SURFACE AND OTHER EFFECTS 419 



tures of interest have a sufficiently small ratio of thickness to distance 

 between regions of support that cavitation will occur for shock waves, 

 and consideration of cavitation effects is therefore important. 



If cavitation forms in front of a plate, the water ceases to exert a 

 significant load. If the cavitation persists, the plate is effectively de- 

 tached from the water after the time 6^ and is brought to rest only by 

 its resistance to plastic deformation. The plastic work done on the 

 plate should then be very nearly the whole of the kinetic energy ac- 

 quired before cavitation occurred, as none of this energy is returned to 

 the water. If the cavitation time is short, a first approximation to the 

 kinetic energy acquired can be obtained by assuming that the plate acts 

 as a free plate without significant resistance. The velocity of a free 

 plate at time dc for an exponential wave Pmer^^^ is readily shown from 

 Eq. (10.7) of section 10.4 to be 



(dz\ 2Pm ^T^ , 



{- = ^ ^ wher 



\at/0^ PoCo 



e = ^ 

 m 



The total kinetic energy for a plate of radius a is then 

 (10.22) KB = i-irahn (^\ = ?^^ Tra^ ^^^ 



and is thus proportional to the shock wave energy density Pm'^d/2poC„, a 

 fraction of which is absorbed. If the plate is assumed to act as a mem- 

 brane under tension aoh, the plastic work done for a central deflection Zc 

 is given by TvaohZc^ from section 10.5. The final deflection Zcitm) is ob- 

 tained when the plastic work is equal to the kinetic energy acquired, 

 and is proportional to the square root of shock wave energy. The value 

 of Zcitm) from Eq. (10.22) is 



_ Prna I 



OnCn \ 



2m ^y' 



12 



(10.23) z,{Q = ^^ J— I ^'-^ = 2. -^ /3i-^ aE'i 



PoCo ^ (Toll > (Tofl 



where E is the shock wave energy density for the assumed acoustic 

 plane wave. 



This result is evidently rather crude, as it neglects diffraction effects 

 or the possibility that, in later stages of the motion, the plate may be 

 reloaded by disappearance of cavitation as the plate decelerates. It is 

 therefore not surprising that deformations calculated from Eq. (10.23) 

 come out to be considerably smaller than observed values. As an ex- 

 ample, experiments made by Goranson (41) may be cited, in which steel 

 diaphragms 21 inches in diameter and 0.11 inches thick were deformed 

 by the pressure wave from 1 pound TNT charges. The central de- 

 pressions computed from Eq. (10.23) w^ere of the order of half the ob- 



