MODES OF MOTION IN QUARTZ CRYSTALS 65 



6.3v3 Shear Resonanl Frequencies 



As shown in seclion 6.23 the low frequency face tyjic shear mode results 

 in a doubly infinite series of frequencies due to the manner in which the plate 

 may break up into reversals of phase along its length and width. While 

 a solution for the low frequency shear motion that satisfies the boundary 

 condition of a free edge has not yet been accomplished, several approximate 

 solutions for the frequencies are available. A modification of the equation 

 developed by Mason will give results which verify experimental data. 





where p — density 



Sjj — shear modulus in dv plane 

 m, n = 1, 2, 3, etc. 

 ^ = length of plate 

 IV — width of i)late 



The value of k so far remains exi)erimental and for low orders of m and it 

 may be assumed unity. Its use is mainly for high orders of m and n where 

 Young's modulus is different in the ^and w directions. Experimental data 

 in the case of BT plates indicates that it should be 1.036 to account for the 

 difference in velocity in the two directions. When m or n is large the velocity 



/T Z^- 



component, namely A/ — - should be replaced by A/ — for reasons ex- 



V P^ii V P 



plained for the extensional case. Equation 6.8 holds for the case of a plate 



vibrating in low frequency shear in regions where no highly coupled exten- 

 sional or jflexural resonant frequencies exist. As will be shown later, these 

 regions are few. By assuming the frequencies are given by these equations 

 and then applying the normal correction for coupled modes, a fairly accurate 

 result will be obtained. 



The high frequency case of a plale vibrating in shear is somewhat similar 

 to the face shear or low fre(|uency case with the exception that three dimen- 

 sions must be considered since two are large compared to the third (the main 

 frequency controlling dimension). An experimental formula hn" this case 

 is given by 



/=2t^VF + ^;^ + *'^^^' 



where Cjj = shear modulus in plane of motion 

 p = density 

 ■C,w,t= length, width and thickness 



'' "Electrical Wave Filters Kniplm inj,' Ouartz Crystals as Elements," W. 1'. Mason, 

 B.S.TJ. July, 1934. 



