vertical oscillations can create only waves with amplitudes which are of higher than 

 first order in the thickness of the blade. Since the only mechanism for energy dissipa- 

 tion is the creation of waves which carry off energy to infinity, it is clear that no damp- 

 ing of such oscillations is to be expected. Since these modes of oscillation seem actually 

 to be damped when observations are made for actual ship's hulls, it is indicated that 

 Michell's assumptions are not always the appropriate ones, even though they do lead to 

 fairly reasonable results for the wave resistance. However, two other forms of our 

 theory are available, both of which would lead to damping in the pitching and heaving 

 modes of oscillation. 



Unfortunately, only for ships of Michell's type are there explicit solutions avail- 

 able for the complicated boundary problems in potential theory posed by our basic 

 theory — and only then for the restricted type of motions mentioned above. We have 

 formulated the problems in terms of integral equations over the hull of the ship, but 

 unfortunately it turns out that the integral equations have singular kernels, and are in 

 general of unconventional types so that even their numerical solution with modern 

 computing equipment presents problems of some difficulty. 



R. F. Chisnell 



I wish to describe an unsuccessful attempt to use a Ferranti Mark I computer 

 at Manchester on the problem of wave resistance of non-thin ships. As this problem is 

 still one of the unsolved problems urgently requiring attention, I mention this attempt 

 as firstly, there may well be others contemplating a similar program who would like to 

 hear of the difficulties encountered; and secondly, with the expected arrival in Man- 

 chester of a Ferranti Mark II machine the attack on this problem may be continued, 

 and we would be pleased to hear from people interested in this work. 



A ship of arbitrary hull shape is assumed to move with constant velocity into 

 an undisturbed ideal fluid. The free surface condition is linearized, and may be ex- 

 pressed in terms of a velocity potential function as, 



1 



<$>** + — 4> y = 0, on y = 0, (92) 



where x is the direction of motion, y normal to the plane of the free surface and F the 

 Froude number of the motion. On the hull the normal velocity <£„ must satisfy 



<f> n = F cos (n,x) (93) 



n being normal to the hull. 



It is the existence of these two separate boundary conditions that makes the 

 problem of finding the appropriate solution of Laplace's equation so difficult. 



The method employed was to represent the ship as a distribution of Kelvin 

 sources over its wetted area. A Kelvin source has a source-type singularity and also 

 satisfies the free surface condition ( 1 ) . The second boundary condition leads to an 

 integral equation to be satisfied by the strength of the Kelvin source distribution. 



For numerical work the continuous distribution is replaced by a sufficiently fine 

 Kelvin point source distribution, the integral equation becoming a set of linear equa- 

 tions for the strengths of the point sources. The majority of terms in these equations 

 are velocities due to Kelvin sources at the location of another of the sources. Part of 

 the modification necessary to convert the velocity potential of a source into that of a 

 Kelvin source is a double integration over a semi-infinite rectangle. The integrand is 

 fierce, containing a curve of singularities. One to two hours computing time was en- 

 visaged for each integral and of the order of a hundred integrals. The reliability of the 

 machine, which was the first production model, was not great enough to perform these 

 integrations. The possibility of performing the integration in several smaller stages was 

 considered, but as it would result in an appreciable increase in running time it was de- 

 cided to postpone the work till the arrival of the Mark II machine. 



136 



