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Abstract

An incoming wave on an infinite string, that has uniform density except for one or two jump discontinuities, splits into transmitted and reflected waves. These waves can explicitly be described in terms of the incoming wave with changes in the amplitude and speed. But when a string or membrane has continuous inhomogeneity in a finite region the waves can only be approximated or described asymptotically. Here, we study the cases of monochromatic waves along a nonuniform density string and plane waves along a membrane with nonuniform density. In both cases the speed of the physical system is assumed to tend to a constant when the spatial variable gets very large. In the case of a string with local inhomogeneity it is possible to find solutions that are asymptotic to sinusoidal waves involving the limiting speed of the string. On the other hand when the coefficients in the equation of the vibrating string are small deviations from a constant, we use a special Green’s function to approximate the solution. We also find a finite series of sinusoidal waves with constant speeds, that are asymptotic to the solutions of a vibrating membrane problem. The number of waves in the series depends on the width of the membrane, the limiting speed of the waves and the time frequency of the sinusoidal waves. The technique we use to show asymptotic approximations involves reducing the pde equations of the physical models to a first-order system of ode’s.

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