73 



87 



1.0 



0.8 



0.6 



n 



TTO* 



0.4 



0.2 



0.2 



0.4 



0.6 



0.8 



1.0 

 R 



1.2 



1.4 



1.6 



1.8 



2.0 



Figure 32 - Plot to Represent the Function rj 



For a piston n is defined in Equation [122]; for a paraboloidal diaphragm, in Equation [119]. 

 The radius of the diaphragm is a, and the distance from its center is R, The curves represent 

 r)/)r«2; the straight lines represent linear approximations having the same area under them as 

 the curves and also the same moment about the axis, ^ = 0, 



If this expression multiplied by na^ Is substituted for the Integral In [n6], 

 and also 



n pc a A 



npc aB 



- h A^ _ T 



M ' M 



Equation [Tl6] becomes the difference-differential equation 



■i,(«) +H,(t)-6[2,(0-2,(t -T)]= ^^'^ * 



[125] 



This equation Is more easily handled than the more accurate Integrodlfferen- 

 tlal equation; In simple cases It can be solved completely. 



THE NON-COMPRESSIVE CASE WITH PROPORTIONAL CONSTRAINT 



Let z\ change so slowly with time that It changes only by a negli- 

 gible amount during a time D/c. Then In Equation [111] or [11 6] z^ can be 

 treated as Independent of s arid can be taken out from under the Integral sign, 

 with the result that 



(M + M;)z; = 2F, + « 



[126] 



M, = :^fjf(x,y)dxdyffjf{x;y')dx'dy' = pjv(s)ds [127] 



In which « denotes the distance between the elements dxdy and dx'dy'. 



