ELECTRICAL WAVE FILTERS 437 



frequency increases as the optical axis dimension is increased and 

 about the only type of motion which does this is a flexural motion as 

 shown by Fig. 29. This figure shows the second type of motion possi- 

 ble to a bar in flexure rather than the first for experiments by Harrison ^^ 

 show that the frequency for the first type of motion is too low to ac- 

 count for this vibration./ Harrison has also measured the frequencies 

 of a bar in its second flexural mode and the solid line, D, of Fig. 28 is an 

 actual plot of these measured frequencies up to a ratio of hllm = 0.25, 

 which is as far as Harrison carries his measurements. The rest of the 

 curve is obtained by extrapolation. There is no doubt then that a 

 flexural motion is involved in this coupling. The mechanism by which 

 the bar is driven in flexure will be evident if we observe what happens 

 to a square on the crystal in the unstrained state. As shown by Fig. 

 29, its deformation is similar to that of a shear deformation. The 

 amount of shear depends on the distance from the nodes of the crystal. 

 Some of the shear is in one direction and some in the other but the 

 two amounts are not balanced and hence a pure shear in one direction 

 can excite a flexural motion of the crystal. 



The strength of the coupling from the mechanical axis motion jy to 

 the shear motion y^, and the extensional motion along the optical axis 

 Zz are indicated by the coupling compliances SuHs'^iSa and SizHsi^Sss, 

 respectively. From the values of these constants we find that the 

 shear motion is more closely coupled than the z extensional motion, 

 and this is indicated experimentally by the greater width of the shear 

 line. 



Effect of a Rotation of the Longest Axis with Respect to the Electrical 



Axis on the Resonances of a Crystal 



From the qualitative explanation of the secondary resonances 



given above, it is possible to predict how these resonances will be 



affected by any change in the crystal which changes the constants 



determining the three modes of motion and their coupling coefficients. 



One method for varying these constants is to change the direction for 



cutting the crystal slab from the natural crystal. In the present paper 



consideration is limited to those crystals which have their major faces 



perpendicular to an electrical axis, i.e., a perpendicularly cut crystal 



with its longest direction rotated by an angle 6 from the direction of the 



mechanical axis. The convention is adopted that a positive angle is a 



clockwise rotation of the principal axis for a right handed crystal, 



when the electrically positive face (determined by a squeeze) is up. 



15 " Piezo-Electric Resonance and Oscillatory Phenomena with Flexural Vibration 

 in Quartz Plates," J. R. Harrison, /. R. E., December, 1927. 



