Submerged Two -Dimensional Bodies 



second-order wave resistance and neglects the free-surface nonlinear term. It 

 would be of interest to see results obtained by applying this method to the shape 

 considered. An ideal situation, however, would result if this method were joined 

 to the method used in this paper, since the result would then be an entirely con- 

 sistent second-order theory. 



DISCUSSION 



E. O. Tuck 



David Taylor Model Basin 



Washington, D.C. 



This is a most admirable piece of work, and is precisely the kind of more 

 realistic study I hoped would be prompted by my work on the admittedly aca- 

 demic case of a circular cylinder. It is unfortunate that in the version presented 

 the author has not carried the theory through to consistent second-order by sat- 

 isfying the body boundary condition to this order, but since he has remedied this 

 situation in more recent work, this objection is somewhat beside the point. It is 

 important to note that in my work I did not recommend that the free -surface 

 second-order effect be included while neglecting the less important body- 

 condition second-order effect; I found a factor of 3 or 4 between the former and 

 the latter for circular cylinders but do not consider this high enough to neglect 

 the latter. Indeed, from the author's Fig. 13, it is quite clear that the body- 

 induced second-order effect is vital for more streamlined bodies at higher 

 speeds, this being presumably a consequence of the Kutta condition, which gives 

 a second-order contribution at zero angle of attack. 



I should comment on the author's statements in his introduction about my 

 work. First, the second-order wave resistance for a circular cylinder does not 

 tend to infinity as the Froude number tends to zero; it tends to zero, but more 

 slowly than the linear resistance, so that the ratio between the two tends to in- 

 finity. Second, it is of course true that the computed results I presented are for 

 a circular cylinder too close to the free surface for the validity of a perturba- 

 tion series, and indeed I would expect gross nonlinearity including breaking 

 waves under these conditions. While one can easily make the series converge 

 by submerging the cylinder deeper and deeper, the relative order of magnitude 

 of the two second-order effects is unchanged. It was therefore convenient to 

 present for illustrative purposes a case where the second-order effects are 

 rather too large for convergence but which shows more clearly the relative 

 orders of magnitude. This argument applies with equal force to the present 

 work; thus. Fig. 16 is valuable in spite of the fact that breaking was observed to 

 occur at the intermediate speeds. The really remarkable feature of the present 

 work is that in some cases (e.g., the lowest speed in Fig. 12) where the second- 

 order wave effect is substantially greater than the first -order, the total never- 

 theless agrees well with the author's experiments. 



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