Evaluation of Motions of Marine Craft in Irregular Seas 



Some reflection on the parametric dependence of K on r and ^ imbedded 

 in (54) will reveal that the action of the fluid between the two points is to filter 

 the wave-induced motion as an inverse function of the complex "distance" z 

 which means that the filtering effect depends on (^^ + l,^)^^^ and the "aspect" of 

 the point defined by the angle tan' ^ (<f/0 • 



Evaluation for ^ = yields the same result as given by Davis and Zarnick 

 [14], viz., 



Vt;,^-^„,0) 



cos[— (at) 



+ C(at) 



sm (y (at)2 



S(at) 



(55) 



where 



- V^ 



(^-^o) 



C(at) 



r 



cos{— fi^] d^ ; 



S(at) 



/ 



■"" 2 J 



sm — /J, d/j. 



The functions C and S are Fresnel integrals which are tabulated. 



Curves of Kya for ^ = and C = 50 feet are shown in Fig. 1. It is seen 

 that the amount of future time wave record needed at .f^.o to compute the pres- 

 ent time disturbance at ^, (, increases as one moves downward into the fluid. 

 As ^ is made large with respect to ^^, less and less future time record is re- 

 quired as would be expected. For ^ = and large a or ^^^^ 



K„(u; ^-^„,0) ^^ a cos (^ - ^ (cm) 



^/2 



which, being even in t, shows that both future and past information are equally 

 weighted at £,. 



The function K^ as given by (54) collapses to the known, very simple, re- 

 sult when one moves the point ^, L under the point ^^.O. Then the argument of 

 the complementary error function becomes a pure imaginary and its value is 

 then unity leaving 



K.(u;0,O =- 



4C _ ^ 



42 



■7T(/3-U)- 



(56) 



A universal curve of Ky/3 is plotted against /3u (or fit) in Fig. 2. It is seen 

 that this function is symmetric, indicating that the motion at depth requires 

 equal knowledge of future and past waves. 



These results allow one to handle the following problem. Suppose one has a 

 system -K( t ) for a particular mode of motion which has been derived from data 

 in which the wave information was secured at ^,0 and then one wishes to calcu- 

 late the motions of a ship using wave records secured or assumed at some point 

 ^j. One may then do either of two operations: 



475 



