Hydrodynamics and Sound
Providing a detailed study of a wide range of highly simplified model problems,' this book presents the most important theoretical methods that will enable the reader to develop analytical models of practical significance.
TABLE OF CONTENTS Hydrodynamics and Sound
5.10: Ray Theory
5.10 Ray Theory
The propagation of waves over water of continuously variable depth cannot usually be treated analytically. However, when significant changes in the depth occur only over distances that are large compared with the surface wavelength it is possible to obtain an approximate representation of the wave motion by extension of the notion of group velocity to a medium with slowly varying properties. This is the basis of ray theory.The idea is quite general and applicable to waves of arbitrary type (sound, electromagnetic, elastic, etc.) in general, anisotropic wave-bearing media.
5.10.1 Kinematic theory of wave crests
Consider first a description of wave propagation through a uniform, wave-bearing medium in terms of a linear partial differential equation with constant coefficients. Solutions exist in the form of plane waves of the type 
, where the amplitude ? o, the wavenumber k, and frequency ? are each constant. The values of k and ? are related by the dispersion relation
The phase ? = k i x i ? ? t has the obvious properties
We can also put 
in the fully dispersed far field of a localized source of waves where now, however, the amplitude ? o ? ? o( x i , t), wavenumber k ? k( x i , t), and frequency ? ? ?( x i , t) vary slowly with position and time on scales of the wavelength 2
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