ELECTRICAL WAVE FILTERS 447 



For a free edge, i.e. no resulting forces being applied to the crystal, the 

 conditions existing for every point of the boundaries are 



Xt, = Xx cos {v,x) + Xy cos {v,y) + X^ cos {v,z) = 0, 

 Y, = Yx cos {v,x) + Yy cos {v,y) + Y^ cos (j/.z) = 0, (28) 

 Zy = Zx cos {v,x) + Zy cos {y,y) -\- Z^ cos {y^z) = 0, 



where j' is the normal to the boundary under consideration. 



If these equations are combined and completely solved, the motion 

 of a quartz crystal is completely determined. The results which were 

 obtained above in an approximate manner could be rigorously solved. 

 However, on account of the difficulty ^^ of the solution, this is not at- 

 tempted here. In the present section it is simply the purpose to find 

 out what resonances a crystal will have if it is vibrating in a shear mode 

 only. To avoid setting up motion in the other modes of vibration, the 

 coupling elasticities Cu, C24, C34 are assumed zero. Similarly if C\i, 

 Ci3, C23 were set equal to zero we should have the possibility of three 

 extensional modes and one shear mode vibrating simultaneously with 

 no reaction on one another, and the equation of motion would be 



P -^ = ^ (C22yy) + J Lcay,'], (29) 



The displacements u, v, and w would be the sum of the displacements 



caused by the four motions. To find the displacements and resonances 



caused by the shear mode yz, we neglect the other modes and have the 



equations 



d^v ^ d , . 



"^^ ^^ (30) 



d^W d , . 



'-^^"^ '''Vy^^'^' 



•J 



Differentiating the first of equations (30) by — , and the second by 



az 



-r- , and adding, there results, 

 dy 



d"^ / dv dw\ _ 



dy- dz^ 



(31) 



1' For example if motion is limited to the y and s directions, and the coefficient Cu 

 is set equal to zero, the equations reduce approximately to those for a plate bent in 

 flexure, and this case has never been solved for the boundary condition of interest 

 here, namelv all four edges being free to move — see Rayleigh "Theory of Sound," 

 page 372, Vol. I, 1923 edition. 



