214 Long Waves in Canals and Standing Waves in Closed Basins 



For i\ the sum only refers to the even integer numbers n, for v 2 only to the odd numbers. 

 Furthermore, s% = n~—k 2 (see p. 210). In order that the e-power remains real and the disturbance 

 makes itself felt only on a finite portion of the canal, « 2 > k 2 and, as the smallest number of n is 1, 

 k z < 1. This leads to the conditional equations (VI. 115) and (VI. 11 6). To the transverse currents 

 belong longitudinal currents in the form 



Mx = 2j A n e SnX cosny and w 2 = 2j A' n e s " x tinny. 



(VI. 121) 



The constants A„ and A' n can be expressed by the constants C„ and C' n by means of the differential 

 equations (VI. 122), which follow from the equations (VI. 101) by the elimination of r\: 



dy dx o \dx dy 



= , and 



du % 



Ty 



8v 2 

 dx 



a \8x dy 



— hr^+_- =0. 



(VI. 122) 



If we introduce the equations (VI. 120) and (VI. 121) into these equations and if we consider 

 that the coefficients of sinwy and cosHy must vanish, we obtain for the ratios A n : A' n the following 

 values 



A, 



for n even 



A' 



n 



V>n 

 Wn 



n 



fa 



in which v'„ = 



s„c 2 



and for n odd — = 

 A' 



(VI. 123) 



Fig. 91. Total reflection of a Kelvin wave at the closed end of a rectangular bay on a ro- 

 tating earth. Co-tidal lines and amplitudes (Taylor). 

 Fig. 92. Current diagrams corresponding to Fig. 91. 



