Perturbation theory for degenerate acoustic eigenmodes

A general formulation of the perturbation procedure applied to the eigenvalue problem for resonant frequencies in crystal resonators under biasing deformations is given, with explicit treatment of the case wherein an unperturbed resonant frequency is degenerate. A rotated Y-cut quartz resonator with degenerate thickness-shear modes (b-mode/c-mode crossover point) is analyzed as an example. Results indeed show that biasing fields can cause a degenerate frequency to split into slightly different frequencies. This phenomenon may be responsible in part for the jump discontinuities in frequency temperature curves and other frequency jump phenomena.