I know a rectangular waveguide cannot support TEM waves, but supports TE and TM waves. In the TE mode, $E_z=0$ and in the TM mode, $H_z=0$ (where propagation direction is along the $z$-axis). I want to know whether the waveguide supports a mode where $E_z$ and $H_z$ are both non-zero, i.e. both electric field and magnetic field have a longitudinal component.
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Yes that is possible, but it would require an accidental occurrence of a TE mode and a TM mode with the same propagation speed. EDIT: after reading @hyportnex's comment I now see that you explicitly refer to a rectangular wave guide, for which this occurs for all modes except the TE$_{01}$.
In that case you can make any combination of them which can easily give simultaneously $E_z\neq 0$ and $H_z\neq 0$:
$$ \begin{align}
{\bf E} &= \alpha\ {\bf E}_{\large\,\text{TE}_{\displaystyle\small nm}} +
\beta\ {\bf E}_{\large\,\text{TM}_{\displaystyle\small nm}}
\\[6pt]
{\bf H} &= \alpha\ {\bf H}_{\large\,\text{TE}_{\displaystyle\small nm}} +
\beta\ {\bf H}_{\large\,\text{TM}_{\displaystyle\small nm}}
\end{align} $$
and that combination would still move as one wave with that same propagation speed.
If you combine arbitrary TE and TM solutions this would not work, the wave would change shape along the way and then we do not call it a "mode" of the waveguide, because that means an eigenmode of propagation. (Sometimes also eigenmodes of impedance can be defined, especially for multi-transmission lines, but that is not the usual meaning for wave guides.)
answered Jun 29 at 12:59
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$\begingroup$ No, for instance on a simple $a\times b$ rectangular 2D cross-section the functions $\cos(n\pi x/a)\cos(m\pi y/b)$ and $\sin(n\pi x/a)\sin(m\pi y/b)$ give the same eigenvalues for the Laplace operator. So for a rectangular waveguide almost all TE and TM modes come in pairs with equal velocity, only the TE$_{01}$ mode is single. For a circular shape, we would need accidental equality of a zero and a maximum of two Bessel functions of different order, which I think does not occur. But if we morph one of those wave guide shapes into the other there will be intermediate shapes where is does occur. $\endgroup$ Commented Jun 29 at 18:43
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$\begingroup$ you are absolutely right, but by interlacing I did not mean "halfway", just that alternating going from one eigenvalue "Dirichlet" to another "Neumann", etc. $\endgroup$ Commented Jun 29 at 18:59
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$\begingroup$ Yes that sounds like something to expect (and to have been proven by Hermann Weyl after Lorentz failed to do so, but that was something else...) Anyhow, as we see here for a rectangle they appear spot-on at the same positions, if we skip TE$_{01}$. $\endgroup$ Commented Jun 29 at 19:21