Eigenfrequencies of a Truncated Conical Resonator via the Classical and Wentzel–Kramers–Brillouin Methods

The eigenfrequencies within a truncated conical cavity resonator have similarities to the eigenfrequencies within both the cylindrical and spherical cavity resonators. This paper first find the family of eigenfrequencies of the truncated conical cavity resonator by solving the classical boundary value problem in the spherical coordinate system, similar to a spherical cavity resonator. Next, eigenfrequencies for the truncated conical resonator are found via the cylindrical coordinate system using the Wentzel-Kramers-Brillouin (WKB) method by making the approximation that the truncated conical cavity with a small half-cone angle is a cylindrical cavity with linearly sloping walls. While the WKB method is an approximation to the actual eigenfrequencies, the solution yielded by this method is far more simple than the solution yielded by the classical boundary value method. Resonant frequencies are derived via both methods and are compared to each other and to measurements made in an experimental truncated conical cavity platform, displaying a very good agreement.