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Greetings Free Reed enthusiasts.  I posted here about a year ago explaining an analysis I was undertaking on how the Western free reed works, starting from fundamental principles in Fluid Mechanics.  I’m happy to announce that I’ve made progress to the point where resulting calculations reveal much about the physical working of this sound source, and I’d like to present some results here.

 

Very briefly, the analysis starts with the Euler-Bernoulli wave equation for the vibration of the tongue and combines it with a physical model that describes the external forces acting on the tongue.  These forces include the bellows pressure force, the viscous friction acting on the skin of the tongue produced by relative air flow, the aerodynamic “form drag” acting on the tongue, and the effect of the mean airflow through the slot.  The model does not include inertial forces caused by air on the tongue, though it can be shown that these forces are small, especially if the vibration frequency is not too large. 

 

There are many different characteristics of this vibration, far too many to present here, but without getting into too much background or detail, I can present some interesting results.  Many concertina players say they can hear a difference between the sound of brass tongues and steel tongues, and many of us have wondered how different materials would sound.  In the case of Brass vs Steel, I uploaded a .docx file that viewers can view on a cloud storage site, with details below.  This file contains harmonic amplitudes plotted on a log scale, in the same way that a Fourier spectrum is viewed.  The amplitudes are normalized with respect to the fundamental. 

 

In order to understand what these harmonics are, consider first that the vibrating tongue can vibrate in different bending modes, each with its own frequency.  All free reed vibration in musical instruments involve virtually only the first bending mode, in which the tongue bends in a smoothly curved fashion, with only one node, where vibration is zero, at the rivet.  The fundamental frequency of the musical tone we hear is the same as the frequency of the first bending mode, also called a fundamental. These details are sometimes confused.  The second bending mode, which can occur under spurious conditions, has a node at the rivet and a node roughly a third of the length away from the free end, and the tongue bends in snakelike fashion.  Unlike a string, whose modes of vibration are easily excited and are spaced at frequencies that are closely integer multiples of the fundamental, the second mode of the tongue vibration is roughly six times the frequency of the first mode, with succeeding modes separated by growing frequency gaps. 

 

Now, within a given mode of vibration, the tongue can also vibrate with frequencies that are multiple integers – harmonics - of the bending mode frequency.  Let’s focus on a tongue vibrating in its first bending mode, the usual case.  Let’s also consider an isolated tongue, without the slot, vibrating as a simple cantilever.  If the excitation is “gentle” enough, or when any initial “complicated” vibrations die out, the motion will be very close to that of a sine wave.  This is why tuning forks sound so pure.  However, when the tongue is placed over the slot and pushed by a bellows pressure, the excitation is not so gentle.  The external forces on the tongue during its vibration change suddenly, depending where the tongue is in its cycle.  These sudden, changing forces induce harmonics in the general motion, although compared to the fundamental motion, they are usually small.  You can imagine, instead of the relatively gentle sinusoidal swinging, sudden minute lurches and hindrances superimposed on the motion.  Although these adjustments are relatively small, they can be measured, and who can really say how much affect they have on air vibration, and thus on the sound we hear? And if you think about it, how else could brass make a different musical tone than steel, if not by the way it vibrates?  We expect that somewhere in the frequency spectrum of the tongue motion, there are clues to why brass and steel sound differently. 

 

The plots in the .docx file compare the harmonic amplitudes of tongue motion between the usual ASTME 1095 spring steel and Brass 260, characterized only by their Young’s Modulus and density.  Each plot is for a different bellows pressure, indicated on the plot.  The log scale means there’s a factor of ten in displacement amplitude separating each integer on the vertical scale.  For low bellows pressure, the first harmonic greatly dominates, indicating relatively pure sinusoidal motion.  The 2.5 orders of magnitude on the plot is a factor of about 316 in amplitude between the fundamental and first overtone.  Also, differences between steel and brass are unnoticeable.  As bellows pressures increase, higher harmonics play larger roles, and the difference between brass and steel becomes evident.  With minor exception, steel often dominates brass in the higher harmonics, especially for the lowest of those harmonics where differences occur, around the 4th or 5th.  However, the first three harmonics or so never really show much difference.  Because of the relative amplitudes, we are more likely to hear the differences in the 4th to 6th harmonics more than those in the higher harmonics. 

 

From these plots we see a demonstration of the “brighter” sound of steel, or the “mellower” sound of brass, with such differences becoming more pronounced at larger bellows pressure.  But why does that occur?  The answer to that lies in the details of the physical model, to be uncovered by further investigation.  In short, it’s due to the nonlinear ways in which energy dissipation acts in the system.

 

The URL for the .docx file is below, and any visitor should be able to view it.  I’m not releasing it for download because I intend to publish these results in the future and I’d rather wait for public release.

 

https://app.box.com/s/zchj1y5d4l65fttok3n4sofupzx2gsio

 

Best regards,

Tom

www.bluesbox.biz

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Interesting. I read through it, but can’t say I understand it all despite having received an undergraduate degree in biophysics 40+ years ago and a firm grounding in fourier analysis. I don’t think I see anything above that answers this question: While air is flowing through the reed shoe and the reed bends in the direction the air is flowing, why does the reed then come back instead of staying bent until the air flow lets up?

Edited by David Barnert

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12 hours ago, ttonon said:

...I posted here about a year ago explaining an analysis I was undertaking on how the Western free reed

works, starting from fundamental principles in Fluid Mechanics. 

 

See also: Acoustics of free-reed instruments for an introduction to this subject. PDF here.

 

This Fluid Dynamics stuff is mind-boggling - sounds like a job for ARCHER.

 

Roger

Edited by lachenal74693

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3 hours ago, David Barnert said:

this question: While air is flowing through the reed shoe and the reed bends in the direction the air is flowing, why does the reed then come back instead of staying bent until the air flow lets up?

 

David, I think it has not been the purpose of Tom‘s paper to answer that - however, albeit my not having a degree like yours, isn‘t the mystery in the feature of eigenfrequency (combined with springiness)?

 

In some discussions here it had been pointed out that „the reed doesn’t vibrate“ (apparently in order to highlight that free reed acoustics are not at all about amplifying a tiny buzzing or singing tone) - but of course the reed vibrates (otherwise there would not be any tone and pitch), just like when we pluck it, only that the sound generation is entirely through cutting and releasing the air flow (which can only be directed and filtered then; no amplification through resonance of woods asf.).

 

But of course you‘re aware of all that...

 

Best wishes - 🐺

Edited by Wolf Molkentin

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Hi Tom, I am glad you are still on the subject. And i am happy you come to similar conclusions as i did. Some of your statements prove Facts i posted many Years ago. Keep on the work! Best regards Johann

 

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7 hours ago, David Barnert said:

While air is flowing through the reed shoe and the reed bends in the direction the air is flowing, why does the reed then come back instead of staying bent until the air flow lets up?

 

 

6 hours ago, Wolf Molkentin said:

David, I think it has not been the purpose of Tom‘s paper to answer that - however, albeit my not having a degree like yours, isn‘t the mystery in the feature of eigenfrequency (combined with springiness)?

 

 

David, that’s quite a puzzle, isn’t it?  And Wolf, the physical model must be able to answer that, otherwise, it’s not a good model.  Yes, it’s related to eigenfrequency, which is another word for natural (resonance) vibration, but the eigenfrequency is the result, not the cause, or explanation. 

 

The question of just how a free reed vibrates puzzled me for years, and no one I discussed it with could make sense out of it, even university professors.  I discovered the key when I developed the model.  The crux is that the pressure forcing the tongue downward through the slot is twice the force that retards it during its upward motion through the slot. 

 

When the tongue is moving upward, it cuts off the airflow, and the pressure force opposing its motion is essentially equal to the full bellows pressure difference.  The interesting thing is what happens when the tongue is moving downward while it’s in the slot.  Just before it enters the slot, the airflow through the slot is near maximum, because the tongue didn’t yet stop it.  That airflow has momentum, and when the tongue covers the slot, the airflow collides with the tongue, converting its momentum to static pressure.  This is the same phenomenon as “water hammer,” and the analogy is apt because air at those small velocities is incompressible, about like water is.  This “dynamic pressure” on the top surface of the moving tongue adds to the static bellows pressure.  Now, since the airflow through the slot is originally caused by the bellows pressure (difference), its kinetic energy has the same magnitude as the bellows pressure (potential) energy.  This dynamic pressure adds to the actual bellows static pressure, so the total static pressure pushing down on the tongue is twice the static pressure retarding the tongue’s motion upward through the slot. 

 

After I first saw this, I looked for experimental or theoretical confirmation and found it in a paper by Ricot et al, which I posted here a few years ago.  Ricot did not analyze the tongue motion, and was concerned with the acoustic sound field produced by the motion, which he did a marvelous job of representing.  But both his theoretical and experimental data shows this water hammer effect, and the interesting thing is that he didn’t recognize it.

 

So I’m happy to announce to the world here on our forum the basic physical reason how the free reed works.  I hope it’s not a blunder, because my ideas have not yet been subject to peer review. 

 

It would be too much to try to present the details of the physical model here.  I hope to publish it in the coming months and interested people can read about it then.  I will mention that a valid model must give an accurate energy audit in the vibration.  The results show a 100% balance between the net energy put into the motion by the bellows pressure and the energy dissipation due to frictional forces.  In reality, some of the bellows pressure energy must go into the acoustic field, which is not part of the model.  However, the acoustical energy produced by the vibration is a very small part of the energy audit, I think.  That reminds me, I have to verify that. 

 

Best regards,

Tom

 

 

 

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4 hours ago, Johann said:

Hi Tom, I am glad you are still on the subject. And i am happy you come to similar conclusions as i did. Some of your statements prove Facts i posted many Years ago. Keep on the work! Best regards Johann

Hi Johann, thanks.  I recall you posting that you felt that there is more to the sound of the free reed beyond the simple sinusoidal motion of the first bending mode vibration.  At that time, I thought your concern was the tongue's second mode vibration, and not the higher harmonic contribution to the fundamental of the vibration's first mode.  I apologize if I didn't understand you, and at that time, I was unaware that those higher harmonics could play a significant role.  I tried a quick search for your posts, but the search engine broke down.  If you'd like, we can re-visit that issue in another thread.

 

Best regards,

Tom

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12 minutes ago, ttonon said:

the eigenfrequency is the result, not the cause, or explanation. 

 

Tom, this is all fascinating, and I am possibly not up to understanding every aspect, even of a "water hammer", without further investigating. However, what I wanted to point out is that the natural resonance occurs in any event, even with the reed simply attached to any other workpiece, with no airflow, no slot, no anything (in fact, that's how I roughly tune a weighted reed: pressing the reed shoe onto the table, listen to the thin sounds and having the digital tuner's mic very close to the reed.

 

So wouldn't the question be: Why does the reed vibrate like usual (*) albeit the bellows pressure should prevent that?

 

(* of course I'm aware of the fact that the envirement of a reed chamber asf. is able to slightly shift the eigenfrequency)

 

Best wishes - 🐺

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1 hour ago, ttonon said:

The crux is that the pressure forcing the tongue downward through the slot is twice the force that retards it during its upward motion through the slot.

 

I have to assume that you are arbitrarily (and not very clearly) assigning meanings to “upward” and “downward” as follows:

 

downward = away from the side of greater air pressure (whether it be in the bellows or ambient air)

 

upward = toward the side of greater air pressure

 

In other words, in the same sense as “upstream” and “downstream” in the flow of air.

 

1 hour ago, ttonon said:

When the tongue is moving upward, it cuts off the airflow...

 

But my question is what starts it moving “upward”? With that uncertainty, the rest of your explanation loses me.

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2 hours ago, Wolf Molkentin said:

So wouldn't the question be: Why does the reed vibrate like usual (*) albeit the bellows pressure should prevent that?

Hi Wolf, by “like usual” do you mean at its natural frequency (the frequency you hear when you suddenly release it out in open)?  If so, the reason is because the bellows pressure and resulting airflow trigger the vibration at the right moments in the vibration cycle that matches the natural frequency.  It’s like when you push a child on a swing.  You push at the right time to accommodate the way the swing wants to swing. 

 

This kind of vibration is called “self excited,” and is different from the kind when a vibrating system is forced at a frequency different from its natural frequency.  With the tongue, about twice the bellows pressure pushes down on the tongue when it’s passing downward through the slot.  That’s the right time in the cycle, as with the swing.  Any force acting in the direction of motion adds energy to the motion.  As I explained, when the tongue is traveling upward through the slot, it’s being pushed at the wrong time, but the pressure there is only the bellows pressure, about half of what it is when it’s pushed in the right direction.  The downward push dominates.

2 hours ago, Wolf Molkentin said:

(* of course I'm aware of the fact that the envirement of a reed chamber asf. is able to slightly shift the eigenfrequency)

That's correct.  The tongue actually vibrates at a frequency slightly less than the natural frequency, and that's because of dissipation in the system.  Actually, there can be large differences between the natural frequency and the vibration frequency, and the model predicts that very well.  But those details are too much to get into here.

 

Best regards,

Tom

Edited by ttonon
Correct name in address.
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1 hour ago, David Barnert said:

I have to assume that you are arbitrarily (and not very clearly) assigning meanings to “upward” and “downward” as follows:

 

downward = away from the side of greater air pressure (whether it be in the bellows or ambient air)

 

upward = toward the side of greater air pressure

Correct.  Sorry for the confusion.

1 hour ago, David Barnert said:

 

2 hours ago, ttonon said:

When the tongue is moving upward, it cuts off the airflow...

 

But my question is what starts it moving “upward”? With that uncertainty, the rest of your explanation loses me.

When the tongue is moving upward THROUGH THE SLOT, it cuts off the airflow.  Sorry for not being clear.  I'm not sure where your confusion is on what causes upward motion.  That motion is part of the vibration, as is the downward motion.

Tom

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Tom,

your initial post adresses differences in timbre between materials and your answer to David describes how reed shoe amplifies tongue vibration, but nowhere you adress how the vibration itself starts, which is what David asks for and what is the crux of the the question "how free reed works" and a subject of endless debates on this forume since forever. I don't know if you are aware, but couple of first oscilations of the tongue happen above the upper plane of the frame, with movements yet too small to close the gap entirely (I've been away from this forum for nearly 5 years and may have missed some important arguments; I saw there was at least one topic about turbulent flow being the source of initial vibration while I was absent). The tongue is already oscilating when it enters the frame, at which point "exactly like child on a swing" and "water hammer" analogies are adequate illustrations of how already existing oscilation is fed energy and reaches peak volume, but all those descriptions concern free reed at fully developed swing. If the initial pressure rise is too steep the reed chokes, because if you apply "fully developed swing" physics to a stationary tongue (not already oscilating slightly above the shoe) what you get is a simple one way pressure valve, with equilibrium position where the spring force of the tongue equalizes the pressure force applied to it. It is the same if the reed is set without the initial gap at all and from what you have presented here to this point I don't know if your model discerns between the initial setup with and without a gap as it should to be true.

 

No model of the free reed will be complete without explaining first weak cycles before the tongue enters the frame.

 

Your model should also be able to explain why "tongue tip over hole end of the chamber" produces significantly weaker speaking reed than "rivet over hole end of the chamber" configuration with every other parameter remaining the same.

 

You might also want to test your model trying to explain why DIX reeds, regardless of materials used, produce less higher harmonics than non-DIX equivalents by the means of only two tiny holes at the tip corners of the shoe.

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21 hours ago, David Barnert said:

 

But my question is what starts it moving “upward”? With that uncertainty, the rest of your explanation loses me.

 Is comparable with a pendulum (https://en.wikipedia.org/wiki/Pendulum) in a Watsch. The air passing through the slot puts force to the reed tongue, air turbulence makes starting easier. Once the tongue is inside or just nearer  the slot the air is reduced or nearly cut of so less force gets to the tongue and the reed tongue moves back. And the cycle starts again. See also https://en.wikipedia.org/wiki/Inertia. Be aware that because of this massbehavier it will always move beyond zero where the system would be balanced. Sometimes bernoulli effects also will be quoted but this is not really the case because we don't have two passes of air flow. In the reed itself energy is moving from mass to stiffness (pendulum - mass gravitation) and back as in other oscillating systems.  The air is just needed to start and put in the loss off energy to keep the oscillating going. One can easily compare the system with  LC Oscillator. https://en.wikipedia.org/wiki/Electronic_oscillator#LC_oscillator Also keep in mind we talk here abute reed motion not the sound produced.  The actual Sound and the reed motion under real condition is even more complex. The reed is the impulse generator coupled to the vibrating air column in the reed chamber. The reed is infected by the air resonance as well, but to a lass extent as the reed itself. Man resonanz is determined be the reed itself bat the air column adds to this as well. Between air column and reed a resistance is present this resistance can be variable. Also the spot on witch sound power is output to the surrounding environment.  Sound in the environment must be coupled to the air column in the chamber. It does matter on which position of the standing wave in the aircolum the output coupling occurs.  Variating some or all of this parameters result in more or less power output, talk reaktion of the reed, pitch and timbre.  All this can't be varrid without infecting each other. So better tak reaktion will change the timbre, pitch and power as well. And not always the wanted results to accomplish and some compromise has to be made.

Edited by Johann

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12 hours ago, Łukasz Martynowicz said:

 

 

Your model should also be able to explain why "tongue tip over hole end of the chamber" produces significantly weaker speaking reed than "rivet over hole end of the chamber" configuration with every other parameter remaining the same.

 

You might also want to test your model trying to explain why DIX reeds, regardless of materials used, produce less higher harmonics than non-DIX equivalents by the means of only two tiny holes at the tip corners of the shoe.

Both is related to resonance and coupling. We have a combined system of an air column that is actually producing the sound and an impulse generator the reed. So the system is rather complex but in short  it does matter where the resonator has the opening.  Adding holes also affects the Resonance of the air column in a chamber. And moving the position of the hole in the chamber changes the coupling resistance.

 

Edited by Johann

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Back to the actual subject of Tom's statement! 

 

Different materials for the reed result in different length (if we stay with the same stiffness) of reed tongues. Loger reeds will produce different overtones in sum as schoter because the reed is vibrating not only on the fundamental and the according harmoniks. It is also vibrating in other modes and corresponding harmonic overtones. Which mode actually will get more in front to be part of the herald spectrum is difficult to tell. It is not only the different length that will cause more or less of this higher modes, or change the frequency relation to the fundamental. What Tom did not implement is the different shape of reed tongues. Shape is the mayor faktor which changes the frequency relations between fundamental and higher modes. Sure different reed tongue length will have results. For most people this will be very confusing. Because i don't think there are a lot of people who understand what modes are. Read more about modes. The easiest way to get it is to reed about https://en.wikipedia.org/wiki/Normal_mode. But also in this article it is not easy to get it. 

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27 minutes ago, Wolf Molkentin said:

Tom, Johann, what about the thinner sounds of an „inboard“ reed (parallel design)?

It all comes down to resonance and coupling, as sayed before the system is very complex we hear the air column vibrating not the reed. Other parts beside camber, boards and casing all participate more or less. The way the reed is fixed to camber or board all - or how the layout of the reeds are set up on the board effect modes and coupling on board. But this is all not really well researched. 

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5 hours ago, Johann said:

Both is related to resonance and coupling. We have a combined system of an air column that is actually producing the sound and an impulse generator the reed. So the system is rather complex but in short  it does matter where the resonator has the opening.  Adding holes also affects the Resonance of the air column in a chamber. And moving the position of the hole in the chamber changes the coupling resistance.

 

 

I know that resonance and coupling matter in case of chamber layout - I was merely stating the fact, that Tom's work does not include (or it hasn't been clearly stated, that it does) chamber geometry and tongue geometry (as you more broadly point out).

 

And with DIX reeds I don't think it's related to resonance, but to minutia of airflow around the tongue in the initial stage of agitation, restricting development of higher modes of vibration, and I brought this up as an extreme case of reed/shoe geometry influence on the tone - DIX reeds are just a tiny variation on otherwise classic accordion reed design - trapezoid tongue in non-tapered opening.

Edited by Łukasz Martynowicz

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