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
5.6: Initial-Value Problems in Two Surface Dimensions
5.6 Initial-Value Problems in Two Surface Dimensions
The application of the method of stationary phase to dispersive waves in two or in a higher number of dimensions will be discussed by reconsideration of the surface wave problem of 5.5.1, but with the initial disturbance confined to the neighbourhood of the origin x ? ( x, y, z) = 0.
5.6.1 Waves generated by a surface elevation symmetric about the origin
Let the motion start from rest with the initial surface elevation
It may be assumed that the uniform, initial value of the velocity potential ? o = 0. Equations (5.5.4) become
where H ? = ? when t > 0.
Write the solution of Laplace s equation in z < 0 as the Fourier integral
where 
( k, ?) is the Fourier transform of H ? at z = 0. To avoid cumbersome notation it is convenient to adopt the convention that x = x 1 and y = x 2, so that scalar products such as k x can also be written as k i x i. Surface condition (5.6.2) gives
with
Proceeding as in 5.5.1 [noting, in particular, that f o( k) = f o( ? k)], we find
The principle of stationary phase asserts that the main contribution to this integral at large times is from wavenumbers k clustered about the wavenumber k o at which the phase
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