Thickness Vibrations of Piezoelectric Oscillating Crystals

For several years, it has been well known that, in the natural vibration of a piezoelectric oscillating crystal, there is a certain period proportional only to the thickness, but, so far as we are aware, there has been no concrete explanation for this vibration nor how its period is connected with the physical constants of the medium. The present paper shows that such a vibration, named thickness vibration for brevity, is due to the standing wave produced by interference of plane waves incident to and reflected from the plane boundary surfaces of the medium, and verifies the theoretical results by several examples. There are three normal modes in the thickness vibration, and the corresponding frequencies are given in first approximation by a representative formula: P/2π = (q/2a) (c/ρ)1/2, where q is any integer, a the thickness of the plate, ρ the density and c a certain adiabatic elastic constant depending upon the orientation of the plate with respect to the crystallographic axes of the medium. It is not always possible to excite all normal modes of vibration piezoelectrically. The electrically measured natural frequency of the thickness vibration is always a little higher than that calculated by the formula. Adiabatic elastic constants of piezoelectric crystals may be determined in first approximation by measurement of the frequencies of thickness vibrations of plates prepared from given crystals.