174-175] DISTURBING FORCES. 283 



The terminal conditions give B = 0, and 



a-ljc = STT (3), 



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



given by 



A . STTX fsirct \ /yl v 



f = 4,sm-j-cosf -j- + e s j (4), 



where the amplitude A s 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 = J I). The period (2Z/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 , J, J, 

 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 q 2 , ... such that 

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



. STTX 



3,001 -j-; 



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

 capable (subject to the conditions presupposed) is given by 



where q lt q z , ... 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 



(ii), 



and 2F=#p^ [* 



* J o 



where a a = i 



