Problem of Forced Oscillations
The problem of diffraction of a plane wave on a system of wedge-shaped regions with the common edge is the key problem of the mathematical theory of diffraction. Its solution helps us understand the physical mechanism of edge diffraction and can be used for the description of edge-diffracted waves by the method of the geometrical theory of diffraction. Section 1of this chapter considers a special formulation of this problem aimed at obtaining formulas for the edge waves only. We begin with the simplest case-diffraction of an acoustic wave on a system of wedge-shaped regions with conditions of rigid contact at the interfaces. The statement of this problem is based on the development of the Sommerfeld-Malyuzhinetz method (Malyuzhinetz, 1951), which is considered briefly in Section 1. A modification of this approach is considered in Section 2. Using the modified approach, we state the diffraction problem in Section 3 as a problem of forced oscillations. In Section 4 we generalize the approach on a case of diffraction of the vector waves of any physical nature on a system of wedge-shaped regions with the arbitrary linear conditions of contact at the interfaces.
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.