412 Proceedings of Royal Society of Edinburgh. [sess. 
Look now to F, F, F, in the three diagrams of fig. 35, and /, /, 
in diagrams 2 and 3. In diagram 1, F marks a perceptible front 
for the rightward travelling component group. In diagrams 2 and 
3, F, / show ideal points travelling rightwards from it at speeds 
respectively, the half wave-velocity, and the wave-velocity. We 
see a manifest wave-disturbance far in advance of F, F ; and very 
small but still perceptible wave-disturbance in front of/,/. Thus 
the perceptible front travels at speed actually higher than the 
wave-velocity, and this perceptible front becomes more and more 
important relatively to the whole group with the advance of time, 
as we may judge from fig. 9 of § 20 above. 
§ 117. It is interesting to see by these diagrams how nearly the 
hypothetical group-velocity is found in the rears : while the fronts 
advance with much greater and with ever-increasing velocity. 
The more elaborate calculations and graphical constructions of 
§§ 20-29 above led to corresponding conclusions in respect to the 
front and rear of a procession, given initially as an infinitely great 
number of regular sinusoidal waves travelling in one direction. 
The diagrams, figs. 9 and 10, showed respectively, at twenty-five 
periods after a sinusoidal commencement, a front extending 
forward indefinitely, and a perceptible rear lagging scarcely two 
wave-lengths behind a point, travelling from the initial position 
of the rear at a speed of half the wave-velocity. 
(3) The Initiation and Continued Growth of a Train of 
Two-Dimensional Waves due to the Sudden Commence- 
ment of a Stationary, Sinusoidally Varying, Surface- 
Pressure. §§118-158. 
§ 118. A forcive consisting of a finite sinusoidally varying 
pressure is applied, and kept through all time applied, to the 
surface of the water within a finite practically limited space on 
each side of the middle line of the disturbance. In the beginning 
the water was everywhere at rest and its surface horizontal. The 
problem solved is, to find the elevation or depression of the water 
at any distance from the mid-line of the working forcive^ and at 
any time after the forcive began to act. 
