166 BELL SYSTEM TECHNICAL JOURNAL 



the ends of the plate as they should in order to satisfy the boundary condi- 

 tions. As the ratio of - is increased the shear stress becomes of greater 



importance. 



7.52. Thickness Flexures 



The final analysis to be considered in this paper is for thickness flexures 

 along the width or length of a thin plate. These modes are of particular 

 interest in connection with the dimensioning of quartz plates for which it is 

 desirable to utilize the fundamental thickness shear mode. (AT plate, for 

 example.) It is found experimentally that even ordered thickness flexures 

 are coupled to this shear to such a degree that at certain ratios of dimensions 

 the operation of the plate as an oscillator or filter component is impaired. 



The two-dimensional solution derived in the preceding paragraphs can 

 be used to predict certain harmonic thickness flexures ; however, in order to 

 obtain a complete picture it is necessary to extend the theory to three 

 dimensions. This has been done by the author with the following transcen- 

 dental equation as a result (refer to Section 7.93). 



2 _ — 2A^2^a /y 22^ 



~~rh~[aB{(\+a')-^Al\\[(l-a'] ^' ^ 



tan I2 - 



Solutions to this equation are exact in nature for a plate of thickness h 

 and of infinite extent in both the x and 2 directions. The quantity a- is 

 equal to the sum of the squares of k and m which appear in the expressions 

 for displacements as follows: 



u = U fi(y) sin kx cos mz 



V = V f'Ay) cos kx cos mz ^ (7.23) 



10 = W fsiy) cos kx sin mz 



Also in equation (7.22) 



/2 - ff' ^ 



2 



fl2 pOi 



(7.24) 



>1 



The lowest order solution to equation (7.22) with a- positive again cor- 

 responds to flexure vibrations, as in the two dimensional case. Fig. 7.15 

 shows a plot oi 6-b against a-b calculated for a = .3. 



