solid boundaries by suitable equations. We assume that a velocity potential <p(x, y, z, t) 

 exists; here v = grad <p. Then <p must satisfy certain well-known conditions: 



A(p = in the region occupied by fluid (1) 



r) x (x,z,t)if x (x,v(x,z,t) ,z,t) — <p y + 7] z <p z + 7] t = 0. (2) 



gv(x,z,t) + <p t (x,rj(x,z,f),z,t) + ^(grad <pY = 0. (3) 



If the equation of a solid surface is given by F(x, y, z, t) = 0, then on such a surface 



F x <p x + F y< p y + F z <p 3 + F t = 0, (4) 



i.e. The normal velocity of the fluid equals the normal velocity of the body. If the 

 body is fixed, F t — and dy/dn = 0. 



The conditions at infinity are somewhat more troublesome; they should at 

 the same time be physically reasonable and guarantee a unique solution. There are, 

 however, few uniqueness theorems for the general problem. If the body starts from 

 rest in still water, one can assume that 



lim rj(x,z,t) = 0. (5) 



If one wishes to consider a steady-state problem, with, say, 77 — rj{x-ct, z), or even 

 a more general motion where the body has moved in one direction for a long time, 

 one requires that the motion vanish far ahead of the body: 



lim r]{x,z,t) = 0. (6) 



x->oo 



In the case of a steadily oscillating body a "radiation" condition is usually imposed 

 at infinity, but after linearization; this is not, however, always sufficient for a unique 

 solution. At infinite depth one imposes : 



lim grad <p = 0. (7) 



J/— 00 



Let us now suppose that some dimensionless parameter e may be associated 

 with the motion in such a way that when e ~ * the motion approaches a state of 

 rest. Some examples of the choice of e will be given later. We now assume that <p 

 and other functions entering into the description of the motion may be expanded in 

 power series in e, where <p itself starts with the first power. For the present we shall 

 be concerned only with ^ and 77 : 



<p(x,lj,z,t) = e<piO,l/,z,0 + e 2 <po(x,y,z,t) -f- • • • , 



y(x,z,t) = r]o(x,z,t) + e?7iO,2,0 + e 2 rj2(x,Z,t) + • • • . 



If these expansions are substituted in conditions (1) — (3) above and coeffi- 

 cients of the same power of e grouped together and equated separately to zero, one 

 obtains conditions to be satisfied by each of the ^ and 77,. One finds immediately: 

 r) = 0. Then from the coefficients of e one gets the usual equations for a potential 

 function satisfying the linearized free-surface boundary condition: 



A<pi = for y < 0. (8) 



— <pi y (x,0,z,t) + r]it(x,z,t) = 0. (9) 



g V i(x,z,t) + <Pu(x,0,z,t) = 0. (10) 

 The last two conditions combine in the usual way to give : 



g<Pi v (x,0,z,t) + <put(x,Q,z,t) = 0. (11) 



One notes that it is a result of the method that the boundary condition on ^ is imposed 

 on the plane yzO and not on the actual free surface. 



110 



