MODES OF MOTION IN QUARTZ CRYSTALS 63 



effect. These values are given bv 



-i[(--i^"(i-)y-f(i-)] " 



A' = 



where Cj, is the shear constant in the plane of motion su is the elastic constant 

 in the direction of propagation. While it is true that these values will 

 result in a lower value of w than those associated with equation 6.2 and 



hence fit the actual measured results more closely for bars wider than — = 



.5, there is some doubt in the minds of various investigators as to the actual 

 amount of correction necessary to apply to compensate for the shear. The 

 solution of equation 6.2 using the constants of equation 6.3 is a lengthy 

 process and could only be applied to a given orientation since the elastic 

 constants vary with direction in quartz. While the results of Jacobsen's 



work are difficult to handle for intermediate values of — where the correc- 

 tion of rotary and lateral inertia do not tit the measured results it does imply 

 that for large values of — that the lie.xure frequencies will be mainly a 



function of the length alone. Therefore when we are concerned with very 

 high orders of He.xure in plates such as the case of high frequency A T and BT 

 shear crystals we may assume the interfering modes due to flexures will be 

 essentially harmonic in nature. Restating the general problem of determin- 

 ing flexure frequencies in quartz rods or plates we may assume that the 

 ratio of width to length is the controlling factor in deciding which method of 



nw 

 attack is to be employed. For values of — less than .1 equation 6.1 will 



. tiiv 

 give quite accurate results. For values of — up to .5 equation 6.1, using the 



values of m determined by equation 6.2 will give satisfactory results. While 



the values of m determined by using equation 6.3 will give more accurate 



results for the range .4 to .6, it is not desirable to carry it further because, 



while 6.2 does take into consideration the effect of shear it does not account 



till' 

 for coupling to the shear mode of motion. Hence for values of — > .6 



