauchy problem in a neighborhood of the wave front.On this basis, the inverse problem is reduced to a problem of integral geometry on a family of straight lines with a given weight function.The weight function, as well as the family of straight lines, is invariant under all rotations around a fixed point.This property allows us to establish a uniqueness theorem for a solution to the inverse problem and to propose a convenient and effective method for constructing its solution.