444 BELL SYSTEM TECHNICAL JOURNAL 



we solve the network of Fig. ^2)B to find the frequencies of zero im- 

 pedance for either an appUed Yy force or an applied Z-, force. The 

 result is two frequencies /i and/2 given by the coupled circuit equations 



(17) 



where 



IwyLyCy liryLzCg 



Then /a and Jb represent the natural frequencies along the Y and Z 

 axis respectively when these two motions are not coupled together. 



Two limiting cases of interest are obtainable from these relations. 

 If /b is much larger than /a, the equations reduce to 



1 



/i =/aV1 -K' = 



f^ = fj^ = ^ 



2k^pszz'[_\ — S'iz'^-jsii.'szz''] 



upon substituting the value of the constants given before. The first 

 equation shows that for a long thin rod the frequency depends on 

 the elastic constant 522', which is the inverse of Young's modulus. 

 For the frequency J2, which corresponds to that of a thin plate, a 

 different elastic constant appears. Upon evaluating the expression 

 ■^33'[1 — S2z''^js22'szz'^ in terms of the elastic constants which express 

 the forces in terms of the strains — see equation (25) — we find that 

 5i3'(l — Siz'^lsii'szz') = I/C33. C33 measures the ratio of force to strain 

 when all the other coupling coefficients are set equal to zero, and 

 corresponds to the frequency of one mode vibrating by itself without 

 coupling to other modes. Hence the frequency of a thin plate should 

 be 



f-^T' ^'") 



where c„„ represents the elastic coefficient for the mode of motion con- 

 sidered, and t is the thickness of the plate. This deduction has been 

 verified by experimental tests on thin plates. 



Let us consider now the curves for the —18.5 degree cut crystal 

 shown by Fig. 31. The values of the elastic constants for this case are 



522' = 144 X 10-1^ cm.Vdynes; 523' = - 21.0 X IQ-i^ 



533' = 92.5 X 10-1^ (21) 



