2. The Amplitudes 



If the spectrum of the waves is narrow, the probability distribution 

 of the amplitudes is known (Rice, 1944). As in figure Z, a sufficient 

 number of amplitudes read from the record will have a known proba- 

 bility distribution function. If the spectrum is wide, the distribution 

 is unknown. However, it would appear from the theoretical results 

 of Neumann (1953) that even a fully developed sea wave record will 

 be approximately distributed in amplitudes according to this known 

 distribution. 



Given, then a wave record and a set of wave amplitude observations 

 which are from a long enough record, the amplitudes will be distributed 

 according to the law given by equation (1) 



g(x)dx=^V''^^ dx (1) 



for 0<x<co, 



which means that the probability that a given amplitude, say, ^ , will 

 have a value between x and x + dx is given by equation (1). This 

 probability distribution is often called the target distribution and also 

 the Rayleigh distribution. It is related strongly to the Chi- square 

 distribution. 



The mean wave amplitude is found from equation (1) as in equation (2) 



/.cD^^z-x /E -xVeioo na> -xVe r r 



1 -e' ^'--'^ |//e dx=4^E= 0.886 ^E 



. - E 



(2.) 



The number E has the dimensions of square feet, and it represents 

 the sum of the squares of all of the amplitudes of the infinite number 

 of infinitesimally high sine waves which add together to make up the 

 total wave record. The average amplitude of all the waves is equal to 

 0.886v^. 



The second moment about the origin of equation (1) can also be found. 

 It is given by equation (3) since the integral from zero to infinity of 

 equation (1) is equal to one. 



