Long Waves in Canals and Standing Waves in Closed Basins 179 



in which T x = 4/j/j (ghx) represents the period of the free oscillation of the 

 bay closed at one end, and corrected for the opening. 



Neumann has tested his theory by interesting experiments in a wave tank, 

 which, in spite of inherent difficulties have shown a very satisfactory 

 agreement between the theory and the observation. Applications of these 

 formulas are given in the following paragraph. 

 (b) The Impedance Theory of Neumann 



It is very difficult to deal with the theory of combinations of basins such 

 as they occur very often in nature. It, therefore, was an excellent idea of 

 Neumann (1944, p. 65, 193) to apply the impedance of oscillating systems, 

 which was proved to be so valuable in the theory of electrical and acoustic 

 oscillations, also to the oscillations of water-masses. In this manner, it is 

 possible to derive in an elegant way the equations for the period of the various 

 combinations of lakes and bays. It is true that the formulas thus derived 

 apply only for rectangular basins of constant width and depth, but they can 

 also be applied, with small changes, to lakes and bays of irregular shape. 

 We will give here only the fundamentals of the theory; the applications and 

 the proof of its manifold uses will be presented later (p. 186), along with 

 a number of examples based on observations. 



The general periodical motion of water-masses in lakes and bays follows 

 the equation (VI. 36) if the frictional influences are also considered. If we 

 introduce |~e'* v and if we assume that the oscillations are produced by 

 a force X acting in the horizontal direction we obtain as the fundamental 

 equation of the forced oscillations of a system (see Defant, 1916, p. 29) 



in which xc = a is the natural frequency of the system. 



We will now put X = Ae kut , in which to is the frequency of the exciting 

 force, and if we assume 



dh 1 



u = — = Ue iwt , £ = — Ue ia " 

 dt ito 



then from (VI. 88) follows 



n 



U[ia) + p + -)=A. (VI.89) 



1(0 



In analogy with Ohm's law for alternating currents and in accordance 

 with the definition in the theory of acoustical filters, the quantity 



Z = £ = p+Uo + * (VI.90) 



U ico 



is called the impedance of the hydrodynamical oscillatory system. It can be 



12* 



