Oscillations of Hollow Quartz Cylinders Cut along the Optic Axis

Hollow quartz cylinders cut along the optic axis have been excited to vibrate at five fundamental frequencies, corresponding to radial, circumferential, longitudinal, torsional, and transverse vibrations. Empirical relations connecting the dimensions of the cylinders and the frequencies of the different modes of vibrations are formulated. All of these modes of vibrations have been confirmed by examination under polarized light and with lycopodium powder.