Sing^ular Perturbation Problems in Ship Eydrodynamios 



^^^ + g^z ~ - 2Ui|;j, - ZUXyil^ty - UXyyipt on z = 0. (3-42) 



The orders of magnitude are noted, again on the basis of information 

 not derived here. This condition can be compared with the linear 

 condition used in the far field, (3-37). The two terms on the left 

 here are obviously the same as the terms (ico) ^ + gij^z in (3-37), and 

 the first term on the right here, - ZUijjj^, is the same as the term 

 2iwU4Jx '^^ (3-37), The other two terms on the right-hand side here 

 are basically nonlinear in origin; they involve interactions between 

 the oscillation and the steady perturbation of the incident stream. 

 The term U \\ty^^ which appears in (3-37) is missing here because 

 it is 0(c^'^6) in the near field by our reckoning. 



Again it is worthwhile to compare this boundary condition 

 with its nearest equivalent In other versions of slender-body theory 

 or strip theory of ship motions. If we did not assume that frequency 

 Is very large, slender-body theory would require In the first approxi- 

 mation that i|j2 = 0, since the other terms are all higher order. This 

 Is just the free-surface boundary condition obtained In this problem 

 by Newman and Tuck [l 964] and by Maruo [ 1967] . Higher order 

 approximations would Involve nonhomogeneous Neumann conditions 

 on z = 0, On the other hand. In most derivations of strip theory. 

 It Is assumed that the free-surface condition Is: ij^tt "^ Z^z = on 

 z = 0, This agrees with the lowest-order condition obtained by 

 Ogllvle and Tuck, as given above. However, the assumption of this 

 boundary condition In the usual strip-theory derivation Is quite 

 arbitrary, and no means Is available to extend It to higher- order 

 approximations. The assumptions made by Ogllvle and Tuck were 

 chosen explicitly so that the simplest approximation would be just 

 strip theory, and we see here that that goal was achieved. This 

 basis for choosing assumptions was selected only because strip 

 theory had proven to be the most accurate procedure available for 

 predicting ship motions. 



The method of solution used by Ogllvle and Tuck Is to find 

 several functions each of which satisfies some part of the nonhomo- 

 geneous conditions. In particular, let the solution be expressed In 

 the following form: 



i|;(x,y,z,t) =2^ [Iw^j + U*j + (l(o)^Unj]ej (t) , (3-43) 



* 

 In other words, we stopped fretting about how Irrational strip theory 



was and set out to derive It formally! 



763 



