ELECTRICAL WAVE FILTERS 



449 



We can conclude, therefore, that the solution for a shear vibration 

 in a quartz crystal will be given by equation (34). It is obvious from 

 Fig. 34 that the shear vibration will have a strong coupling to a bar 



Fig. 34— Form of crystal in shear vibration. 



bent in its second mode of flexure, since the form of the bent bar, as 

 shown by Fig. 29, is very closely the same as a given displacement line 

 in the crystal vibrating in shear. Little coupling should exist between 

 the shear mode and a bar in its first flexure mode, since this mode of 

 flexure requires a displacement which is symmetrical on both sides of 

 the central line whereas the bar vibrating in shear has a motion in 

 which the displacement on one side of the center line is the opposite 

 of the displacement on the other side of the center line. 



The Elastic and Piezo-Electric Constants of Quartz for Rotated Crystals -^ 

 W. Voigt ^^ gives for the stress strain and piezo-electric relation in a 

 quartz crystal, for the three extensions and one shear found above, 



— Xx = SxiXx + SiiYy + SxzZi + SuY^, 



— jy = siiXx + 522^2, + snZz + SiiY:, 



— Zz = SizXx + SizYy + SzzZz 4" SziYz, 



— Jz = ^14-^1 + S2iYy + SziZz + Si^Yz, 



— Px = diiXx + diiYy + dnZz + dxiYz, 



(40) 



(41) 



2" The material of this section was first derived by Mr. R. A. Sykes of the Bell 

 Telephone Laboratories. 



^' W. V'cigt, Lehrbach Der Kristallphysik. 



