# Is the argument of zero tangential field components in wave guide walls a fallacy? Or: How EM waves "magically" find their correct transverse modes

I just came across a nice answer on Quora, explaining visually how wave guide modes can be constructed from the condition of zero tangential fields in the wave guide walls.

In mathematical terms, this corresponds to a certain set of boundary conditions for the Maxwell equations resulting in a certain set of solutions. The physical reasoning is that for perfectly conductive walls, any non-zero tangential fields would induce infinite currents in them - therefore, the waves somehow "know" which modes to choose and which to avoid, or, as the author of the cited answer writes,

"One nice thing about the wave guide is that you don’t have to design the waves. You just stick an antenna in the wave guide and the field just finds the correct angle for each wave train. The waves simply propagate in a mode that they can."

Obviously, it wasn't the author's intent to personify waves as sentient beings who test out different modes and select only those consistent with physical laws. Still, both the mathematical as well as physical arguments kind of put the cart before the horse:

1. The boundary conditions are not axiomatically derived, but a conscious choice made by whoever sets up a theoretical calculation or simulation. They do not follow from the differential equations, but are chosen as parameters (correct me if I'm wrong).

2. Similarly, the physical argument against infinite currents inside the wave guide walls is a justification, but not an explanation of fundamental causes or underlying physical processes.

The question is therefore:

What is the real physical mechanism behind the emergence of distinct propagation modes inside wave guides? What rather than who decides in which modes waves can propagate and in which they cannot?

• Could it be that the "zero tangential fields" argument is factually wrong?

Meaning: If we (quote) "stick an antenna in the wave guide", could it be that there are indeed high-frequency currents induced inside the walls - or rather: in the wall surfaces - such that any modal components, whose tangential fields don't perfectly cancel out, simply dissipate as heat (resistive/absorptive loss) or leak out (radiative loss)?

Note that wave guides are in general fabricated from either highly conductive materials (usually metals) or have at least a highly conductive inner layer/coating (copper or even gold), both of which actually facilitates an effective induction of currents!*

A definite answer from someone with either experimental (e.g. RF/microwave research) or theoretical experience (e.g. FDTD simulation) would be much appreciated.

*Addendum: The choice of material actually depends on the frequency of the electromagnetic radiation and doesn't necessarily have to be a metal, but can be anything that is sufficiently conductive at the desired frequencies.

• The waveguide actually is in charge of the waves and tells them whose allowed where. Seriously though, the boundary conditions (ie waveguide geometry) of your PDE's, here Maxwells equations determine this. The boundary condition says there's an equipotential along the inner metallic wall of the waveguide (ie zero E-field) Feb 9 at 2:42
• for the equipotential the material needs to be conductive, which leads to reflection, and antimony's comment. Wood doesn't hardly restrict E-Fields therein, but a conductive material spreads charge out equally throughout the whole wall. try reading about TM, TE, and TEM modes for for detail on this. Feb 9 at 3:27
• Yes, ultimately zero tangental is technically the electrostatic approximation, because the relaxation time is so quick, we often use it in dynamics as well Feb 9 at 5:36
• @sarahslvmn Find a copy of Jacksons Classical Electrodynamics 3rd edition if you have access. section 8.1 (4 pages) will answer all of your questions. (I was just thumbing through mine, it's been a while Was my book for a fun filled year long grad class ). It specifically goes through the failure of the tangental E field and normal B-field to vanish in a waveguide. Feb 9 at 5:47
• The reflection actually happens over a depth in the conductor. you could think of it as jumping on a trampoline, the wave pushes into the metal, but ultimately the metal (charge distribution) catches up and pushes it back out. Now that I'm thinking about it, passive radars are designed to detect even people moving around by our reflections of (now ambient) waves like cell signals. Feb 9 at 6:07

You wrote: "stick an antenna in the wave guide", could it be that there are indeed high-frequency currents induced inside the walls - or rather: in the wall surfaces - such that any modal components, whose tangential fields don't perfectly cancel out, simply dissipate as heat (resistive/absorptive loss) or leak out (radiative loss)?

In a "normal", ie., copper or plated copper waveguide of high wall conductivities the dissipation is insignificant for your question. What is significant, however, is that all those non-propagating but excited modes, say in a fundamental mode guide, form a reactive field around launcher and increase its reflectivity back to the generator. In other words, you have a generator connected, for example, to a coaxial line whose outer sheath is soldered to the top wall of rectangular waveguide and its inner conductor is inserted as a wave launcher, antenna probe, inside the guide otherwise insulated from everything else.

Assume that the generator's source impedance $$Z_s$$ is equal to the wave impedance $$R_w$$ of the coaxial line: $$Z_s=R_w$$ and thus if the coax is terminated at its other end with a load impedance $$Z_L$$ such that $$Z_L=R_w$$ the there is no reflected wave propagating on it in either direction. Now connect the waveguide whose one end, the far end from the probe, is also terminated with a perfect absorber designed so that is matched for modes in the waveguide, in practice it could be a tapered longish resistive cone or pentahedron structure attached to the end wall. So in the direction towards its load the waveguide is well-matched.

But inserting the probe you notice that depending on what you to do at its near end of the waveguide the reflectivity in the coax changes. Usually the simplest thing to do is to have a movable metal "piston" placed about a quarter wavelength (measured in the waveguide $$\lambda_g$$) away from the probe. The round trip reflection from the probe towards this back wall will be a about a half wavelength, so would come back essentially in the opposite phase but there is also a phase flip on the shorted back wall, so instead of canceling reflected wave it as actually an "electrical" open circuit (ie.,no load) on the probe. Now by adjusting the short wall's distance relative to the probe with the probe's depth it is possible to eliminate all reflections going toward generator from the probe on the coaxial line. In other words, if the magnetic field is stronger around probe than its electric field, that is if the probe is "inductive", than we we add a little capacitance, electric energy, in parallel to it.

The actual distance where this happens depends on the probe's depth in the waveguide but it is roughly around a quarter wavelength as measured in the guide, ie., in terms of propagating mode wavelength.

Now, what does this really mean? It means that $$\ell_s \approx \lambda_g/4$$ at which the probe is matched, that is when the movable short has length $$\ell_s$$. Then there is no reflection measured in the coaxial line towards the generator from the probe, and essentially all energy from the generator is transferred to the waveguide in its propagating mode going toward the far end matched load. Every discontinuity in a homogeneous transmission line/waveguide will excite an infinity of modes, some propagating, some localized, but they are there. By properly adjusting the depth of the launcher together with the length of movable short we will have prevented those excited waves to be launched back to towards the generator and "focused" them be coherent with the propagating mode directed towards the load. There is still EM reactive energy staying around the probe, just there would in an LC filter in its passband, but their only observable effect at a single frequency is to shift the phase of the incident wave from the generator as it passes through it. By properly adjusting the depth of the probe with the back short distance one can eliminate all reflections. It matters what the actual depth of the probe/back short distance if you care to operate at a wider frequency band along the radius and detailed shape of the probe becomes important, etc. Some combinations are better than others, and as you can imagine there is a science associated with designing the probe, especially the shape of the tip for higher powers, etc.

• Incredible description! re. "non-propagating but excited modes" is there a name for these modes? Thanks! Feb 10 at 1:57
• @antinomy evanescent modes Feb 10 at 1:58
• Thanks! So the presence of these reactive fields around the launcher indicate imperfect impedance match of the launcher? Since if it coupled 100% power into the fundamental mode that would be a perfect impedance match. But since it also couples some power into non-propagating modes, the impedance match by definition is not 100%? Consequently the input(?) impedance of the launcher has changed? Feb 10 at 2:06
• Which in turn leads to some power being coupled into backwards propagating modes, which propagate back towards the generator? Feb 10 at 2:13
• That is correct. Around a discontinuity all the modes of the waveguide will be launched by some amplitude and phase. Usually waveguides operate at a frequency so that only a single mode can propagate, and the other modes have "imaginary" wavelengths, thus exponentially decay. They represent a reactive load and oscillating energy in the sense of a lumped element component in a circuit, L and C. The effective impedance of the launcher and the waveguide following is the result form the reactive energy stored in the evanescent modes and the wave impedance of the propagating mode down the guide. Feb 10 at 2:28