420 SURFACE AND OTHER EFFECTS 



served values. These varied from 1.2 inqhes at 6 feet to 4.2 inches at 

 2.5 feet, indicating that the plates must actually have absorbed 4 times 

 as much energy from the shock wave as is accounted for by the kinetic 

 energy at the time of cavitation. Goranson showed in fact that the 

 energy absorbed was very nearly equal to the total available shock wave 

 energy, by a calculation of plastic work similar to the one outlined here. 

 This approximate equality suggests the existence of some mechanism 

 by which the kinetic energy of cavitated water in front of the plate can 

 be delivered to the plate and increase the deformation by doing further 

 plastic work on it. 



A mechanism for providing this reloading has been suggested and 

 examined by Kirkwood (62). The physical basis of the theory lies in 

 the fact that, as the diaphragm decelerates, the pressure in front of it 

 increases in a manner calculable from its equation of motion. When 

 this pressure becomes larger than the cavitation pressure, the cavitation 

 is destroyed and a reflected wave of compression moves back into the 

 cavitated regions. The water between the plate and the front of recom- 

 pression is moving with the plate in the noncompressive approximation, 

 and the kinetic energy of both this layer and the plate is ultimately 

 dissipated as plastic work. If the reloading wave is idealized to be a 

 plane wave front, moving into the cavitated water with forward velocity 

 at essentially zero pressure, its velocity and hence the thickness of the 

 reloading layer of water can be computed from the pressure behind the 

 front and the Rankine-Hugoniot condition at the front. 



Kirkwood's application of these considerations gives the result that 

 the deformation predicted by Eq. (10.23) should be increased by a fac- 

 tor (1 -f- ^/^y^, his equation being 



Zc{tm) 



poCo y aoli \ 4/ 



This result, when applied to the 21 inch diaphragms for which jS = 6.7, 

 gives central deflections in much better agreement with experiment for 

 an assumed yield stress (To = 45,000 Ib./in.^, but still somewhat smaller. 

 Kirkwood has suggested that the secondar}^ bubble pulse ma}'- account 

 for the increased damage. If this is to be true, the peak pressure P„/ 

 in the secondary pulse must be greater than the normal stress corre- 

 sponding to the final shock wave deflection Zr{tm), as otherwise the later 

 displacement will be elastic without permanent set. The necessary 

 pressure is therefore given by PJ = {^doh/a^) z,{Un), which for this case 

 gives the result that PJ must be greater than 6 per cent of the shock 

 wave peak pressure. Secondary pressures of at least this magnitude 

 would be expected in this case, and the explanation is thus a tenable one. 

 An actual calculation of the supplementary damage would of course 



