88 7** 



For a circular diaphragm of radius o, substitution of [119] or 

 [122] for T] in [127] and evaluation of the integral gives, for the paraboloi- 

 dal and the piston-like motions, respectively, 



M, = 0.813 pa^ M, = |pa^ [I28a, b] 



o 



A third type of motion that is of some interest Is described by 



z=z^{t)f{x,y) = z^{t)(l-^j for r < o [129] 



The integral in [102] becomes in this case, when z varies slowly enough with 

 the time. 



j\z,_,dS = z^j\(l-^ydS [130] 



Now this last integral represents the electrical potential at any 

 point of a disk due to a density of charge on it equal to (1 - rVo^)'*; and 

 it Is a known theorem in electrostatics that a surface density varying in 

 this manner produces a constant potential over the disk. The constant value 

 of the integral is easily found by evaluating it for a point at the center, 

 where « = r and dS may be replaced by 2jrrdr, so that 



/7(l-S''^^=/(l-I^r2-rf'- = -^a [131] 



With the use of this result, the integral for Af, in [127] is easily 

 evaluated, thus: 



^' = i7/V^) '2-rdr^\l-^)'2ndr- 



a 2 - A 



= 7r^paf(l-^) rdr = n^pa^ [132] 



" 



Furthermore, substitution from [130] and [131] for the Integral in 

 [103] gives 



g-paz^ = 2p,. + p(, -p [133] 



This result will hold for the water surface exposed in a circular opening of 

 radius o in a plate lying against the water, when, with the exposed surface 

 initially plane and stationary, a comparatively steady pressure equal to 

 2p. + p, is generated in the water back of the hole while the pressure on the 

 exposed surface is p. Then [129] represents the displacement of the water 



