Spectral and Asymptotic Analysis of Acoustic Wave Propagation

Abstract

Classical theories of acoustic wave propagation provide a wealth of examples of boundary value problems for evolution partial differential equations. These problems may be described categorically as initial-boundary value problems for certain systems of linear hyperbolic partial differential equations with variable coefficients. However, the known existence, uniqueness and regularity theorems for these problems are only a first step toward understanding the structure of the solutions. To obtain a deeper insight it is essential to discover how the nature of the solutions changes with the geometry of the boundary and with the coefficients. An examination of recent scientific literature on acoustics reveals a great variety of physically distinct phenomena. Examples include phenomena associated with acoustic wave propagation in stratified fluids, anisotropic solids such as crystals and man-made composites, open and closed waveguides, periodic media and many others. A theory which treats all of these phenomena on the same footing can provide only the most superficial information about the structure of acoustic waves.