Variational formulation of the Stevens-Tiersten equation and application in the analysis of rectangular trapped-energy quartz resonators.

The two-dimensional scalar differential equation for transversely varying thickness modes in quartz crystal resonators operating with thickness-shear modes is formulated into variational form for trapped-energy resonators with both electroded and unelectroded regions. A theoretical analysis of rectangular trapped-energy resonators of singly rotated quartz is performed using the Ritz method based on the variational formulation. Free vibration resonant frequencies and modes are obtained. The results show the existence of trapped modes under the electrodes. The effects of various geometric and physical parameters on the trapped modes are examined. It is also found that the classical frequency prediction given by Tiersten and Smythe from an approximate analysis using the scalar differential equation has an inaccuracy on the order of 100 ppm for the fundamental mode, significant in resonator design.