High Frequency Vibrations of Thin Crystal Plates

The boundary conditions of free vibration can be satisfied on the major surfaces of a plane-parallel plate if the displacement components are assumed to be products of trigonometric functions. In addition, the boundary conditions can be approximately satisfied on the minor surfaces when the plate is thin. The theory leads to a frequency equation $\nu =\frac{1}{2}{\left(\frac{c}{\rho }\right)}^{\frac{1}{2}}{[{\left(\frac{n}{2b}\right)}^2+k{\left(\frac{m}{2a}\right)}^2]}^{\frac{1}{2}}$ which has been found empirically to satisfy observations. The theoretical values of the constant $k$ are 3.7 and 1.8 for the $\mathrm{AT}$ and $\mathrm{BT}$ quartz plates, respectively, while the observed values are 3.9 and 1.7, respectively.