Long Waves in Canals and Standing Waves in Closed Basins 215 



In order that, for a definite, but still unknown value x = x x the difference between the longitudinal 

 currents of both solutions be always zero, we must have for all values — %n =S v ^ +\n, 



axx 

 sin — cosh ay 

 c 



n = i 



oo 



cos — sinhcry = 7 A n si 



cosny , 



sxnny . 



(VI. 124a) 



With equations (VI. 123) and (VI. 124a) we can determine the constants A„, A' n and x%. Since the 

 rapidly converging series contain an infinite number of coefficients, we may accomplish 

 every desired accuracy during the fulfilment of these conditions. For instance if we choose 

 the five points y = 0, = ±n/4 and = ±n/2, at which (VI. 124a) should be satisfied, we obtain 

 from (VI. 124a) by use of (VI. 123) a number of equations sufficient to determine the unknowns 

 A x , A 2 , A 3 , A x as well as A[, A 2 , A' 3 and x x : 



Ax + A 2 +A 3 + A t 



ax x 



(1) 



-Ao + A. = sin — cosha - , (2) 



c 2 



A 1 -A 3 -l-4l5A i = l-415sin — cosha-, (3) 



c 4 



, , oxx n 



A t —A 3 = cos — sinha - , 



c 2 



, , , oxi n 



Ay\- 1-415 A 2 +A 3 = l-415cos — sinha - , 



c 4 



Ax-yxA'x = 0, 



2 , 

 ,4„_ -A 2 = 0, 



A 3 - -4 = 0. 



(4) 



(5) 

 (6) 

 (7) 



(8) 



(VI. 1246) 



If we combine these equations in the following way, (1 — 2)+ (2+ 3), then (4+ 5)+ (6, 7) and 

 (4+ (6 and 8)+ (1 — 2), they will only have as unknowns A x —A 2 and x x . The elimination of A x 

 and A 2 gives an equation for tang (axjc). This procedure will fix all constants and the place in the 

 canal where the incoming wave is reflected. If k = 0-5 and a = 0-7 which, with a period of 12 h 

 (semi-diurnal tidal period), corresponds to a bay with a rectangular cross-section having a width 

 of 465 km and a uniform depth of 74 m at approximately 53° of latitude, we get, according to the 

 above method, tanipxjc) = 0-383 against 0-385, as found by Taylor (1920) by another, more 

 accurate way. 



If one wishes to know more about oscillations in such a canal, closed at 

 one end, he must compute numerically some special cases. Taylor has 

 done this for a bay having the same dimensions as mentioned in the previous 

 paragraph and which corresponds to the North Sea and computed the distri- 

 bution of the amplitudes and the phases, as well as the currents inside the 



