Frequency Spectra of AT-Cut Quartz Plates with Electrodes of Unequal Thickness

We study vibrations of an AT-cut quartz plate with electrodes of unequal thickness. Mindlin's first-order plate theory is used. Dispersion relations for straight-crested waves in an unbounded plate and frequency spectra of modes in a finite plate are obtained. Results show that, because of the unequal thickness of the electrodes, the two originally uncoupled groups of modes are now coupled. One group consists of thickness-shear, flexure, and face-shear. The other has thickness-twist and extension. This coupling by asymmetric electrodes changes the frequency spectra and affects resonator performance. To avoid the effects of this coupling, a series of discrete values of the length/thickness ratio of the crystal plate need to be excluded.