Mathematical Sciences: Asymptotic Pertubation and Numerical Studies of Wave Propagation

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This project is concerned with mathematical methods in the studies of acoustic waves. Wave propagation is a means of moving energy from one region to another. When the wave amplitudes are sufficiently small, as occurs in many problems, then the motion is adequately described by a linearized theory of wave propagation. Thus, the classical theories of wave propagation in acoustics,optics, electromagnetics, elastic solids, etc. are linearized theories. When the geometry, material properties and source structures are simple it may then be possible to solve wave propagation problems explicitly. However, for realistic situations and hence more complicated problems it is necessary to employ asymptotic, perturbation, numerical and other approximate methods to solve wave propagation problems. The purposes of this research project are to develop and apply asymptotic, perturbation and numerical methods to wave propagation problems of scientific and engineering significance, and to study specific physical problems that are basic to future progress in wave propagation. Some of the present effort represents a continuation and elaboration on past work. Professor Reiss is a well known authority on wave propagation and Professor Kriegsmann is his younger talented partner. They constitute an excellent, very productive team. This research falls into the general area of asymptotic methods and perturbation theory in applied mathematics, which constitute an active and practical trend in applied mathematics research. Many of the results obtained by these researchers have direct engineering applications in acoustics, radar, sonar and in the studies of electromagnetic wave propagation.