Ship Resistance. 491 
where gic | 16 ens BOE We | LO sine (6) 
The mean energy per unit area of the wave motion given by (5) is 
2«*(6?+?)/yp. Now the head of the disturbance advances with velocity », 
while the rate of flow of energy in the train of waves is the group velocity 40; 
hence the net rate of gain of energy per unit area is $v times the above 
expression for the energy. If we equate this product to Rv, then R may be 
called the wave-making resistance per unit breadth ; and we have 
R= 2 ($+ V)/9p. (7) 
We have in each case to evaluate the complex integral 
x= Grin =| /Oertae. 8) 
In the examples which follow, the integral has a finite, definite value 
which can be obtained in Cauchy’s manner by integrating round a closed 
simple contour in the plane of the complex variable & The function /(&) 
is such that (i) it has no critical points other than simple poles in the 
semi-infinite plane situated above the real axis for &; (ii) it has no critical 
points on the real axis; and (iii) its value tends to zero as & becomes infinite. 
Further, the quantity « is restricted to real, positive values. Under these 
conditions it can be shown* that 
| 76 ext dE = 2mZA, 
where 2A is the sum of the residues of the integrand at the poles of /(&) 
situated above the real axis. If a is a pole, A is given by the value of 
(E—a)f(&)e"* when E =a. Alternatively, in the following examples /(&) is 
of the form F(&)/G(é), none of the zeros of G(E) coinciding with those of 
F(&), and A is given by F(a)e**/G‘(a). 
3. For the sake of comparison the results which have been obtained 
previously may be repeated briefly. If 
p=fO= wee (9) 
the poles are at €=-+7i«, of which the positive one alone concerns us. 
Hence we have 
~~ Aexé . | Aeixé aT 
xX => | eee => 2770 Ein io => z Ae 6 (10) 
22 2,A2 
Hence from (7) i aa = a e-2ga/v2.t (11) 
* Jordan, ‘Cours d’Analyse,’ vol. 2, § 270. 
Ge Lamb, ‘Hydrodynamics,’ 1932 edn. p. 415. 
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