Analysis of Wave Propagation in 1D Inhomogeneous Media

In this paper we consider the one dimensional inhomogeneous wave equation with particular focus on its spectral asymptotic properties and its numerical resolution. In a first part of the paper we analyze the asymptotic nodal point distribution of high frequency eigenfunctions, which, in turn gives further information about the asymptotic behavior of eigenvalues and eigenfunctions. We then turn to the behavior of eigenfunctions in the high and low frequency limit. In the latter case we derive an homogenization limit whereas in the first we show that a sort of self-homogenization occurs at high frequencies. We also remark on the structure of the solution operator and its relation to desired properties of any numerical approximation. We subsequently shift our focus to the latter and present a Galerkin scheme based on a spectral integral representation of the propagator in combination with Gaussian quadrature in the spectral variable with a frequency-dependent measure. The proposed scheme yields accurate resolution of both high and low frequency components of the solution and as a result proves to be more accurate than available schemes at large time steps for both smooth and non-smooth speeds of propagation. Department of Mathematics, University of California at Irvine, Irvine, CA 92697-3875, partially supported by NSF CMS0330470 Department of Petroleum Engineering, Stanford University, Stanford, CA 94305-2220 Department of Mathematics, University of California at Irvine, Irvine, CA 92697-3875, supported by NSF DMS0112416 and DMS0307011.