Ogilvie 



The wave- excitation problem can be formulated as a singular 

 perturbation problem, but such a problem has never been satis- 

 factorily solved, even for the zero-speed case. Fortunately, another 

 approach is available for obtaining the wave excitation; this is the 

 very elegant theorem proven by Khaskind [ 1957] . It allows one to 

 compute the wave excitation force , including the effects of the diffrac- 

 tion wave, without solving the diffraction problem. Since we thus 

 avoid the singular perturbation problem altogether, only the final 

 results are presented here. (Reference may be made to Newman 

 [1963] for details of the zero-speed case.) Let the incident wave 

 have the velocity potential: 



^o(x,z,t) = -2_ e 



CO 



the corresponding wave shape is given by: 



^Q(x,t) = he 



This is the head-seas case. For an arbitrary body, the heave force 

 due to the incident waves is: 



F*(t) = pghe'"'* r dS e^'-'^^Kl - v^glnj + iv^^^n, } . 



If the body is a slender ship, with axis parallel to the wave-propa- 

 gation direction, this formula simplifies to the following: 



Fglt) ^ pghe'*^* r dx e''"* r di n3e''^(i - v^, (3-34a) 



The corresponding expression for pitch moment on a slender bady is: 



F^lt) '^ pghe'*^* r dxe"''^(-x)r di n^e^^d - v4>J. (3-34b) 



5 Jl Jc{x) ^ ^ 



In the expression (I- v^j^) in the integrand, the first term leads to the 

 force (moment) which would exist if the presence of the ship did not 

 alter the pressure distribution in the wave; in other words, it gives 

 the so-called "Froude-Krylov" excitation. This fact can be proven 

 by applying Gauss' theorem to the integral. Dynamic effects in the 

 wave ("Smith effect") are properly accounted for. The second term 

 gives all effects of the diffraction wave. 



A final rewriting of the wave-force formula is worthwhile. 

 The above approximate expression for F3(t) can be manipulated 

 into the following: 



756 



