Ogilvie 



; = 0(€^). 



The kinematic condition then reduces to: 



^, = on z = 0; (3-11) 



thus, $1 represents the flow which would occur in the presence of 

 a rigid wall at z = 0. From the dyneunic free-surface condition, 

 we can compute the first approximation to the wave shape: 



gUx,y) ~ - (U$,^ +|$f^)|^^^. (3-12) 



It mtay appear to be a paradox that we have a flow without waves , 

 from which we compute a wave shape! But, like all paradoxes, it 

 is a matter of interpretation and understanding. We shall return 

 to this point presently. 



Since $| satisfies the Laplace equation in two dimensions and 

 a rigid-wall condition on z = 0, it can be continued analytically into 

 the upper half space as an even function of z. All of the arguments 

 used in the infinite -fluid problem can then be carried over directly. 

 In particular, at large distance from the origin, we can write, as 

 in (2-64), 



A ~ C +^log r +0(l/r), as r — oo. 



The two-term inner expansion can be matched to the two-term outer 

 expansion. We obtain 



A,o = 2(r(x); 



C, = - i^f(x) - g(x). 



Note that there is again a factor of 2 difference from the infinite- 

 fluid results, (2-66). Of course, the term g(x) is new here. 



We can again determine A|q and thus c in terms of body 

 shape, without the necessity of solving the near-field hydrodynamic 

 problem. By the simple flux argument, we find that: 



A,o= 2Us'(x), (3- 13a) 



where s(x) is the cross-sectional area of the submerged part of 

 the hull. With this convention, we find that 



736 



