= Transverse electric mode of a rectangular waveguide
{title2=$\mathrm{TE}_{mn}$}
For a transverse electric mode, $E_z=0$ and $B_z=B_0(x,y)e^{i(kz-\omega t)}$. Put $\Delta=(\omega/c)^2-k^2$. <Maxwell equations> give $B_x=ik\partial_xB_z/\Delta$, $B_y=ik\partial_yB_z/\Delta$, $E_x=i\omega\partial_yB_z/\Delta$, and $E_y=-i\omega\partial_xB_z/\Delta$. The wall conditions become <Neumann boundary conditions> for $B_0$, whose separated solutions are $\cos(m\pi x/a)\cos(n\pi y/b)$. The constant $(0,0)$ candidate is excluded by Faraday's equation because its corresponding electric field vanishes.
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