430 BELL SYSTEM TECHNICAL JOURNAL 



to both the optical and electrical axes. In this appendix a study is 

 made of the modes of motion of a perpendicularly cut crystal. The 

 effect has been studied of rotating the direction of the principal axis 

 while still maintaining the principal surfaces perpendicular to the 

 electrical axis. Such a crystal is designated as a perpendicularly cut 

 crystal with an angle of rotation 6. 



The perpendicularly cut crystal has received considerable theoret- 

 ical and experimental consideration especially from Cady,^ Van Dyke,^ 

 Dye ^ and Vigoreux.^ They have assumed that the crystal has a plane 

 wave vibration, and have calculated the frequencies of resonance in 

 terms of the elastic constants and the density of the crystal, and have 

 derived equivalent electrical networks for giving their electrical im- 

 pedance. Such representations indicate that there should be one 

 resonance for the crystal, the frequency of which is inversely propor- 

 tional to the length and independent of the width of the crystal. As 

 long as the length of the mechanical axis is large compared to that of 

 any other axis, this prediction agrees with the experiment, but when 

 the length of the other axes become comparable with that of the 

 mechanical axis, the prediction is no longer fulfilled by experiment. 

 It has long been recognized that this deviation is due to the failure of 

 the plane wave assumption. Rayleigh ^^ has given a correction for 

 taking account of lateral motion, which is applicable to an isotropic 

 medium. In a crystal, shear vibrations may be set up as well and 

 for this case Rayleigh 's correction can only be regarded as qualitative. 

 Also if the other sets of resonance frequencies are to be investigated, 

 account must be taken of the resonances of the other modes of vibra- 

 tion, and their reaction on the mode to be studied. 



In this appendix experimental results have been obtained showing 

 the frequencies of resonance found in perpendicularly cut crystals of 

 various shapes and orientations. These frequencies are correlated 

 with the elastic constants of the crystal and are shown to be com- 

 pletely accounted for by them. A coupled circuit representation is 

 developed which is capable of predicting the main features of the 

 principal vibration, including the change of frequency with the shape 

 and orientation of the crystal, and the temperature-»Goefficient curves. 



Experimental Determination of the Resonant Frequencies 

 In order to investigate the modes of motion in a perpendicularly 

 cut crystal in which the main axis coincides with the mechanical axis 

 of the crystal, a set of measurements has been made on crystals whose 



2.3.4.5LoC_ Cit. 



^^ Rayleigh, "Theory of Sound," Vol. I, Chapter VII, page 252. 



