Method of propagating waves for a one-dimensional inhomogeneous medium
type: Item
instance of: scholarly article
title: Method of propagating waves for a one-dimensional inhomogeneous medium
zbMATH Open document ID: 1145.35077
author: A. V. Borovskikh
published in: Journal of Mathematical Sciences (New York)
publication date: 2007-08-07T00:00:00Z
review text: The author of this interesting paper establishes a method of propagating waves. The main idea here is a wave to be considered as a changing state of a medium. Given the wave equation \[ a(x){\partial^2u\over \partial t^2}={\partial \over \partial x} \biggl[ b(x){\partial u\over\partial x}\biggr] \quad(x\in \mathbb R^1). \tag{i} \] It is known that the equation under consideration can be reduced to the canonical form \(k(s)u_{tt}=(k(s)u_s)_s\) (\(z_s\equiv \partial z/\partial s\), \(t\)-time, \(s\)-space) by the transformation \(s=\int_{0}^{x}\sqrt{a(\sigma )/b(\sigma )}d\sigma \) of the space variable \(s\), where \(k^2(s)=(ab)(x(s))\). Therefore the equation (i) should be equivalent to Klein-Gordon equation (or telegraph equation in case \(\Phi'(s)+\Phi^2(s) =\) const) \[ z_{tt}=z_{ss}-\bigl[ \Phi'(s)+\Phi^2(s) \bigr]z , \tag{ii} \] where \(z(t,s)=\sqrt{k(s)}u(t,s)\), \(\Phi (s)=k'(s)/2k(s)\), \(k(s)\) is twice continuously differentiable and positive. It turns out that the method of solving offered by author here allows us to represent a solution of the equation (ii) in an inhomogeneous medium as the sum of two constantly deformed waves, that is, the ``right wave'' and ``left wave'', transported from point to point with coefficients depending on the points and the transport time. It is shown that there exist explicit formulas for solutions (in classical sense) of the mixed problem with homogeneous and nonhomogeneous boundary conditions and solutions of Goursat problem as well.
reviewed by: Dimitar A. Kolev
Mathematics Subject Classification ID: 35L10
Mathematics Subject Classification ID: 35L05
Mathematics Subject Classification ID: 35L20
zbMATH DE Number: 5177650
zbMATH Keywords: waves
zbMATH Keywords: wave equation
zbMATH Keywords: Klein-Gordon equation
zbMATH Keywords: telegraph equation
zbMATH Keywords: Goursat problem
MaRDI profile type: Publication
full work available at URL: s10958-005-0174-3
OpenAlex ID: W2166994047
cites work: A traveling wave formula in a one-dimensional inhomogeneous medium
cites work: Q4002592
cites work: Q3678037
cites work: A general approach to splitting and invariant embedding for linear wave equations
cites work: Factorization of the wave equation in higher dimensions
DOI: 10.1007/S10958-005-0174-3
description: scientific article; zbMATH DE number 5177650