Exact solutions for wave equations of two-layered media with smooth transition

Bluman, George; Kumei, Sukeyuki
January 1988
Journal of Mathematical Physics;Jan1988, Vol. 29 Issue 1, p86
Academic Journal
The wave equation c2(x)uxx-utt=0 is solved for wave speeds c(x) corresponding to two-layered media with smooth transition from layer to layer. The wave speed c(x) has four free parameters to fit a given medium. Solutions are constructed from invariant solutions of a related system of first-order partial differential equations that admit a four-parameter symmetry group. These solutions are superposed to solve general initial value problems for data with compact support; the computation of the superposition coefficients uses elementary Fourier analysis. Solutions are illustrated for various initial conditions.


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