100 86 



The value of i^ given by [l6l] represents the velocity as calcu- 

 lated for the time ty from non-compresslve theory, except that In the equali- 

 zation of velocities by impact as represented by the first two terms on the 

 right the velocity of the liquid surface is taken, not at the time of Impact 

 t^i , but at a somewhat earlier time (J;, 



So far nothing has been said as to fluid back of the plate. If the 

 plate, or plate and baffle, lie between fluids in which the density and speed 

 of sound are, respectively, p,, Cj, and p^, c^, then all of the results in 

 this section will hold good provided p and c are replaced by p^ and c^, with 

 the understanding that ^ or * includes an allowance for the release pressure 

 in the second fluid. More explicit formulas can be obtained by substituting 

 for <l> ov <P (but not <P ,) from [1^3] or [l44]. 



SOME SWING TIMES 



Suppose that a plate, mounted in a fixed plane baffle and con- 

 strained to move proportionally, is free from incident pressure, and that the 

 motion is slow enough so that the water or whatever liquid is in contact with 

 its faces can be treated as Incompressible. Furthermore, let the motion be 

 small enough so that its component parallel to the plane of the diaphragm can 

 be ignored. Then [126] becomes 



(M + M,)i, = * [165] 



This can be integrated after multiplication by z^dt: 



(M + Af,)z^z^dt = (Pz^dt = 4>dz^ 

 whence 



■|-(M + M,)i/ =/*d2, [16U] 



From a knowledge of z^ as a function of z^ the swing time can be found as 



taken between the limits z^ = and the first value of z, at which z^ = 0. 



The most important case is that of a circular diaphragm of radius o 

 and uniform thickness h, constrained to move in symmetrical paraboloidal form 

 or according to [n8]. For the small motions considered here, the difference 

 between a paraboloid and a sphere can also be ignored; the diaphragm can be 

 assumed, therefore, to behave as a spherical membrane under uniform tension. 

 Elementary theory then gives, as in the deduction of [93], for the contribu- 

 tion of the stresses to *, 



