TEM Waves

Now we can start looking at waveforms in various cavities. Suppose we let $E_z = B_z = 0$. Then the wave in the cavity is a pure transverse electromagnetic (TEM) wave just like a plane wave, except that it has to satisfy the boundary conditions of a perfect conductor at the cavity boundary!

Note from the equations above that:

$$\mbox{\boldmath$\nabla$}_\perp\cdot\mbox{\boldmath$E$}_\perp = 0$$

$$\mbox{\boldmath$\nabla$}_\perp \times \vec{E}_\perp = 0$$

from which we can immediately see that:

$$\nabla_\perp^2\mbox{\boldmath$E$}_\perp = 0$$ (10.46)

and that

$$\mbox{\boldmath$E$}_\perp = -\mbox{\boldmath$\nabla$}\phi$$ (10.47)

for some suitable potential that satisfies $\nabla_\perp^2\phi = 0$. The solution looks like a propagating electrostatic wave. From the wave equation we see that:

$$\mu\epsilon\omega^2 = k^2$$ (10.48)

or

$$k = \pm \omega\sqrt{\mu\epsilon}$$ (10.49)

which is just like a plane wave (which can propagate in either direction, recall).

Again referring to our list of mutilated Maxwell equations above, we see that:

$$ik\mbox{\boldmath$E$}_\perp = -i\omega(\hat{\mbox{\boldmath$z$}} \times \mbox{\boldmath$B$}_\perp)$$

$$\mbox{\boldmath$D$}_\perp = -\frac{\omega\mu\epsilon}{k} (\hat{\mbox{\boldmath$z$}} \times \mbox{\boldmath$H$}_\perp)$$

$$\mbox{\boldmath$D$}_\perp = \pm \sqrt{\mu\epsilon} (\hat{\mbox{\boldmath$z$}} \times \mbox{\boldmath$H$}_\perp)$$ (10.50)

or working the other way, that:

$$\mbox{\boldmath$B$}_\perp = \pm \sqrt{\mu\epsilon} (\hat{\mbox{\boldmath$z$}} \times \mbox{\boldmath$E$}_\perp)$$ (10.51)

so we can easily find one from the other.

TEM waves cannot be sustained in a cylinder because the surrounding (perfect, recall) conductor is equipotential. Therefore $\mbox{\boldmath$E$}_\perp$ is zero as is $\mbox{\boldmath$B$}_\perp$. However, they are the dominant way energy is transmitted down a coaxial cable, where a potential difference is maintained between the central conductor and the coaxial sheathe. In this case the fields are very simple, as the $\mbox{\boldmath$E$}$ is purely radial and the $\mbox{\boldmath$B$}$ field circles the conductor (so the energy goes which way?) with no $z$ components.

Finally, note that all frequencies are permitted for a TEM wave. It is not ``quantized'' by the appearance of eigenvalues due to a constraining boundary value problem.