= Solution
Boundary conditions quantize the transverse <wavevector> of an <electromagnetic waveguide>. For an ideal hollow rectangular conducting guide of sides $a,b$, filled with a nondispersive medium of wave speed $u$, a <transverse electric mode of a rectangular waveguide> or <transverse magnetic mode of a rectangular waveguide> obeys
$$
\boxed{\omega^2=\omega_c^2+u^2\beta^2},\qquad
\omega_c=u\pi\sqrt{(m/a)^2+(n/b)^2}.
$$
The allowed integer pairs differ: TE permits one zero index, but not both; TM requires both positive. Here $\beta$ is the longitudinal propagation constant. Below cutoff $\beta$ is imaginary, so the mode is evanescent. Above cutoff,
$$
v_p=\frac\omega\beta=\frac{u}{\sqrt{1-(\omega_c/\omega)^2}},\qquad
v_g=\frac{d\omega}{d\beta}=u\sqrt{1-(\omega_c/\omega)^2},\qquad
\boxed{v_pv_g=u^2.}
$$
The positive branch approaches the straight line $\omega=u\beta$ far above cutoff, while near cutoff its slope tends to zero. Different <frequencies> acquire different group delays and a pulse can spread. A <wave phase> speed above $u$ does not convey superluminal information; in this ideal guide <energy> transport follows the <group velocity>. A single hollow conductor cannot support a TEM mode; two-conductor guides can have a zero-cutoff TEM branch. Material dispersion, dielectric fibres and lossy boundaries require their actual <dispersion relation> and need not preserve the product rule.
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