U3 57 



represent the resultant velocity as given by the usual formula for an inelas- 

 tic impact between masses M and Mj, 



If the liquid surface is not constrained in shape, as in reality it 

 is not, expressions comparable to these are hard to obtain. It appears, how- 

 ever, that at least the order of magnitude of the effects to be expected may 

 be ascertained in a given case by assuming a reasonable form of proportional 

 constraint for both plate and liquid surface and employing Equations [62] and 

 [64], The equations will hold so long as no further short-lived pressure 

 waves arrive to cause temporary departures from the non-compressive motion. 



In using the equations it may be possible to fix the value of z/ 

 or 2^', only within certain limits, but this may be sufficient for practical 

 purposes. 



PART 4. DAMAGE TO A DIAPHRAGM 

 A PEW SWING TIMES 



It is often desired to estimate the swing time of a plate or dia- 

 phragm. A rough estimate can be based upon the formula for the following 

 special case; see the Appendix, Equation [173]. 



Consider a circular diaphragm of radius a and uniform thickness, 

 held rigidly at the edge, and thin enough so that bending resistance can be 

 neglected. Assume that the elastic range is negligible, that the yield 

 stress has the constant value cr, that the diaphragm, initially flat, remains 

 symmetrical and paraboloidal in form during its motion, and that thinning may 

 be neglected. Then the swing time, or time for the diaphragm to swing free- 

 ly through a short distance from the flat position and come to rest at its 

 maximum deflection, if there is gas at equal pressure on both sides, is 



where p^ is the density of the material. If the density is 0.283 pounds per 

 cubic inch, as for steel, so that in dynamical units p^ = 0.283/386, if <7 = 

 80,000 pounds per square inch, which may be a reasonable nominal estimate for 

 mild steel under high strain rate, and if a is in inches and T, in milli- 

 seconds, 



T,=n^^0.06la [66] 



Va 



If the diaphragm js mounted in a rigid baffle with liquid of den- 

 sity p^ on one side, the hydrostatic pressure in the liquid being the same 

 as the pressure of the gas on the opposite face, then the swing time is in- 

 creased, as a result of loading by the liquid, to 



