434 BELL SYSTEM TECHNICAL JOURNAL 



motion y^. Under static conditions, then, the motion at any point in 

 the crystal is given as the sum of four elementary motions, three ex- 

 tensional motions and one shear motion. Moreover, these motions are 

 coupled ^^ as is shown by the fact that a force along one mode produces 

 displacements in other modes of motion. Figure 26 shows how a 

 perpendicularly cut crystal will be distorted for an applied Yy force. 



Suppose now that an alternating force is applied to the crystal. 

 The simplest assumption that we can make regarding the motion is 

 that the motion of any point is composed of four separate plane wave 

 motions of the four types of vibration and that these react on each 

 other in the way coupled vibrations are known to act in other mechan- 

 ical ^^ or electrical circuits. For the present purpose we can neglect 

 motion along the X or electrical axis since this axis has been assumed 

 small. The three remaining motions if existing alone will have reso- 

 nances as shown by the solid lines of Fig. 27. That along the mechan- 

 ical axis will have a constant frequency, since the mechanical axis is 

 assumed constant, and is shown by the line C. The extensional motion 

 along the optical axis will have a frequency inversely proportional to 

 the length of the optical axis and will be represented by the line A of 

 the figure. The shear vibration y^, as shown by the section on the 

 resonance frequency of a crystal vibrating in a shear mode, will have 

 a frequency varying with dimension as shown by the line B. 



In view of the coupling between the motions, the actual measured 

 frequencies will be as shown by the dotted lines in agreement with well 

 known coupled theory results. 



If we compare these hypothetical curves with the actual measured 

 values some degree of agreement is apparent. The main resonant 

 frequency except in the region 0.2 < Ujlm < 0.3 follows the dotted 

 curve drawn. Also, the extensional motion along the optical axis has a 

 frequency agreeing with that of Fig. 25. The shear vibration, however, 

 has an entirely different curve from that conjectured. What is happen- 



" The idea of elementary motions in the crystal being coupled together appears 

 to have been first suggested in a paper by Lack "Observations on Modes of Vibration 

 and Temperature Coefficients of Quartz Crystal Plates," B. S. T. J., July, 1929, 

 and was used by him to explain the effect of one mode of motion on the temperature 

 coefficient of another mode and vice versa. The idea of associating this coupling 

 with the elastic constants of the crystal occurred to the writer in 1930 but was not 

 published at that time. It is, however, incorporated in a patent applied for some 

 time ago on the advantages of crystals cut at certain orientations. More recently 

 the same idea is given in a paper by E. Giebe and E. Bleckschmidt, Annalen der 

 Physik, Oct. 16, 1933, Vol. 18, No. 4. They have extended their numerical calcula- 

 tions to include three modes of motion. 



'* This coupling is shown clearly for a mechanical system by one of the few 

 rigorously solved cases of mechanical motion for two degrees of freedom — the vibra- 

 tion of a thin cylindrical shell — given by Love in "The Mathematical Theory of 

 Elasticity," Fourth Edition, page 546. 



