1909. ] The Wave-making Resistance of Ships. 295 
Also if w is the group-velocity for wave-length 2z/« and wave-velocity V, 
we have, in this case, 
ad d 3 y 
— dn tN? ae {xv—Kf («)} = v—{f(«) +f (x)}. 
Hence, since in the final value /(«) = 0, we have «f’(«) equal to v—w. Thus 
if « is the wave-length of the regular wave-trains in the rear of the 
disturbance, we find that they are given by 
” = const. x ©) SIN Kz, (26) 
ae 
9 
where v= Je tanh Kh), u=thv (1 +a): 
Hence for the amplitude a we have 
BSOx) (|(1 
~ sinh 2eh 
2eh ) 
Substituting now in (21) we obtain for the wave-making resistance, I 
proportional to 
(6(9)*|(1- ee). 
sinh 2«h 
If we take the same distribution of pressure in the travelling disturbance, 
namely, F(a) = Pa/a(a?+2*), we have (x) = Pe; further, we may 
again assume that the pressure P varies as v, so that we have the resistance 
in the form 
R= Atote-|(1— Zale ) 
sinh 2«h 
: tanhkh — v? 
tl pe are 
with = a (27) 
Considering R given as a function of v by these two equations, we see 
that R increases slowly at first and then rapidly up to a limiting value at 
the critical velocity ,/(gh); after this point R is zero, for there is no value of 
« satisfying the second equation with v/gh>1. 
Further, the limiting value of R at the critical velocity is finite, for we 
have 
272 
: wh 
Lim 
S25 Cos ED eo 
We see that the R, v curve given by (27) is of the type sketched in fig. 9, 
We may compare this with some of the curves given by Scott Russell for 
canal boats. The continuous curve in fig. 10 is an experimental curve of 
53 
