84 BELL SYSTEM TECHNICAL JOURNAL 



These modes have been measured in AT and ^T-cut crystals. Fig. 6.18 

 shows the points at which these modes intersect the fundamental high fre- 

 quency shear mode in A T-cut plates. This is the case for high orders along 

 the Z' or width axis. A similar set of resonances can be shown to exist 

 when the X or length axis is varied. Experiment has shown that these 

 frequencies of coincidence between high order plate shear modes and the 

 fundamental high frequency Xy' shear mode for the case of .4r-cut plates is 

 given by 



254 2 

 fxs = -^TT- ftxa kilocycles 6.24 



fz'a = —^ «z'« kilocycles 6.25 



where X and Z' are given in centimeters. Only odd orders are strongly coupled 

 if the crystal plate has a symmetrical contour with respect to an applied 

 equipotential electrode. Upon substitution of the value of C55 for an .4 T-cut 

 crystal in equation 6.22 there results 



/. X / = ^ /^-^ = \ j/^^^l^ = 251.0 kilocycle - cm. 6.26 



which is within 1 per cent of that found experimentally. Since Young's 



modulus is nearly the same along the A" and Z' axis the value of k in equation 



6.23 is essentially unity. Fig. 6.19 shows measured values of high order Z^ 



shear modes near the high frequency A^- shear mode in a i^T-cut crystal 



for various values of the width or Z' axis. More detailed measurements 



have been made of the high order Z^ plate shear modes in ^T-cut plates 



along the X axis. Fig. 6.20 shows both the shear and flexure modes along 



the X axis near the vicinity of the high frequency Xy' shear mode. Since the 



frequency constant for the Z^ shear modes is different from that for the Xy' 



flexures there are regions where, if no coupling existed, all three modes would 



be at the same frequency. It is obvious from Fig. 6.20 that this is not the 



case. Therefore, we must assume that not only are the high order Zx 



shears and Xy> flexures coupled to the high frequency Xy- shear but that they 



are coupled to each other. 



While it is difficult to see from Fig. 6.20 the relative coupling of flexures 



to the Xj^' shear, experiment has shown the flexure modes along A' to have 



X . 

 the greater coupling to the A'^' shear. This is true when the ratio — , is 



such that the flexure modes along A' and high order Zx shear modes along A" 

 have their maximum separation. When these modes approach each other 



