MODES OF MOTION IN QUARTZ CRYSTALS 75 



except where they are coexistant. We can then state generally that ev^en 

 though there is coupling between particular modes of motion, if the difference 

 in order is great, the approximate frequencies may be computed as though 

 they were isolated. This is more clearly shown in the case of thickness 

 shear modes. The modes that are shown coupled to the face shear mode 

 are Zx flexures propagated in the direction of the length or X axis. The lower 

 orders can be shown to follow the general frequency equation discussed 



w . . 

 in section 6.3 but the higher orders for a given — , it will be noticed, are regu- 

 larly spaced in frequenc}' and show the eflfect of shear. The Xy flexure modes 

 determined by the length and thickness are shown as nearly horizontal 

 lines since only the width was changed. Since these two groups of flexure 

 modes are propagated in the same direction, it would be expected that the 



(-■f=0 



difference in frequency for the same ratio of dimension ( i.e., -7=7) would 



be due to the differences of the shear coefficients in the two planes of motion. 

 The vertical dotted line indicates the ratio of thickness to length. When 

 the ratio of width to length is equal to this value it can be seen that the 

 flexure modes in the width-length plane are in all cases higher than the same 

 order flexures in the thickness-length plane. An examination of Fig. 6.7 

 shows that for an .4C-cut crystal the shear modulus in the width-length 



plane ( /-j- \ is greater than that in the thickness-length plane ( /-;- J . 



This is in agreement with the observation made above. One other generality 

 may be drawn from the experimental data shown in Fig. 6.12. The coupling 

 between flexure modes and shear modes in planes at right angles to each 

 other is very small in comparison with that between modes in the same 

 plane. 



As mentioned before the eflfect of coupling between modes of motion is 

 greatest when the orders are more nearly similar. In this particular crystal 

 this effect can be shown between the fundamental width-length Zx shear and 

 the second order width-length Z^ flexure. This is shown in Fig. 6.13 which 

 is an extension of the data shown in Fig. 6.12 for a crystal nearly square 

 and shows the frequency range covered only by the second flexure and 

 the fundamental plate shear. A computation of the uncoupled second flex- 

 ure mode propagated along the length and the first plate shear mode are 

 shown by the solid lines // and /« respectively. Inserting the appropriate 

 constants the formulae of section 6.3 become 



. 1 ^ /7.85 X 10" 2Z' ... 



•^^ = 2^ y 12 X 2.65 ^^ ^''' 



, 1 /71.8 X 101° /-I— f 



^' = 2V 2.65 VX^ + Z^ '-'' 



