The general differential equations which describe the vibrations of plates of quartz or tourmaline are stated together with the corresponding boundary conditions for a free plate. Solutions are presented for the longitudinal modes in which pure compressional waves are propagated along X, Y or Z in quartz or tourmaline. Experiment shows that these modes are purest (i.e., their displacements most nearly limited to the direction of propagation) in rectangular plates whose greatest dimension is that along which the compressional wave trains are propagated. As this dimension is shortened, the longitudinal modes decrease in purity and finally fail to appear. The purest longitudinal mode in the usual X, Y or Z cuts of quartz is that in which compressional waves are propagated along X in Y or Z cuts. The vibrational patterns which correspond to the mode in which compressional waves are propagated along Y in X or Z cuts of quartz are so complicated that the correspondence is open to considerable doubt. These latter longitudinal modes were found in tourmaline, and appeared to be simpler than the corresponding modes in quartz. The elastic constants c11 and c33 were obtained with an experimental error of but 1/3 percent. These measurements differ by 3 to 7 percent from those made by Voigt. The simple shear, pure shear, transverse and the yz, zx and xy shear modes are defined and distinguished. The differential equations of motion are not satisfied by free periodic vibrations of the simple shear or the pure shear types. No experimental evidence was found for the existence of any type of free simple or pure shear modes in a large variety of X, Y and Z cuts of quartz. Whereas forced simple shear vibrations are theoretically possible in a Y cut, the free simple shear vibrations are not. Solutions are presented for the various types of transverse vibrations. Experimental evidence for these transverse vibrations was not obtained. More general theoretical considerations indicate that the free transverse modes do not exist. The yz, zx and xy shear modes were observed in suitably oriented Y cuts of α quartz. The observations upon the latter modes are compared with the observations of Mason and Lack, Willard and Fair. It is concluded that the mode which was employed by Lack, Willard and Fair was not a transverse mode, but rather, the xy shear mode. The predominant modes of vibration are of longitudinal, flexural or torsional nature. Crystals of β quartz are active oscillators even at 800°C and will be shown to be piezoelectric.

Osterberg, H., & Cookson, J. W. (1935). Longitudinal, Shear and Transverse Modes of Vibration in Quartz and Tourmaline. Physics, 6(7), 246–256.