288 TIDAL WAVES. [CHAP. VIII 



x increases by 2?ra. It may therefore be expressed, by Fourier s 

 Theorem, in the form 



(2). 



Substituting in (1), with the last term omitted, it is found that P s 

 and Q s must satisfy the equation 



The motion, in any normal mode, is therefore simple-harmonic, of 

 period 2?ra/5C. 



For the forced waves, or tides, we find 



c^H ( x \ 



whence y = - r^cos2 [n fH --- he ............... (5). 



2 c 2 n 2 a 2 V a J 



The tide is therefore semi-diurnal (the lunar day being of course 

 understood), and is direct or inverted, i.e. there is high or low 



water beneath the moon, according as c ria, in other words 



according as the velocity, relative to the earth s surface, of a point 

 which moves so as to be always vertically beneath the moon, is 

 less or greater than that of a free wave. In the actual case of the 

 earth we have 



(?/n *a* = (gln *a) . (h/a) = 311 h/a, 



so that unless the depth of the canal were to greatly exceed such 

 depths as actually occur in the ocean, the tides would be inverted. 



This result, which is sometimes felt as a paradox, comes under 

 a general principle referred to in Art. 165. It is a consequence 

 of the comparative slowness of the free oscillations in an equatorial 

 canal of moderate depth. It appears from the rough numerical 

 table on p. 274 that with a depth of 11250 feet a free wave would 

 take about 30 hours to describe the earth s semi-circumference, 

 whereas the period of the tidal disturbing force is only a little 

 over 12 hours. 



The formula (5) is, in fact, a particular case of Art. 165 (12), for 

 it may be written 



