60 



46 



. Incjdant Prassura 



Oisplocament of 



Tima 



Figure 25 - Illustration of Case 1 , 

 Relatively Long Swing Time 



The swing time of the diaphragm, T,, Is much longer 

 than either the time constant T,„ of the Incident 

 pressure wave or the diffraction time 7j. 



where Zj is the deflection calcu- 

 lated from E with neglect of the 

 elastic range, that is, from E = 

 (Tfc4i4j where 4Aj corresponds to z , , 

 or by putting z = z^ in Equation 

 [70a, b, c]. It is assumed here 

 that the same shape occurs at all 

 deflections mentioned. 



In a few special cases, 

 formulas for the deflection pro- 

 duced in a diaphragm by a shock 

 wave can now be obtained by bring- 

 ing forward suitable formulas for 

 E. 



CASE 1: Relatively Long Swing Time, No Cavitation; i.e., T,» T^ and T, » T^, 

 or the swing time of the diaphragm several times longer than either the dif- 

 fraction time or the time constant of the wave, as Illustrated in Figure 25. 

 These conditions as to the times are usually satisfied in practical test as- 

 semblies because of the thinness of the diaphragms. 



Let the diaphragm be mounted in a fixed plane baffle. Then, if it 

 is assumed to be proportionally constrained in its motion, in the sense de- 

 fined on page 26, its center will acquire a velocity 



"cm = 



2JF, 



dt 



M + M, 



:73] 



This equation results from integration of Equation [35] in case Td«r„, so 

 that non-compressive theory holds; otherwise it follows from the reduction 

 principle as expressed in Equation [50b]. It is only necessary that stresses 

 in the structure have little effect on the diaphragm until the hydrodynamic 

 action is completed. 



The combined kinetic energy of diaphragm and water will then be 

 converted into plastic work, so that 



E - t(M + Af,)«i - 



{1^. 



dt 



M + M, 



[74] 



For a circular diaphragm of radius o deflected parabololdally. 



