one-third of the wave heights in the distribution. This value was selected 

 since it appeared to approximate the results of wave heights as reported by 

 experienced observers making visual estimates of wave heights in the ocean 

 (Sverdrup and Munk , 1947). In addition, since from equation (B-1), E is 

 proportional to H2 , most of the energy is carried by the higher waves. So 

 Hg is a better representative wave height than the mean wave height in most 

 coastal engineering problems. 



However, the significant wave height, Hq , does not equal the rms wave 

 height, Hp^g, so Hg cannot be used to evaluate E or P^ directly. Thus, 

 the relation between Hg and H^^s must be determined. This can be done for 

 restrictive conditions following Longuet-Higgens ' (1952) work (see also Kinsman, 

 1965 (Sec. 3.4) and Harris, 1973). 



To develop a relation between Hg and i\>rns > ^^ ^^ assumed that the total 

 wave energy density, E, as given in equation (B-1), depends on the potential 

 and kinetic energy densities, Ep and E^, respectively, both given by 



Ep = % = T^ h2 (B-4) 



where H = H^^^g . For this to hold, the following conditions must occur: 



(a) The waves are linear and have small amplitudes. 



(b) All waves of a single frequency arrive from the same direction. 



(Multidirectional waves will not seriously affect the result as long as waves 

 at a given frequency do not come from more than one direction.) 



The wave heights will form a Rayleigh distribution if the following condi- 

 tions are also assumed: 



(a) The wave spectrum contains a single, narrow band of frequencies. 



(b) The wave energy comes from a large number of different sources 

 of random phase. 



Let n(t), the departure of the water surface from the mean sea level with 

 respect to time, t, be expressed as the Fourier integral 



n(t) = f_2 A(co) e^'^* doj (B-5) 



where A(a)) is the spectrum function (possibly complex though only the real 

 part of the integral in eq . B-5 is taken) of the amplitude of n(t) with 

 respect to the frequency oi. At any time, twice the amplitude equals the wave 

 height. Hi , measured from trough to crest (see Fig. B-1). 



26 



