f=o , CD 



where E is the siam of the potential and kinetic energies excluding 

 thermal energy and the kinetic energy of turbulent eddies. The 

 expression for E contains y which is then determined to satisfy 

 equation (1). In the computations of potential and kinetic energies, 

 any components which are constant with time, such as the potential 

 energy of balanced weights or the kinetic energy of rotating parts, 

 can be included in undetermined constants and therefore can be dis- 

 regarded. The potential and kinetic energies are determined from 

 the mass, elevation, and speed of solid members, and by the inte- 

 grals over the entire voliime of water of pgz and pv^/2, where p is 

 the density of water, g is the acceleration due to gravity, z is the 

 elevation over any convenient level, and v is the magnitude of the 

 velocity vector. 



Following Figure 3 all motions are governed by 



s = S sin at , C2) 



and by the conditions of continuity: 



A 



\--r' (3) 



r 



where s and Sp are the displacements of the pistons and free surface 

 in the direction of increasing'^ from their (common) elevation at 

 t = 0, and where V[l) is the average velocity over a cross section 

 in the direction of i. Taking the elevation of the active counter- 

 weight to be y cos 2at, and the variable potential energy of the 

 water to be its potential energy at any time, minus its potential 

 energy when s = 0, the total variable potential energy of the sys- 

 tem is found to be: 





Py = M^ g y cos lot + -^ I 1 + ir^ I s2 • (5) 



With the approximation that over a cross section the flow 

 velocity is constant and equal to v(£) , the variable kinetic 

 energy of the system is expressed in the form: 



14 



