MODES OF MOTION IX QUARTZ CRYSTALS 61 



required. The motion associated witli this crystal is similar to that shown 

 for the +5° cut of Fig. 6.6. The low temperature coefficient is obtained 

 through coupling to, and the effects of, a shear mode of positive temperature 

 coefficient. Its use has been mainly for pilot channel filters of rather 

 narrow frequency bands. 



6.3 Frequency Equations for Flexurel, Extensional and 

 SHE.A.R Motions 



In determining the motion and resonant frequencies of a particular type 

 of vibrating system it is customary to consider an isolated type of motion in 

 order that the solution shall be in a simple enough form to be practical even 

 though it may not be too accurate. The more accurate type of solution 

 is often so complex that its use for practical solutions might be small. Since 

 any solutions so far obtained are not complete in every detail, it is usually 

 necessary to resort to experimentall}' determined frequencies in any case, 

 and the solution can only be regarded as a guide to the complete result. In 

 the following treatment it will be assumed that the frequency equations are 

 given for isolated modes of motion and it will be later shown which of these 

 forms are coupled and the effect of the coupling. 



6.31 Flexural Resonant Frequencies 



The simplest equation relating the resonant frequencies of a rod vibrating 

 in tlexure is given by 



7H^ k 



2^? 



f-'^J.' 6.1 



where v = velocity of extensional propagation = -x/Fq/p 



k = radius of gyration of cross section 



Yq = Youngs modulus 



^ = length 



w — (n + l/2)7r for free-free modes 



= (n — l/2)7r for clamp-free modes (» > 1) 



n = order of mode (1, 2, 3, etc.) 

 This equation holds only for the case of a long thin rod. Measurements 

 of the resonant frequencies of a quartz crystal vibrating with both ends free 

 has shown the above equation to be true where m is defined approximately 



as {n + l/2)x provided — is less than .1. For values greater than this the 



measured values are somewhat lower than that predicted. When the di- 

 mension in the direction of vibration is appreciable in comparison with the 



1 Raj'leigh, Theory of Sound, Vol. 1, Chapter VIII. 



