singular Pevtuvhation Problems in Ship Hydrodynamics 



In a not-yet published paper, Newman has applied the 

 Khaskind- Newman relations in the forward-speed problem by arbi- 

 trarily ignoring the J2j functions in the forced-motion potential 

 function. He finds for the heave excitation force: 



Fj(t) ^ pgh(l + Ucoo/g)e"^* \ dx e''"*^ \ di n e 



where, as before, u) is the frequency of oscillation (that is, the 

 frequency of encounter) and v = co /g; the frequency measured in an 

 earth- fixed reference frame is denoted by coq. and we define 

 Vq = u^o/g. The actual wave length of the incident waves ^s \ = Zir/v^ . 

 The two frequencies are related as follows: co = cOq + UcoQ/g. These 

 formulas are all valid for the head-seas case only. 



This formula should be compared with (3-34a), which was 

 the corresponding result in the zero-speed problem. The first 

 term in brackets yields the Froude-Krylov force, and the second 

 term yields a pure-strip-theory prediction of the diffraction wave 

 force, which can be interpreted approximately in terms of the 

 relative- motion hypothesis. The remaining terms represent an 

 interaction between forward speed and the incident waves. 



Again, it should be pointed out that more than just nonlinear 

 effects have been neglected in setting Qi] equal to zero. In fact, 

 the usual linear free-surface condition for ship-motions problems 

 can be written: 



4Jtt "•"84^2=" ^^"^tx " ^^^xx' on z = 0. 



(Cf. (3-37) and (3-47).) Even the inclusion of the Q, \ terms still 

 omits some effects usually considered as linear, namely, the effects 

 of the term - U ^i^x '^^ this boundary condition. These effects are 

 higher order in the theory presented here solely because of the high- 

 frequency assumption. 



IV. THIN-SHIP THEORY AS AN OUTER EXPANSION 



It has already been shown how one can view a symmetrical 

 thin-body problem in terms of inner and outer expansions; the usual 

 description of the flow around such a body is really just the first 

 term of an outer or far- field expansion. It was not at all obvious that 

 one had to use such a powerful method on such a problem, but it was 

 clear that one could do this. Probably the only advantage of doing 



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