Long Waves in Canals and Standing Waves in Closed Basins 149 



when a is the half width of the canal (origin of co-ordinates in the middle of the canal). We assume 

 a solution of the equation (VI. 15), in which Z, U, V must only be functions of y. If we introduce 

 these values into (VI. 16), these functions must fulfill the following equations (c = ajtt) 



u = g -z, 



c 



g dZ d dZ 

 - — , - [h — 

 c dy dy\ dy 



x 2 h\Z = 0. 



The boundary condition (VI. 17) becomes 



dZ 

 h — = for v=±o 

 dy 



If we consider a canal of parabolical cross-section, then 



(VI. 18) 



(VI. 19) 



(VI. 20) 



in which h is the maximum depth in the middle of the canal. The mean depth of the water is 

 h = f/? . The equations (VI. 18) and (VI. 19) now become 



d 

 ~dy 



X 2 1- 



gh \ 



0, 



and 



y 2 \dZ 



- — J — = (for y = ± a) 

 a 2 dy 



A symmetrical solution can be represented in the form of an infinite series 



(V1.21) 



y* y" 



Z= A Q +A 2 -+...+A n - 

 a 2 a" 



(VI.22) 



(/? even whole numbers). The substitution gives for the coefficients 



and 



2A i +\ — -xW ^o = 



\gh 



(n + 2)(n+ l)A n+2 - («- l)A„+ I — xV }A n +x 2 a 2 A„_ 2 = . 



(VI.23) 



The determination of the coefficients A results in continued fractions similar to those developed 



by Laplace in his theory of tides, and to a relation between a and x, which determines the wave 



velocity c. We then obtain 



3 a 2 / 1 a 2 a 2 \ 



x 2 = 1+ . (VI.24) 



2gh \ 30 ghj 



The average depth of the canal being h = f h we find 



4ti 2 u 2 



= V(gh) • 



1- 



90 



if we put a 



(VI. 25) 



Consequently, in first approximation, the wave velocity \/{gh) which checks with the rule of 

 Kelland according to which the Lagrangian equation applies, when we take the mean depth of the 

 canal as the water depth. If A is the wave amplitude in the middle of the canal, we have 



