413 Dr. T. H. Havelock. The Propagation of [Aug. 26, 
The same method can be used for waves on water of depth h, due to a 
travelling impulse system. For in the integrals (43) we should have 
fO=eN ace , (2 tanh Kh). (47) 
K 
The group with respect to « would give a term proportional to 
cos {uwx?f" (x) +47}, (48) 
where « has the value given by 
—a/u=f(k)+Kf («). (49) 
We then select the group with respect to u by 
d / 
Fh uxf’ («)} = 0. (50) 
Using (49) we find this leads to* 
= tanh xh 
Fie) =0, or Vae= (gh) 4/(2E) (51) 
Since tanh «h/«xh diminishes continually from 1 to 0 as xh increases from 
0 to 20, there is only a real solution of (51) when c? is less than gh. In this 
case we have regular waves of length suitable to velocity ¢ following in the 
rear of the impulse; when c is greater than the maximum wave-velocity there 
is no recular wave form. 
§ 10. Capillary Surface Waves. 
In order to illustrate the propagation of an element of the Fourier 
expression as a limited travelling group of undulations, we consider another 
form of velocity function. If waves are propagated over the surface of a 
liquid of density p under the action of the surface tension T, it can be shown 
that the velocity of simple waves of length 27/« is 
V = /(Te/p). (52) 
Hence in this case the group-velocity is 
U = $V/(Te/p) = $V; 
thus the group-velocity is greater than the wave-velocity, and we shall see 
how this affects some of the previous results. 
(a) Initial elevation consisting of (2n+ 4) simple oscillations of wave-length 
27 /«’'—If we consider the same problem as in § 7 we have 
AN ( (et me cs (e— Vt) de. (53) 
12 
0 K 
* Of. Lord Rayleigh, ‘ Phil. Mag.,’ vol. 10, p. 407 (1905). 
16 
