Impedance of Conical Sections

If one of the waveguide sections is a cone, then impedance is actually frequency dependent.

For spherical pressure waves in conical tubes, waves popagating away from the cone apex (denoted by the $+$ superscript), the impedance is given by

$$Z_n^+(r; \omega) = \frac{\rho c}{S}\frac{j\omega}{j\omega + c/r},$$ (1)

where $r$ is the distance from the observation point to the cone apex.

For spherical waves propagating toward the cone apex, the impedance is given by

$$Z_n^-(r; \omega) = \frac{\rho c}{S}\frac{j\omega}{j\omega - c/r} = Z_n^{+*}(r; \omega).$$ (2)