Ogilvie 



lead to a non-zero normal velocity component on the body, and this 

 would have to be offset by a flow which does not change the condition 

 at the plane y = 0. Presumably, all higher- order approximations 

 would be solutions of problems which are identical in form to this 

 one. 



A variation on this approach has been discussed several 

 times by Professor L. Landweber, although he has not published 

 the work. He points out that the usual linearized free -surface 

 condition, 



? + ^-o^ =0. on z = 0, where K = g/XJ^, 



becomes the rigid- wall condition when U — *• , and so one might try 

 an iteration scheme in which <}> is expanded in a series, <^ ~ V ^ , 

 and the terms are obtained as the solutions in an iteration scheme: 



In order to test the scheme, Professor Landweber proposed trying 

 to obtain the potential function for a Havelock source in this way; this 

 obviates the need to satisfy a body boundary condition, and the known 

 potential for the source can be expanded in a series in terms of i/ K. 



Neither of the above schemes appears very promising to me. 

 Salves en's findings about the singular low- speed behavior seem to 

 condemn any approach which overlooks the peculiar nature of the 

 free- surface problem at low speeds. The next section should make 

 clear why I am pessimistic about these approaches. It should be 

 obvious even now that the wave -like nature of the problems has been 

 lost, but the difficulty is more serious than that. 



5.42. A Dual-Scale Expansion. According to linearized 

 wave theory, the wave-like nature of a free surface disturbance 

 loses its identity exponentially with depth. A disturbance created 

 at the free surface is attenuated rapidly with depth, and a disturbance 

 created at some depth causes a free-surface disturbance which de- 

 creases with the depth of the cause. The depth effect is essentially 

 proportional to e'^^, where, as above, K = g/U and y is measured 

 as positive in the upward direction. 



As U approaches zero, this depth- attenuation factor ap- 

 proaches zero for any fixed y. In other words, the free-surface 

 effects are restricted to a thin layer which approaches zero thickness 

 as U -*- 0. We might say that the free-surface is separated from 

 the main body of the fluid by this "boundary layer" in which there is 



798 



