570 CABTWBIGHT [CHAP. 15 



increasing number of waves of nearly constant period and slowly varying 

 amplitude, confirming the persistence of swell from a distant storm. The 

 Fourier Integral, in the Cauchy-Poisson form (Lamb, 1932), can also be applied 

 to waves from a highly localized disturbance such as a wind gust (Munk et al. , 

 1956) or seismic eruption (Ichiye, 1950). However, for general purposes the 

 method is of little use on account of the indefinitiveness of the initial conditions, 

 and we are forced to use a more generalized statistical model as discussed below. 

 Non-linear periodic solutions of varying degrees of approximation are also 

 known (e.g. Davies, 1951), but are not easily adaptable to general types of 

 wave motion with varying amplitude and phase, chiefly because non-linear 

 solutions are non-additive. When the amplitude is small, the non-linear solu- 

 tions tend to the sinusoidal form (4) with small corrections which can be 

 expanded as a power series in ka. Since ka is usually less than about 10~i for 

 ordinary deep-water sea waves, the linear solution, generalized by some sort 

 of integral, is usually quite a good approximation. Statistical treatment has, 

 however, been applied to motions of order ka, and will be mentioned in section 

 8 of this Chapter (page 586). 



3. Statistical Formulation 



There have been several attempts to define a statistical model of sea waves 

 by logical derivation from first principles. None is entirely convincing, but they 

 all possess the unique feature of arriving at essentially the same result ; a 

 random, moving surface defined by a stationary! Gaussian process. The fact 

 that this result and various statistical properties derived from it agree fairly 

 well with most observations justifies its adoption until serious evidence of its 

 failure is produced. For this reason we shall not elaborate any rigorous statisti- 

 cal arguments, which the interested reader may find in the references cited, but 

 merely indicate the various lines of approach and adopt a consistent formula- 

 tion as premise for the analysis of the sea surface to follow. 



We shall first outline a rough physical argument. Waves from any small 

 element of the generating disturbance spread outwards with circular crests 

 which, over a distant limited area, appear practically straight. By means of 

 Fourier Integrals these waves can be resolved linearly into a continuum of 

 waves of type (4) covering a range of frequencies. If we now integrate the effect 

 of all elements of the disturbance over the whole generating area, the net 

 result is a distribution of long-crested waves of different frequencies coming 

 from all directions within the angle subtended by the generating area. 

 The mixture of frequencies accounts for the varying amplitude and wavelength, 

 and the mixture of directions of long-crested waves accounts for the sum total 

 being in fact short -crested. The components in any small band of frequency 

 and direction will result from many sources extended randomly in time and 

 space (after the manner of raindrops on a pond), and so one would not expect 



1 "Stationary" in the statistical sense ; i.e. probability of any given event is constant 

 over a long period of time and over a large area. 



