284 TIDAL WAVES. [CHAP. VIII 



It is to be noted that the coefficients of stability (c g ) increase with the 

 depth. 



Conversely, if we assume from Fourier's theorem that (i) is a sufficiently 

 general expression for the value of at any instant, the calculation just 

 indicated shews that the coefficients q a are the normal coordinates ; and the 

 frequencies can then be found from the general formula (8) of Art. 165 ; viz. 

 we have 



<r a 



in agreement with (3). 



176. As an example of forced waves we take the case of a 

 uniform longitudinal force 



X=f cos (erf + e) ........................ (5). 



This will illustrate, to a certain extent, the generation of tides in 

 a land-locked sea of small dimensions. Assuming that f varies as 

 cos(o- + e), and omitting the time-factor, the equation (1) becomes 



d^ a* f 



_ z _i __ _:_ 



d^C^~ C 2 ' 



the solution of which is 



(6). 



d" C C 



The terminal conditions give 



E=l, sin^(l-cos<L<)/ ............ (7). 



Hence, unless sin o-l/c = 0, we have D =fjo & . tan <rl/2c, so that 



2/ . ax . or (I x) , . 



* , sin -^- sin ^ . cos (crt + e), 



'* cos 5c 



and t] = , sin - -= -- - . cos (at + e) 



(8)- 



If the period of the disturbing force be large compared with 

 that of the slowest free mode, cr/2c will be small, and the formula 

 for the elevation becomes 



.................. (9), 



Jy 



approximately, exactly as if the water were devoid of inertia. The 



