Breslin, Savitsky, and Tsakonas 

 Q(aj) = -A(aj)^ sin [cp(a;)] 



1 - — 



2V 



tan" 



2^' 7- 



1 - — 



(68) 



Hence the impulsive response function for the submerged body in roll relative 

 to the wave orbital motions at depth is evaluated by the following equation de- 

 rived from the Fourier integral 



Kp(t) 



a 



[P(aj) cos a)t + Q(aj) sin wt] doi 



(69) 



where P(&j) and Q(w) are given by Eqs. (67) and (68). It is interesting to note 

 that K(t) given in Eq. (69) is independent of submergence. This is to be ex- 

 pected so long as the submerged body motions are related to wave orbital ve- 

 locities at body depth. 



Since the desired deterministic solution involves relating the time history 

 of the surface wave profile to the time history of the roll motions of the sub- 

 merged body, it is necessary to first know the wave motion at depth in terms of 

 the wave motion at the surface. Equation (59) shows this relation to be: 



CO 



4Ti 



dr 



where 



T7o(t) = time history of surface wave profile, 



77j^( t ) = time history of orbital motion at depth h , and 



h = depth of submergence to center of vertical fin. 



The time history of roll motion in terms of the surface wave profile can now be 

 obtained by use of the convolution integral: 



e{t) 



00 



= J Kg(r') Vt-r') dr' 



(70) 



where Kg(r') is given by Eq. (69) and -q^it) is given by Eq. (59) in terms of the 

 surface wave elevation rj^if). The expression for e(t) can be rewritten as: 



486 



