Abstract
We study thickness-shear vibrations of a piezoelectric plate of AT-cut quartz with two pairs of electrodes. The plate represents a monolithic, two-pole acoustic wave filter. The scalar differential equations by Tiersten and Smythe for thickness-shear vibrations of electroded and unelectroded quartz plates are employed. Based on the variational formulation of the scalar differential equations established in a previous paper and the Ritz method with trigonometric functions as basis functions, free vibration resonant frequencies and thickness-shear modes of the plate are obtained. For a structurally symmetric filter, the modes can be separated into symmetric and antisymmetric ones. Trapped modes with vibrations mainly under the electrodes are presented. The effects of the electrode inertia and the distance between the two pairs of electrodes are examined. It is also found that the classical frequency prediction given by Tiersten from an approximate analysis has an inaccuracy of tens of parts per million, significant in filter design and application.