Understanding and Prediction of Ship Motions 

 C,/^e) = l^(^e'^e) [^z('^e'^e) <^/^e'^e) + ^z(^e'^e) ^/^e-^e)] ^^e (1^) 



Qz/^e) = /S(^e-^e) [^z(^e'^e) ^/^e'^e) " ^zC^e'^e) V^e- ^e)] ^^e • (20) 



The response amplitude operator defined by St. Denis and Pierson is 

 seen to be given by Eq. (21) in terms of the spectrum for the heaving motion and 

 the spectrum of the seaway of encounter. This is simply the square of the re- 

 sponse of the vessel to a unit sine wave at a particular frequency of encovmter 

 and direction of encounter. 



The coherency for short crested waves takes on a new and essentially dif- 

 ferent feature, however. Consider, for example, the coherency between the 

 forcing waves and the heave. It is given by Eq. (22). 



[C^,(^e)] ' + [Q.z(-e)] ' (22) 



This can be rewritten in full as Eq. (23), and the top expression can be 

 shown to satisfy the relationship given by Eq. (24). 



[;Se(-e-^e)^z(-e-^e)d^e]' + [JS ^(c^, 6 ^) q ^( co^, 6 ^) dd ^^ 



K /a;J = P = (23) 



;S^(-,,e,)d^^.JS,(a,^,^,) [{c,(^,,9,))' + (q.(-e-^e))'] ^^e 



77Z>> e' 



[!S^(co^,0^)c^(a:^,0^)d0^y S!S^(a>^,6^)ddJS^(c.^,e^) [c ,(c.,, 6 ^)]' d0 ^ . (24) 



For a particular ship headed in a particular direction through a particular 

 forcing seaway it is quite possible for the numerator of the expression for the 

 coherency to be zero. The function under the integrand which is integrated over 

 direction through in radians can change sign in such a way that the integral is 

 very small or zero. The denominator of this expression for the coherency is 

 composed of terms that are everywhere positive, and it must be large. 



In St. Denis and Pierson (1953), these "phase relationships" were not con- 

 sidered, and since cross spectra were not considered, the effects just discussed 

 did not enter into the problem. The spectra of heave, pitch, and roll were prop- 

 erly predicted. 



In 1957, the writer wrote the following three paragraphs (with a change of 

 notation to conform to these comments) in connection with the interpretation of 

 coherency and spectra and cross spectra for ships in short crested waves: 



Consider a ship in head seas such that S(co^,e^) is exactly sym- 

 metrical, i.e., Sg(a)^, 0^) = Sg(Wg, -0^). Under these conditions, 77^(t), 



85 



