358 Prof. T. H. Havelock. Wave Resistance: Some 
also the part due to the regular waves in rear of the wave front 
x cos 6+y sin = 0 is given by 
ce) it) > yl 2 
4 ako? A sec? pe-*o/ser + | dy’ | eyed 
= 477? A2Ky? sec? pe-*ofser*?, (15) 
Collecting these results we have, from (11) and (15), 
[2 
R = (4m/gp) Aa? | sect qh e~ 0120" ds (16) 
0 
We may express R in terms of known functions in two convenient forms. 
Tf Wi,m(«) is the confluent hypergeometric function defined, under certain 
conditions, by* 
4 e742 ak 0 Eee Ra eee 
Wi, m(4) = toa. u (1+u/«) e“du, (17) 
and if K,(«) is the Bessel function for whicht 
iKo(@) = (SIP | * ¢7ac03h 4 cosh nap dyp, (18) 
i) 
we find, after some reduction, that 
R = (a? /4gpf?) A2a3e~ 4? Wy, 1 (a) (19) 
= (1/89pf?) A?ake— 2? {Ko(a/2)—(1+1/a) Ki («/2)}, (20) 
where « = 2kf = 2gf/¢. 
In a previous paper,t the same function of velocity, except for the con- 
stant factors, was found for the wave resistance of a submerged sphere; the 
result was given in the form (19), and a graph was drawn to show Rasa 
function of c. The resistance rises to a maximum in the neighbourhood of 
¢ =/(gf), and then falls asymptotically to zero. 
Although there are few tables available for the functions K, in general,’ Ky 
and K, are given in ‘ Funktionentafeln’ (Jahnke u. Emde) under the form of 
(im /2) Ho (ix) and (7/2) Hi (az) respectively. 
5. Reference has been made to the wave resistance of a sphere submerged 
at a depth /, large compared with the radius a ; this was calculated directly 
as the resultant horizontal pressure on the sphere. The connection with the 
present analysis is easily shown. 
In the paper referred to, the approximate solution for a submerged body 
was found directly, following Prof. Lamb’s method for a cylinder. It is con- 
* Whittaker and Watson, ‘ Modern Analysis,’ p. 334. 
+ Grey and Mathews, ‘Bessel Functions,’ p. 90. 
t ‘Roy. Soc. Proc.,’ A, vol. 93, p. 530 (1917). 
(Note by Editor: These functions have now been tabulated in G.N. Watson, 
Theory of Bessel Functions (pub. C.U.P., 1922). 
150 
