Rectangular Waveguides

A transmission line needs two conductors — a signal path and a return — to support the transverse electromagnetic (TEM) wave, in which both \(\vec{E}\) and \(\vec{H}\) lie entirely across the line with nothing along the direction of travel. A waveguide is a single hollow conductor; there is no second conductor to hold the return, so a TEM wave is impossible inside it. Yet a hollow copper pipe guides microwaves superbly, with far lower loss and far higher power handling than coax. How?

The wave survives by tilting. Instead of running straight down the guide, it bounces at an angle between the walls, zig-zagging forward. That tilt gives the field a component along the axis — either an axial electric field or an axial magnetic field — and that is exactly what distinguishes the two allowed families of solutions.

Since the axial direction is no longer field-free, we classify solutions by which axial component is present:

• Transverse magnetic (TM) modes: \(H_z=0\), \(E_z\neq0\). The magnetic field is purely transverse; the electric field has an axial part.
• Transverse electric (TE) modes: \(E_z=0\), \(H_z\neq0\). The electric field is purely transverse; the magnetic field has an axial part.

Every field inside the guide can be built from these two families. The trick is that once you know the single axial component (\(E_z\) for TM, \(H_z\) for TE), Maxwell's equations hand you all four transverse components algebraically. So the whole problem reduces to solving one scalar Helmholtz equation for that axial component, subject to the metal-wall boundary conditions.

With a propagation factor \(e^{-j\beta z}\), the axial component obeys the two-dimensional Helmholtz equation \(\nabla_t^2\psi+k_c^2\psi=0\), where \(k_c^2=k^2-\beta^2\) and \(k^2=\omega^2\mu\varepsilon\). Separation of variables on the rectangle, with \(E_z=0\) on the walls (TM) or \(\partial H_z/\partial n=0\) on the walls (TE), gives sinusoidal solutions labelled by two integers:

The indices \(m\) and \(n\) count the half-wave field variations across the width \(a\) and the height \(b\). For TM modes both must be at least one (if either is zero the sine collapses the field to nothing), so the lowest TM mode is \(\text{TM}_{11}\). For TE modes either may be zero but not both, so \(\text{TE}_{10}\) exists. The transverse wavenumber that sets everything is:

The axial phase constant follows from \(\beta^2=k^2-k_c^2\). For the wave to travel, \(\beta\) must be real, which requires \(k>k_c\) — the frequency must exceed a threshold. Below it, \(\beta\) is imaginary, the field decays exponentially, and the mode is evanescent: it does not propagate at all. That threshold is the cutoff frequency:

Each mode has its own cutoff, so a waveguide passes a band of frequencies with a definite set of modes. The dispersion curve makes the behaviour vivid: at cutoff the wave is purely transverse (\(\beta=0\), it bounces straight across without advancing); well above cutoff it hugs the light line and behaves almost like free space.

The mode with the lowest cutoff is the dominant mode. For the usual convention \(a>b\), that is \(\text{TE}_{10}\) — one half-wave across the wide dimension, none across the narrow one. Its cutoff depends only on the wide wall:

This is the workhorse of microwave engineering. By making \(a\) roughly twice \(b\), the next modes — \(\text{TE}_{20}\) at \(u'/a\) and \(\text{TE}_{01}\) at \(u'/2b\) — sit well above \(\text{TE}_{10}\), leaving a clean single-mode band between \(f_{c,10}\) and the next cutoff. Operating there guarantees the field has just one predictable shape, which is essential for couplers, filters, and antennas. Standard guides like WR-90 are dimensioned precisely to place that band where it is needed.

Because the wave travels diagonally, the distance between wavefronts measured along the axis is stretched. Writing \(\beta=k\sqrt{1-(f_c/f)^2}\), the axial wavelength — the guide wavelength — is longer than the wavelength the same frequency would have in the unbounded medium:

The two velocities split apart. The wavefronts' axial sweep — the phase velocity — exceeds \(u'\) (even exceeding \(c\) in an air guide, with no contradiction, since no energy moves that fast). The energy itself travels at the group velocity, which is correspondingly slower:

The clean product \(u_p u_g=u'^2\) is a handy check. As \(f\to f_c\), \(u_g\to0\) (the energy stalls — the wave bounces straight across without advancing); as \(f\gg f_c\), both approach \(u'\) and the guide behaves like open space.

The ratio of the transverse electric to transverse magnetic field — the wave impedance — also depends on frequency, and differs between the two mode families. Relative to the medium's intrinsic impedance \(\eta'=\sqrt{\mu/\varepsilon}\) (377 Ω for air):

So a TE mode looks more inductive than free space and a TM mode more capacitive, both converging to \(\eta'\) far above cutoff and diverging from it (\(\eta_{\text{TE}}\to\infty\), \(\eta_{\text{TM}}\to0\)) as cutoff is approached. These impedances matter when matching a guide to an antenna or joining sections of different size — the same matching logic as transmission lines, carried into the microwave pipe.

For each item, identify the mode and compute \(f_c\) first; whether a signal propagates, and how fast, both follow from the ratio \(f_c/f\). Assume air filling unless stated. Difficulty rises down the list.