516 BELL SYSTEM TECHNICAL JOURNAL 



part combinations of certain fundamental types of vibration. The 

 ultimate frequency stability attained with a given crystal-controlled 

 frequency generator is then a function of the equivalent electrical 

 characteristics of the combination vibration set up in the crystal plate 

 as well as the constants of the rest of the generator circuit. In 

 particular, the frequency change in a crystal oscillator with changes 

 in tube constants or attached load is a function of the equivalent 

 electrical decrement of the vibration which the crystal happens to be 

 executing. Further, the temperature coefficient of frequency of the 

 crystal oscillator depends largely upon the temperature coefficient of 

 frequency of the crystal vibration, which in turn depends upon the 

 change with temperature of the various mechanical elastic constants 

 that are called into play by this vibration. 



The general relation between stress and strain, which in an ordinary 

 isotropic medium involves only two constants, in crystal quartz 

 requires six.^ The choice of a particular constant or constants that 

 enter into a given mode of vibration depends upon the orientation of 

 the plate with respect to the original crystal axes, and the particular 

 type of vibration, whether longitudinal, torsional, etc. 



It is to be expected, therefore, that there will be a variation among 

 the characteristics of the modes of vibration of plates cut in a different 

 fashion, as well as between the different modes of a given plate. In 

 practice we have found considerable difference in the magnitude of the 

 electric and electrothermal constants, between the various modes of 

 vibration of a given crystal plate, even when the vibration frequencies 

 are within a few hundred cycles of each other. 



To secure uniformity of results with respect to frequency stability 

 it becomes necessary, therefore, to study the various possible modes of 

 vibration of these crystal plates in detail, and set up certain criteria 

 by which it will be possible to produce plates that will vibrate in a 

 definite mode whose characteristics are known. 



The theoretical aspects of this problem offer considerable difficulty, 

 for it will be remembered that the classical case of the vibrations of 

 an isotropic plate whose edges are free has as yet only been solved 

 approximately,^ and with the extension of the theory made necessary 

 by the crystalline nature of quartz, the complexity of the problem is 

 considerably increased, with the possibility of a complete solution 

 very remote. 



Using long rods or bars of crystal, instead of plates, other investi- 



^Voigt's "Kristallphysik," pp. 749-755, or Love's "Mathematical Theory of 

 Elasticity," Chap. VI. 



2 Rayleigh, "Theory of Sound," Chap. X and X^. 



