Where a quarter-wave resonator belongs

A machine sends a train of pressure pulses down the pipe, one for every lobe or piston stroke. If nothing sends them back, they travel away and that is the end of it. The pressure at any point rises and falls by the height of one pulse, and no more.

Put something at the far end that reflects — a vessel, a blank, a change in bore — and the returning pulses run back through the outgoing ones. Where the two coincide they add, and where they oppose they cancel. What is left is a pattern that no longer travels along the pipe but stands still in it and swings in place. That is a standing wave.

Two waves passing through one another add, so the standing wave is about twice the height of a single pulse before anything else happens.

Resonance is what happens next, and it is a different thing. If the length of the pipe is such that each returning pulse arrives back exactly in step with the next one the machine sends, the machine keeps adding to what is already there and the pattern grows far beyond twice a single pulse. Every resonance has a standing wave in it; not every standing wave is a resonance.

With a standing wave established in the pipe, the dashed line along the centre represents the average pressure whilst the animated trace indicates the pressure variation. Where the pressure variation crosses the dashed line there would be no net pressure variation and a pressure gauge would hold steady. Where the pressure swings at its maximum the same pressure gauge would swing with the pressure change. These points of minimum and maximum pressure change are considered to be nodal points, and these nodal points occur every quarter of a wavelength along the pipe.

A quarter of a wavelength is a real distance, so where those nodes and antinodes actually fall along the pipe depends on the wavelength, and therefore on the frequency at which the pipe is standing. That frequency is not something we are free to choose. It is settled by what is driving the pulsation at one end and by how the pipe is terminated, or closed off, at the other. The termination is usually the easier of the two to describe. The source is not, and it is where we have to start.

Any analysis of pulsation in pipework has to begin by defining the source, and there are only two physical quantities available to define it with: pressure and flow. Whichever of the two the source sets, the pipework is left to settle the other.

Put simply, a velocity source gives the fluid velocity whilst a pressure source squeezes it. The choice is not a modelling preference, as it has a fundamental effect upon what an analysis of a system will find.

A velocity source therefore presents a hard boundary at the start of the pipe: pistons and blower rotors move, come what may, whatever the gas column might push back at them.

A pressure source presents a soft one, since there is no wall between a large volume and the pipe it is feeding. Which of the two we choose settles the boundary at that end of the pipe, and with it the frequencies at which the pipe will stand.

No real machine is purely one or the other, so it is better to think of a scale with the two definitions at its ends, with turbomachinery between them and rather nearer the volumes, since gas can find its way back through the impeller.

The far end of the pipe does the same job as the machine and has to be described in the same way. It is simply the second boundary, and the pipe stands at whatever frequencies suit the pair of them.

A pulsation dampener belongs with the drums and vessels at the soft end, and for the same reason: it is a volume large enough that gas can come and go without shifting its pressure.

Four runs at the same frequency. The columns are the two kinds of termination; the rows are the two kinds of machine.

The termination anchors the pattern whilst the nature of the machine determines the magnitude of the pressure fluctuation.

The machine end here sits on neither a node nor an antinode, and lands on one only when the run is at a resonance.

Knowing where the antinodes lie is what makes it possible to put a resonator where it will work.

A quarter wave resonator is a closed end branch pipe, usually of the same diameter as the main pipe to which it is attached. It has a closed end, which is a hard termination, and the length is chosen so that a pressure wave entering the resonator has a total return journey length of half a wavelength, which, in turn, means that the pressure waves that enter the resonator arrive back at the main pipe out of step (opposite in sign) to the wave in the main pipe, such that when they recombine the ‘resonator wave’ cancels out the ‘main pipe’ pressure wave.

Such resonators work best when positioned at a pressure antinode.

In essence, a quarter-wave resonator is a device that turns a pressure antinode, where pressure fluctuates, into a pressure node, where it does not.

A perfect resonator would hold the branch point at the mean pressure, with what is left of the initial pressure wave pushed back towards the machine and very little passing beyond. A real one has losses and works over a band rather than at a single frequency, so the cancellation is significant rather than total.

This is the case the worksheet calculates for. It calculates the quarter wavelength, measures that distance from the machine, and places the resonator there, with the resonator branch specified to be a quarter of a wavelength long and at the same diameter as the pipe.

Measuring a quarter wavelength from the machine only lands on an antinode if a node is anchored at the machine, so the rule carries that assumption whether it is stated or not. In this case the assumption happens to hold, which is why the resonator works. Panel 5, though, showed that it is the termination which anchors the pattern, not the machine.

One pipe, one machine, one frequency, and the same resonator shown in two positions: one where the worksheet puts it, a quarter wavelength away from the machine, where it lands on a pressure node; and a second on the nearest pressure antinode.

In the first case, at the pressure node, there is no pressure swing in the pipe, so almost no gas enters the branch and it does nothing at all. In the second case it sits on a pressure antinode, where the pressure swings and gas can flow into the resonator to make it work.

The node and antinode pattern is anchored at the dominant reflector, so to find the position of the resonator, where the run ends in an open end such as a dampener or a vessel, it is necessary to count back an odd number of quarter wavelengths from that reflector — the first, third, fifth, seventh, and so on — then position the resonator at one of those points. This is a change from the worksheet’s established procedure.

It can be seen that the nature of the machine does not really enter into it. Whether it dictates the pressure or the flow, the pattern is anchored in the same place and the resonator goes in the same position, so consequently the worksheet does not really need to make any assumptions about the machine at all.

Where there is no dampener and the discharge simply runs on into pipework, the method is the same but the datum has to be found rather than assumed: identify the strongest reflector in the run and count back from that. There is often no end in the ordinary sense, because the line simply carries on into more pipework, and the reflector is then wherever the bore or the volume changes sharply, which may be a long way from anything anyone would call the end. A vessel, a discharge to atmosphere or a sharp increase in bore is soft and is counted in odd quarter wavelengths. A blind, a closed valve or a sharp reduction in bore is hard and is counted in half wavelengths.

Where nothing in the run stands out as the dominant reflector, no counting rule applies and the position has to be taken from a calculated pressure profile.

The pressure a resonator has to work on falls as the cosine of its distance from the antinode, but the attenuation it gives falls roughly as the square of that, because what the branch removes depends on the energy at its mouth rather than on the amplitude. The curve above is the attenuation, and it is the one that matters.

An eighth of a wavelength out of position costs about half the attenuation. A quarter of a wavelength out costs all of it.

The tolerance is a fraction of the wavelength, so it shrinks as the frequency rises. For a blower discharging air at 80 °C, where the speed of sound is 377 m/s:

So at the lobe passing frequency the resonator has to be placed within about half a metre of the antinode to keep most of its effect, and within a couple of hundred millimetres at the second harmonic. That is a pipework tolerance rather than a nominal position.

The square is the first-order picture rather than an exact law, since a real branch has losses and a finite bore ratio. The bore of the line does not enter into the tolerance at all: it governs how much the resonator can do once it is in the right place, not where that place is.

What a standing wave is

• A standing wave creates a nodal pattern along the pipe.
• There are pressure nodes, where no pressure swing occurs.
• There are pressure antinodes, where the maximum pressure swing occurs.

What determines it

• The position of the nodal pattern is set by the end termination, not by the source.
• The nature of the source determines the magnitude of the wave.
• Whether there is a node or an antinode at the termination itself is set by the nature of that termination.
• A soft termination, such as a vessel, a dampener or an opening to atmosphere, sets a node at itself.
• A hard termination, such as a blank, a closed valve or a sharp reduction in bore, sets an antinode at itself.
• An elbow in constant bore is not a reflector; it adds length to the run but does not anchor the pattern.

Where the resonator goes

• A resonator can cancel out pressure pulsation.
• A quarter-wave resonator must be positioned at a pressure antinode.
• A pressure antinode on which to fit a quarter-wave resonator is found by counting back from the end termination: an odd number of quarter wavelengths from a soft one, half wavelengths from a hard one.
• Where no single reflector dominates, no counting rule applies and the position has to be taken from a calculated pressure profile.
• An eighth of a wavelength out of position costs about half the attenuation, and a quarter of a wavelength out costs all of it. As frequency rises and wavelength reduces, that physical dimension gets smaller.