1908. | Groups of Waves in Dispersive Media, ete. 404 
If f’ (uo) is positive, 
Bo = 4 Girt } (0m) 008 {7 (Ho) — In} 5 a7) 
and if f’’ (a) is negative, 
ti 2ar 4 
| ap = { =) $ (wo) cos { (to) +477}. (18) 
The chief form in which such integrals occur is 
y= | $ (x) cos x (w—Vt) dx, where V =f(c). (19) 
The principal groups are given by the values x such that 
d @  @ ne 
aE {x(a—Vt)} =0, or ae («V) = U. (20) 
The aggregate value of the group can be written down from one of the 
previous forms ; if OU/0« is negative, we should have 
Yo = {= a0 [oe ie co) (Ko) cos {ko (a— —Vt) +47}. (21) 
As an illustrative example we may suppose a disturbance y to be given at 
time ¢ by the expression* 
y= | cos « (e— V#) de. (22) 
0 
When z—V¢ is large, the elementary waves given by (22) reinforce each 
other only for the simple groups given by values «) for which the argument 
of the cosine is stationary, so that 
x—Ut = 0. (23) 
This equation (23) defines a velocity U such that to an observer starting 
from the origin and travelling with this velocity the complex disturbance has 
the appearance of simple waves of length 27/. Or again, we may regard 
(23) as giving the predominant value of x at any position and time in terms 
of zand¢. The features of the disturbance will depend on the form of the 
velocity function V ; we proceed to consider some special forms, 
§5. Initial Line Displacement on Deep Water. 
We consider surface waves on an unlimited sheet of deep water, the only 
bodily forces being those due to gravity. Let the z-axis be in the undisturbed 
horizontal surface, and the y-axis be drawn vertically upwards. Let 7 be the 
elevation of surface waves of small amplitude with parallel crests and troughs 
perpendicular to the zy-plane. It can be shown that for an initial displace- 
* Lord Kelvin, loc. cit. ante. 
