A corrected modal representation of thickness vibrations in quartz plates and its influence on the transversely varying case

A modal representation of the thickness vibrations of rotated Y-cut quartz plates, which was used in the treatment of driven transversely varying thickness modes, is shown to be defective in certain respects. The differential equations and edge conditions for transversely varying thickness modes have been used in the accurate treatment of trapped energy resonators, monolithic crystal filters, and contoured quartz resonators, even though those defects were present. In this work those defects in the thickness solution are corrected along with the influence on the differential equations and edge conditions in the transversely varying case. The corrected modal representation shows that, because in practical applications to the above mentioned devices, the driving frequency is always near a thickness resonant frequency, essentially the same results will be obtained with the corrected representation as were obtained with the defective one, which explains why the results obtained with the defective equations were so accurate.