# Waveguide modes: does a particular mode only propagate for one particular frequency?

I am trying to understand modes in a cylindrical dielectric waveguide. Does a particular mode only propagate for one particular frequency? Or, does a mode exist simply as long as the signal is above the cut-off frequency?

Basically, if I propagate a signal of a single frequency down the waveguide (a frequency that I know corresponds to the peak resonance of some mode), will I excite only that one mode?

Transverse (TE and TM) modes in a recangular waveguide typically have dispersion equations $$\omega(k_z)$$ of the form $$\omega^2= ak_z^2 + \omega^2_{\rm min},$$ where $$a$$ and $$\omega_{\rm min}$$ depend on the mode and the geometry of the waveguide and $$z$$ is the direction down the waveguide. So, yes, once the frequency $$\omega$$ is greater that $$\omega_{\rm min}$$, there will be some real wavenumber $$k_z$$ for which the mode will propagate at the given frequency. For frequencies lower than $$\omega_{\rm min}$$ the wavenumber $$k_z$$ will be imaginary, and so the mode will be evanescent.
• The extent to which a given mode is excited by a source depends on the impedence matching. In TE and TM dispersive modes the impedence does depend strongly on the frequency near $\omega_{\rm min}$ as you can read here:en.wikipedia.org/wiki/Wave_impedance. The impedence does not depend on the frequency in non-dispersive TEM modes such as in a coaxial cable. Dec 16, 2020 at 13:27