Multi-Scaled Short-Wave Asymptotic Solution of the Cauchy Problem to One-Dimensional Wave Equation with Smoothed Jump of the Velocity

Abstract

The paper studies a wave equation whose velocity has a perturbation localized at a point \(x_0\). The initial condition has the form of a rapidly oscillating wave packet whose wavelength is not comparable to the scale of the inhomogeneity. Specifically, the length of the initial wave is of the order of \(\varepsilon\) and the width of the localized inhomogeneity is of the order of \(\varepsilon^{\alpha}\), where \(\varepsilon\) is a small parameter tending to \(0\) and \(\alpha\) is any positive number. The cases \(\alpha<1\) and \(\alpha>1\) are considered separately.