Transient SH-waves in a wedge-shaped layer

Summary. A generalized ray theory for transient SH-waves in a wedge-shaped layer over an elastic half-space is developed in this paper. The ray integrals for multiply reflected waves in the layer are derived in terms of two systems of coordinates and two sets of local wavenumbers, one along the free surface and the other along the sloped interface. All local wavenumbers are then transformed to a common wavenumber in all ray integrals which are evalu-ated by the Cagniard method. Results for the first motion approximation are in agreement with previous investigations. The propagation of steady or transient two-dimensional scalar waves in a wedge-shaped layer has been a subject of numerous studies. For the case of a single wedge where both sides are either stress-free or rigid, the problem can be analysed by applying the Kantorovich-Lebedev transform or the method of homogeneous function (self-similarity). These and other methods are reviewed by Knopoff (1 969). When the wedge is bonded to an elastic half-space to form a dipping structure, none of the aforementioned methods is directly applicable. An approximate solution in terms of ray expansion was first derived by Ishii & Ellis (1970a, b), and later by Hong & Helmberger (1977). Another approximate solution based on the method of self-similarity was derived recently by Wojcik (1979). In the paper by Hong & Helmberger, an approximate solution for a dipping structure was constructed by first expanding the known solution for a single wedge (Hudson 1963) into a series of ray integrals. These integrals are then evaluated by the Cagniard-de Hoop method together with the first motion approximation (Gilbert & Knopoff 1961). This solution fer a single wedge of which one side is free, and the other rigid, is extended to that for a dipping structure, that is, a wedge overlaying an elastic half-space. The extension is made on the basis of three assumptions: (1) The diffracted waves associated with the edge tip are neglected. (2) A high frequency approximation for the Bessel functions is used