Free-field wave solutions in a half-plane exhibiting a special-type of continuous inhomogeneity

Abstract

This work presents closed-form solutions for free-field motions in a continuously inhomogeneous half-plane that include contributions of incident waves as well as of waves reflected from the traction-free horizontal surface. Both pressure and vertically polarized shear waves are considered. Furthermore, two special types of material inhomogeneity are studied, namely (a) a shear modulus that varies quadratically with respect to the depth coordinate and (b) one that varies exponentially with the said coordinate. In all cases, Poisson’s ratio is fixed at one-quarter, while both shear modulus and material density profiles vary proportionally. Next, a series of numerical results serve to validate the aforementioned models, and to show the differences in the wave motion patterns developing in media that are inhomogeneous as compared to a reference homogeneous background. These results clearly show the influence of inhomogeneity, as summarized by a single material parameter, on the free-field motions that develop in the half-plane. It is believed that this type of information is useful within the context of wave propagation studies in non-homogeneous continua, which in turn find applications in fields as diverse as laminated composites, geophysical prospecting, oil exploration and earthquake engineering.