Mathematical Sciences: Asymptotic Pertubation and Numerical Studies of Wave Propagation

这个项目涉及声波研究中的数学方法。波传播是一种将能量从一个区域转移到另一个区域的方法。当波幅足够小时，就像在许多问题中发生的那样，那么运动就可以用波传播的线性化理论来充分描述。因此，声学、光学、电磁学、弹性固体等领域的经典波传播理论都是线性化理论。当几何形状、材料性质和震源结构简单时，就有可能明确地解决波的传播问题。然而，对于现实情况和更复杂的问题，有必要采用渐近、摄动、数值和其他近似方法来求解波的传播问题。本研究项目的目的是发展和应用具有科学和工程意义的波传播问题的渐近、摄动和数值方法，并研究对波传播的未来发展奠定基础的具体物理问题。目前的一些努力是对过去工作的继续和阐述。Reiss教授是波传播领域的知名权威，Kriegsmann教授是他的年轻天才搭档。他们组成了一支优秀的、富有成效的团队。本研究属于应用数学中渐近方法和摄动理论的一般范畴，构成了应用数学研究中一个活跃而实用的趋势。这些研究人员获得的许多结果在声学，雷达，声纳和电磁波传播研究中具有直接的工程应用。

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.