Breslin, Savitsky, and Tsakonas 



K(t) = 1^ j $(0.) e'^'d.. = ^1 Mco) e^f"*-^(">^ dco 



- — A(aj) cos [wt - (p(aj)] dco + — A(aj) sin [wt-cpCw)] dco. 



- 00 - CO 



Since K(t) must be a real function, the last integral must vanish identically for 

 all values of t. This requires that cp(a)) be an odd function of w (and if it is not 

 then something is in error in its determination!). Since A(co) must be an even 

 function of co, then calculate K( t ) from 



CO 



K(t) = ^ A(co) cos [aJt-cp(w)] do;. (42) 



It must be realized that by referencing the force and moment functions to 

 the wave, an arbitrary phase angle is introduced or simply that the phase of the 

 frequency response function is tied to the location of the wave measurement rel- 

 ative to the body. To clarify this, suppose that the heave and pitch of a surface 

 model is recorded at the center of mass and the wave is measured abreast of 

 the center of mass. Then the derived impulsive response function will exhibit 

 features peculiar to this reference point. It will, for example, have values dif- 

 ferent from zero for negative time which then requires that the wave motion 

 forward of amidships be known or, in effect, "future waves" are required in 

 order to compute the response at the present time. If it is desired to reduce the 

 extent of the negative time for which the empirically derived impulsive response 

 function has nonzero values by, say, referring the motion to waves measured 

 forward of the bow, it is necessary to shift the phase of the transfer function by 

 the angle i (aj|aj|x/g) where x is the amount of the horizontal shift (taking care 

 to regard x as positive or negative) so that the modified transfer function be- 



comes 



$(w, x) = A(co) e 



and the modified impulsive response function is 



K(t,x) 



9(^) + a 



(43) 



- A(co) COS [cot - cp(aj) 1 doj . (44) 



In addition, it will be necessary to convolute this shifted function K(x, t) 

 with the wave record at the new point so that the v and the K are consistently 

 referenced to the same point of measurement. If only the wave record abeam of 

 the center of mass is available, this can also be shifted to the same point as is 

 discussed in a later section. 



472 



