242 Proceedings of the Royal Society of Edinburgh. [Sess. 
XI. — On Waves in a Dispersive Medium resulting from a Limited 
Initial Disturbance. By George Green, M.A., B.Sc., Assistant to 
the Professor of Natural Philosophy in the University of Glasgow. 
Communicated by Professor A. Gray, F.R.S. 
(MS. received November 11, 1909. Read December 6, 1909.) 
§ 1. In a former paper “ On Group-Velocity and on the Propagation of 
Waves in a Dispersive Medium” ( Proc . R.S.E., xxix. pp. 445-470, 1909), it 
was shown that group-velocity, or the principle of “stationary phase,” 
provides us with a satisfactory explanation of the modus operandi of 
dispersion ; and the principle was applied to obtain an expression for the 
effect of a single impulse confined to the neighbourhood of a point of 
the medium. The present paper is intended to fulfil a promise given in 
§ 29 of that paper, to show that by means of this principle we can arrive 
at the general features of the wave-system in a dispersive medium resulting 
from any limited initial disturbance. 
It is proposed to examine the effect of the same initial disturbance in 
several dispersive media in which the group-velocity is positive, that is, 
always in the same direction as the individual waves travel. Using the 
notation of the paper referred to above, we take V as the velocity of an 
infinite train of waves of wave-length X, and period 2i t/JcY, where 7c = 27 r/X, 
and where V =/(&), since the medium is such that the wave-velocity varies 
with the wave-length. For convenience the investigation is restricted to 
media for which the wave-velocity is given by V = A Jc n . The group-velocity 
U, corresponding to 1c, is then given by equation (16) of the former paper, 
namely, U = (l + 72 /) AAf; so that we have the further restriction, n> — 1 , to 
make this always of the same sign as the wave-velocity. 
§ 2. The wave-system resulting from a given initial displacement has 
already been investigated for several different forms of initial displacement in 
the particular cases where n — — J, which corresponds to deep-water waves, 
and where n — 1, which corresponds to flexural waves in an elastic rod. 
Professor Schuster, in his paper on “ The Propagation of Waves through 
Dispersive Media,” * has also dealt with the cases included in the formula 
Y = a + bk~ 1 , where a and b are constants. The form of initial displacement 
* Boltzmann, Festschrift , 1904. 
