442 BELL SYSTEM TECHNICAL JOURNAL 



bar. This is contrary to actual conditions, since expansions or con- 

 tractions proceed in the form of a wave from the ends of the bar toward 

 the center. However, if consideration is Hmited to low frequencies, 

 i.e. frequencies which do not exceed by much the first resonance of the 

 bar, the approximation is good and a considerable simplification in the 

 analysis is made. To take account of wave motion, the representation 

 has to be an electric line as was pointed out in connection with acoustic 

 filters.16 



To represent two separate modes of motion and their coupling, the 

 circuit shown by Fig. 2>d)B is employed. A little consideration shows 

 that the type of coupling existing in a crystal is capacitative since an 

 extension along the mechanical axis produces a contraction along the 

 optical axis, and vice versa. Since strains in mechanical terms are 

 equivalent to charges in electrical terms, this type of coupling can be 

 represented only by a capacitative network. This representation is 

 entirely analogous to the T network representation for a transformer.^^ 

 The constants of the network can be evaluated in terms of the elastic 

 constants of the crystal as follows: For a —18.5 degree cut crystal, 

 we can write the stress strain equation (7) as 



Jy = 522'F, + 523%, (8) 



since we are neglecting motion along the X axis and since ^24', the 

 coupling coefficient of the shear to the Y' axis is zero. No Y^ force is 

 assumed acting. If we work out the equation for the charges on the 

 condensers of the equivalent representation shown in Fig. ZZB we have, 

 with the charges and voltages directed as shown 



Qi = 1 ^. + e. 



1 - K^ ' M - X^ ' 



, (9) 



„ ^ ey\CyC,K e^Cz 

 ^' 1 - i^2 -t- ^ _ ^2 , 



where K the coupling factor between the two modes of motion, is de- 

 fined by the relation 



K = ^^ . (10) 



Associating Qi with jy, the displacement per unit length, Q^ with 



1^ See "Regular Combination of Acoustic Elements," W. P. Mason, B. S. T. /., 

 April, 1927, p. 258. 



1^ See, for example, p. 281 in the book "Transmission Circuits for Telephonic 

 Communication" by K. S. Johnson. 



