Thickness-shear and thickness-twist vibrations of centrally partially electroded circular AT-cut quartz plate

This paper analyses free vibration frequencies and modes for thickness-shear and thickness-twist vibrations of centrally partially electroded circular AT-cut quartz plate. The elliptical coordinate is introduced for both un electroded and electrode part. Exact solutions of Mathieu equations which are transferred from the two-dimensional scalar differential equation derived by Tiersten and Smythe are obtained. Trapped modes are found whose vibrations are confined mainly under the circular electrodes.