411 



to be neglected in comparison with the membrane stresses. Since the former 

 is proportional to the cube of the thickness while the latter is propor- 

 tional to the first power, the above assumption is applicable to thin dis- 

 phragms. There is another assumption (seldom mentioned because it is ob- 

 viously valid for practically all cases of interest) that the time of 

 travel of a wave front through the thickness of the diaphragm is very small 

 compared with the decay time of the incident shock wave. 



Taking advantage of the axial sj^mmetry we consider the forces act- 

 ing on an infinitesimal segment of the circumferential strip included be- 

 tween f,r-vJir , ^ f and ^ -* ^ ^ . We make use of the unit basis vec- 

 tors 1 , "1.^ t"*^ } snd C» ^MMSHHttili vriiich are related to each other as 

 follows: _-, , _^ 



-» "-» i:* • ^ ^^-^^ 



where ^ is the angle between "«? and ^1,^ and is given by 



i5>- = -t^-t (ax/ar;^ . (A-2) 



Before proceeding further^we assume that two of the principail stresses (TJ" 

 and ^ are in the meridional (c) and circumferential (exv<- ) directions 

 respectively and that the third principal stress in the normal direction 

 ( ft? ) may be neglected in comparison with the first two. The force acting 

 on the element included between »■ , r+«lir, <^ ^ and ^ 4 d^ , due to cir- 

 cumferential t ension is 



cJi5"^<A4>=- -1^ 04.c-^,a (J-^ Ar ci4» • (A-3) 



The force due to the incident pressure p is 



92 



