The node that is not standing still

A wave sent at a wall comes back, and the outgoing and returning waves add to something that does not travel: a standing wave, with nodes where the two always cancel and antinodes where they always agree. That is where discreteness comes from — only the wavelengths whose nodes land on the ends are allowed — and it is one of the most productive pictures in the subject.

It also assumes something that is almost never true. A perfect standing wave needs a perfect reflection, and most boundaries reflect only part of what arrives.

What a partial reflection gives is a partial standing wave: a pattern with the same spacing and the same positions, whose minima are not zeros. Everything useful about the arrangement is in how deep it goes.

The number, and what it is worth #

One number — the ratio of the maximum to the minimum — carries the whole pattern. It is measurable by sliding a small detector along the line and reading two values, and it inverts exactly rather than by calibration, which is what makes it an instrument rather than an indicator.

Two things come out of the inversion. The magnitude of the reflection follows immediately, and its square is the fraction of the power sent back. And the position of a minimum, measured in wavelengths from the load, gives the phase of the reflection — so the two readings together determine the reflection completely, and therefore the load that produced it.

That last step is the one that made the method a laboratory standard. The load may be inaccessible, or too small to attach anything to, or at a frequency where nothing can be measured directly. What is accessible is the line leading to it, and a pattern on that line determines what is at the end of it. It is an inverse measurement of a familiar kind: the same logic that reads a spectrum out of an interferogram, applied to position rather than to delay.

The steepness of the curve at the left is the practical difficulty. Almost the whole useful range is crowded into ratios below two, so a measurement of a good match needs the two readings to differ by a few per cent and needs them accurate to a fraction of that. A bad match is easy to measure and a good one is not, which is the wrong way round.

The sliding is worth its own figure because it is the half of the technique that turns a magnitude into an identification.

A reflection is a complex number: a magnitude and a phase. The pattern’s depth gives the magnitude. Its position — how far the first minimum sits from the load, measured in wavelengths — gives the phase, because the two waves’ relative phase at the load is what decides where along the line they first oppose each other.

So two readings taken with one sliding detector determine the reflection completely, and from the reflection the load follows by one algebraic step. A short circuit puts a minimum at the load; an open circuit puts one a quarter of a wavelength away; a matched load gives no pattern at all and no position to read. Everything in between is somewhere on a circle of possibilities parameterised by the phase, which is the picture the Smith chart was drawn to make usable.

Nothing is standing still #

The flat line is the fact that separates a partial standing wave from a complete one, and it is more surprising than it looks.

The power crossing a point is the difference of the two waves’ powers, and a difference of powers contains no cross term — the interference is in the amplitude, and it cancels out of the flux. So the flow is uniform along the line and equal to the fraction transmitted, while the amplitude varies by the standing-wave ratio and its square by the ratio squared.

Energy is therefore passing steadily through the minima. A minimum is not a place where nothing is happening; it is a place where the two waves nearly cancel and the small remainder is carrying the whole of the transmitted power. That is why a detector that reads amplitude and a detector that reads flow give completely different pictures of the same line, and why an argument about where energy is going has to be made with the second.

In the perfectly reflecting case the flow is exactly zero everywhere, which is consistent and is the source of the confusion: there the amplitude nodes really are places where nothing happens, and the intuition built there does not survive the general case.

There is a second thing the figure settles. The pattern does not move. A partial standing wave is often described as a standing wave with a travelling wave superposed, drifting past it — and that is a legitimate decomposition, but the pattern of amplitude is stationary, at fixed positions, however small the reflection. What travels is energy, not the pattern.

What a deep pattern is for #

A large reflection is called a poor match and is usually a fault. It is worth saying where it is the object.

A resonant cavity is two nearly perfect reflectors facing each other, and the whole of its usefulness is that the wave bounces many times before leaving. Its quality factor is set by how close the reflections are to complete, and a standing-wave ratio of a thousand is a good cavity rather than a broken line.

An antenna trap is a deliberately mismatched section that keeps a current from going past a point, so that the same wire is a different length at different frequencies. Its design target is a reflection magnitude near one at one frequency and near zero at another.

And a quarter-wave transformer uses two deliberate reflections that cancel each other: the section is mismatched at both ends, and the two reflections arrive back a half-wavelength apart in phase and destroy one another. Looked at as a pattern, the line before it carries no standing wave at all while the section itself carries a large one.

So the honest general statement is that a deep pattern means the two waves are of comparable size, and whether that is wanted depends entirely on what the line is for. The measurement is neutral; the interpretation is not.

What the same pattern is in other subjects #

The arrangement is not about any one kind of wave, and the versions worth knowing use it in different directions.

In acoustics, an impedance tube measures the absorption of a material by exactly this method: a loudspeaker at one end, the sample at the other, a microphone that slides, and the standing-wave ratio read directly. The absorption coefficient is one minus the square of the reflection, and the measurement is the international standard for the quantity.

In optics, a partially reflecting surface makes the same pattern in the light in front of it, with a spacing of half a wavelength — a few hundred nanometres — which is far too fine to probe with anything inserted. Wiener demonstrated it in 1890 by laying a photographic film at a slight angle across the pattern in front of a mirror, so that the film crossed several fringes and recorded them stretched out. The developed plate showed dark bands, and their position settled that it is the electric field rather than the magnetic one that darkens a photographic emulsion.

In quantum mechanics the same pattern appears in front of a step in potential: a partially reflected wavefunction, minima that are not zeros, and a probability current that is uniform and equal to the transmitted fraction. The reading of the flat line there is the same reading — the current is what is conserved, and the density is not what is flowing.

Three subjects with the same figure and different quantities on the axes, which is what makes the argument worth learning once.

Where the pattern comes from, drawn as a boundary #

It is worth closing the loop back to where reflections come from, because it says what a measured pattern is a measurement of.

A reflection is produced by a change in the medium, and how much is produced depends on how abrupt the change is compared with the wavelength. An abrupt step gives the full mismatch reflection; a gradual one gives almost nothing, because each thin slice reflects a little and the slices’ contributions arrive back out of step with one another and cancel.

So a deep standing-wave pattern reports two things at once — that the media differ, and that they change over a distance short compared with the wavelength — and it cannot separate them. A measurement at one frequency that finds a large reflection is consistent with a big abrupt change and with a bigger gradual one, and telling those apart requires sweeping the frequency and watching the reflection fall as the wavelength shrinks.

That is the taper’s whole principle read backwards, and it is the reason a reflection measurement is a broadband measurement or nothing.

Where the model stops #

The line is assumed lossless. With loss, the backward wave has been attenuated more than the forward one by the time both reach a given point, so the measured ratio falls as the distance from the load grows — a pattern that shallows out along the line rather than repeating. Reading a reflection off it then needs the loss to be known, and a badly lossy line reports a good match wherever it is measured far enough from the load.

One wave is assumed in each direction. Where the line supports several modes, each has its own reflection and its own pattern, and what a detector reads is their sum — which need not have a period of half a wavelength or a well-defined ratio at all. That is why the method belongs to single-mode lines and guides.

The measurement is at one frequency. A load’s reflection depends on frequency, and a single ratio says nothing about the bandwidth over which it holds. Two loads with the same reflection at one frequency can be completely different over a band, which is why the modern instrument sweeps.

The pattern is assumed to be established. It is a steady state, reached after the wave has travelled to the load and back a few times, and a measurement made faster than that sees a travelling wave with no pattern in it at all. For a metre of line that is a few nanoseconds; for a room it is tens of milliseconds, which is why an acoustic measurement of this kind needs the source to have been running.

And the detector is assumed not to disturb the line. Anything inserted to sample the field is itself a discontinuity, so it reflects a little, and the pattern being measured includes the measurement. Keeping that below the reflection under study is most of the practical difficulty of the classical technique.

Why a good match is hard to measure #

The steepness noted earlier deserves a number, because it decides how the measurement is actually done.

At a reflection magnitude of 0.05 — a quarter of a per cent of the power returned, which is a good match — the standing-wave ratio is 1.105. The maximum and the minimum differ by ten per cent, so reading the reflection to within ten per cent of itself needs the two amplitudes to a per cent. That is demanding for a sliding detector whose coupling changes as it moves, and it is why the classical method is poor at exactly the cases anybody cares about.

The modern answer is not a better detector but a different measurement. A directional coupler separates the forward and backward waves physically rather than reading their sum, so the reflected wave is measured on its own against the incident one and there is no near-cancellation to resolve. The improvement is not in precision but in conditioning: a ratio of two comparable numbers replaces a difference of two nearly equal ones.

That is a general lesson about measurements and worth carrying. A quantity extracted from the difference of two nearly equal readings is measured badly however good the readings are, and the fix is almost always to find an arrangement in which the quantity appears directly rather than as a residue.

What the pictures cannot show #

Every figure here draws a magnitude and none draws a phase, and the phase is half the information. Two loads with the same reflection magnitude and different phases give patterns identical in shape and shifted along the line, and it is the shift that says which load it is. A figure at one phase looks like a complete description and is a slice through a two-parameter family.

Nor does any figure show what happens in time. The pattern is a standing envelope, and inside it the field oscillates — at a maximum with the full amplitude, at a minimum with the residue — and the phase of that oscillation slides steadily along the line rather than jumping by half a turn at each node as it would in a complete standing wave. That sliding phase is the travelling part, and it is exactly what carries the power the flat line reports.

The line as a component #

There is one more consequence of the sliding, and it is the reason a length of cable is a circuit element rather than a wire.

What a source connected to the line sees is not the load; it is the load transformed by however much line is in the way. Because the pattern slides with distance, the ratio of amplitude to flow — the impedance — looking into the line changes with position, repeating every half wavelength. A short circuit a quarter of a wavelength away looks like an open circuit. An open circuit an eighth of a wavelength away looks like a pure reactance.

So a length of line is a transformer whose ratio is set by its length, and a stub of line with a short at the end is an adjustable component with no components in it. Both are standard, both are used at frequencies where discrete parts have stopped behaving, and both are consequences of the pattern being stationary in space while the phase slides along it.

The same statement has a limit worth naming. Everything here repeats every half wavelength, so a line short compared with a wavelength transforms nothing and is just a wire. The frequency at which a piece of wiring stops being a wire and starts being a line is set by its length against the wavelength, and it is the same threshold that decides whether a circuit’s own inductance matters.

Where the ladder goes next #

The standing-wave ladder began with only some notes fitting, went through the box that allows only some energies, the drum that has no harmonics, how many ways there are to vibrate and the count that cannot be cheated. This rung asks what happens when the reflection is not complete. The rungs after it: the impedance seen looking into a line, which is what the pattern is really measuring and which rotates with distance in a way that makes a length of line into a component; the multi-port case, where the same measurement made in both directions separates what a device transmits from what it reflects; and the time-domain version, where a pulse is sent instead of a tone and the reflections come back separated by their distances rather than mixed into a pattern.

The habit worth carrying away is to check which quantity an argument is about. Amplitude, energy density and energy flux are proportional in a single travelling wave and in nothing else, and a pattern of minima that look like nodes is exactly the situation in which treating them as the same thing gives a confident wrong answer.