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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 obeysThe 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,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.

For a hollow, air-filled rectangular electromagnetic waveguide with conducting walls, transverse confinement givesA transverse electric mode of a rectangular waveguide permits one zero index, but not two; a transverse magnetic mode of a rectangular waveguide requires both indices positive. Take $w=20\mathrm{mm}$ and $h=10\mathrm{mm}$. The dominant mode is ${\mathrm{TE}}_{10}$, withThe waveguide cutoff frequency is the limiting value: positive axial group velocity and propagating power require $f>f_c$. There is no transverse electromagnetic mode in this single hollow conducting pipe; otherwise a zero cutoff would incorrectly follow.

For the usual hollow rectangular electromagnetic waveguide, let the propagation axis be $z$ and the interior be $0<x<a$, $0<y<b$. A transverse magnetic polarization has $B_z=0$ and $E_z\ne 0$. At a perfect conductor wall the tangential electric field vanishes. Since $E_z$ is tangential to all four side walls, it satisfies a homogeneous Dirichlet boundary condition around the rectangle.

Separating the transverse Helmholtz equation therefore gives the transverse magnetic mode of a rectangular waveguideA zero value of either index makes $E_z$ identically zero, rather than producing a nonzero field uniform along that direction. The transverse electric and magnetic fields of this TM family are obtained from transverse derivatives of $E_z$, so they also vanish. Thus both TM indices must be nonzero: neither ${\mathrm{TM}}_{n0}$ nor ${\mathrm{TM}}_{0m}$ exists in this rectangular hollow-guide model. In contrast, transverse electric modes obey a Neumann boundary condition on $B_z$ and may have one zero index. The claim is about this waveguide geometry, not all possible guides.