Perturbation formulas for linear dispersive waves in weak spatially non-uniform media

Abstract

First-order perturbation formulas for frequency, wave number, group velocity, amplitude and phase are obtained for linear dispersive waves propagating in weak spatially (but otherwise general) non-uniform media. The formulas are derived from Whitham's theory under the assumptions of slow wave modulations and weak (i.e. small scale and smooth) spatial nonuniformity. The analysis presented here indicates that the perturbations at any location of the medium depend on both the local heterogeneity and its spatial average over the propagation path. We illustrate the theory with the well-known example of flexural waves along a non-uniform beam to demonstrate that our theory is identical to a joint WKBJ-stationary phase representation of the solution.