OgiZvie 



and we suppose that all conditions except forward speed are identical 

 in all experiments. We shall discuss what happens when "U-* 0," 

 and we shall understand by the limit operation that we are passing 

 through the sequence of experiments toward the limit case in which 

 there is no forward speed at all. In each experiment, U is a 

 constant. 



As U — ^ , we certainly expect all fluid motion to vanish. 

 But we would like to know to what extent the velocity field vanishes 

 in proportion to U (that is, what part is 0(U)), what part vanishes 

 more rapidly than U (that is, what part is o(U)), and what part, 

 if any, vanishes less rapidly than U. 



In an infinite fluid, the velocity everywhere is exactly pro- 

 portional to U. Far away, the velocity approaches zero; it drops 

 off like l/r if there is a circulation around the body, and it drops 

 off like l/r^ if there is no circulation. But in both cases the 

 constant of proportionality is 0(U). No matter how distant our 

 point of observation is from the body, the velocity is 0(U) as 

 U— 0. 



At very low speed, one expects that gravity will force the 

 free surface to remain plane. The constant-pressure condition will 

 be violated to the extent that the magnitude of the fluid velocity on 

 that plane is not quite constant, but the error in satisfying the dy- 

 namic condition will be proportional to the square of the fluid 

 velocity magnitude. The kinematic condition will be satisfied in a 

 trivial manner. Accordingly, it seems quite reasonable to assume 

 that the free-surface disturbance is 0(U ) as U -* , and so the 

 velocity potential in the first approximation is the same as if the 

 free surface were replaced by a rigid wall. Let the rigid-wall 

 velocity potential be denoted by (j>Q(x,y). Clearly, it is true that: 



«^o(x,y) = 0(U). 



This follows by the same arguments as those used in the preceding 

 paragraph. The more important problem is to determine the order 

 of magnitude of [ 4)(x,y) - ^o^x,y)] , where ^(x,y) is the exact 

 velocity potential for the case of the body moving at speed U under 

 the free surface. 



In order to be specific now, let 4)o(x,y) be the velocity 

 potential in two dimensions which satisfies the conditions: 



^= 0, on body; | ({>o - Ux| — , as x ^ - oo; ^ = on y = 0. 



This is the same point that I belabored in the last paragraph of 

 Section 1.2. Again, I apologize to those to whom it is obvious. 



794 



