Impulsive spherical-wave reflection against a planar absorptive and dispersive dirichlet-to-neumann boundary - an extension of the modified cagniard method

Abstract

A closed-form analytic time-domain expression is obtained for the scalar wave function associated with the reflection of a point-source excited impulsive spherical wave by a planar boundary with absorptive and dispersive properties. The physical properties of the boundary are modeled as a local Dirichlet-to-Neumann boundary condition in the form of a time-convolution integral, the kernel function in which (denoted as the boundary's time-domain admittance) meets the conditions for linear, time-invariant, causal, passive behavior. Parametrizations of the boundary admittance function are put forward that have the property of showing up explicitly, and in a relatively simple manner, in the expression for the reflected wave field. The partial fraction representation of the time Lapplace-transform domain boundary admittance and the plane-wave admittance representation are shown to serve the purpose. The expression for the reflected wave field is constructed through an appropriate combination of the modified Cagniard method for analyzing wave propagation in layered media, Lerch's uniqueness theorem of the unilateral Laplace transformation, and the Schouten-Van der Pol theorem pertaining to a change of transform parameter in this transformation. The result can serve as a benchmark solutions to the modeling of transient wave reflection against absorptive and dispersive boundaries in more complicated geometries where numerical methods are the tool of analysis. The obtained analytical expressions show that, the configuration under consideration, no true surface waves occur along the boundary.