1905-6.] Lord Kelvin on the Growth of a Train of Waves. 427 
certainly still more nearly zero throughout the first period, though 
we have no curve to represent it, we see that the negatives of the 
tangents of the slopes in the curves for S^(8 , 1 , t) and S^(8 , 1 , t) re- 
present very nearly the values of the applied surface-pressures during 
the whole of the first period.* Look now to fig. 33 ; see how near 
to zero is if/(8 ,1,0), and how far from zero is <£(8 ,1,0); and 
we see dynamically how it is that S^(8, 1 , t) is very nearly zero 
throughout the first period, and S<^(8 , 1 , t) is very far from zero, 
and is somewhat near to being sinusoidal. 
§ 141. We have also a very instructive comparison between 
£$(8 , 1 , t) and S^(8 , 1 , t). In the </> case, for values of x as 
large as 8, or larger, we approach somewhat nearly to the case of 
a sinusoidally varying uniform surface-pressure over an infinite 
plane area of water, in which there would he no surface displace- 
ment, and the pressure at and below the surface would be at 
every instant equal to the applied surface-pressure plus the 
gravitational augmentation of pressure below the surface. Thus 
we see why it is that, with a great periodic variation of applied 
surface-pressure, at a; =8, there is scarcely any rise and fall of 
the surface level there, until after a period and a half from the 
beginning of the motion, as shown in the curves for £^(8,1, t). 
§ 142. The second, fourth, and sixth, curves of fig. 37 represent 
the arrival of three classes of disturbance, , £^ , S^, , at x= 32, 
four wave-lengths from the origin. If the front of the disturbance 
travelled at exactly the wave-velocity, the disturbances of the 
different kinds would all commence suddenly at the end of period 
4. In the cases of S^,(32 , 1 , £) and £</>(32 , 1 , t) the diagram 
shows that they are quite imperceptible at the end of period 4, 
and begin to he considerable at the end of period 8, which would 
be the exact time of arrival if there was a definite “ group- 
velocity ” equal to half the wave-velocity. The largeness of 
S^,(32 , 1 , t), approximately uniform throughout the first four 
periods, is explained in § 141. Its gradual augmentation through 
periods 5, 6, 7, 8, depends on the wave propagation of disturb- 
ances from the origin, as shown for S^(32 , 1 , t) and £^(32,1 ,t) 
in the second and fourth curves. 
* Remember that downward ordinates in all the curves of figs. 36, 37, 38, 
39, correspond to positive values of the quantities represented. 
