Inverse Solutions for One-Dimensional Seismic Waves in Elastic, Inhomogeneous Media

An inverse procedure is developed for obtaining exact solutions to the one-dimensional inhomogeneous wave equation. Transformations of the independent spatial variable and the dependent variable are introduced so that the wave equation assumes the form associated with a homogeneous material. The resulting transformation relations are nonlinear but of such a nature that they can be easily integrated if the reciprocal of the wave speed distribution can be expressed in terms of elementary functions. One functional form that yields realistic values for material properties of soil layers is investigated in detail. Amplification factors for a sinusoidal seismic shear wave in inhomogeneous and homogeneous layers are derived and illustrations of significantly different characteristics for the two types of layers are shown.