Method of propagating waves for a one-dimensional inhomogeneous medium
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.
scientific article; zbMATH DE number 5177650