= Rectangular electromagnetic waveguide
Take a straight hollow <rectangular electromagnetic waveguide> with <vacuum> interior $0<x<a$, $0<y<b$ and axis $z$, where $a>b>0$. Its four walls are ideal <perfect conductors>. The time-dependent <Maxwell equations> require zero tangential <electric field> and zero normal oscillating <magnetic field> at these walls. Real metal has finite <electrical conductivity>, so real guides lose <electromagnetic energy> by <Joule heating>; the perfect-wall model isolates confinement and <wave dispersion> from that loss.
The simplest propagating <transverse electric mode of a rectangular waveguide> is $\mathrm{TE}_{10}$. “Transverse electric” means $E_z=0$, not that the <magnetic field> is also wholly transverse. With real peak scale $E_0$, its <complex amplitudes>, including the common axial factor, can be written
$$
\begin{aligned}
E_y&=E_0\sin(\pi x/a)e^{i\beta z},& E_x&=E_z=0,\\
B_x&=-\frac{\beta}{\omega}E_0\sin(\pi x/a)e^{i\beta z},&B_y&=0,\\
B_z&=-\frac{i\pi}{a\omega}E_0\cos(\pi x/a)e^{i\beta z}.&&
\end{aligned}
$$
The physical fields are the <real parts> after multiplication by $e^{-i\omega t}$. In particular, $E_y$ is zero at $x=0,a$, maximal at $x=a/2$, and independent of $y$. At $y=0,b$ it is normal to the metal, so it need not vanish there. The $\mathrm{TE}_{10}$ <electric field> crosses the narrow gap, while its strength varies across the broad dimension.
\Image[/rectangular-waveguide-te10-field-geometry.png]
{title=Original vacuum-filled rectangular guide: the TE10 electric-field profile across a 30 mm by 12 mm cross-section}
{description=The arrows show the <electric field> at an axial plane and time where its common <wave phase> factor is positive; their length follows $\sin(\pi x/a)$. They are field directions, not particle trajectories. The profile is uniform along $y$ and vanishes at the side walls. The guide axis $+z$ points out of the cross-section. All dimensions and the operating <frequency> are chosen model values.}
For an original example choose $a=30\,\mathrm{mm}$, $b=12\,\mathrm{mm}$ and $f=1.25f_c$, where $f_c=c/(2a)$. With $c=299792458\,\mathrm{m/s}$, this gives
$$
\begin{aligned}
f_c&=4.99654097\,\mathrm{GHz},& f&=6.24567621\,\mathrm{GHz},\\
\lambda_0&=48\,\mathrm{mm},&\beta&=25\pi\,\mathrm{m}^{-1},&\lambda_g&=80\,\mathrm{mm}.
\end{aligned}
$$
Here $\lambda_0=c/f$ is <vacuum> <wavelength> and $\lambda_g=2\pi/\beta$ is <guided wavelength>. The next cutoff is $\mathrm{TE}_{20}$ at $2f_c$; $\mathrm{TE}_{01}$ starts at $(a/b)f_c=2.5f_c$. Thus only $\mathrm{TE}_{10}$ can propagate at the chosen <frequency>, although a source or discontinuity can additionally excite local <evanescent waves>. These are calculated values, not specifications or measurements of a commercial guide.
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