One-Dimensional Elastic Waves in Inhomogeneous Media

Exact solutions to the inhomogeneous wave equation are obtained through the use of an inverse procedure in which the determination of the functional form of the inhomogeneity constitutes part of the problem. The result is a restriction on the form in which the wave speed changes with the spatial variable, but the restriction is not overly severe for the purpose of illustrating general features, many of which are applicable to stress waves propagating downward into the near-surface region of the earth. Solutions are given for the step, Dirac delta, and exponentially decaying pressure-time functions. Of particular interest are the conditions under which a tensile stress can exist and the subsequent amplification of the peak stress and the negative impulse per unit area.