MODES OF MOTION L\ QUARTZ CRYSTALS 71 



mode along the length (the Vy mode) is unafTected only in the case of the 

 — 18° cut. The effect of coupling between the extensional and shear is 

 clearly shown in the case of the 0° cut by the change in the length-extensional 

 frequency. This is more pronounced in the -f-18° case not because of more 

 coupling but because the frequency constants of the two modes are more 

 nearly alike as indicated in Fig. 6.8. 



The mode of motion associated with the line intersecting the e.xtensional 

 Xy mode is that due to the second length-width flexure mode. As, mentioned 

 before it is strongly coupled to the shear mode in the same plane. The 

 coupling between this flexure and the extensional mode is directly related 

 to the coupling between the shear and the extensional mode. This is borne 

 out by Fig. 6.9, for in the case of the — 18° cut, 504 is zero and as can be seen 

 the change in frequency of the extensional mode is very slight even when the 

 flexure mode is nearly identical in frequency. 



We may state generally that the change in frequency of a particular mode 

 of motion from that of its uncoupled state is dependant on two factors; 

 the coupling to and the proximity to other forms of motion. This follows 

 well established mathematical procedures but to solve the case just discussed 

 would require the solution of a four mesh network with mutual impedances 

 the values of some of which are at best only approximate. This will serve 

 to illustrate that the use of formulae such as given in section 6.3 may be used 

 more as a guide in establishing certain modes of motion rather than for accu- 

 rate determinations of resonant frequencies. 



6.42 Flexure to Shear Coupling 



1. Lon' Frequency Shear 



As previously indicated there is no simple means of mathematically 

 determining the coupling between flexure and shear types of motion as there 

 is between the extensional and extensional to shear modes. Here we must 

 base our assumptions upon observed experimental evidence and simple rea- 

 soning. The relation between flexure motion and shear motion can be illus- 

 trated by the figures associated with Fig. 6.10. The forces that are necessary 

 to produce flexure and shear motion are shown by arrows in Fig. 6.10. 

 When the two arrows point toward each other, it indicates a compression 

 and when the arrows point away from each other, it indicates tension. The 

 diagrams on the left of Fig. 6.10 illustrate the conditions for flexure motion 

 and the diagrams on the right indicate the conditions for shear motion. 

 Notice that in the case of the first flexure and the second shear that the 

 forces applied to the top and bottom of the plate are similar. Also in the 

 case of the second flexure and third shear, they are similar. Here again we 

 have certain similarities which in this case are important to remember. 



