Ogilvie 

 . = X + vt 



(2) 



Such a coordinate system moving with this velocity would record or see a sea 

 surface given by Eq. (3) as a function of position and time with reference to the 

 moving origin. 



•00 «77- 



T7(x_,y„, t) = I cos 



nil 

 „ -1 



(x COS + y sin 8) - [co v cos (9) t - e 



(3) 



For region I of St. Denis and Pierson, the spectrum, S.{oj,d) becomes the 

 spectrum of encounter as given by 



/l - v/l -4aj^v cos e^/g \ 



S \ r -z ; (X> ,d ] 



\ 2aj V cos e Jg e e/ /.v 



S (OJ ,5 )t = ^ ' 



e^ e' e'l / — ; — 



Vl - 4aJe V COS 0p/g 



and the seaway of encounter is given by 



-^elC^e'Ve't) = J J ^os |(aj^t+e) - [(^(oj^)) yg] (x^cos^^ + y^ sin 0^)| 



Region I 



X \/2S^i(a;^,0^)da;^d0^ . (5) 



The above steps can be repeated with appropriate modifications for regions 

 II and III with the result that 



^eC'^e'Ve't) = V ^^(^ ^,y ^, t^ + 77^ j j( X ^, y ^, t ) + ^el I I^'^e- Ve' ^ ) • (6) 



If the center of gravity of a ship is located at the point x^ = y^ = o Eq. (5) 

 can be written as 



7?3l(t) = Z2 COS (a;^t+ e) ^2S^(oo^,e^) Ao;^ M^ (7) 



where a partial sum is indicated as an approximation to (5). The important 

 point to note at this stage of the derivation is that the same frequency of encoun- 

 ter, oj^, can result from many different waves coming from many directions, 6^, 

 and can be associated with many different wavelengths. 



In practice for a given 6^^ and w^j, given the wavelength, the region can be 

 determined. One single term in the double partial sum of (7) is thus sensed as 

 in Eq. (8) by a wave recorder located at the moving origin. 



•?7gj(t) = a cos (w^jt + e) . (8) 



82 



