# 



Integration gives ; 



'^,(t) = 4 ^^-^^ ' 



TM NOo 377 



(A=i^5) 



The second derivative of ^ (.Tjis given by: 



The derivative term in (A~39) inay be neglected if; 



Since -^ - STT (T^)"-^ (where T^ is the period of the oscillation)^ and -i^ = 

 Ki(l)-1 = (Tj.)~l| then from {A-k7)t 



I » ^lU^ ^ 



As shown in chapter IV, Tr for the wave meters was rotjghly 50 ijiilliseconds. 

 If the dominant period of driving oscillations of wind waves is about ^000 

 milliseconds (k sec)^ then; 



Thus, if the auto^ccvariance function is of the form given by (A-^S)* then the 

 error in the estimate of <fo^(T) i^i equation (A"39) will be about k percent too 

 small. 



This simple example is given only to demonstrate that the relative magni=> 

 tudes of the instrument frequency response and the dominant frequencies of the 

 driving motions must be considered in providing an accurate sta'tistical analy- 

 sis of the motions, Thiis^ both the spectrum function and auto-covariance of 

 the driving functions can be evaluated to an accuracy which is unavoidably 

 dependent upon the amount by which the instrument response exceed the ambient 

 driving frequencies « 



Amplitude Pi-obability Distribution of Sinusoidal Waves 



Consider a wavy ocean surface made up of an ensemble of FoTirier components 

 of equal amplitude and frequency^ but of i-andom phase. Two of the waves are 

 pictured in figure A-l<, The random parameter in the waves is the phase angle 

 with respect to time t == 0. Letting o( be the random phase angle of the wave 

 with an angular frequency J2y ^ and having an amplitude A, the random amplitude 

 ^ of the wave at t = tj^ is expressed by. 



A-9 



