ELECTRICAL WAVE FILTERS 



443 



z^, By with Yy and e^ with Z., we have on comparing (9) with (8) 



522 



c. 



1 - K^ 



; ^23 — 



1 - K^ 



; ^33 



a 



1 - K^ 



or inversely 



Cj/ — 522 



1 - 



523 



522 533 



;a 



533 



1 



523 



522 533 



\K = 



523 



A 522 533 



(11) 



(12) 



If, now, alternating forces are appUed to the crystal, another reac- 

 tion to the applied force enters, namely the mass reaction of the crystal 

 due to the inertia of the different parts. To take account of this reac- 

 tion, the inductances are added to the two meshes representing mass 

 reaction for the two modes of motion. To determine the value of the 

 inductance, consider first the representation for one mode of motion 

 shown by Fig. i^A. The resonant frequency of the system is given by 



fr 



Itt^LC 



On the other hand, the resonant frequency of a bar is given by 



f^_ 1_, 



(13) 



(14) 



where / is the length of the bar, 5 its compliance, and p its density. 

 But in the above representation the capacitance C is the compliance 

 constant 5 so that, on comparing (13) and (14) we find 



TT" 



(15) 



In a similar manner for the coupled circuit. Fig. 33B, there results 



T -Hp.T - 



(16) 



where ly is the length of the crystal in centimeters along the y axis, and 

 Iz the length of the crystal in centimeters along the z axis. Hence all 

 of the constants of Fig. 33B, which represents the crystal for mechanical 

 vibrations subject to the restrictions noted above, are determined and 

 we should be able to predict all of the quantities which depend only 

 on the mechanical constants of the crystal. 



Of these the most important are the resonance frequencies of the 

 crystal and their dependence on dimension, temperature coefficient and 

 the like. To determine the natural mechanical resonance of a crystal, 



