16 THEORY OF SHIP WAVES AND WAVE RESISTANCE. 
From the definition of wave resistance given in the text, assuming the slope of the 
ey ao) See UE ao ea Pee end aaa 
the integral being taken over the whole surface. ‘The particular case for which the 
surface to be small, we have 
calculations have been made is 
3 
j= IN) BAGG? sk 79) : : : : « (5) 
A and f being constants. It is found that the integral (4) reduces to 
T 
> 
= (47 9? atin | secd pe ACE) EE? dp . 6 0 . (6) 
This integral can be expressed in terms of Bessel functions, of which tables are avail- 
able, in the form 
mT? A2pse Pp.) . ee 
R= EAE {ine 122? im}, 
where p= gf/c?. This is the expression whose ne is given in Fig. 1. (Proc. Roy. 
Soc. 4, 95, p. 354. (1919). 
2.—With the same notation, and with h as the depth of water, instead of (6) we 
now have 
R= 
p g sec” (c?—gh sec? d) + K*cth ° e 2 ‘ p (8) 
Po 
where x satisfies the equation 
7 
47 A2c8 | 2 Ke 2 sec pad 
g 
Ke? = g sec’ & tanh xh. 
The lower limit d, is to be taken zero if c’<gh, and to be the value of are 
cos Vghic? if c’>gh. The integral (8) was evaluated by graphical methods, the 
integrand being graphed on a certain base and areas taken by an Amsler planimeter. 
The process was carried out for the different values of the ratio h/f shown in Fig. 2. 
(Proc. Roy. Soc. A., 100, p. 499. 1922.) 
3.—With h as the distance between the centres of the two pressure systems, the 
integral for the wave resistance is 
Tv 
y 2 
R = (16mg?A*/pc*) | ; sect g e— “(Af lc?) sect cos? | (gh/c?) sec \ hyo o © 
The particuiar case shown in Fig. 3 is for h = 2/, the integral being evaluated by 
numerical methods. (Reference as in Note 1.) 
4.—For a study of some cases, with further references, see Proc. Roy. Soc. A., 89, 
p. 489. 1914. 
5.—The general expression for any distribution of sources and sinks is found by 
beginning with a doublet of given moment at a given depth in the liquid, with its axis 
parallel to Ox. The results are generalized by integration for any continuous dis. 
tribution of such doublets in the plane y = 0, the moment per unit area in this plane 
being (2, z); this gives for the wave resistance the expression 
T 
[) 
0 0 ) i 
R= 16matpe—* | a | “J ue dx} . bW)dz . bY/52'. 
x sco'p e[ IE + HIE | #2 ® cog oa | | gl - xry/ct { seo |» 5 (00) 
262 
