Effects of a liquid layer on thickness-shear vibrations of rectangular AT-cut quartz plates

Thickness-shear vibrations of a rectangular AT-cut quartz plate with one face in contact with a layer of Newtonian (linearly viscous and compressible) fluid are studied. The governing equations for vibrations of piezoelectric crystal plates given in a previous article by Lee, Yu and Lin are employed in the present study. One-dimensional solutions for the normal and shearing stresses at the bottom of a liquid layer are used as approximations to the stresses of the liquid layer exerting on the crystal surface in the plate equations. Closed form solutions are obtained for both free and piezoelectrically forced thickness-shear vibrations of a finite AT-cut quartz plate in contact with a liquid layer. From the present solutions, a simple and explicit formula is deduced which includes the effect of compressional wave in the liquid layer and that of the thickness-to-length ratio of the plate. The formula reduces to the well-known frequency equation obtained by many previous investigators for infinite plates. The resonance frequency of a rectangular AT-cut quartz resonator, computed as a function of the thickness of the liquid layer, agrees closely with the experimental data measured by Schneider and Martin.