446 BELL SYSTEM TECHNICAL JOURNAL 



measurements and have been found to be, in parts per million per 

 degree centigrade: 



r.^^ = + 13; r,., = - 1230; T.^^ = - 347; (23) 



r.^, = + 130; r.33 = + 213; T,^ = + 172. 



Using these values and neglecting the extensional motion Z^, the tem- 

 perature coefficients calculated from equation (22) for a degree cut 

 crystal are shown on the dotted line of Fig. 5 and agree quite well with 

 the measured values. 



The Resonance Frequencies of a Crystal Vibrating in a Shear Mode 

 The equations of motion for any aelotropic body are 

 d^u ^ dX^ dXj dX. 



df dx By dz 



(24) 



where u, v, w are the displacements of any point in the crystal along the 

 X, y, z axes respectively and X., etc. are the six applied stresses. The 

 strains have been expressed in terms of the stresses by equation (7). 

 It is more advantageous for the present purpose to express the stresses 

 in terms of the strains, which can be done by the following equations: 



Xx = CnXx + Cuyy + Cisz^ + Cuyz, 



Yy = Ci2X^ + C22yy + CisZ^ + C24>'2, 



Z^ = CisXx + Cosyy + CssZ^, (25) 



Yz = CuXx -\- C24yy + Cuyz, 



z^ X ^ c^iZx ~r CiiXy, 



Xy = CliZx + 2(^11 ~ Cl2)Xy, 



where the c's are the elastic constants and the strains Xx, etc., are given 

 in terms of the displacements u, v, w by the equations 



Bu Bv Bw I Bv , Bw 



Bx'-"" By' ' Bz ' ^' \Bz^ By 



-^ + -^j;x, - y^ + ^j- 



(26) 



In equation (24) there exist the reciprocal relations 



Xy = Yx; X, = Zx\ F, = Zy. ill) 



