300 TIDAL WAVES. [CHAP. VIII 



the solution of which, consistent with (14), is 



rj = a cos or (*--), M = coso-U--) ................... (ii). 



\ GJ C \ C/ 



For a second approximation we substitute these values of ?/ and u in (1) and 



(3), and obtain 



du drt 



dn 7 du go-a 2 . ... / x\ 



:r = ~ h -j- ~ *-*- sm 2o &quot; ( * ~ ~ ) 

 dt dx c 2 \ C J 



(iii). 



Integrating these by the usual methods, we find, as the solution consistent 

 with (14), 



= a cos 0- 



/ x\ 

 U--J 



. a 

 --3- x sin 2o- 



A #\ \ 

 It ) , 



x sin 2o- 



H) 



.(iv). 



The figure shews, with, of course, exaggerated amplitude, the profile of 

 the waves in a particular case, as determined by the first of these equations. 

 It is to be rioted that if we fix our attention on a particular point of the canal, 

 the rise and fall of the water do not take place symmetrically, the fall 

 occupying a longer time than the rise. 



When analysed, as in (iv), into a series of simple -harmonic functions of 

 the time, the expression for the elevation of the water at any particular place 

 (#) consists of two terms, of which the second represents an over-tide, or 

 tide of the second order, being proportional to a 2 ; its frequency is double 

 that of the primary disturbance (14). If we were to continue the approxi 

 mation we should obtain tides of higher orders, whose frequencies are 

 3, 4, ... times that of the primary. 



If, in place of (14), the disturbance at the mouth of the canal were given 



by 



= a cos &amp;lt;rt + a cos (a- t + e), 



it is easily seen that in the second approximation we should obtain tides of 

 periods 2w/(&amp;lt;r + &amp;lt;r ) and 2ir/(cr &amp;lt;r ); these are called compound tides. They 

 are closely analogous to the combination-tones in Acoustics which were 

 first investigated by von Helmholtz*. 



* &quot;Ueber Combinationstone,&quot; Berl. Monatsber., May 22, 1856, Ges. Abh., t. i., 

 p. 256, and &quot; Theorie der Luftschwingungen in Kohren mit offenen Enden,&quot; 

 Crelle, t. Ivii., p. 14 (1859), Ges. Abh., t. i., p. 318. 



