ELECTRICAL WAVE FILTERS 



445 



Hence from equations (17), (18) and (21) one should be able to check 

 the measured frequency curves of Fig. 31. The result is shown on the 

 dotted lines of these curves. The agreement is quite good although a 

 slightly better agreement would be obtained if 523 had a smaller value. 

 Since these constants have never been measured with great accuracy, 

 it is possible that they deviate somewhat from the curves of Fig. 30. 



This theory can be applied also to a = + 18.5 degree cut crystal 

 since the extensional coupling coefficient vanishes for this angle. The 

 agreement is quite good if the frequency for the uncoupled mode given 

 by the section on vibration in shear mode (page 446) is used in place of 

 equation (14). The resonances for the = 0° cut crystal shown by 

 Fig. 25 cannot be accounted for quantitatively by the simple theory 

 given here since there are three modes of motion operating. The 

 shear mode of motion is more closely coupled to the principal mode 

 than is the Z^ extensional mode and hence a fair approximation is ob- 

 tained by considering only the shear mode. However, for complete 

 agreement the theory should be extended to a triply coupled circuit and 

 that is not done in this paper. 



Another phenomenon of interest which can be accounted for by the 

 circuit of Fig. 335 is the temperature coefficient of the crystal and its 

 variation with different ratios of axes and different angles of rotation. 

 To obtain the relation, we assume that each of the vibrations may have 

 a temperature coefficient of its own as may also the coefficient of 

 coupling K. If a small change of temperature occurs, /i will change 

 to /i(l + T^T), /a to /^(l + TaAT), Jb to /^(l + TbAT) and K to 

 K{1 + Tk^T). Assuming AT" small so that its squares and higher 

 powers can be neglected, we find from equation (17) that 



r = ^ 



TaJa' 



1 + 



fsW -2K') -/a' 



X 



//(I + 2K' 



+ Tb/b' 



Ia' 



<Ub' 



fA^y + ^KjAjs'^ 

 2TKfAjB' 



^{fB'-fA'y-i-^KjAjB'} 



(22) 



The temperature coefficients of the six elastic constants have been 

 measured at the Laboratories ^^ by measuring the frequency tempera- 

 ture coefficients of variously oriented crystals. The temperature 

 coefficients of the six elastic constants can be calculated from these 



'* These coefficients have been evaluated in cooperation with Messrs. F. R. 

 Lack, G. W. Willard and I. E. Fair and their work is discussed in detail in their 

 compani6n paper in this issue of the B. S. T. J. 



