Ogilvie 



gti(x) +-i[^x + 4] --tU^ = 0' on y = ^(x); (5-9) 



)(^x-#y=0' on y = 'n(x); (5-io) 



1^= 0, on the body; (5-11) 

 on 



(x,y) - Ux -♦0, as X — ^ -co; (5-12) 



The rigid-wall potential, <j)Q(x,y), satisfies (5-11) and (5-12) too, 

 but it does not satisfy the free-surface conditions, of course; 

 instead, we have 



^ = 0, on y = 0. (5-13) 



Now we introduce one more potential function, the difference between 

 the above two potentials: 



$(x,y) = ^(x,y) - <^o(X'y) • (5-14) 



It must satisfy the body boundary condition, of course, and it 

 vanishes far upstream. On the free surface, which we now define as: 



y = Ti(x) = r|o(x) + H(x) , (5-15) 



where T)q(x) is defined as in (5-7), the new potential satisfies the 

 two conditions: 



= gH(x) - I 4^x,0) 



+ -| [ 4x + H + 2^0x^x + 2^0y*y + ^x + ^y] ly=T7(x) > (5-16) 



= [tio(x) +H'(x)][40x + *xl ly,^(,) - [*Oy+^y] ly,,(,) • (5-17) 



These conditions are still exact. An obvious approach to 

 solving for ^(x,y) and H(x) is to re-express these conditions on 

 y = •n(x) as conditions on y = 0. Here I shall assume that this can 

 be done in the usual way.* Then it follows from the exact conditions 



This is the crucial point which distinguishes this section from the 

 next section. 



796 



