An analysis of contoured quartz resonators with beveled cylindrical edges using the correct variation of thickness

All previous works on transversely varying thickness modes in contoured quartz resonators employ an expansion in the variable thickness squared in which terms only of lowest order in the radial position squared are retained. In this work that expansion is not made and the radial position operator that occurs naturally in the equation is retained. Calculations are performed for the fundamental and third harmonic families of modes and comparisons are made with the results of earlier calculations