449 
1908-9.] On Group- Velocity and Propagation of Waves. 
clear that we cannot again have agreement in phase of all the trains at 
any single point. But, on the other hand, it can be shown that if at any 
time we have an infinite number of trains agreeing in phase at any point, 
and these trains move at nearly the same velocity, there will always be a 
point at which an infinite number of these trains, though not the whole 
number, will continue to agree in phase at a different phase. For 
example, taking into consideration only trains whose wave-lengths and 
wave-velocities are nearly equal, the space separating points of equal phase 
on any two of the trains increases continuously as we pass away on either 
side from the point where all the trains are in agreement of phase. Con- 
sequently, on one side of the point of agreement of phase all points on the 
trains having any specified phase must be approaching each other, while 
on the other side all points of equal phase are moving more and more 
apart. This is illustrated in the accompanying diagram, which shows two 
trains T 1 T 1 and T 2 T 2 agreeing in phase at phase zero at point 0. Each 
train moves rightwards at its own wave-velocity. If the wave-velocity for 
the trains gradually increases with increasing wave-length, then all points 
of equal phase on the left-hand side of 0 come in turn into coincidence as 
time goes on ; and if the wave-velocity for the trains gradually diminishes 
with increasing wave-length, then all points of equal phase on the right- 
hand side of O come in turn into coincidence, and then continue to move 
farther and farther apart forever. It follows, therefore, that for the infinite 
number of trains which we are considering, all nearly of the same velocity 
and wave-length, there will always be a point at which an infinite number 
of them will agree in phase, provided one such point exists initially. In 
general, the phase at which the agreement occurs, and the point of the 
medium at which the agreement occurs, alter continuously with the time. 
§ 9. Now when an infinite number of trains of nearly equal velocity 
are in agreement of phase at any point, the sum of their effects must 
determine very approximately the displacement of the medium at that 
point. For the remaining trains are infinite in number and of all possible 
phases, and we shall therefore assume for the present that their effects 
counterbalance each other, as is the case initially with all the trains at 
vol. xxix. 29 
