The bell decides what gets out

Every essay on this ladder has treated the open end of a tube as a boundary condition: the place where the pressure has to be zero, which is what quantises the modes and decides whether the even partials exist. The end correction rung refined the position of that boundary and found it to be a few millimetres outside the physical opening.

None of them asked the obvious next question. If the wave turns round at the opening, how does any sound get out?

The answer is that not all of it turns round, and the fraction that does not is the instrument’s entire output. Which means the two things a wind instrument has to do — hold a standing wave and make a noise — are in direct competition, and one number settles both.

One number, and it is a size against a wavelength #

The quantity that decides it is ka — the wavenumber times the radius, which is the circumference of the opening measured in wavelengths. When it is small the air just outside the opening moves as an incompressible lump and nothing radiates; when it is large the opening behaves like a piston in a wall and radiates freely.

Because k is proportional to frequency, the crossover frequency goes as one over the radius. That is the whole of why the four curves are where they are, and it is why the widest and narrowest openings drawn here are nearly sixteen times apart.

The trade, stated exactly #

Now put that beside what a resonator needs.

A standing wave exists in a tube because the wave reflects at both ends and comes back. The amplitude it builds to depends on how much survives each round trip: reflect ninety-nine per cent and the resonance is sharp and strong, reflect fifty per cent and it is broad and weak.

Energy that leaves is energy that did not reflect. So the same curve read upward is the radiation and read downward is the reflection, and an instrument cannot have both.

That is not a design flaw and it is not a compromise anybody chose. It is the statement that a resonator with no output is inaudible and a resonator with perfect output is not a resonator, and every wind instrument in existence is somewhere between.

A brass instrument’s low partials stay in and its high partials come out. The first is what makes the standing wave strong enough to hold a note against a player’s lips; the second is what makes the sound bright. A single curve, read at two ends.

That also explains an oddity of brass playing that otherwise needs a separate account. The pedal note — the fundamental of the tube — is weak and hard to produce on a trumpet, and the reason is on the curve: at 116 hertz almost nothing escapes, so there is very little to hear even when the standing wave is present. What the listener supplies instead is a fundamental inferred from the partials that do get out, which is the residue mechanism this collection has measured four times.

What a bell is for #

A bell does not remove the boundary. It moves it, by making the opening bigger, and the crossover moves as one over the radius.

So a bell buys loudness and brightness together, at a cost that is written on the same curve: the partials it lets out are partials that are no longer reflecting, so those modes are weaker resonances than they were.

A flare is a gradual version of the same move, and this is where the piston model used here stops being adequate. A trumpet’s bell is not a hole in a wall; it is a horn, whose cross-section grows continuously so that the impedance change from tube to room is spread over a length rather than concentrated at a discontinuity. A gradual change reflects less than an abrupt one, which is the same principle as an acoustic taper anywhere else.

What the model does get right is the position of the transition, which goes as one over the radius and is the whole argument.

And it points, for the same reason #

The third consequence of ka is the one this collection has already drawn from a different direction.

Those two crossovers are the same crossover, and “within a few per cent” understates how exactly. Both are thresholds on ka and nothing else, so their ratio is a pure number with no radius in it:

The same 1.11 in every row, to three figures. It is not an agreement that happens to be close for these four openings; it is one quantity read at two thresholds, and the two would track each other exactly on an opening of any size, in any gas, at any temperature. A source radiates efficiently when it is comparable with a wavelength, and a source that is comparable with a wavelength also beams. So loud, bright and directional are one parameter with three names, and the instrument that is loudest is also the one that is most obviously pointed at somebody.

How much of the resonance the radiation is actually costing #

The trade this essay is built on — energy that leaves is energy that did not reflect — is exactly true and is a smaller effect than it reads as, because the fraction is per round trip and a resonance is measured per cycle.

A mode makes 1/n round trips per cycle, so a rough quality factor for mode n is 2πn divided by the fraction radiated at that mode’s frequency. Evaluated on four instruments:

Measured input-impedance peaks on real brass instruments have quality factors of a few tens. These are one to two orders of magnitude higher, which means radiation is not what limits the resonance — viscothermal friction at the bore wall is, throughout the range, and by a wide margin at the bottom of it.

That does not overturn the trade; it bounds it. The competition between holding a wave and making a noise is real, it is exactly as described, and at low frequencies it is a competition over one or two per cent of the loss. Where it becomes a live constraint is high in the range, where the radiated fraction has risen by an order of magnitude and the wall loss has not — which is another way of saying that the bell’s effect on the strength of a mode is concentrated exactly where its effect on the spectrum is.

The relative column is the safer reading and it survives every crudity in the estimate. A flute’s radiation-limited Q is eight times a trumpet’s at the fundamental, purely because its opening is a sixth the radius. Whatever the absolute numbers are, a narrow opening is holding on to its wave far harder than a wide one, and that ratio is the same 1/a that runs through the whole essay.

Which gives a compact account of a fact every orchestrator knows. A horn is turned away from the audience and is famously the instrument most dependent on the room; a trumpet is pointed at it and is the loudest thing on the stage. The horn’s bell is the largest in the orchestra, its crossover is the lowest at 403 hertz, and it is therefore directional over almost its whole range — so almost none of its direct sound reaches a listener who is not in front of it.

The measurement that would break it #

The account makes a prediction that is easy to state and would be easy to falsify: an instrument’s spectral balance should follow its opening’s crossover and nothing else about it.

Two instruments of the same bore and length with different bells should differ in brightness by the amount their crossovers differ, and two instruments with the same bell should not differ however else they are made. That is very nearly the experiment cornet and trumpet makers have been running for two centuries, and the results go the right way — a cornet’s more gradual, wider flare is duller, and it is duller in the region the curve says.

The failure condition is real and there is a known partial failure. At high playing levels the spectral balance changes enormously with dynamic on a trumpet and hardly at all on a flute, and no radiation curve says anything about level. That effect is the nonlinear steepening named below, it lives in the bore rather than at the bell, and it is a genuine limit on how much of the brightness the argument here can claim.

The other half of the same impedance #

There is a unification available here that is worth stating, because it makes the end correction rung and this one two halves of one quantity rather than two separate facts.

The air just outside an opening presents an impedance to the wave inside, and an impedance has two parts. The reactive part is the air that moves with the wave without carrying energy away — the lump of air that has to be pushed back and forth — and it behaves exactly like a little extra length of tube. That is the end correction: 0.6133 times the radius, which is a distance.

The resistive part is the air that carries energy away and does not come back. That is the radiation, and it is the curve at the top of this essay.

The reactive half of the same boundary is an added length: about eighteen millimetres at every note for a thirty-millimetre opening, which is a real fraction of a short tube and a negligible one of a long tube. It is the same physics as the radiation curve seen from the other side — what does not leave is what is reflected, and what is reflected is what lengthens the tube.

So the two rungs are measuring the imaginary and the real parts of the same number, and the reason a wide bore is hard to keep in tune and the reason it is loud are the same reason.

What the room does with it #

The consequences do not stop at the instrument, because the frequencies that escape best are the frequencies a room absorbs fastest.

Beyond the critical distance — where a hall’s reverberant field overtakes the direct sound — a listener is hearing the room rather than the instrument, and the room has no directivity at all. So everything in this essay is a statement about the direct field, which is most of what a player hears of themselves and a decreasing share of what an audience hears the further back it sits.

That is the same conclusion the room ladder reached from the hall’s side, and the pairing is worth noticing: an instrument’s brightness is manufactured at its open end and is the first thing a distance takes away.

The same argument, on things that are not tubes #

The crossover is a statement about a radiator’s size against a wavelength, so it applies to anything that radiates, and this collection has already met it twice under other names.

A violin’s body is a radiator a few tens of centimetres across, so its crossover is a few hundred hertz — which is why the body’s response is what reaches the room and why the string, being a thin wire, radiates essentially nothing on its own.

A drum’s membrane is large, so it radiates its low modes reasonably well, and what a drum is doing instead of having a fundamental has a radiation term in it as well as a modal one.

And a singer’s mouth, at two and a half centimetres, has its crossover at a few kilohertz — which is why the singer’s formant is directional and the fundamental is not, and why a singer turning away loses the part of the sound that was carrying.

Four objects, four sizes, one curve. What varies is the radius, and the radius is what a maker chooses.

Whose instruments, and when #

Brass instruments got their bells for reasons that had nothing to do with acoustics as a discipline, and the shapes were arrived at by two centuries of makers listening.

What is worth recording is that the acoustics ran ahead of the explanation and behind the practice. Bessel-horn profiles fit measured brass bells well and were fitted to them rather than derived; the horn equation is Webster’s, published in 1919, long after the instruments; and the impedance measurements that make the account above quantitative are from the second half of the twentieth century.

Two caveats about the repertoire belong here rather than in the model.

A brass instrument’s brightness is not all bell. At high playing levels the wave inside a long narrow bore steepens as it travels — the compressions travel slightly faster than the rarefactions — and the steepening generates high partials inside the tube before the bell has done anything. That nonlinear brassiness is a large part of what a fortissimo trombone sounds like, and it is not in any of the curves here.

It is worth noticing that the two mechanisms are distinguishable by exactly the test this essay proposes, because they depend on different variables. The bell’s contribution is a fixed multiplication: the same curve applies at every dynamic, so a quiet note and a loud one leave through the same filter and their radiated spectra differ only by a scale factor. The steepening is a function of amplitude and of the distance travelled, so it grows with the dynamic and with the length of the bore. A spectral balance that changes with level is not the bell, and a trumpet’s does while a flute’s does not — which is the failure condition named above, and it is a failure of scope rather than of the account.

And a horn player’s hand is a variable bell. Putting a hand into the bell narrows the opening, which raises the crossover and takes the top off the sound, and closing it further changes the effective length as well. Hand-stopping is the one place in the orchestra where a player is moving the boundary this whole essay is about, continuously, during a note.

What the picture cannot show #

The model is a piston in an infinite baffle and an instrument’s bell is neither a piston nor in a wall. What it gets right is the position of the transition, which goes as one over the radius; what it gets wrong is the detail of the ripple above the crossover and the behaviour of a flared profile, which reflects less than the abrupt end assumed here.

Nothing here is a horn equation. A real bell’s profile is a continuous flare with a cutoff of its own, below which it will not transmit at all, and that cutoff is not the same number as the piston crossover. The two agree in order of magnitude for the instruments drawn and the essay does not need more than that.

And there are no wall losses anywhere in it, which is the largest omission and the reason the quality factors above come out in the thousands where a measured brass instrument’s are in the tens. Viscothermal friction in the boundary layer at the bore wall grows as the square root of frequency and as one over the bore radius, and on a long narrow tube at a low frequency it is the whole of the loss. Every statement here about radiation is unaffected — the fraction that escapes is the fraction that escapes — but every statement that reads it as a statement about how strongly a mode resonates needs the other channel beside it, and the other channel is larger.

The radiated fraction is per round trip and not per second. How much energy actually leaves in a given time depends on how often the wave reaches the end, which depends on the tube’s length. The section above converts the fraction into a quality factor on that basis and finds radiation to be a minority loss channel — so a long instrument at the same crossover is losing less per second than a short one, and the correction is small compared with the wall losses neither figure contains.

And nothing about the mouthpiece is in the model. A brass mouthpiece is a Helmholtz resonator in series with the bore and it substantially reshapes which modes are strong; it is the other end of the instrument and it deserves the rung this ladder has not yet written.

The ladder from here #

Six rungs, and every one is about a tube: what its ends do, where they are, what drives it, what temperature does to it and what escapes. What is missing is the mouthpiece at the other end — a small resonator in series with a large one, which moves the tube’s modes into a harmonic relation they would not otherwise have — and the tone holes, which are a ladder of their own and are the reason a wind instrument can play more than one note at all.