174-175] DISTURBING FORCES. 283 



The terminal conditions give .5 = 0, and 



<rl/c = S7r .............................. (3), 



where s is integral. Hence the normal mode of order s is 

 given by 



f. . . S7TX fSTTCt \ /A . 



f = ^sm-j-cos( j-+e*J .................. (4), 



where the amplitude A 8 and epoch e 8 are arbitrary. 



In the slowest oscillation (s = 1), the water sways to and fro, 

 heaping itself up alternately at the two ends, and there is a node 

 at the middle (x %l). The period (2l/c) is equal to the time a 

 progressive wave would take to traverse twice the length of the 

 canal. 



The periods of the higher modes are respectively ,,,... 

 of this, but it must be remembered, in this and in other similar 

 problems, that our theory ceases to be applicable when the length 

 l/s of a semi-undulation becomes comparable with the depth h. 



On comparison with the general theory of Art. 165, it appears that the 

 normal coordinates of the present system are quantities q lt <? 2 , ... such that 

 when the system is displaced according to any one of them, say q 8 , we have 



and we infer that the most general displacement of which the system is 

 capable (subject to the conditions presupposed) is given by 



(i), 



where q lt <? 2 , ... are arbitrary. This is in accordance with Fourier's Theorem. 



When expressed in terms of the normal velocities and the normal co- 

 ordinates, the expressions for T and V must reduce to sums of squares. 

 This is easily verified, in the present case, from the formula (i). Thus if S 

 denote the sectional area of the canal, we find 



and *P^ 



where a, = $ P Sl t c 9 *=&ir*gphSll ........................ (iv). 



