468 Proceedings of the Royal Society of Edinburgh. [Sess. 
similar features will appear in the case of any medium in which the wave- 
velocity is greater than the group-velocity for each wave-length and in- 
creases with increasing wave-length (V = Ak~ n , n< 1). But it is shown in an 
earlier paper (see Proc. Roy. Soc. Edin., vol. xxix., 1909) that all the curves 
shown for water-waves may be used as illustrations of waves in an elastic 
rod, which is a medium in which the group-velocity exceeds the wave- 
velocity and the wave-velocity diminishes with increasing wave-length. Our 
diagrams therefore illustrate the process of dispersion for two distinct laws 
of dispersion, namely : Y = A-Jc~ h , U = J A x Ar* ; and V = A 2 /c, U = 2A 2 &. For 
the first, each diagram shows the displacement at each point of the medium 
at a given time ; for the second, each diagram shows the displacement at a 
given point of the medium from t == 0 to t = co . It is sufficient to point out 
the distinguishing features of the second case illustrated by the diagrams, 
which may be taken as typical of any medium in which V = A h n , n> 0. 
These are : the continuous increase in wave-length of the disturbance at each 
point of the medium as time goes on, and the continuously increasing impor- 
tance of the rear of any disturbance as compared to the front, owing to the 
individual waves lagging behind the main group and retaining part of its 
energy. One interesting point of difference between the two cases illustrated 
lies in the manner of formation of additional waves. Thus in fig. 34 we 
see from the numbering of the zeros that they are continually being formed 
in pairs near the point x = 1 : the outer zero travels outwards with con- 
stantly increasing speed, and the inner zero travels inwards with constantly 
diminishing speed. From a? = 0 to cc = l we have an ever increasing number 
of waves, each moving inwards with diminishing length and therefore also 
diminishing speed, and from x — 1 to x=oo we have an equal number of 
waves moving outward with increasing length and speed; each wave in the 
entire system moving at each instant with the velocity corresponding to its 
length. Only at t = oo is the number of water-waves infinite. The illustra- 
tions when applied to waves in an elastic rod, however, show that each zero 
is formed at infinity ; and an infinite number are formed in quick succession 
at the commencement. For a short time the waves formed all move inwards 
from infinity towards the part of the medium initially disturbed, lengthening 
and slowing down as they come inwards. Very soon the inward motion 
ceases, and all continue for ever afterwards moving outwards with in- 
creasing length and diminishing speed. At all times the wave-length 
diminishes from the middle outwards, and at each point it increases 
continuously with the time : in both respects exactly the reverse occurs in 
the case of water- waves; In both cases the flow of energy is outwards from 
the place of the original disturbance at all times and places, even when the 
