The dispersion relations and vibration modes of infinite quartz crystal plates at higher frequencies

High frequency vibrations of quartz crystal plates have been subjected to extensive studies for applications in the design and analysis of quartz crystal resonators functioning at the fundamental thickness-shear mode and its overtones. One important result needed in studies and product design is the exact dispersion relations of quartz crystal plates to be used for the validation of approximate theories, notably the Mindlin plate theory and Lee plate theory, for the vibration analysis of finite plates which are essential in the proper selection of resonator configurations. Earlier analyses have been focused on the fundamental thickness-shear vibrations, which have been studied with the first-order plate equations. The analysis of quartz crystal resonators vibrating at overtone modes of the fundamental thickness-shear mode requires the employment of higher-order plate equations, which pose a challenge in the validation before them can be used for solutions of frequency and vibration modes. Clearly, in addition to the improvement of plate equations involving successive higher-order displacements through correction factors, validation of these equations with accurate dispersion relations from the three-dimensional analysis of vibrations of infinite plates are essential. The computational procedure for the accurate dispersion relations of infinite plates have been established, but available numerical results have been restricted to lower frequencies due to the confined interests at resonators of the fundamental thickness-shear mode. To extend the known methods and theory for the analysis of quartz crystal resonators, complete dispersion relations in a large frequency range will be important in deriving the approximate equations suitable for the analysis of vibrations at the higher-order overtones. The piezoelectric effect can also be considered in the dispersion relations under the current implementation. The complicated dispersion relations and patterns of mode intersection can also be used in the prediction of suitable overtone modes with better performance.