Book excerpt: A previous treatment of overtone modes in trapped energy resonators is extended to the case of plates with slowly varying thickness. The resulting single scalar equation is applied in the analysis of plano-convex contoured quartz crystal resonators and a lumped parameter representation of the admittance, which is valid in the vicinity of a resonance, is obtained. The analysis holds for the fundamental and anharmonic overtones of the fundamental and each harmonic overtone thickness mode. The influence of piezoelectric stiffening, electrode mass loading and electrical shorting is included in the analysis. No adjustable parameters are required in the theory. Although the basic piezoelectric differential equation employed here is quite different from the ones that have been employed in similar applications heretofore, the analysis accounting for the contouring has appeared in the literature. It is shown that calculations based on the analysis agree extremely well with experimental results obtained with contoured AT-cut quartz crystal resonators. (Author).