51 



65 



M,2, 



l«-«.)(i^) - 



1^. 



t2 



(ft 



:82] 



'/' M + Mi 



This energy represents a fraction M/(M+ M,) of the energy of the moving wa- 

 ter, whose total magnitude is 



i M, K' = 





M, 



[831 



The fraction M;/(Af + M,) will, however, be close to unity in practical cases; 

 and if the diaphragm is moving at the time of impact, it will take on a still 

 larger fraction of the kinetic energy of the water. 



The kinetic energy of water and diaphragm will then be converted 

 into additional plastic work. The total work should thus be at least as 

 large as 



rr ■'2 

 1 2 ^ U ^•''' 



M + M, 



Inserting again the values for the paraboloidal circular diaphragm and v„ 

 from Equation [7], noting that, if p^ = p„e~"' 



by Equation [5a]» and equating the value found for P to nahz^, it is found 

 that Equation [Bl] is replaced by 



^.-^r¥-^(i+ 



1 + 0.776 -^ f 



^) 



■ih] 



for an incident wave of exponential form. In these formulas it might be more 

 nearly correct to omit M, or the 1 in the denominator under x^ in the last 

 equation. 



Comparison of Equation [BU] with Equation [81] shows that the re- 

 loading increases the deflection in the ratio 



(-! 



^) 



1 + 0.776 -^ f 

 Pd h 



Since x increases with l/a, or with the duration of the wave, it appears from 

 the considerations advanced in the discussion of Case 1 that the deflection 



