The frequency equation of thickness-shear vibrations of SC-cut quartz crystal plates

The thickness-shear vibrations of a quartz crystal plate serve as the functioning mode of a resonator with strong couplings to many other modes of vibrations, which can affect the frequency and mode shape, consequently the resonator properties. For applications, it is always desired to have pure thickness-shear vibrations that only exist in an infinite plate. With the three-dimensional equations of elasticity in Cartesian coordinates, boundary conditions are approximated and the frequency equation of the thickness-shear mode is given in terms of materials constants and plate parameters. This relation shows that the thickness-shear frequency can be determined with known parameters of plates. The frequency equation then is compared with the frequency spectra from the Mindlin plate equations for the determination of some key coefficients. This is an extension of our earlier study on the frequency equation of AT-cut quartz crystal plates and it will be used in the design of SC-cut quartz crystal resonators of rectangular shape.