= Solution
Guided electromagnetic waves propagate along a structure whose boundaries select transverse field patterns. At a perfect conductor the tangential <electric field> vanishes. In a hollow rectangular guide, TE modes have $E_z=0$ and TM modes have $H_z=0$. Their longitudinal <wavenumber> obeys
$$
k_z^2=\frac{\omega^2}{v^2}-k_\perp^2,
\qquad k_\perp^2=\left(\frac{m\pi}{a}\right)^2+
\left(\frac{n\pi}{b}\right)^2.
$$
Below the <waveguide cutoff frequency> $\omega_c=vk_\perp$, that mode is evanescent rather than freely propagating. TM modes require both indices positive; TE modes may have one zero but not both. For $a>b$, the lowest cutoff is the TE$_{10}$ mode, with $f_c=v/(2a)$. A hollow single-conductor guide has no TEM mode, whereas a two-conductor transmission line can support one without this geometrical cutoff.<Electromagnetic Fields and Energy by Hermann A. Haus and James R. Melcher/Rectangular Waveguide Modes>[§13.4, mode boundary conditions, index restrictions and lowest-cutoff discussion]{ref}
For a propagating TE or TM mode, $v_p=v/\sqrt{1-\omega_c^2/\omega^2}$ and $v_g=v\sqrt{1-\omega_c^2/\omega^2}$. The <phase velocity> can exceed $v$ without superluminal information transfer. Frequency-dependent group delay can broaden pulses; excitation of several modes adds intermodal <dispersion>.
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