Exact solutions of the one-dimensional acoustic wave equations for several new velocity profiles: Transmission and reflection coefficients

Abstract

After reviewing the relationship of the acoustic equation to the Schrödinger equation, solutions of the latter are used to construct new indices of refraction [or equivalently sound velocity profiles (SVP)] for which the solutions of the acoustic equation are expressed in terms of elementary functions. The new indices of refraction can be used to form layers to model more general inhomogeneous media, just as piecewise constant slabs or other indices have been used in the past. Four of the new indices of refraction also provide especially simple closed-form examples which can be used to describe how waves are scattered in media with such indices. More general scattering problems are discussed. In particular, the notion of outgoing wave is gone into. The five indices given in the present paper are the first of several which have been obtained from Schrödinger equations with potentials for which the eigenfunctions can be found in closed form. Others will be given in later papers.