44 



30 



If the plate moves like a piston, the shape factor In Equation [28] 

 becomes /(x,y) =1. If the plate is circular and of radius a, it is found, 

 as in Equation [128b] in the Appendix, that 



M, 



o 3 



T pa 



:37] 



Furthermore, if m or p,, respectively, is uniform over the plate, It is ob- 

 vious from Equations [30], [3Ta], and [3'^a, b] that 



2 2 



M = nma , F- = na p^ 



M.=M, B = Tva' 



[38a, b] 

 [39a, b] 



Piston-like motion involves, however, a discontinuity at the edge. 



A simple type in which there ig no discontinuity is the parabololdal 

 form, 



„2 , „2, 



f{x,y) =1 2 . ^ 



4'-W 



[UOa, b] 



where r denotes distance from the center and r = a represents the fixed rim. 

 A spherical shape is scarcely different so long as the curvature remains 

 small. In this case, as in Equation [128a] of the Appendix, 



M, = 0.813 pa'* [41] 



and if m or p,, respectively, is uniform. Equations [30], [31a] and [3^3, b] 

 give 



TT 2 



Q 1 



[U2a, b] 

 [43a, b] 



see Appendix, Equations [120] and [121]. 



Approximately spherical or parabololdal shapes are produced by 

 static pressure, but under explosive loading more pointed shapes appear to be 

 commoner; see Figure l8. 



The results just cited suggest that in general the formula 



M, = 0.8 -^JW 



■l|4' 



Figure l8 - Typical Profiles of a Diaphragm Deflected by a Non-Contact 

 Underwater Explosion (Left) or by Static Pressure (Right) 



