286 TIDAL WAVES. [CHAP. VIII 



Another very simple case of forced oscillations, of some interest 

 in connection with tidal theory, is that of a canal closed at one end 

 and communicating at the other with an open sea in which a 

 periodic oscillation 



rj = a cos (<rt + e) ....................... (11) 



is maintained. If the origin be taken at the closed end, the 

 solution is obviously 



(12), 



I denoting the length. If crljc be small the tide has sensibly the 

 same amplitude at all points of the canal. For particular values 

 of I, (determined by cos al/c = 0), the solution fails through the 

 amplitude becoming infinite. 



Canal Theory of the Tides. 



177. The theory of forced oscillations in canals, or on open 

 sheets of water, owes most of its interest to its bearing on the 

 phenomena of the tides. The 'canal theory/ in particular, has 

 been treated very fully by Airy*. We will consider one or two of 

 the more interesting problems. 



The calculation of the disturbing effect of a distant body on 

 the waters of the ocean is placed for convenience in an Appendix 

 at the end of this Chapter. It appears that the disturbing effect 

 of the moon, for example, at a point P of the earth's surface, may 

 be represented by a potential O whose approximate value is 



(1), 



where M denotes the mass of the moon, D its distance from 

 the earth's centre, a the earth's radius, 7 the ' constant of gravi- 

 tation,' and ^ the moon's zenith distance at the place P. This 

 gives a horizontal acceleration dfl/ad&, or 



/sin 2^ ................................. (2), 



* Encycl. Metrop., "Tides and Waves," Section vi. (1845). Several of the 

 leading features of the theory had been made out, by very simple methods, by 

 Young in 1813 and 1823 ; Miscellaneous Works, t. ii. pp. 262, 291, 



