Discrete orthogonal wavelets are a family of functions with compact support which form a basis on a bounded domain. Use of these wavelet families as Galerkin trial functions for solving partial differential equations (PDE's) has been a topic of interest for the last decade, though research has primarily focused on equations with constant parameters. In the current paper the wavelet-Galerkin method is extended to allow spatial variation of equation parameters. A representative example from the field of vibration illustrates the method: compression waves in a bar with varying elastic modulus. The computed natural frequencies and modeshapes are compared to finite element solutions and show excellent correspondence. The wavelet-Galerkin method is also shown to be an efficient and convenient solution method as the majority of the calculations are performed a priori and can be stored for use in solving future PDE's. This efficiency is displayed by performing a stochastic analysis of elastic modulus variation to determine the effect on the frequency response function.