Understanding and Prediction of Ship Motions 



in waves. They can also be compared with the much simpler case of long 

 crested head seas in which for heaving motions, for example, 



Tj^Ct) = 1 COS (a;^t+ e) ^25(0;^) A^^ 



(25) 



z^(t) = 2 [cos (ojgt+e) c^(Wg) + sin (w^+e) q^C^^)] 725(0;^) Isoj^ . 



The cross spectra are given by 



(26) 



Qz(^e) = S(a;^) q^(co^) . 



It therefore follows that the coherency is one. 



These conclusions about the behavior of ships in both short crested and 

 long crested waves bring up one of the main points that needs to be made. The 

 method of Fuchs and MacCamy has not yet been extended to the point where it 

 can describe the motions of actual ships in actual waves. In fact, the simple 

 measurement of the forcing waves at one point as a function of time is insuffi- 

 cient for the prediction of the complete behavior of a ship in these waves. This 

 is true whether or not the ship is underway just as long as the waves are short 

 crested. 



The essential reason for this difference between short crested waves and 

 long crested waves is that in the above relationships an integration over direc- 

 tion is still involved in the definition of each function of frequency that is ob- 

 tained by the analysis of a time history. An extra dimension is added to the 

 problem when short crested waves occur. This dimension can conceivably be 

 removed by recording the forcing waves along a line as a fimction of time with 

 this line moving with the vessel. Or, a sufficiently dense network of points sur- 

 rounding the moving ship should provide that kind of data that would make it 

 possible to recover functions such as c^(co^,e^). However, the problem is one 

 of a double Fourier analysis and not a single Fourier analysis properly gener- 

 alized in terms of time series and probabilistic concepts. Some suggestions 

 were given by Pierson (1957) as to how to do this. 



Nevertheless, the spectral theory is complete for ships in short crested 

 waves. It has not been fully exploited. Experimentation with actual ships in 

 short crested waves should provide useful design information today in this 

 connection. 



COHERENCY AND RESOLVABILITY OF SPECTRAL 

 AND CROSS SPECTRAL SHAPES 



Poor resolution of rapidly varying cross spectra is likely to be reflected in 

 a computation of low coherency. Such apparent low coherencies are not actually 

 low coherencies, and great care must be taken in interpreting experimental re- 

 sults for long and (especially) short crested waves in this connection. Low 



87 



