251 
1909-10.] On Waves in a Dispersive Medium. 
which shows that the amplitudes fall off according to x~ h , or according to 
where U is the group- velocity corresponding to the wave-length 
considered. From (32) we can obtain the relation between energy and 
wave-length at any fixed place of observation. 
§ 14. We can now trace the general course of the disturbance in any 
medium at a chosen place of observation, x ; and we can also obtain an idea 
of the state of the disturbance throughout the medium at any particular time, 
t. It is convenient for this purpose to distinguish between media for which n 
is positive and those for which n is negative. In the case of n positive, the 
wave-velocity is greater the shorter the wave-length, and the group-velocity 
for any chosen wave-length exceeds the corresponding wave- velocity ; in 
the case of n negative, the wave- velocity is greater the greater the wave- 
length, and the group-velocity is less than the corresponding wave-velocity. 
In both cases the general course of the disturbance at a very distant point 
x is evident from the following statements. When t is very small, we see 
