SURFACE AND OTHER EFFECTS 415 



where for an infinite baffle P{t) is given by Eq. (10.17). If the plastic 

 deformation is approximated by membrane tension, the value of o-n 

 is approximately (iaoh/a^)Zc from Eq. (10.11) (this result is strictly for 

 small spherical deformations, but the difference in profile for small values 

 of Zc/a is negligible) . Substitution in the equation of motion gives 



(10.20) m^' + ^2. = 2P,(0 



— poCo 



■f+l-" 



-e,) -^ (zMdt\ 



The deformation of a plate actually involves thinning, as it is ap- 

 proximately true that the density p remains unchanged in plastic flow, 

 and hence neither m nor h is strictly constant. If this thinning is neg- 

 lected, Eq. (10.20) is a linear integro-differential difference equation 

 which is most readily solved for known Px{t) by use of Laplace trans- 

 form theory. Kirkwood (61) has obtained the solution for the case of 

 an exponential incident wave of the form Pi(0 = P^e~'/^, and has also 

 outlined a procedure by which empirical pressure-time curves can be 

 used for Piif), if the incompressive approximation of Eq. (10.19) for 

 P\{t) is used. The solution for Zc{t) so obtained can apply only up to 

 the time tm of maximum deflection for which dzc/dt = 0. The final 

 solution using the more exact expression for the diffracted wave given 

 by Kirkwood is rather complicated. The physical nature of the solu- 

 tions can, however, be seen by considering the solutions obtained in the 

 limiting cases of no diffraction effect at the center, and of noncompres- 

 sive action. 



For times less than dd the last two terms on the right of Eq. (10.20) 

 can to a first approximation be neglected. The inertial term for suf- 

 ficiently thin plates is relatively small, and if it is dropped, the equation 

 of motion for an exponential wave P(t) = Pme~^^^ is 



dzc , 4:aoh 2Pm ,,^ 



at poCoa^ PoCo 



The solution of this equation is 



Zc = ^^— [e ^'^ — e ^/^pj, where 6^ = - — — 



PoCo 6 — dp 4(7o/i 



The characteristic time dp is the time for the plate to deflect to a fraction 

 1 — 1/e = 0.63 of its final set in response to a step pressure, and is 

 called the plastic time by Kirkwood. The final deflection Zc{tm) for 

 which dZc/dt = is given by 



Zcitm) = — — e-^/0^ , where tm = - — ^ log ( ^ ) 



PoCo O — Up \Up/ 



