Reflection and Transmission of Elastic-Waves by the Spatially Periodic Interface between 2 Solids (Theory of the Integral-Equation Method)

The linear theory of two-dimensional reflection and transmission of time-harmonic, elastic waves by the spatially periodic interface between two perfectly elastic media is developed. A given phase progression of the incident wave in the direction of periodicity induces a modal structure in the elastodynamic field and leads to the introduction of the so-called spectral orders. The main tools in the analysis are the elastodynamic Green-type integral relations. They follow from the two-dimensional form of the elastodynamic field reciprocity theorem, where in the latter a Green state adjusted to the periodicity of the structure at hand is used. One of these relations is a vectorial integral equation from which the elastodynamic field quantities can be determined. The consequences of field reciprocity in the structure and of the conservation of energy are developed in view of their serving as a check on numercal results to be obtained from the relevant integral equations. The formalism thus developed applies to profiles, if periodic, of arbitrary shape and size and can without too serious difficulties be implemented on a computer. The major difficulty in this respect is the relevant Green function, the series representation of it being slowly convergent. Its evaluation becomes tractable after an appropriate technique for accelerating the convergence. The only practical limitations are then put by the speed of the computer and its storage capacity.