Wave propagation in a restricted class of orthotropic inhomogeneous half-planes

This work presents closed-form solutions for free-field motions in a certain type of orthotropic, continuously inhomogeneous half-planes that include both incident and reflected waves from the traction-free horizontal surface. A state of plane strain holds and both pressure and vertically polarized shear waves are considered. The methodology is restricted to media with orthotropic material characteristics that vary quadratically with respect to the depth coordinate. The method of solution is a hybrid approach based on the plane wave decomposition technique, augmented by appropriate functional transformation relations for the displacement vector. The existence of a unique analytical solution as superposition of incident P- or SV-waves and the corresponding reflected P- and SV-waves is proved under certain restrictions on the incident wave direction. Finally, a numerical study reveals the influence of material inhomogeneity and orthotropy on the displacement free-field motion.