Thickness-shear modes of an elliptical, contoured at-cut quartz resonator

We study free vibrations of an elliptical crystal resonator of AT-cut quartz with an optimal ratio between the semi-major and semi-minor axes as defined by Mindlin. The resonator is contoured with a quadratic thickness variation. The scalar equation for thickness-shear modes in an AT-cut quartz plate by Tiersten and Smythe is used. Analytical solutions for the frequencies and modes to the scalar equation are obtained using a power series expansion that converges rapidly. The frequencies and modes are exact in the sense that they can satisfy the scalar differential equation and the free edge condition to any desired accuracy. They are simple and can be used conveniently for further studies on other effects on frequencies and modes of contoured resonators.