Yim 



where a = x cos 6 + y sin ^, and fi^ is the doublet strength. The wave height due 

 to the quadrupole in x direction at (o,o,-Zj) is 



^a = T-77 Ke dkd6' , (32) 



^""^ J-^-^o k-ko sec2 6? - i/^ sec 



where V is the quadrupole strength. By combination with k^K - /Xj we obtain in 

 the limit ^i ^ 



^q + ^dz - - ^K^ ( \ 2ik^ COS e'^'""''^ dkd^. (33) 



-'-77 -^0 



This indicates that the local disturbance (= /x ^q), is due only to the doublet at 

 (o,0,Zj) and the mirror image. The total flow is produced by the singularity 

 system shown in Fig. 2, which satisfies the free surface boundary condition. 

 Namely, the x component of the perturbation velocity on the free surface is ex- 

 actly the same as that due to the doublet and its mirror image only. Since the 

 sign of the image of the quadrupole is opposite from the real one, these do not 

 contribute to the horizontal velocity on the free surface. It is interesting to 

 have such a simple set of singularities and images which satisfies the free- 

 surface boundary condition. 



(J) + k * 



X = QUADRUPOLE STRENGTH 

 H? = DOUBLET STRENGTH 



Fig. 2 - Image system 

 of a no-wave singularity 



In this manner, we can obtain the simple image system for any order no- 

 wave point singularities, namely, for all the derivatives as well as integrals of 

 the first-order no-wave point singularity mentioned above. For example, if we 

 uniformly distribute the first-order no-wave singularity from x to x = infin- 

 ity, we obtain the zeroth-order no-wave singularity which is composed of a dou- 

 blet in the x direction and a lift element at (o,0,Zj) with the image composed of 

 the doublet and the lift element in the opposite direction at the position of the 

 mirror image. M. P. Tulin, in a seminar at Hydronautics, Incorporated, has 

 brought attention to this particular no-wave singularity system. He pointed out 

 that the existence of this system indicates the theoretical possibility of a finite 

 three-dimensional ship without wave resistance (although, as he also pointed 

 out, the buoyancy of the ship would be canceled by its negative dynamic lift). He 

 has also discussed the application of these singularities to both two-dimensional 

 and three-dimensional cavity flows beneath free surfaces. 



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