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BELL SYSTEM TECHNICAL JOURNAL 



Since— + -7- = jz, this reduces to 

 dz dy 





(32) 



where c^ = CuIp- 



For a simple harmonic vibration, of frequency /, the equation 

 reduces to 



2 "I" 110.2 



_ dy 



dz^ 





0, 



{S3) 



where p = It/. The solution of this equation consistent with the 

 boundary conditions (28) is 



ZZA, 



. miry . nwz 



sm sm — J— 



a 



cos pt, 



(34) 



where a is the length of the crystal in the y direction, h the length of the 

 crystal in the z direction, and m and n are integers. Substituting this 

 equation in the equation (32), we find that it is a solution provided 





(2^fy 



Hence the resonant frequencies of the crystal in shear vibration are 



(35) 





(36) 



To find the shape of the deformed crystal, we have from (30) for 

 simple harmonic vibrationNl, 



V = — 



W = rs 



p^ dz 

 c2 dyz 



- a'^P 



dyz 



m'^TT^b'^ + n^TT^a^ dz 



- a'b^ dy. 



(37) 



m 



p^ dy m^TrW + n^TT^a^ dy 



The cases m = 0, n = 1 and m = 1, n = require a stress known 

 as a simple shear to excite them, whereas the stress applied by the 

 piezo-electric effect is a pure shear. Hence the case m = I, n = 1 

 provides the lowest frequency solution. The displacements v and w 

 for this case are by equations (34), (37) and (38) 



— a^bir I . -wy TTZ 

 ^ ~ 9 9 I — ^r^ sm -^ cos -7- 



ab^TT / Try . TTZ 



"^ — T~r~i — 513 cos -^ sm -7- 



ir^a^ + T^ b~ \ a b 



(39) 



The resulting distortion of the crystal is shown by Fig. 34. 



