Singulav Perturbation Problems in Ship Hydrodynamics 



F^(t)- Zpgj dxb{x,0);Q(x,t) Tl -,^^-^y dz e'''b(x,z) 



+ ipco \ dx ^0 (x,t) \ di n-^^e"'. 

 Jl °t Jc{x) 3^3 



The first term shows the Froude-Krylov force quite explicitly; the 

 product of t,Q{x.,t) and the quantity in brackets is often called an 

 "effective waveheight," the second factor being a quantitative repre- 

 sentation of the Smith effect. The second integral term has been 

 expressed in terms of the vertical speed of the wave surface, 

 ^0*(x,t). This term should be compared with the force expression 

 for the calm-water problem, (3-30), For a slender body, the hydro- 

 dynamic part of the latter can be written, for j = 3, 



-ipw^ dSn3[e3(t)V^5(t)<^5] - -ipooj dx [ e3(t)-xe5(t)] ^ di njC^j. 



The last quantity in brackets is the vertical speed of the cross section 

 at any particular x. Comparison with the second term of F*'(t) 

 shows that the latter is almost exactly the same as the hydroaynamic 

 force that we would predict if each section of the ship had a vertical 

 speed - C,o{x,t), This analogy would be exact, in fact, if the expo- 

 nential factor, e"^ , were not present in the F3(t) formula. 



Except for that factor, what we have found is that Korvin- 

 Kroukovsky's well-known "relative- velocity hypothesis" is approxi- 

 mately correct according to the analysis above. The hypothesis is 

 particularly accurate for very long waves, in which case e^^ ta. i 

 over the depth of the ship, but it is less accurate for short waves. 

 Again, it should be noted that we have no absolute basis for saying 

 whether a particular wave is short or long in this respect. In com- 

 puting the Froude-Krylov part of the force, it is well-known that the 

 exponential- decay factor must be included in practically all cases of 

 practical interest; this has been amply demonstrated experimentally. 

 It suggests that one should be wary of dropping the exponential factor 

 in the diffraction- wave force expression. 



Summary . In the far field, we assumed that the effects of 

 the heaving/pitching ship could be represented by a line of pulsating 

 singularities located at the intersection of the ship centerplane and 

 the undisturbed free surface. For a first approximation, we tried 

 using just sources , and these were siifficient to allow matching with 

 the near-field solution. In particular, the inner expansion of the 

 outer expansion showed that the near-field expansion would satisfy a 

 two-dimensional outgoing-wave radiation condition, at least in the 

 first approximation. With this fact established, we formulated the 

 near-field problem; it reduced ultimately to the determination of a 

 velocity potential in two dimensions , the potential satisfying a linear 



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