One can clearly satisfy (2) by a distribution of sources over S with density 

 —c£ x /2tt. However, (1), (3), and (4) must also be satisfied. This can be done if 

 one can find a function H{x, y, z; £, -q, £) = r^ 1 + h, r- — (x— £) 2 -f (y—rj) 2 

 + (z— £) 2 , where H satisfies (1), (3), and (4) and h is harmonic for y < 0. This 

 function is occasionally called a Havelock source of strength — 1. A distribution of 

 Havelock sources over S with density — c^ x /2tt will satisfy all conditions. 



The determination of H may be made in several ways. Timman and Vossers 

 [1, 2] use double Fourier transforms, and indeed most methods seem to use them in 

 some way. Perhaps the greatest difference in methods comes from the way in which 

 (3) is satisfied. Havelock [6] and Sretenskii [3] introduce a "fictitious viscosity", a 

 device originally used by Rayleigh [v. Lamb, Hydrodynamics, p 399]. Although Lamb's 

 statement [loc. cit.], "This law of friction does not profess to be altogether a natural 

 one; but it serves to represent in a rough way the effect of dissipative forces", is mis- 

 leading in that it seems to imply that the fictitious viscosity has some connection with 

 a physical force, its use does require one to interpret certain improper integrals pro- 

 perly, so as to obtain the desired solution. T. Y. Wu and C. R. DePrima in recent 

 unpublished work have shown why this gives the correct solution, and indeed give a 

 physical interpretation to the fictitious viscosity. Timman and Vossers [loc. cit.] and 

 Kochin [1] confront (3) directly. Havelock in several papers [e.g., 11] has derived 

 the function H by letting the source start from rest at time t — and move with 

 constant velocity; he then finds the limit as f ^ oo. This method has the virtue that 

 (3) is automatically satisfied. In any case, one obtains finally the result: 



1 1 f*'t 



H(x,y,z£,ri,£) = \v J e Hy+,,) ^ sin [v(x - £) sec 6] 



r r' Jo 



cos j>0 - f ) sin fl sec 2 0] sec20 dd (33) 



4„ f*/2 /-co eM (2/+l) co g [ M ( x _ |) CO g Q] cos [^ ( 2 _ £) s i n ff] 



dd I dfx, 



7T Jo Jq fl COS" — V 



where 



(34) 



r 2 = (x - |) 2 +(y- v y +{z- f) 2 , r' 2 = (x - |) 2 + (y + v y 4- (z - f) 2 , 



v = g/c 2 , 



and the principal value is to be taken in the second integral. The velocity potential is 

 then given by: 



<p(x,y,z) = J J — { x (£,ri)H(x,y,z,%ri,0)d£dri. (35) 



In substituting in the formula for the wave resistance, with <p t — — c<p x , only the 

 first integral in the expression for H leads to a nonzero term. It gives the well-known 

 Michell's integral: 



(36) 

 4<? 2 p ff ff C^~ 



R = / / dxdy I / dZdtf x (x,y){ x (£,v) I d0sec 3 0e" ( ^+" )sec!( ' cos [v(x - £) sec 6]. 



* c ~ J J J J Jo 



s s 



o o 



For convenience let us denote the last integral with respect to 8 by K(v(x — £), 

 v(y + r))). The function K occurs in the literature in various forms. For example, 

 letting A = sec 6, one gets 



/•« X 2 



K(x,y) = / d\ — e*y cos \x. (37) 



J x V X 2 - 1 



114 



