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Generalizations are made for three types of well-known and widely used twodimensional scalar differential equations in the literature describing transversely varying thickness modes in piezoelectric plate resonators. They are for singly-rotated quartz plates, doubly-rotated quartz plates, and plates of crystals of class 6mm with the c-axis along the plate thickness, respectively. The purpose of the generalizations is to include the effects of surface mechanical loads such as mass layers or fluids for resonator -based acoustic wave sensor applications. Surface acoustic impedance is introduced to take into account various surface loads in a general manner for time-harmonic motions. Both unelectroded and electorded plates are treated. For electroded plates, both free and electrically-forced vibrations under a time-harmonic driving voltage are considered. Simple two-dimensional scalar differential equations are constructed from the asymptotic dispersion relations quadratic in the small wave numbers of transversely varying thickness waves at long wavelengths. The equations obtained can be reduced to the equations in the literature in the special case when the surface acoustic impedance is set to zero. As illustrations of the usefulness and effectiveness of the equations obtained, simple examples of pure thickness vibrations of unbounded plates with surface loads, propagation of long thickness waves in unbounded plates with surface loads, and vibrations of finite plates with surface loads are presented. It is expected that a lot of theoretical results can be obtained in the future using the equations derived in this dissertation for piezoelectric plate acoustic wave sensors.