49 €3 



projected with a common velocity v„, and the entire diaphragm will acquire 

 kinetic energy of magnitude 



E = \hp,Av^ [19] 



where A is its area. This energy will then be converted into elastic and 

 plastic work, and if the relation between this work and the deflection is 

 known, the deflection can be calculated. 



Equation [72a], with Zj calculated from [70a] and [79]. thus gives, 

 for a circular diaphragm of radius a, deflected into a spherical shape. 



z 



2 = z,2 - iz.2, z, = «''m/^ [80a, b] 



For a shock wave of exponential form, p. = P„e""', the maximum ve- 

 locity is given by Equation [7], and 



k¥ 



«Pm i/2p7 ^r^ 



[81 



' pc f a 



where x = pc/ap^h, in which p is the density of water in dynamical units and 

 c the speed of sound in it. 



If the diaphragm is deformed into a more pointed shape, as commonly 

 happens, z will be somewhat greater; for a conical form, Zj would be greater 

 in the ratio V2. On the other hand, the actual maximum velocity will prob- 

 ably be somewhat less than v„ as given by Equation [7], because cavitation 

 will probably not occur until the pressure has sunk more or less below the 

 hydrostatic value; z will be correspondingly reduced. 



These equations predict nearly the same variation of z with dis- 

 tance R from the charge as was inferred for Case 1, but, for ordinary thin 

 diaphragms, a somewhat slower variation with charge weight W. The difference 



arises from a decreased influence of the duration of the wave. This in- 



1 

 fluence is represented, for an exponential wave, by the factor i i-^ In Equa- 

 tion [81], Since I = pc/am, x increases in proportion to l/a and hence in 

 the same ratio as does the factor Ip^dt in Equation [77]; but in practical 



cases X lies between some such limits as 2 to 10, and a glance at Figure U 



1 

 on page 5 shows that in this range x 1-* increases much less rapidly than 



does X. 



The deflection z and the projection velocity v^ may vary, there- 

 fore, in this case, either a little more rapidly or a little less rapidly 



than as l/K; they should vary more rapidly than as W3, but not so rapidly 



2 1 



as WS. Both Zj and v might happen to be nearly proportional to W5. 



