284 TIDAL WAVES. [CHAP. VIII 



It is to be noted that the coefficients of stability (c 8 ) 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 



ov, = (c./a.)* = 

 in agreement with (3). 



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

 uniform longitudinal force 



X=fcos(o-t + 6) ........................ (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(cr + e), and omitting the time-factor, the equation (1) becomes 



^ + -% = _/ 

 dx^ c 2 * c 2 



the solution of which is 



f T. . orx T-. (TX 

 = -*-+DSUL + #cos ................. (6). 



(7 2 C C 



The terminal conditions give 



Hence, unless sin crlfc = 0, we have D =f/& 2 . tan cr//2c, so that 

 2/ ,.: aso ;_ &amp;lt;r(l a) 



&amp;lt;7* 



and 



hf o-(2a:-0 ...... 



i] = . sin - -^ - - . cos (crt + e) 



If the period of the disturbing force be large compared with 

 that of the slowest free mode, &amp;lt;rl/2c will be small, and the formula 

 for the elevation becomes 



(9), 

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



