MODES OF MOTION IX QUARTZ CRYSTALS 



83 



becomes appreciable in comparison with a half wave length along w or / 

 it becomes necessary to use the c constants. When considering high orders 

 of the low frequency shear equation 6.8 is modified to 



/ = 





6.21 



Equation 6.21 shows that high orders of the low frequency or plate shear are 

 dependent upon both the length and width dimensions and it might be as- 

 sumed that this would lead to very complicated results in so far as analysis 

 of experimental data is concerned. The coupling between these modes and 

 the high frequency shear is a result of coupling in the mechanical as well as 



5 1 640 



> 1560 



FUND. XljSHEAR \ 



I I \ 



i-J, \ \ l\ \ \l \ i.l 



t=Y = .293 MM. 

 W= Z' 



\- X- 11.16 MM 



\ 





i. 



^ 



\ \ 



\ \ 

 \ 



'\^ 



\ \ 

 \ \ 

 \ 



\ 





\ 



\ I \ \ 



* V \ 



\ \ 

 \ \ 



\ \ 



\ \ 

 \ \ 



-\- 



\ 

 \ 

 \ 



FT 

 \ 

 \ 

 \ 



\ \ 





\ \ \ 

 \ \ \ 



\ \ \i 

 \ \ 



\ 





to 14 18 22 26 30 34 38 42 46 



J/ 



y 



Fig. 6.18 — High frequency shear resonances in an .4r-cut plate. 



the electrical systems. The strongest coupling with reference to the length 

 axis would then be for high odd orders of w and unity for n with successively 

 smaller coupling for higher orders for n if the driving potential extends over 

 the complete surface of the crystal. In a similar manner when considering 

 high orders of plate shear along the width axis the highest coupling will 

 result from unit order for m. Based on these assumptions then to a first 

 approximation we can assume these modes to be functions of length and 

 width alone. Equation 6.21 then reduces to 



ffs - 



1 



f =^ 



P 1 



Cjj tlwa 

 p W 



6.22 



6.23 



where Ugf = order of shear mode along ^ axis, 

 nsw = order of shear mode along w axis. 



