86 72 



vis) - J « 



(2-3R')cos-'f+(\R +^R'-j-R')V4^^^ 



[119] 



2 \2 3 24 

 where R = s/a. Furthermore, from [112], if m is uniform over the plate, 



M = yma^ [120] 



and, if the incident pressure p^ is also uniform, from [113aj, 



F,=^7ra'p, [121] 



Or, if the diaphragm moves like a piston, except for a negligible 

 ring at the edge, f{x,y) = 1 and 



77(s) = a'(2cos"'| -fj/r^^) [122] 



M = nma'^, F = jra^p, [123a, b] 



The curves for 77(3) corresponding to these two formulas do not vary- 

 much from straight lines of the form 



77(s) = 7ra'(^'-B'|) [124] 



If the constants A' and B' are determined so as to give correct values to the 

 two integrals 



J Tjis) ds, J s ri(s)ds 



then, for paraboloidal constraint. A' = 0.357. B' = 0.246; for the piston- 

 like constraint. A' = O.961, B' = O.^hh. The curves for ri/na^ and the cor- 

 responding lines are shown in Figure 32. 



If an expression for 77 of the form of [124] is substituted in [II6], 

 the integral can be evaluated. For the upper limit, however, 2a must be re- 

 placed by 8 = A'a/B', at which r; as given by [12U] vanishes. A dot over 

 Zj(t - s/c) is equivalent to differentiation with respect to the argument 

 (( - a/c), hence, at fixed t, in analogy with [104], 



Hence, integrating by parts, 



a' a J _ A'a 



V--'>iy'.{'-7)'"-i-'{^-'^-i)'.('-i)*i^-4'-i)]'" 



- «''■*.'" + T«l=.(' - Iv) - ''"'1 



/ 



