OgiZvie 



Physically, the situation may be described in the following 

 way: If U is small enough, the body extends over a distance of 

 many wave lengths of the surface disturbance. The initial dis- 

 turbance is caused by the body, of course; this is the "rigid-wall" 

 motion, and its dimensions are characteristic of the body. It 

 causes a free-surface disturbance, with the result that waves are 

 created. But these waves are very, very short, whereas the initial 

 disturbance from the body appears to be just a slight nonuniformity 

 in flow conditions when viewed on the scale comparable to the wave 

 length. The miethod is, in fact, quite similar to classical methods 

 such as the W-K-B method. 



When the assumptions listed above are actually applied, we 

 find that the approximate free -surface conditions given in (5-18) 

 and (5-19) must be replaced by the following: 



gH(x) + ^o,(x,0)^,(x,TTo(x)) - 0; (5-22) 



^y(x,r|o(x)) - (^o^(x,0)H'(x) - p'(x)} 



the function p(x) is the same that was given in (5-20). Note that ^ 

 in both conditions here is to be evaluated on y = •no(x) , rather than 

 on y = 0. The reason is the same that was given in Section 3.2 in 

 the near -field problem.: If we tried in the usual way to expand 

 ^(x,r|o), say as follows: 



^(x,Tio) = $(x,0) + Tio*y(x,0) + i riQ<J>yy(x,0) + . . . , 



we would find that every term on the right-hand side is the same 

 order of magnitude according to my assumptions. In particular, 

 T|o= O(U^), and, symbolically, we have: d/By = 0(1 /U ). So this 

 expansion procedure is not useful. 



The two conditions above can be combined consistently into 

 the following: 



*y(x,Tio(x)) +-|4bx(^'0)*xx(^'^oW)=^ P'(^)- <5-23) 



This is remarkably similar to the free-surface condition for another 

 problem. In the ordinary linearized theory of gravity waves, sup- 

 pose that a pressure distribution, p(x) , is travelling at a speed U. 

 The free- surface condition would be: 



^y(x,0) +^<J>,,(x,0) =p'(x) , 



800 



