178 Long Waves in Canals and Standing Waves in Closed Basins 



of this basin itself. From the equations (VI. 8) and (VI. 6) we obtain, when 

 considering the boundary condition x = and neglecting the time factor e' w \ 



if if 



I, =— — r-sin^A- and m = —lux- T cosxx (VI. 82) 



smxl x sinji/j v J 



with xc x = co. 



The second equation gives for the right side x = l x the second relation 



between If and ??f, in the form 



7th x co\7ck£ + 7$ = . (VI.83) 



The two equations (VI. 80) and (VI.83) can only exist when their de- 

 terminant vanishes, which gives the equation for the frequency co of the 

 connected system. After a few transformations, we obtain 



coUl x =^-(\-^\. (VI. 84) 



a 2 \ co 2 / 



The period T of the system is given if we put w 3 = 2n\T z and T x = 2l x /y (gh x ) 

 (period of I when completely closed at the right): 



«**?» = **. I (l_?3. (VI. 85) 



T c x a 2 T\ Tsl 



This formula applies only on the condition that in the connecting canal 

 there will be compensating currents and no oscillations. This is identical 

 with the condition that the cross-section of the canal be small, compared 

 with S 2 , which is the cross-section of the lake. Equation (VI. 85) can also 

 be applied for irregularly shaped lakes provided that first T x be determined 

 considering the width and depth according to one of the preceding methods, 

 and then T be corrected by equation (VI. 85). 



If xl x is small (l x small compared to the wave length of the total oscillation), 

 we can substitute in the equation (VI. 84) \/(xl x ) for cotxl x and after a few 

 tranformations we obtain 



Ml 



Ag F x +F 3 



This is the equation derived by Honda and his collaborators. Equa- 

 tion (VI. 86) corresponds with the relations which have been derived in 

 hydraulics for the oscillations in U- shaped tubes and for oscillations in 

 a "Wasserschloss", see Forchheimer (1924, p. 344) who also considers 

 the damping through friction. This method can also be applied to Seiches. 



One can derive in the same way the period of a bay connected with a small 

 basin through a narrow canal. If the opening of the bay into the open ocean 

 lies at x = 0, the equation for its period will be 



to ,-.^ = _^I(l_^ > (VI.87) 



IT c x a 2 T \ T s f 



Vlst 



