245 
1909-10.] On Waves in a Dispersive Medium. 
§ 5. Going back to equation (2), we assume that it is permissible to 
reverse the order of the integrations, in which case the equation may be 
written in either of the forms : 
+ oo 
oo 
■ 
l jdk cos k(x - aq) cos kY t 
J 
-oo 
0 
+ oo 
oo 
- dx 1 f(x 1 ) / <ift{cos k(x — x l -Yt) + cos k(x - x 1 + Vtf)} 
-00 
0 
( 10 ). 
In either of these forms the first integration gives the effect at point x, at 
time t, of a single initial disturbance confined to the neighbourhood of a 
point x v and the second integration then gives the sum of the effects of 
all the initial disturbances at all points of the medium, which together 
comprise the total initial disturbance represented by f(x). From the 
second of equations (10), we can write down the effect of a single initial 
impulse confined to the neighbourhood of the origin in the form : 
t — 
c — 
1 °° 
J dk{ cos k(x 
Yt) + cos k(x + V^)} 
( 11 ), 
of which the right-hand member is the integral evaluated by Lord Kelvin 
in his paper of 1887,* and used in my former paper referred to in § 1 above. 
From equation (11) we see that half the total number of constituent wave- 
trains move in the positive direction and half in the negative ; and it is 
clear from the argument adopted in my former paper that the displacement 
at any point whose x is positive is due to positively moving trains alone, 
and the displacement at any point whose x is negative is due entirely to 
negatively moving trains. Equation (1) of my former paper should in fact 
be replaced by equation (11) above, as it represents a single impulse at the 
d£ 
origin, with ^ — 0 when t = 0, only under the restriction V —f(k) = — /( — k). 
The principle underlying the evaluation given by Lord Kelvin applies to a 
single impulse with or without this restriction, giving, as it does, the value 
of £ in 
00 
£=4 p»osft(a; + Y#) (12), 
27 r! 
o 
or in the corresponding equation with sin instead of cos, where the negative 
sign is taken for the positively moving trains (x positive) and the positive 
sign is taken for the negatively moving trains (x negative). 
* Phil. Mag., vol. xxiii., March 1887. 
