singular Perturbation Problems in Ship Hydrodynamios 



where T)(x,y) Is the free-surface shape In the steady-motion prob- 

 lem (the ^(x,y) of Section 3.1), and 9(x,y,t) is whatever must be 

 added so that ^(x,y»t) is the complete free- surface deformation. 



The body surface is defined mathematically just as in Section 

 3,3 for the zero-speed problem; see (3-18) and (3-19), The same 

 assumptions are made about orders of magnitude: 



ei(t) = 0(e6) ; oo=0(e-'^). 



From these assumptions and the subsequent analysis, it turns out 

 that 



4;(x,y,t) = 0(e^2§)^ e(x,y,t) = 0(e6), 



as either e or 6—^0, We can look on the complete solution as a 

 double expansion in e and 6. From this point of view, the expan- 

 sion for the potential can be written: 



^(x,y,z,t) = {Ux + Uxi (x,y,z) + .., } 

 0(6°e°) 0(5V) 



+ {ijj|(x,y,z,t) + 4;2(x,y,z,t) + .,. } + o(6). (3.36) 

 0(6'c^^^) 0(6'€^) 



The order of magnitude of the term UX|(x,y,z) was found in Section 

 3.1, The order of magnitude of ij^i n^ay be somewhat surprising* 

 Physically, it im.plies that the effects of ship oscillations dominate 

 the effects of steady forward motion --in the first approximation. 

 These orders of magnitude were derived by Ogilvie and Tuck. Here, 

 I shall not prove them, but I hope to make them appear plausible. 

 It should be noted that the high frequency assumption was rnade just 

 so that the orders of magnitude would conae out this way . (Cf. the 

 discussion in Section 2,3, in which it was pointed out that the for- 

 malism for the steady-motion slender-body problem is established to 

 force certain expected results to come out of the analysis. We are 

 doing the same here, forcing strip theory to come out as the first 

 approximation. ) 



The linearity of the i|j| problem permits us to assume that 

 the time dependence of ^y and of the corresponding first term in a 

 9 expansion can be represented by a factor e *^ , 



In order to find any effects of interaction between steady 

 motion and oscillatory motion, it is necessary to solve for the term 



759 



