192 BELL SYSTEM TECHNICAL JOURNAL 



undesired resonance. However, to get a reasonably high frequency out 

 of a length vibrating type of cut requires too small a length to be practical. 

 It was not long before crystal oscillators were controlled by thickness 

 vibrating crystals whose frequencies were determined by the thickness of 

 the crystals or by their smallest dimension. Referring to Fig. 1.6A, we see 

 that the same X cut type of crystal will generate a vibration along the elec- 

 trical or X axis when a field is applied along this axis. Since the thickness 

 dimension can be made very small, a high frequency is obtainable. How- 

 ever, when the smallest dimension is used to control the frequency, a diffi- 

 culty arises not present when the largest dimension is used to control the 

 frequency, namely, that harmonics and overtone modes of all the lower 

 frequency types of motion produce frequencies near the frequency of the 

 thickness mode and it is difficult to pick out the desired mode. This was 



6X - A COS '^ COS 2nnfi 



a 



C|l-C|2 



f-^ {W I "=''^'^ ■> ^^^^ 



2t " P J ■■ ''--'-^ > "^ 2 



Fig. 1.10 — High frequency shear mode of motion 



especially true for the thickness vibrating .Y cut crystal and led to its 

 abandonment in favor of V cut crystals vibrating in shear. 



As seen from Fig. 1.6B, when a voltage is applied along the Y or mechan- 

 ical axis, a shear vibration is produced which tends to change a square into 

 a rhombus. For a large plate in which the edge dimensions are large com- 

 pared to the thickness, the motion occurs as shown by Fig. 1.10. For such 

 a plate the motion is perpendicular to the thickness, which is the direction 

 of transmission of the wave, and hence a shear wave is sometimes called a 

 transverse wave. The frequency of such a wave can be shown to be 



/ = I, |/^; (1.6) 



where / is the thickness of the plate, Cee is the shear stiffness constant and p 

 the density. The use of Y cut plates considerably improved the per- 



