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BELL SYSTEM TECHNICAL JOURNAL 



It is easily seen that u = cos kx is a solution to this equation if ^ = w 



/l 



E 



If now the boundary condition that the stress Xx must become zero at the 



ends of the bar (i.e., x = 0, x — ^ — refer to Eq. (7.5)), is fulfilled, the 



du 

 solution wiU be complete. At :>; = 0, X^ = E—- will always equal zero. 



ox 



TT . IT 



- or any whole number multiple of - the extensional 



Furthermore, if k 



stress will likewise reduce to zero at x = ^. The desired solution will then 

 be as follows, / being the resonant frequencies. 



U = cos CO a/ -^ 



1* 



0) = 27r/ = 



WT 



VI 



s^ 



m = 1, 2, 3, etc. 



(7.8) 



-* t 



Fig. 7.5 — Thin plate 



The plate of Fig. 7.5 will now be considered. Here it can no longer be 

 assumed that the Xx stress is the only one of importance. Instead, the 

 displacements v and w will be considered zero and the displacement u a 

 function of x only. This means that the shear stresses Xy , Xz , Yz vanish, 

 so that the equilibrium equations 7.2 reduce to 



or 



. 5 M , „5 W 2 



d U — pu> U 



dx^ 



A -{- B 



(7.9) 



(7.10) 



This is seen to be of the same form as equation (7.7) previously discussed, 



and will again have the solution w = cos kx with k = o) A/ - — j — - 



y A -\- Jj 



The 



