Breslin, Savitsky, and Tsakonas 



This function (plotted in Fig. 2) is seen to be a symmetrical function of time and 

 hence requires some future time knowledge of surface wave profile in order to 

 predict the wave profile at depth. Using the above kernel in a convolution inte- 

 gral will provide a time history of the orbital motions at depth v(^) in terms of 

 the time history of the irregular surface wave profile, Vo(^) immediately above 

 the test point. 



?(t-r)2 



(59) 



'' - I .v^)^yf 



^(t) = r^Jr) ^ U—e '^ dr. 



Predicted Heave Time Histories 



Considering the heave motion of the neutrally buoyant submerged body, it 

 was shown in Ref . 5 that the body behaves in heave and sway like a water parti- 

 cle at depth, i.e., it has identical amplitude A(a;) - 1 and zero phase <:p(aj) - 0, 

 relative to a submerged wave particle. The heave transfer function of the sub- 

 merged body (relative to water motion at depth) is then 



0(0)) = A(a;) e-^'P^") = (1) e"° = 1 . (60) 



The impulsive response function of this mechanical system is then written 



00 CO 



K(t)^, = ^ j cDCc.) e'^^'dc -- ±- I e'^'dc = S(t) (61) 



- CO - CO 



where S is the Dirac delta function. Operating with the delta function on a 

 bounded and continuous function f(t) , it can be shown that 



OD 



j f(t) S(t- t„)dt = f(t^) . (62) 



- CO 



Thus, convolving the body heave impulsive response function [Eq. (61)] with the 

 wave motion at depth, Eq. (59), gives the time history of heave motion z(t) of 

 the submerged body in terms of the surface wave profile (77^) to be 



CO CO _ ?.(t-t' ) 



Z(t) = J J V^(r') -J ]/^ e ^^ dT'S(t-T)dT 



- CO - CO 



so that 



;(t-r')^ 



z( 



t) = J Vjr-) ly^e '^ dr' 



(63) 



The above integral was evaluated by Dalzell and resulted in a computed 

 heave time history of submerged body. The analytical results are compared 



482 



