84 70 



PLATE OR DIAPHRAGM PROPORTIONALLY CONSTRAINED AND MOUNTED 

 IN AN INFINITE PLANE BAFFLE 



Let it be assumed that in the displacement of the plate from its 



initial plane position all elements move in fixed proportion, so that it is . 



possible to write 



z = z^{t)f(x,y) [no] 



where z^ is a function of the time whereas f{x,y) is a fixed function of po- 

 sition on the initial plane; z^ may represent the displacement of some point 

 on the plate, such as the center, at which then f{x,y) = 1. Let the baffle 

 be Immovable. 



Then [102] becomes, with aS replaced by dx'dy', 



mzSt)f(x,y) =2p, + <A - J^ \\ \ K^^ - ^) f(x\y) dx dy' 



Here z^, in contrast with z in [102], is a function of time alone, and 

 ic(t - s/c) denotes the value of d^z^/dt^ at a time ( - s/c. By multiplying 

 through by f[x,y) and integrating again over the whole area of the plate, a 

 convenient ordinary integrodifferential equation is obtained for z^ : 



M-i^ = 2F + <P - -^jjf(x,y)dxdyjj^z,(t - ^)f(x\y)dx'dy' [111] 

 where 



M= l\m[f{x,y)^ dxdy [112] 



F,=\\vJ{x,y)dxdy, <P = jj (/> f(x,y) dxdy [113a. b] 



Since, from [110], 



i = ijix,y) [nu] 



the kinetic energy of the plate is 



K = jj^miUxdy = |z/J/m[/(x,j/)] dxdy = ^MiJ' [115] 



A more useful form for the Integral in [111] is obtained if x'y' are 

 replaced by polars s,6, with origin at the movable point x,y, but with the 

 axis in a fixed direction, so that dx'dy' is replaced by sdOds. Here s and S 

 may be defined by the equations 



X ' — X = s cos ^, y' — y = ssinO 



see Figure 31. Then, after changing the order of Integration, [111] can be 

 written 



