Long Waves in Canals and Standing Waves in Closed Basins 175 



makes an angle nn with the centre axis of the canal (see Fig. 74) equations 

 (VI. 75) is replaced by 



in which 



r i— a— o" 



A. = »J £ , ' dt. (VI. 77) 







The influence of the shape of the opening proves to be very great. Thus we 

 find for 



According to Rayleigh's 

 equation 

 n = 1/4 1/2 3/4 1/2 



rm = 45° 90° 135° 90° 



Correction factor (1+e) = 1285 1-259 1233 1265. 



For a canal of the length / open on both ends we obtain again as period 

 of the longest oscillation T = 21/]/ (gh) ; nodal lines are present at both ends 

 and, except these, this oscillation has no further nodal lines. 



The longest possible period of a standing oscillation is therefore the same 

 as the period of a lake of similar shape, but in a lake antinodes are located 

 at the ends of a lake, whereas in a canal open on both ends, nodes are found 

 at the ends. 



The computation of the period of the free oscillation of ocean bays and 

 canals (straits) of irregular shape can be made after the methods explained 

 previously; particularly well suited for this is the method of a step by step 

 approximation, because it can also be applied in the case of very complicated 

 bottom configuration. For ocean bays the computation is started at the inner 

 end with an arbitrary value of rj and £ = and, if the period is selected 

 correctly, we must obtain ry = at the opening. However, we can also start 

 at the opening with £ arbitrary and r\ = 0, and we then obtain, if T is cor- 

 rectly selected, I = at the inner end. 



3. The Character of the Oscillation of Connected Systems 



(a) Free Oscillations of Connected Systems 



The determination of the period of the free oscillation of the lakes and 

 ocean bays with a complicated- configuration becomes difficult with the 

 methods previously mentioned, especially if the cross-sections become so 

 narrow that the whole area of oscillation can no longer be adequately re- 

 presented by one and the same oscillatory function of time and place. In such 

 cases, the total oscillatory system is split up into separate parts, which are 

 considered as separate areas for theoretical oscillatory problems. Together 

 they form the total oscillatory system. The following example will illustrate 



