Reflection of sound waves by sound-speed inhomogeneities

Abstract

An analysis is given of the influence of the rate of change of sound speed on reflection and transmission in a perfect gas. Asymptotic formulas, valid for both low and high frequencies, are developed to compute the reflection and transmission coefficients for one-dimensional waves propagating through variable-speed layers. The sound speed may have a discontinuous first derivative. It is shown that local reflection effects are proportional to the square of the derivative of the logarithm of the sound speed. The method predicts reflections for gradually varying sound-speed profiles having continuous derivatives of all orders. In the special case where the sound speed is a piecewise linear function, the general method produces an exact solution in addition to the asymptotic formulas. This exact solution is valid for arbitrarily large sound-speed gradient and reduces to the classical result for discontinuous sound speed in the limit of an infinite gradient over an infinitesimal distance. The exact solution produces an effective numerical method when an arbitrary sound-speed profile is modeled by a mixed sequence of discontinuities and ramp functions.