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Thickness-shear vibrations of rectangular AT-cut quartz with one face in contact with a layer of Newtonian (linearly viscous and compressible) fluid are studied. The two-dimensional (2D) governing equations for vibrations of piezoelectric crystal plates given previously are used in the present study. The solutions for 1D shear wave and compressional wave in a liquid layer are obtained, and the stresses at the bottom of the liquid layer are used as approximations to the stresses exerted 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, rectangular AT-cut quartz plate in contact with a liquid layer of finite thickness. From the present solutions, a simple and explicit formula is deduced for the resonance frequency of the fundamental thickness-shear mode, which includes the effects of both shear and compressional waves in the liquid layer and the effect of the thickness-to-length ratio of the crystal plate. The formula reduces to the widely used frequency equation obtained by many previous investigators for infinite plates. The resonance frequency of a rectangular AT-cut quartz, computed as a function of the thickness of the adjacent liquid layer, agrees closely with the experimental data measured by Schneider and Martin (Anal. Chem., vol. 67, pp. 3324-3335, 1995).