180 Long Waves in Canals and Standing Waves in Closed Basins 



shown that for this quantity the following equation can be used as its 

 definition : 



Z= Po = amplitude of pressure 



S(d£/dr) max area x amplitude of velocity ^ ' ' 



If this quantity is known, the natural frequencies co of the system can be 

 computed from Z = 0, if frictional influences are disregarded (otherwise 

 from Z = minimum). 



The correctness of this argument can be proved from (VI. 90). Without 

 friction /? = 0, Z = only when co = a, which means the system oscillates 

 with its natural frequency. 



In a canal closed on one end (head of the canal x = 0, opening x — — /, depth h, width b) 

 there follows from (VI. 8) 



ax . [d$\ a I 



£ = £ sin — e iat and I — ) = — /Vr£ sin — (for x = — /) . 



From (VI. 91) follows 



and as, according to (VI. 7), 



with S = bh 



= gg*7m ax 



a ox 



r\ = — /?-| cos — 



ipc jI 



Z=-^cotg-. (VI. 92) 



5 c 



4/ 

 gives as the period of the free oscillation the well-known formula T = 



V(Qh) 



Neumann has computed for the various parts of an oscillatory system 

 their impedance Z. The most important cases are (using the customary 

 designations, q = bh cross-sectional surface of a connecting canal, 5 = bh 

 that of a lake, q density of water and X wavelength: 



(1) Basin closed on one end 



Z = -^cot^ (VI. 93) 



with / = \X. For a canal closed at both ends the formula remains the same, 

 but we have / = J A. 



(2) Basin open at both ends 



Z = ^tan a/ (VI.94) 



o c 



with I = \l. 



(3) Narrow flow-off opening or narrow canal (Cross-section q, length /' *) 



Z = ^ . (VI. 95) 



* An effective length /' = I+a can be introduced in lieu of the geometrical length / to take 

 into consideration the co-oscillation of water-masses of the open ocean near the entrance. 



