Transverse electric mode of a rectangular waveguide (${\mathrm{TE}}_{mn}$)

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 $B_x=ik{\partial }_x{B}_z/\Delta$, $B_y=ik{\partial }_y{B}_z/\Delta$, $E_x=i\omega {\partial }_y{B}_z/\Delta$, and $E_y=-i\omega {\partial }_x{B}_z/\Delta$. 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.