Lord Kelvin on the Groirtli of a Train of Waves. 27 



complete history o£ velocity-potential and of surface dis- 

 placement through all time from the beginning of application 

 of pressure to the surface. The very approximately accurate 

 sinnsoidality of each of these three curves through periods 

 6, 7. 8, shows that the continuation through endless time is 

 in each case sinusoidal. 



In remarkable contrast with the initial agreement between 

 S*(0, 1, and S (O, 1, 0. io whicn we alluded in § 139, 

 we find very instructively a remarkable contrast between 

 8^(8, 1, t) and Scf,(8, 1, t) throughout the whole of the first 

 period. Remembering that in a liquid of unit density the 

 pressure is equal to minus the rate of augmentation of the 

 velocity-potential per unit of time, and remarking that the 

 displacement ^(0, 1, t) is, as is shown in its curve, very 

 nearly zero throughout the first period, and that 5//(0, 1, t) 

 is 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) represent 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 

 ^(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<a(8, 1, f) is very far from 

 zero, and is somewhat near to being sinusoidal. 



§ 141. We have also a very instructive comparison between 

 5^(8, 1, f) and 8^(8, 1, t). In the cf> case, for values of as 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 be no 

 surface displacement, 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 #=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). 



§ 112. The second, fourth, and sixth, curves of fig. 87 

 represent the arrival of three classes of disturbance, S^, J^, 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 I. In the cases of 8-4/(32, 1, t) 



* Remember that downward ordinates in all tlie curves of figs. 36, 37 

 38, 39, correspond to positive values of the quantities represented. 



