Ogilvie 



The right-hand side of this equation takes account of the nonlinearity 

 of the free-surface conditions , since obviously it involves just the 

 potential function from the previous cycle of the iteration. r\\ is the 

 free-surface elevation from the previous approximation; it is given 

 by: 



Tl,(x) = - (U/g) Re {F^Q+F/j}, 



with the right-hand side evaluated on y = b. One might try to cut 

 corners in (5-6) in either of two ways, namely, 1) ignore the right- 

 hand side by setting it equal to zero, 2) Drop the terms involving 

 Fb| on the left-hand side. The first is equivalent to retaining just 

 a linear free-surface condition. The second is equivalent to neglect- 

 ing the effect of the second-order body correction at the free surface; 

 this is the "inconsistent" second-order theory to which Fig. (5-6) 

 refers . 



Apparently, Salvesen did not prove one important step in his 

 development, namely, his claim that FbQ i-s 0(1) near the body 

 and 0(e) far away from the body. In fact, with his definition of 

 C = t/b, it appears that the statement is wrong. The potential FbQ 

 represents just a thickness effect, since it is the solution of the 

 problem of a symmetrical body in a uniform stream. Although the 

 body can be replaced by a distribution of sources, the disturbance 

 will appear from far away to have been caused by a dipole, and 

 so it must have the form: Fbo~ C/z. If the body were a circular 

 cylinder, we could evaluate C: C = Ut , where t is the radius of 

 the cylinder. The complex fluid velocity on the free surface caused 

 by the body is, in the first approximation, - C/z = 0(e ), since 

 z = X + ib on the level of the undisturbed free surface. This con- 

 clusion contradicts Salvesen's assumption that the free-surface 

 disturbance is 0(e), but perhaps it does not matter. At this point, 

 the results would presumably be just the same if he had defined: 

 C = (t/b)'' , (The argument above for a circular cylinder agrees 

 with Tuck's conclusions.) 



When the first free-surface correction is found, namely, 

 Ff , its effect in the neighborhood of the body is not diminished by 

 an order of magnitude, since at least one part of Ff, involves an 

 exponential decay with depth, the exponent being /c(y - b). Near the 

 body, y «^ , and so the exponential -decay factor is e , and it has 

 been assumed that /cb is 0(1). (See Salvesen's paper.) 



2 

 Since Ff is 0(€ ) near the body, the order of magnitude 



of the next correction term, Fb., must be the same. This time, 



however, the nature of the body disturbance is quite different from 



a dipole disturbance. The effective incident flow corresponding to 



Ff, is not a uniform stream, and so the presence of a sharp trailing 



edge on the body requires that a Kutta condition be imposed, and 



then a circulation flow occurs. From far away, it appears that Fbj 



790 



