274 



TIDAL WAVES. 



[CHAP, viii 



The interpretation of these results is simple. Take first the 

 motion represented by the first term in (9), alone. Since F(x ct) 

 is unaltered when t and x are increased by r and CT, respectively, 

 it is plain that the disturbance which existed at the point x 

 at time t has been transferred at time t + T to the point x + CT. 

 Hence the disturbance advances unchanged with a constant 

 velocity c in space. In other words we have a ' progressive wave ' 

 travelling with constant velocity c in the direction of ^-positive. 

 In the same way the second term of (9) represents a progressive 

 wave travelling with velocity c in the direction of ^-negative. 

 And it appears, since (9) is the complete solution of (6), that any 

 motion whatever of the fluid, which is subject to the conditions 

 laid down in the preceding Art., may be regarded as made up of 

 waves of these two kinds. 



The velocity (c) of propagation is, by (8), that ' due to ' half 

 the depth of the undisturbed fluid*. 



The following table, giving in round numbers the velocity of 

 wave-propagation for various depths, will be of interest, later, in 

 connection with the theory of the tides. 



The last column gives the time a wave would take to travel 

 over a distance equal to the earth's circumference (2-Tra). In order 

 that a * long ' wave should traverse this distance in 24 hours, the 

 depth would have to be about 14 miles. It must be borne in 

 mind that these numerical results are only applicable to waves 

 satisfying the conditions above postulated. The meaning of these 

 conditions will be examined more particularly in Art. 169. 



* Lagrange, Nouv. m6m. de VAcad. de Berlin, 1781, Oeuvres, t. i. p. 747. 

 t This is probably comparable in order of magnitude with the mean depth of 

 the ocean t 



