Submerged Two -Dimensional Bodies 



It appears that for the problem of a symmetrical foil the author chose a new 

 parameter 



s = -Ls 



in terms of one-half the maximum foil thickness S = 0.187 ft. It follows then for 

 the lowest to the highest speeds tested, u = 2.5 to 5.5 ft/sec, the corresponding 

 values of the perturbation parameter s ranges from 0.92 to 0.20, while the 

 Froude numbers defined in a customary manner by use of the full chord length 

 correspond from 0.43 to 0.92. 



In the asymptotic series given as Eq. (Dl), fg represents a uniform stream; 

 f 1 represents the image of the uniform flow within a circle and its reflection 

 with respect to the free surface; f 2 consists of the image of the first -order 

 free -surface disturbance within the circle, its reflection with respect to the 

 free surface, and the nonlinear correction term corresponding to the particular 

 solution of the inhomogeneous equation describing the second-order free-surface 

 condition, and so on. 



To complete a true second-order solution of the problem, the author indi- 

 cated that missing parts of the solution in f 2 are now being sought by a process 

 similar to that employed by Giesing and Smith for the two-dimensional problem, 

 as cited in Giesing's discussion, by the preceding footnote. For the configura- 

 tion of a circle or a sphere, the disturbed kinematic condition due to the intro- 

 duction of the external singularities can readily be adjusted by invoking the 

 circle or sphere theorem. However, for a general geometry, the violation of 

 the kinematic condition can be arrested by the solution of the Neumann's exterior 

 problem via an integral equation in which the density of the surface-distributed 

 sources appears as the unknown, with the induced normal velocity due to the 

 external singularity being given over the surface of the body. 



Application of Tuck's procedure to the three-dimensional problem of a 

 sphere, which is under progress at Davidson Laboratory, is found to be rather 

 complicated due to the appearance of the wave-angle integral in the analysis, in 

 which case the velocity potential can be expanded in an asymptotic series of the 

 form 







here a being the perturbation parameter, the actual radius of the sphere a di- 

 vided by the typical length u^/g and where, for example, one finds the first- 

 order potential as 



1-2 



^Re 



i f cos ^d^if^^ /i.^^r , X n 



2;fJ 2 ^2^] (qi)^ucosd?d^+ 2j (q2)^u cos t? 



•Tr -77 --rr/O 



with u - y + i(x cos (9 + z sin 0), 



+ 0(a^u2) 



635 



