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P. Borejko, F. Ziegler:
"Pulsed Asymmetric Point Force Loading of a Layered Half-Space";
in: "Acoustic Interactions with Submerged Elastic Structures, Part 4: Non-Destructive Testing, Acoustic Wave Propagation and Scattering", A. Guran, A. Boström, O. Leroy, G. Maze (ed.); World Scientific, New Jersey, London, Singapore, Hong Kong, 2002, ISBN: 981-02-4271-9, 307 - 388.

Problems for transient sources in a multilayered elastic medium may be treated by the theory of generalized ray. In this theory the Laplace-transformed multiply reflected and/or transmitted cylindrical/spherical wave, known as a ray integral, is constructed by linear superposition of the Laplace-transformed plane waves. The potential reflection and transmission coefficients for plane waves are basic to such a superposition. The well-known potential reflection and transmission coefficients are applicable only to the formulations of the theory of generalized ray for which the total reflected/transmitted wave motion is constructed by linear superposition of the two-dimensional plane waves, the coupled and waves, or the waves. These formulations were developed for the following problems: (1) plane strain (two-dimensional) problems in the Cartesian coordinates (e.g., a line of vertical forces emitting the cylindrical and waves), (2) antiplane strain (two-dimensional) problems in the Cartesian coordinates (e.g., a line of horizontal forces emitting the cylindrical wave), (3) axisymmetric (three-dimensional) problems in the cylindrical coordinates (e.g., a vertical force emitting the spherical and waves, or a torque with vertical axis emitting the spherical wave), and (4) asymmetric (three-dimensional) problems in the cylindrical coordinates (e.g., an oblique force emitting the spherical , , and waves). In the present paper we develop the theory of generalized ray for two three-dimensional problems in the Cartesian coordinates, the asymmetric point load source in a horizontally layered elastic medium, and the same source in a dipping elastic layer overlying an elastic half-space. For the two problems, the total wave motion at an interface is composed of the coupled spherical and waves, and the multiply reflected and/or transmitted ray integrals are therefore constructed by linear superposition of the Laplace-transformed three-dimensional coupled plane and waves. The potential reflection and transmission coefficients for the three-dimensional coupled plane and waves are basic to such a superposition. The inversion of the ray integrals is treated by the modified Cagniard-de Hoop method. A discussion of the numerical integration of the ray integrals together with numerical examples is presented in the paper. The paper also discusses how a transient response of a viscoelastic half-space perturbed by the buried asymmetric point load source can be obtained from the solution of the elastic problem by the use of the numerical version of the elastic-viscoelastic correspondence principle.