92 78 



Comparison with [126] and with the result of integrating this equa- 

 tion from t^ to tf, respectively, shows that the values of z^ and z^ given by 

 [13^] and [1351 or [156] are equal to the values obtained from non-compressive 

 theory except for the initial correction due to the first integral in [135] 

 or the substitution of t/ for t^ in [136]. 



INITIAL MOTION OF A PROPORTIONALLY CONSTRAINED PLATE 



After a proportionally constrained plate has been either at rest or 

 moving uniformly for a time greater than D/c, let a wave of pressure p- sud- 

 denly begin to fall upon it, at time t = 0. Then, in [111], z^(t - s/c) will 

 at first differ from zero only for small s, for which f(x',y') may be replaced 

 by f{x,y) and taken out from under the integral sign. The integration with 

 respect to dx'dy' or dS' can then be carried out In analogy with [106]: 



N'-f)^=27rcMt) [139] 



provided 2^(- «) = o. Thus [111], becomes, approximately, for a short time, 



M'i^+ pcAz^ = 2F^ + <P [140] 



where 



A^\l[f{x,y^dxdy [l4l ] 



EFFECT OF FLUID ON BOTH SIDES OF THE PLATE 



If there is fluid of appreciable density behind the plate as well 

 as in front of it, a release pressure will be developed on both sides. That 

 in front will be, from [99], 



P- = ~^/7'^-^'^^ 



1 



where p^ is the density of the fluid in front and c^ is the speed of sound in 

 this fluid. The release pressure behind the plate will be similarly, 



where p^ and c^ refer to the fluid behind the plate. The reversal of sign 

 here arises from the fact that in obtaining the formula for the release pres- 

 sure z was assumed to be measured positively away from the fluid, whereas 

 here the positive direction for z is taken always toward the back side of the 

 plate. The total pressure on the back face is then 



P2 " Po2 ^' Pi2 



