Rectangular Waveguide Modes

A hollow rectangular waveguide carries only discrete TE and TM modes. Each has a cutoff frequency $f_c = (c/2)\sqrt{(m/a)^2 + (n/b)^2}$; above it the mode propagates at the guide wavelength $\lambda_g = 2\pi / \beta$ (longer than free space), below it $\beta$ is imaginary and the field is evanescent, carrying no power. The primary scene is physical: the transverse field map of the chosen mode in the $a \times b$ cross-section and a longitudinal strip showing the wave travelling down the guide or decaying when below cutoff. The side panel is the mode-cutoff spectrum with the operating frequency, so single-mode operation is visible.