Solution of the Wave Equation in the Boundary Layer
where the wave velocity depends on point M of an inhomogeneous medium. We look for high-frequency solutions of this equation in a narrow vicinity of the shadow boundary of an individual reflected/transmitted wave, where the geometrical theory of diffraction fails. In Section V.3.2 we identified such a domain as the boundary layer. Here we extend this concept to the case of 3-D inhomogeneous media. To do so, we have to consider a geometrical scheme of rays and wave fronts that follows from the kinematic law of edge diffraction (see Chapter I, Section 6).
Let us take two families of rays, determined by two corresponding sets of parameters individual ray of the first family. Each pair of gives a ray of the second family. We call the families and the congruencies of reflected/transmitted and diffracted rays, respectively.
Any of the expressions gives a surface, formed by the rays of the congruence. Any of the expressions gives a surface, formed by the rays of the congruence. We will consider the specific case when the congruencies under consideration have acommonsurface.Wesuppose the latter can be given simultaneously
Figures & Tables
This book is devoted to one important aspect of development of physical foundations of the seismic method — the theory of edge diffraction phenomena. Thoese phenomena occur when conditions of the regular wave reflection/transmission change sharply. Though these phenomena drew the attention of many scientist for many decades, their real influence on the resolution ability of the seismic method was truly understood rather recently due to interpretation of seismic data in block structures. Clearly, to develop seismic method for investigation of such structures without developing the theory of edge diffraction phenomena is impossible. The latter is the aim of this book.
The seismic method is based on the fundamental laws of continuum mechanics. These laws describe the behavior of wavefields on the microscopic level, i.e., in the form of differential equations of motion. Integrating these equations under some initial conditions or boundary conditions, makes possible acquisition of all necessary information on the wavefield in the given situation. However, the working base of the seismic method consists of not only the differential equations of motion themselves but of some general and simple enough consequences of their solutions, which are formulated in the form of physical principles and l aws. The latter include the concepts of wave, Fermat’s principle, the law of conservation of the energy flux, and the reflection/transmission laws. Essentially these laws and principles must form a system of concepts sufficient for the solution of some class of typical interpretation problems. In fact, these principles and laws form the physical fo ndation of the seismic method.