2 _ A2 L ,^ 2 



„ _ „ r^^^i ^ pa c ^ds,- .,. 



where L is the length of the test section and 



_ A 



I +s 

 f o r 



d£ 



A(£) 



(7) 



where the path of integration extends from the pistons to the 

 free surface, with Iq the total path length when s = 0. Since 

 Ac and Ar, the values of A(£) at the lower and upper limits of 

 the integral, differ slightly, a depends slightly upon s. Dif- 

 ferentiation with respect to s provides, with equation (3), 



£da^Aa^£A/\l_ 1_\ 



a ds a a L I A A^ " A i * ^^J 



Then, by the dimensions of the tunnel, and since s cannot 

 exceed Sm, Aot/a cannot exceed 0.5 percent. Neglecting this 

 small variability, the integral in equation (7) can be 

 evaluated for s = 0, and a regarded as a constant. 



The total energy E is the sum of the variable Py and Ky 

 plus an undetermined constant. Obtaining s^ and (ds/dt)^ 

 from equation (2), using the identities: 



. 9 1 

 sm^at = jCl - cos 2at) , (9) 



9 1 



cos^at H -(1 + cos 2ot) , (10) 



and defining 



K-«- 



the addition of equations C5) and C6) provides: 



15 



