ym = 



-7- 289 



{3 77)-^ i P ^ 



R' —2 — . (box-TOdet) (76) 



Which Decomes larg? for lar:;e R. In Doth these cases it can be verified that the spring 

 approximation ij *^3t i sfactory because the time to reach maximum deflection is proportional to 



I ^'l R 

 - , that is to - , whereas tht timi; of travel of the plastic waves is proportional only to -. 

 \ N ■.' 



(3.2) Motion of pl3tc assuming cavitation . 



It i=. shown in I that up tj the time of cavitation the plate acquires a Kinetic energy 



per unit area of \^J2^ ~\"^^x where a = *"'" _■ If the plate acquires no further energy 



p '■. ^ p ' 9 



tither from "follow-up" cr fliffraction, then the final deflection is given iimply Dy equating this 



to the plastic energy " cr.^ '* j 



^ y "• ^e tnen have: 



ym = n ^^ - ^'' « -^"—-4 (ship) (Ba) 



? " K% '^'' 



The transition to the Dox-mooel is made siTiply by putting N = 1. The doubling of the 

 pressure is here retained in both models i3ecsuse for a large plate diffraction plays no part in the 

 motion of the plate up to cavitation. We have therefore: 



1 ♦_g^ gi 



ym = g ^^ ' °"^ R !? 1 (box-model) (8b) 



2' (^0^0 ^"'' 



If we mane the alternat i ve assumption that follow-up occurs but there is still no siynificani 

 contrioution from diffraction, {sap section 2 for j discussion of this) then wo have just thi.- case 

 treated in I. There we found that a rough rule was that about two-thirds of the incident energy 

 .vas evr-ntually converted into plastic work. The energy absorbed per unit area is now 



r. ^ n ^ * ° i 



P ^ '}' \ — r T / 1 \ 



-!3 and the factor n ' ' in equations ("la and b) is replaced by/-i- 1 . w.^ now have 



3Po ^ ? U^ / 



to examine the validity of the "spring" assumptior. Equations (8a and b) are based strictly on 



energy considerations and it is immaterial whether the energy communicated to the plate is absorbed 



by a spring or by plastic waves. The theory of t, however, assumes a uniform spring and a failure 



:f thi? assurption might modify the factor cf ^ . ^^91^ ^^^ theory of I we find that the time to 



re-ACh maximum deflection is of the order of magnitude » where S is given by 



/a a N^ / ph 



-/ 2 / . (It should be noted that in I we use a for thickness of plate while in 



/ p r2 / p^C 



2 



II and this report we use h. we nave also introduced the factor N to allow for subdivision of the 



Lianel). The ttine for the plastic wave to reach '.ne centre of the sub-panel is tt divided by the 



velocity r^o of plastic wfives. We thus see thnt, for the "spring" approximation to be valid we 

 / p 



■nust have 8a less than unity. Since a is not usually greater than O.i in practical cases and is 



often less, it appears that the spring assumption probably docs not lead to serious error in this 



case, though the error does not diminish with increasing R as i t does in the incompressible model. 



(3'3) Motion of plate assuming that water can stjnd tension . 



In II (Appendix, equation A.3) it was shown that for a piston moving in a rigid wall, the 

 rigorous expression for the force on the unit area of plate due to the hydrodyn.-jmic forces is 



-^o^/f ^(^X^;^)^_^. -.-re^^^j 



defined 



