390 



at last break on some shore situated at the distance of per 

 haps hundreds of miles from the region where they were 

 first excited.&quot; 



When the wave is very long it is easy to give a short 

 demonstration of its most useful properties. And this is 

 the more advantageous, for, as we shall presently see, the 

 tide is nothing more than an exceedingly long wave. 



Suppose we have a rectilinear channel of uniform section ; 

 let y be the elevation of the water above the mean level at a 

 time t, and at a distance x measured along the channel from 

 the origin of measurement. Let k be the greatest value of 

 y. Let h be the depth of the channel, K the length of the 

 wave travelling along the canal. Then by hypothesis A is 

 very great. Suppose the particle of water which when 

 undisturbed to have been at a distance x from the origin, 

 to be at the distance x + X at the time t. First let there 

 be two imaginary planes placed at the beginning and end 

 ing of the long wave. The mass * of water elevated above 

 the mean level will be comparable with xk, while the water 

 which has passed the two imaginary planes will be com 

 parable with h X. These two quantities, then, must be of 

 the same order ; hence 



7 ~\r 



k is of the order ; 



A 



and, as \ is very great, this must be very much smaller than 

 X. Now X is itself supposed a small quantity ; hence k 

 may be altogether rejected when compared with X, and 

 we must suppose the elevations performed by the horizontal 

 motions of the fluid pressing together some parts, and thus 

 raising it above the general level. So far as the vertical 

 motion is concerned, we may consider the fluid in equili 

 brium, and hence may use the ordinary equations of hydro 

 statics to find the pressure at any depth. 



We shall now show that if we assume the displacement 



* Camb. Dub. Mathematical Journal, Nov. 1849. 



