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Equations of motion accommodating thickness-twist overtones of thickness-shear and flexural vibrations of rectangular, AT-cut, quartz plates are obtained. The equations are solved in closed form for plates with all four edges free. The predicted frequencies are compared with measured resonances. X-ray topographs of typical thickness-twist overtone modes are exhibited which verify that the mode shapes correspond to those predicted.
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Mindlin et al. (1967) studied this question.
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