668 BARBER AND TUCKER [CHAP. 10 



However, the waves on a stormy day are obviously different in some systematic 

 way from those on a calm day, and some way must be found of describing this 

 difference. It is, of course, the statistical properties of the wave pattern which 

 are significant, and the most obvious ones to use are average wave height, 

 period and direction of travel, defined in some suitable way. Most early methods 

 of prediction aimed at predicting these properties, and for many engineering 

 purposes a knowledge of these is sufficient. They do not contain all the sig- 

 nificant information about the wave system, however, and more detailed 

 information is sometimes required. It has been found in practice that the most 

 generally useful way of describing the wave pattern is by its "power spectrum", 

 and from this any of the other statistical properties may be derived. 



The concept of the wave power-spectrum is based on the assumption that 

 the wave pattern may be regarded as a simple superposition of a large number 

 of low, sinusoidal, long-crested wave trains such as is described in section 1 of 

 this Chapter, each component having a different period and a different direction 

 of travel. The interference of these components, sometimes reinforcing one 

 another and sometimes cancelling one another, produces the complicated wave 

 pattern observed. If 8E is the sum of the energies per unit area of the sea 

 surface of all those component wave trains whose angular frequencies ( = 27r/ 

 wave period) lie between ct and a+8a and whose directions of travel lie between 

 6 and 6 + hd, then one may choose to describe the spectrum by a spectral 

 density function E'{g, 6) defined by : 



E'{a, d) = limit as Sct -> and hO -> of S^/(8ct hd). 



This is sometimes known as the two-dimensional spectrum of the sea surface 

 since it can be pictured as a contour plot using a and 6 as co-ordinates. Some 

 writers prefer to use wave number rather than wave frequency. There would 

 seem to be some advantage in using polar co-ordinates rather than rectangular 

 ones. 



In practice, S^" is taken as the mean square elevation of the sea surface ; 

 strictly speaking, the energy is pg SE per unit area. 



In both theory and practice, E'{a, 6) is found to be continuous with frequency 

 and direction ; that is, there are in effect an infinite number of infinitely low 

 wave trains differing infinitesimally in frequency and direction. 



For many purposes the direction of travel of the component wave trains is 

 either not important or is too difficult to measure. In these cases the "one- 

 dimensional" spectrum is used. This is defined as 



E'{a) = f " E'{cy, 6) de. 



(5) 



This is the frequency spectrum of the output of a recorder which records the 

 height of the water surface above a fixed point on the sea-bed. Up to the 

 present time, practical methods of wave prediction give this spectrum, since 

 knowledge of the directional properties of sea- waves is scanty (see section 10 of 

 this Chapter). 



