1908. | Groups of Waves in Dispersive Media, ete. 402 
where A, B are constants which need not be specified further in the present 
connection. 
These integrals are of the type (6), and represent infinite wave-trains 
travelling in the positive and negative directions respectively. We see from 
(8) that when pw is small, the amplitude factor $(«) consists practically of a 
single well-defined peak in the neighbourhood of the value «’. Hence, when 
the damping coefficient ~ is small, the wave-trains in question may be 
considered as travelling in the form of a group «’ of unchanging waves of 
this specified structure. 
This example serves to illustrate the propagation of a very long train of 
simple harmonic waves subsiding as they travel owing to a small damping 
coefficient, and is of interest in connection with Lord Rayleigh’s general 
proof that the group-velocity U is the velocity with which energy is being 
propagated.* A small damping coefficient ~ is introduced by him, so that 
the energy transmitted is determined by the energy dissipated; the argument, 
which of course loses its meaning if w is actually zero, shows that when yp is 
diminished indefinitely the rate of transmission of energy approaches U as a 
limiting value. Similarly, although the Fourier transformation is inapplicable 
when yp is actually zero, we infer from the above analysis that when yu is 
diminished indefinitely, the disturbance is representable as a simple group of 
unchanging waves of definite structure. 
(b) Interrupted simple wave-train—Consider an initial disturbance 
defined by 
i (@) = (-—d<a<d) 
= ¢#* sin «’ (2—d), (a > d), 
= —cH sink’ (z@+d), (a < —d). 
Then the disturbance is given by an expression of the form (9), in which 
foe) 
26 (x) = af e * sin k'(w —d) cos kw dw 
Be (e+x’) cos K d=pam wd (K’—k) cos cs app Sa K a (10) 
pe+(K+ Ke) pet (« —K) 
Now suppose » and d very sinall, so that the initial disturbance approxi- 
mates to an infinite simple harmonic form with a narrow range of 
discontinuity ; we see that the graph of the amplitude factor ¢ («) is then 
reduced to a single peak in the vicinity of the value x’. We infer from this 
example that a very long simple harmonic wave-train which is interrupted 
for a short interval is kinematically equivalent to a group of unchanging 
waves, of definite structure ranging round the value 27/x’ of the wave- 
length. 
* Lord Rayleigh, ‘ Proc. Lond. Math. Soc.,’ vol. 9, p. 24 (1877). 
