The pipe that will not carry a low note

A rectangular metal pipe 23 mm across will carry a radio wave at 10 GHz. It will not carry one at 5 GHz — not weakly, not with loss, but not at all. Fed at 5 GHz the pipe fills with field that decays away along its length, and nothing arrives at the far end however long the transmitter is left on. Somewhere between the two frequencies is a sharp edge, and its position depends on the width of the pipe and on nothing else.

The edge is not a property of the metal, of the air inside, or of the transmitter. It is a property of the width, and the reason is that a wave which has been told what it must do across a pipe has only so much of itself left over to travel along it.

What a mode is #

A wave in free space has a wavevector: a direction and a magnitude, with the magnitude fixed by the frequency through $|k| = \omega/c$. Put walls on two sides and the field has to vanish at them, which is exactly the condition that gives a string its harmonics.

A string held at both ends is the same condition with one dimension fewer. Only a whole number of half-waves fits between the ends, so only certain wavelengths exist and the longest of them is set by the length — and everything a guide does across its width, a string does along its length. What is different, and it is the whole difference, is that the guided wave is still free to travel in a direction the boundary says nothing about. A string’s modes are a list of frequencies; a guide’s are a list of frequencies each of which carries a wave away down the pipe.

So a mode of the guide has a transverse wavenumber $k_\perp$ fixed by the width — $\pi/a$ for the lowest one in a pipe of width $a$ — and a longitudinal wavenumber $k$ that is whatever is left:

$$k^2 = \frac{\omega^2}{c^2} - k_\perp^2 .$$

That is the whole subject. Rearranged it is $\omega^2 = \omega_c^2 + c^2k^2$ with $\omega_c = ck_\perp$, and read as a graph it is the hyperbola in the first figure.

The picture behind the algebra is worth having, because it makes the cutoff obvious. A guided mode is two plane waves, each travelling at the full speed $c$, crossing the pipe at equal and opposite angles and bouncing off the walls. Their interference makes the transverse standing pattern; their common progress makes the travel along the guide. Neither wave is doing anything a free wave would not do — each front advances as its own wavelets say it should — and the guiding is entirely in what the walls forbid.

Two sets of plane wavefronts crossing at an angle make exactly this: a pattern that is stationary across the crossing and moving along it, with the spacing across fixed by the angle and by nothing else. Everything peculiar about a guide — the cutoff, the dispersion, the phase velocity above $c$ — is a property of that geometry rather than of anything the waves are travelling through, which is why a guide full of vacuum is dispersive and a slab of glass full of nothing is not.

The wall’s part in it is one line of arithmetic and it is worth seeing as a picture. A pulse arriving at a fixed end comes back inverted, because the field has to vanish there and the only way an arriving wave can arrange that is to send back one that cancels it exactly at the wall. A guide is a pair of such walls facing each other, and a mode is what survives an unlimited number of those reflections without changing shape — the one transverse pattern that reproduces itself, bounce after bounce, forever.

Now lower the frequency. The wavelength grows, but the transverse condition is fixed: the wave must still fit the pipe across. So the two plane waves must cross at a steeper angle. Keep going and the angle reaches ninety degrees — the waves are bouncing straight across, back and forth, with no component along the guide at all. That is the cutoff, and below it the geometry has no solution: there is no real angle at which a wave of that wavelength fits the pipe.

Above the cutoff: two speeds, and their product #

The hyperbola has an immediate and startling consequence. The phase velocity is $\omega/k$, the slope of a line from the origin to the point on the curve, and the curve lies above the diagonal everywhere — so the phase velocity exceeds $c$ at every frequency, without exception, and diverges as the cutoff is approached.

The group velocity is $d\omega/dk$, the slope of the curve itself, and the curve is everywhere shallower than the diagonal — so the group velocity is below $c$ at every frequency, and goes to zero at cutoff. Differentiating $\omega^2 = \omega_c^2 + c^2k^2$ gives $\omega\,d\omega = c^2 k\,dk$, so

$$v_\phi v_g = \frac{\omega}{k}\cdot\frac{c^2 k}{\omega} = c^2,$$

exactly, at every frequency, for every guide. The figures measure both slopes off the drawn curve and find the product one to four decimal places.

A relation of a quite different kind makes the point that this shape is not a property of a medium. Deep water disperses too — $\omega = \sqrt{gk}$, a curve that lies below the diagonal, with the packet travelling at half the speed of the crests — and there the dispersion belongs to the water: to gravity, to the depth, to the fact that a deep-water wave’s orbits die out downward. The guide’s curve is a similar-looking graph with no material in it at all. Take the metal away and leave the shape and the dispersion is unchanged, because the only thing producing it is the requirement that the wave fit across.

The phase velocity above $c$ is not a difficulty and it is worth saying exactly why. What travels at $v_\phi$ is the intersection line of the two crossing wavefronts, sliding along the wall. Nothing is at that intersection; it is a place where two things coincide, and such places can move arbitrarily fast, as the speed that carries no signal sets out at length. The energy, and anything modulated onto the wave, travels at $v_g$, and $v_g$ is below $c$ by exactly the amount that keeps the product fixed.

The dispersion is real and it has consequences. A pulse launched into a guide contains a band of frequencies, each with a different group velocity, so it arrives spread out — in a hollow pipe with a vacuum inside. Drawn on the guide’s own relation, the two speeds come apart visibly in a single frame: the envelope and the crests inside it do not move together, and the crests slide forward through the packet as it goes.

How fast a pulse comes apart depends on how wide its spectrum is, and the dependence runs the way that first seems backwards.

That is the practical shape of the problem: a radar wanting fine range resolution wants a short pulse, a short pulse is a wide spectrum, and a wide spectrum is exactly what a dispersive guide mistreats. The spreading is the same arithmetic that widens any packet whose components disagree about speed, applied to a medium that is not there. Radar engineers compensate for it; so do the designers of the waveguide runs between a particle accelerator’s klystrons and its cavities, where a nanosecond of spread is a phase error on a bunch.

Below the cutoff: a field that goes nowhere #

Below $\omega_c$ the expression for $k^2$ is negative, so $k$ is imaginary and $e^{ikz}$ becomes $e^{-|k|z}$. The field does not oscillate along the guide; it decays. And it decays without dissipating anything: the walls in this idealisation are perfect, nothing is absorbed, and the power that goes in comes back out of the input.

The same solution appears at a boundary light cannot cross. Past the critical angle the transmitted wavevector is imaginary, the field on the far side falls off exponentially instead of travelling, and the depth it reaches to diverges as the critical angle is approached from beyond. An undersized guide does exactly that against frequency rather than against angle, and the divergence at the cutoff is the same divergence: just below $\omega_c$ the decay length is enormous, and it shortens to about the width of the pipe well before the frequency has halved.

This is the third place in this collection where the same solution appears, and the three are worth naming together. Past a critical angle at a boundary the transmitted wave is evanescent. Inside a barrier a particle has not enough energy for the wavefunction is evanescent. Below a guide’s cutoff the mode is evanescent. In every case the wave equation has been handed a region where the local wavenumber is imaginary, and it has produced the only thing it can.

What that buys is a length-controlled attenuator. Transmission through an evanescent region falls by a fixed factor per unit length, so the transmitted power falls exponentially with how much forbidden region there is — a straight line on a logarithmic plot, with a slope set by the decay length and an intercept set by the mismatch at each end. A short section of undersized guide behaves that way and is sold as one: the loss is set by a distance, which can be machined to a micron, rather than by a material, which cannot be trusted to a decibel across a production run.

The practical form of this is that a below-cutoff pipe is not a wall. Make it short enough and a usable fraction gets through, and the fraction is controllable to a decibel by moving a probe along it. Make it long enough — a few pipe diameters is usually plenty, since the decay length below cutoff is of order the width — and the attenuation is enormous. The perforated screen on a microwave oven door is a sheet of very short below-cutoff guides: each hole is a pipe whose cutoff is far above 2.45 GHz, so the microwaves see an evanescent field a millimetre deep, while visible light, at a wavelength ten thousand times shorter, is nowhere near cutoff and passes freely.

The shapes the cutoff depends on #

The lowest cutoff belongs to the mode with the largest transverse wavelength, which is the one with the simplest pattern across the pipe. Every other mode has more structure, a larger $k_\perp$, and therefore a higher cutoff of its own.

The numbers are worth putting down, because they are what makes a guide a piece of hardware rather than an idea. The lowest cutoff of a rectangular guide of width $a$ is at a free-space wavelength of $2a$: the guide carries anything whose wavelength is shorter than twice its wide dimension and nothing whose wavelength is longer. Standard WR-90 guide is 22.9 mm across, so its cutoff is 6.56 GHz, and it is sold for the band from 8.2 to 12.4 GHz — comfortably above its own cutoff and comfortably below the second mode’s. A guide for a 50 Hz power line, by the same rule, would have to be three thousand kilometres wide, which is the reason mains electricity travels on wires and radar does not.

That interval is the reason waveguides have the dimensions they do. Run a rectangular guide between its first and second cutoffs and only one field pattern exists, so the guide has a definite characteristic impedance — the quantity a mismatch at a junction is computed from — a definite group velocity, and a signal that arrives in one piece. Run it higher and several modes propagate at different speeds, a bend converts some of one into some of another, and the output is a sum of things that took different times to arrive. Standard guide sizes are catalogues of that single-mode band.

The same boundary-value problem turns up once more with a wavefunction in it. A particle in a box has a lowest energy it cannot go below, for exactly the reason a guide has a lowest frequency: confinement fixes a transverse wavenumber, and a fixed wavenumber is an energy. The motion that cannot be taken away by cooling and the frequency that cannot be reached by turning a transmitter down are the same statement in two vocabularies, and both are consequences of a wall rather than of anything the wave is made of.

The loss, and where it is least #

The perfect walls are the idealisation this essay leans on hardest, and relaxing them produces a curve with a minimum in an interesting place.

A real wall lets the field into a skin depth, and the current flowing in that skin dissipates. The resulting attenuation depends on frequency in two competing ways. Near the cutoff the group velocity is almost zero, so the energy takes a very long time — and a very large number of bounces off the walls — to travel a metre, and the loss per metre runs away. Far above the cutoff the skin depth falls as the inverse square root of frequency, so the surface resistance rises and the loss climbs again.

Between the two is a minimum, and for the ordinary rectangular mode it sits at something like two to three times the cutoff frequency. That is one reason the recommended band of a standard guide begins where it does: not merely to stay single-moded, but to sit near the bottom of the loss curve.

The numbers are respectable rather than remarkable. A centimetre-band copper guide loses of order a tenth of a decibel per metre — far better than a coaxial cable of comparable size at the same frequency, which is why guide is used at all despite being rigid, bulky and expensive to bend. What it is not is lossless, and a run of a hundred metres is a run with a real budget attached.

The frequency dependence is also a warning about idealisation. Everything else in this essay is geometry, exact and material-free; the loss is the one quantity that depends on what the pipe is made of, and it is the quantity that decides whether a guide is worth building.

What is bought by guiding at all #

A wave let loose spreads, and its intensity falls with distance in a way that depends only on the geometry.

A spherical wave’s intensity falls as the inverse square of distance and a cylindrical one as the inverse first power, and a wave confined in both transverse directions does not fall at all. Over the three decades between a 23 mm guide and a 23 m run that is the difference between delivering everything and delivering a millionth: the energy in a guide has nowhere to spread to, so a perfect guide hands over at the far end exactly what was put in at the near one.

That is the whole argument for a guide and it is a strong one. A transmitting antenna radiating into space is throwing away all but a vanishing fraction of what it emits; a guide throws away only what the walls absorb. The price is the cutoff, the dispersion and the fixed cross-section — a guide is a device that has been told what wavelengths it is for, and it cannot be told otherwise afterwards.

The attenuator that is a length #

The below-cutoff decay has a use that turns it into a standard, and the reason is that its rate is calculable from a drawing.

Well below cutoff, the decay constant stops depending on frequency at all — it approaches $k_\perp$, the transverse wavenumber, which is fixed by the geometry. For a circular pipe in its lowest mode that gives an attenuation of very nearly 32 decibels for every pipe diameter travelled, whatever the frequency, whatever the metal, and whatever the power.

Which makes an attenuator out of a piece of tube and a ruler. Put a launching loop at one end and a pickup loop on a sliding piston, and the attenuation between them is 32 dB per diameter of separation, computed rather than measured. No calibration against a reference is needed, because there is no material property in the answer: the only quantities are a length and a diameter, both measurable with a micrometer.

That is a rare thing in measurement, and it is why the piston attenuator was for decades the primary standard of microwave attenuation. Almost every other attenuator has to be compared against something — its loss depends on a resistivity, a film thickness, a temperature. This one depends on the arithmetic of an evanescent field in a pipe of known size, and its accuracy is the accuracy of the machining.

The same insensitivity is what makes the microwave-oven door work over the whole band without tuning, and what makes a screened enclosure’s ventilation holes effective without anybody specifying a material for them.

The mode that got better with frequency #

One mode behaves unlike every other and is worth recording, because for a while it was going to carry the world’s telephone calls.

The wall loss described above rises at high frequency for the usual modes, because the surface currents grow with the skin resistance. In a circular guide there is one family of modes — the ones whose electric field is entirely azimuthal, so that the current in the wall runs only around the circumference and never along it — whose loss does the opposite: it falls as the frequency rises, without limit.

That was an extraordinarily attractive property in the 1950s and 1960s. A single circular pipe, a few centimetres across, running at tens of gigahertz, with attenuation improving as the band was pushed higher: the projected capacity was enormous, and long test installations were built and worked.

It was abandoned, and the reasons are instructive. The useful mode is not the lowest one, so every other mode in the pipe is also allowed to propagate — and any bend, any dent, any imperfection in the wall converts some of the wanted mode into an unwanted one that then travels at a different speed and arrives as noise. Keeping a guide straight and round to the tolerance required over kilometres turned out to be the whole problem, and helical or dielectric-lined walls were developed specifically to suppress the competing modes.

Then optical fibre arrived, offering a guide with a smaller cross-section, an enormously larger bandwidth and the ability to be bent round a corner, and the millimetre-wave pipe was left as a completed technology that nobody needed. It is a fair example of an engineering programme defeated not by its own physics — the loss really did fall with frequency, exactly as predicted — but by a rival whose constraints were different in kind.

Where the model stops #

The walls are taken to be perfect conductors and are not. A real metal admits a field to the skin depth, and the resulting loss rises as the square root of frequency for most modes. That is a small effect at centimetre wavelengths and a fatal one at optical ones, which is why optical fibres guide by total internal reflection rather than by a conducting wall — the same cutoff mathematics, the same evanescent field outside, and no metal anywhere.

Only one transverse dimension has been discussed. A real rectangular guide has two, so its modes carry two indices, and a circular one has a Bessel-function eigenvalue problem in place of a sine. The structure of the argument is untouched: there is a discrete set of transverse patterns, each with its own cutoff, and the guide is useful between the first two.

And “no travelling solution below cutoff” is a statement about an infinite guide. A finite one has ends, the ends have reflections, and a resonance can exist below the cutoff of the pipe it is made from. That is what a cavity is, and a cavity’s resonant frequency is not the guide’s cutoff — it involves the length as well, and can be either side of it.

What the pictures cannot show #

The hero figure draws a relation between two numbers and cannot draw the field it is about. The thing that would make the cutoff intuitive is an animation of the two crossing plane waves as the frequency is lowered and the angle steepens toward ninety degrees; a static figure of that has to be read as a sequence, and the moment of cutoff — where the longitudinal progress reaches zero — is precisely the frame in which nothing appears to be happening.

Nor can any figure show the evanescent region as anything other than a decaying curve. What is peculiar about it is not the shape, which is an ordinary exponential, but that it carries no power along the guide at all while having a perfectly ordinary energy density. Nothing about a drawing distinguishes a field that is going somewhere from one that is not.

Where this ladder goes next #

This is the first rung of a new ladder, and the anchor is guiding rather than waves generally: what changes when a wave is not allowed to spread. The rungs above it are the ones this essay kept pointing at and declining. The impedance of a guide, which is where the reflection at a discontinuity comes from and is not the same quantity as the impedance of an unbounded medium. Guiding by refraction instead of by walls, which is the optical fibre and brings its own dispersion, of two kinds that can be made to cancel. And the cavity, which is a guide with both ends closed and is the object that turns a cutoff into a resonance.

The habit worth carrying away is short. A constraint across a wave’s motion becomes a floor under its frequency. It happens to light in a pipe, to sound in a duct, to an electron in a wire, and to a phonon in a thin film — and in every case the floor is the free-space speed divided by the size of the constraint, which is why the number is always a wavelength comparison and never a material property.