ELECTRICAL WAVE FILTERS 



433 



optical to mechanical axes are greater than 0.2, it becomes a matter of 

 some importance to investigate the causes of the additional resonances. 



Interpretation of the Measured Resonance Frequency Curves of a Per- 

 pendicularly Cut Crystal 

 The plane wave assumption is valid for crystals whose width is 

 less than 1/5 of their length, but it fails for wider crystals. It fails to 

 represent a rectangular crystal because it does not allow for a wave 

 motion in any other direction. That such a motion will occur is readily 

 found by inspecting the stress-strain equations of a quartz crystal, 

 given by equation (7). 



- Xj, = SnX^ + SiiYy + SuZ, + SuY„ 



- Jy = S12X:, + SuYy + SnZ^ — 5i4Fj, 



- z, = Sx^X^ -\- SuYy -\- SzzZ,, (7) 



—■ y, = SnXjr — SnYy + -^44^2, 



2^ = SuZx -{- SliXy, 



Xy = SuZx -T 2(-^ll Sl2)Xy, 



where Xx, yy, Zz are the three components of extensional strain, and 

 Jzy Zx, Xy the three components of shearing strains. X^, Yy, Z^, Y^, Z^, 

 and Xy are the applied stresses and 5ii, etc. are the six elastic compli- 

 ances of the crystal. Their values are not determined accurately but 

 the best known values are given in equation (42). In this equation the 

 X axis coincides with the electrical axis of the crystal, the Faxis with 

 the mechanical axis, and the Z axis with the optical axis. 



Z AXIS 



Y AXIS 



Fig. 26 — Form of crystal distorted by an applied Yy force. 



Limiting ourselves now to an X or perpendicularly cut crystal the 

 only stresses applied by the piezo-electric effect are an X^, a Yy, and 

 a Fj, stress. Hence for such a crystal only four of the six possible types 

 of motion are excited, three extensional motions Xx, yy, Zz and one shear 



