214 BELL SYSTEM TECHNICAL JOURNAL 



For a completely plated crystal such as we are considering, the potential 

 gradient Ex will be independent of the y direction, since any charge distribu- 

 tion will be equalized with the speed of light which is much higher than 

 the speed of sound in the crystal. Then equation (A.34) when differentiated 

 by y becomes 



_ ^« = _ ^^ = ^fi ^ . (A.37) 



dy dy- dy 



Introducing equation (A.36), the equation of motion for a plated crystal 

 becomes 



For simple harmonic motion the variation of | with time can be written in 

 the usual form 



^ = ^e'"\ (A.39) 



so that for simple harmonic motion equation (A.38) becomes 



where v the velocity of sound in the plated crystal is given by the formula 



v'=^. (A.41) 



psn 



A solution of equation (A. 40) with two arbitrary boundary conditions is 

 ^ = ^ cos - y -f 5 sin ^ . (A.42) 



V V 



To determine the constants A and B, use is made of equation (A.34), 

 Differentiating (A.42) 



dy V 



.4 sin - y — 5 cos -^ = Sn Vy + duEx. (AAS] 

 V J 



V 



When y = and y — I the bar length 



Yy = Yy, and F, = F,, (A.44) 



provided the crystal is driving a load. For most electrical cases the only 

 load driven is an air load and this is usually very small so that it is customary 

 to set Yy^ = Yy^ = 0. Under these conditions 



— - B = dnEx and - A sin — — B cos — = di\ E^. (A.45) 



