MODES OF MOTION IN QUARTZ CRYSTALS 77 



by experiment. Applying a coupling coefficient of 35% and computing the 

 values of fi and /j from equation 6.17 the results are the dotted curves 

 shown in Fig. 6.13. The observed points follow the computed values to a 

 fair degree of accuracy for all frequencies below 180 kilocycles. Above this 

 range there is a strong coupling to the fourth flexure and this would require 

 separate consideration. Based upon these results the equation for the low 

 frequency or face shear given in section 6.3 would not give the observed 

 results for a nearly square plate because of the high coupling to the second 

 flexure mode. For an approximately square plate, cut near the ylC-cut 

 the plate shear frequency including the effect of coupling would be given by 



f=-?^./X, 6.18 



2(/ Y pS56 

 where 



d = i(A^ + ^0 



and .849 is the factor resulting from the use of equation 6.17. For crystal 

 cuts far different from the above it would be necessary to consider the flexure 

 and shear as uncoupled and then apply equation 6.17 to determine the 

 appropriate factor for square plates. 



2. High Frequency Shear 



The motion associated with flexure has been shown in Fig. 6.1 and in 

 order to determine the frequency of higher order flexures, measurements 

 were made on an ylC-cut crystal. The results of these measurements are 

 shown in Fig. 6.12. The first flexure motion to be expected with this 

 crystal would be a flexure in the plane of the length and width. The various 

 orders of these flexures are shown by the curved lines labeled second z'x 

 fourth, sixth, etc., all radiating from zero frequency (Primed values of z 

 and y indicate that these are not crystallographic axes). The equation 

 commonly determining the frequency of flexure states that the frequency 

 should be proportional to the width and inversely proportional to the 

 square of the length. If this were true, these curved lines representing the 

 resonances of this type flexure shown on Fig. 6.12 would then be straight 

 lines. Since the actual conditions show a wide departure from this, we must 

 assume that this departure is due to rotary and lateral inertia and the 

 effects of shear. It will be noticed that as we progressively increase the 

 order of the harmonic, that the actual frequency spacing for a given value of 



— is very nearly linear instead of a square law. This point is more clearly 



seen when we examine the frequency of higher orders of the flexures in the 

 length thickness or xy' plane. As shown on Fig. 6.12 these frequencies 



