Optical studies of longitudinal vibrations in circular z-cut quartz discs

Abstract

A historical review of the use of powder patterns, polarized light and intorferometry to examine small vibrations is given. Suitably cut quartz crystals are used for an examination by these methods of the longitudinal vibrations of thin isotropic discs. Previous calculations of the frequencies and displacements for these modes of vibration are extended and the distribution of stress is computed numerically. The theory underlying the use of the three methods of observation Is discussed with particular reference to the examination of these stresses and displacements. It Is found that for sufficiently thin discs the primary stresses and displacements are substantially in agreement with theory, except for the presence in all cases of a coupled flexural mode having a particular symmetry dependent upon the symmetry of the longitudinal mode. Moreover, for those modes which do not have circular symmetry about the disc axis, there are always two frequencies of vibration for which the orientations of the vibratory systems have a particular relation, again dependent upon the particular modal symmetry. A theory is given whichadequately explains the coupling phenomena by reference to the cross shear strains which are due to the particular crystal symmetry of quartz and are not possible for isotropic media. In connection with the polarized light experiments, a theory of the dependence of the apparent rotatory power of quartz upon stress is developed and agrees well with an experimental test; some phenomena observed by earlier workers are explained on the basis of this theory. It is shown that there is good correlation between the three methods of observation but that the information yielded by any one method separately is likely to give an oversimplified representation of the state of vibration.