= Transverse magnetic mode of a rectangular waveguide
{title2=$E_z\propto\sin(n\pi x/a)\sin(m\pi y/b)$}
In a hollow <rectangular electromagnetic waveguide> with <perfect conductor> walls and axis $z$, a <transverse magnetic polarization> has $B_z=0$ and a nonzero longitudinal <electric field> $E_z$. The wall <boundary conditions> impose $E_z=0$ around the transverse rectangle. <Separation of variables> therefore gives $E_z=E_0\sin(n\pi x/a)\sin(m\pi y/b)e^{i(k_z z-\omega t)}$, with $n,m\geq1$. Setting either index to zero makes the entire field vanish. Thus there is no $\mathrm{TM}_{n0}$ or $\mathrm{TM}_{0m}$ branch. The transverse <eigenvalue> is $k_\perp^2=(n\pi/a)^2+(m\pi/b)^2$, with <dispersion relation> $\omega^2/c^2=k_z^2+k_\perp^2$. This exclusion is specific to this geometry and these <boundary conditions>.
Back to article page
