Several one-dimensional impact problems for inhomogeneous elastic media are reduced by suitable transformations of variables to three different canonical forms. One of them is the impact problem for the wave equation with variable sound speed. Another, which we call Problem I, is the impact problem for the Klein-Gordon equation with variable coefficients. Several methods of approximating the solution of Problem I are discussed. The wave front approximation is obtained by assuming that sufficiently near the discontinuity propagating from the impacted boundary the solution has a Taylor series in time. The coefficients in the series are the time derivatives of the solution evaluated on the discontinuity. They are obtained from an analysis of the propagation of the discontinuity using the theory of weak solutions. A formal asymptotic expansion of the solution is obtained for oscillatory impact data. Reflections from boundaries are also considered. Perturbation methods for media with slowly varying inhomogeneities and asymptotic methods for media with rapidly varying inhomogeneities are briefly discussed.
Wave propagation in one dimensional inhomogeneous elastic media.
