Ogilvie 



6. The only forward-speed terms not yet discussed are those 

 in Case 3 which involve the integral I. They arise from the inclusion 

 of the functions S2j in the potential function, as in (3-43), and the 

 necessity for including those functions is a consequence of the fact 

 that the right-hand side of (3-42), the free-surface condition, is not 

 zero. Now, the right-hand side of (3-42) represents an interaction 

 between the forward motion and the oscillation. One might try to 

 simplify matters by assuming that one can neglect the effects of x > 

 the perturbation of the incident stream by the body. But this reduces 

 (3-42) to the following: 



4jtt + g4>z^ - 2Ui|jt,, on z=0. (3-47) 



0(c'^^5) 0(e6) 



The right-hand side is still not zero, and we would still have the 

 flj functions to contend with. In fact, it may be recalled that this 

 remaining term on the right-hand side was the one that caused the 

 major trouble in interpreting the Q,] problems. Neglect of the x 

 terms leads to the condition on S2j (cf. (3-46)): 



flj^ - vJ2j = - (2/g)$j^, on z = 0, (3-48) 



and it is the one remaining right-hand term which causes the solution 

 for Q\ to diverge at infinity. The usual procedure at this point is to 

 set Oj = , turn the other way, and just ignore these problems. The 

 results are in remarkably good agreement with experimental obser- 

 vations, and one still wonders how this can be rationalized mathe- 

 matically. 



Finally, we should at least mention the problem of predicting 

 wave excitations in the forward-speed problem. The singular per- 

 turbation problem involved in solving for the diffraction waves has 

 not been satisfactorily worked out yet, at least, not in a manner 

 compatible with the approach presented above. 



One might hope to avoid the diffraction problem by using the 

 Khaskind relations, as in the zero-speed problem. (See Section 3.3.) 

 In fact, Newman [ 1965] has derived what I call the Khaskind- Newman 

 relations. These provide a generalization of Khaskind's formula, 

 relating the wave excitation on a moving ship to the problem of forced 

 oscillations of the ship when the ship is moving in the reverse direc- 

 tion. Unfortunately for our purposes, Newman's derivation is based 

 on an a priori linearization of the free-surface, in the sense that our 

 terms involving x can be neglected. Therefore, the appropriate 

 diffraction problem cannot really be avoided in this way. Also, it is 

 necessary to have available the potential function for the forced- 

 motion problem, and this includes at least a part of the f2j functions 

 even if the x dependence is ignored. 



768 



I 



