singular Perturbation Problems in Ship Hydrodynamics 



"pressure distribution" is periodic in time, and it is also periodic in 

 y as |y| ~^ oo; the latter comes from the term containing ^Jx« 

 Furthermore, the time and space periodicities are related to each 

 other in just the way that one would expect for a plane gravity wave. 

 This can be proven by studying the boundary- value problem for $j . 

 Thus, there is an effective pressure distribution over an infinite 

 area, and it excites waves at just the right combination of frequency 

 and wave length so that we have a resonance response. In an ordi- 

 nary two-dimensional problem, there would be no solution satisfying 

 all of these conditions. However, our solution need not be regular 

 at infinity; it must only match the far-field expansion. And the far- 

 field expansion predicts an appropriate singular behavior at infinity. 

 It is shown by Ogilvie and Tuck that the solution of this inner prob- 

 lem does exactly match the above far-field solution. The way the 

 pieces of the puzzle all fit together is rather typical of the method 

 of matched asymptotic expansions, and it indicates at least that the 

 manipulations of asymptotic relations were probably done correctly! 

 (It still says nothing about the correctness of the assumptions.) 



There is no benefit to be derived by repeating here the solu- 

 tion of the above detailed problems. Rather, we jump to the results 

 for the heave force and the pitch moment, and we do little more than 

 compare these results with the comparable formulas in two previous 

 problems : 



CASE 1: The oscillating slender body, with forward speed, 

 in an infinite fluid (Section 2.32) 



CASE 2: The oscillating slender body (ship), at zero forward 

 speed, on a free surface (Section 3.3) 



In all cases, let the force (moment) be expressed in the form: 



FJ"(t) = - 2^ [(ico)^aji +(lcu)bjj +Cji]ei(t) . 

 i=3,5 



We define cjj to be independent of frequency and of forward speed. 

 (We must make some such arbitrary convention, or the separation 

 into ajj and Cjj components is not unique.) With this convention, 

 Cjj represents just the buoyancy restoring force (moment). Thus, 

 Cjj = for all j,i in case 1; in cases 2 and 3, Cij is given by: 



[Cjj=2pgy dx {l,-x}(/Jb(x,0). 



765 



