668 



CARTWKIGHT 



[chap. 15 



indicated, and these can mostly be isolated by using recording instruments 

 sensitive only to a limited band of frequencies, or by using filters, digital or 

 otherwise, in the subsequent analysis. In this general discussion of statistics we 

 shall assume that some such filtering has been applied, so that we have only 

 to deal with a spectrum which can be considered zero below a certain frequency 

 and which converges suitably on integration to infinity. This avoids certain 

 difficulties of minor importance which would otherwise detract from the main 

 argument. As such the analysis will apply equally well to instrumental records 

 from almost any part of the spectrum, though its most important application 

 is to wind waves, which have received most attention in the literature. Excep- 

 tions are the tides, which have line-spectra with well-defined phases, and 

 transient phenomena, such as storm surges or waves from instantaneous 

 explosions or eruptions. Other methods are available for dealing with such 

 wave forms. The special problems associated with various bands of the wave 

 spectrum are discussed in subsequent chapters. 



^ 100 



> IQ-' 



i 10-^ 



u 



10 ^ 



>- 



fr 10" * 



0.1 I 10 100 1000 10000 



FREQUENCY, c/ks 



Fig. 1. Spectrum of sea-surface level above a point at La Jolla, California. (After Munk 

 et al., 1957, Fig. 2.) 



2. Fundamental Equations of Wave Motion 



We take rectangular axes x and y in the mean water surface and z vertically 

 upwards. For irrotational motion, which we shall always assume to hold (and 

 which is always valid in linear approximations), the velocity potential, 

 <f>{x, y, z, t), must satisfy the equations (Lamb, 1932) : 



dx^ dy^ 8z^ 



0, 



(1) 



(2) 



where p is the pressure and p the density, assumed constant. (For non-uniform 

 density see Chapter 22.) Since we are not here concerned with problems of 

 generation, we may take the boundary equation to be ^ = at the free surface, 

 z = t,{x, y, t), and d(f>ldz = Q at a horizontal or very deep bottom, z= —h. (See 



