Reed math, a question?

[ttonon]

Thanks tom for the great reply. You made a lot of stuff clear that probably would be to those the paper was written for. One question I have is when they are referring to reynolds number, what aspects are they applying that to? The window? the gap at neutral, (that varies from zero at the root of the reed to .4 mm at the tip,) or the gaps between the reed and the window as the reed enters which is typically around .019-.025mm. Presumably if they are interested in starting forces then the reed hasn't reached that point yet. But my experience with reeds is that they do not begin with a complete cycle, but build over a period of 20 or 30 partial cycles depending on the pitch of the reed in a decidedly non linear fashion, starting with very small oscillations that increase gradually to a point and then jump dramatically in power / travel at the point where the main harmonics begin to become visible in the sound output. This doesn't seem the sort of thing you would see if a I am curious about this jump where it seems the forces acting on the reed increase in effectiveness if not in amount as well. This would suggest to me that aspects of the geometry that are not reached initially as the reed begins to oscillate have important effects that can't easily be shown by a static flow of dye.

Reynolds number, which can range from zero to well over millions, is the ratio of inertial forces (usually in steady state) to viscous forces. For large Reynolds Number, viscosity plays little effect, and for very small Reynolds number, viscosity cannot be ignored, and may dominate the motion. Numerical values of Re have different significance in different types of flow problems. For internal flows, as in a pipe, it’s most likely that the flow will be laminar for Re less than about 2400 and turbulent for higher values. For external flows, as for flows around an airfoil, Re can be very large, and different flow regimes are encountered throughout. Complications arise when the flow becomes supersonic (shocks). For subsonic flows, viscosity can be neglected except for a very small region near the boundary of the foil, known as the boundary layer. I hope you don’t mind my digressions. Re = UL/v, where U is the “characteristic” velocity, and L the “characteristic length, and v the kinematic viscosity of the fluid (which is equal to the dynamic viscosity divided by the fluid density). In their problem, they choose the width of the reed slot as the characteristic length, and I would agree with this choice. The velocity to use in this evaluation is also straightforward, being that which the fluid attains when, in steady flow passes through the slot. I am, however, uneasy at simply looking at the Re and concluding that the air flow and water flow cases are similar, because of the oscillatory nature of the flow. But I’m not going to object. In this game, unless you have something concrete to show, or a better analysis to present, it’s best to first see what the results are. Until I can come up with something better, I look to this paper as a milestone.

 

I think you have valid points concerning the start transient of the motion, but again, I say that the start transient is not relevant to this paper. Ricot simply uses his formulations to calculate a periodic motion that is (presumably) independent of any kind of start transient. I agree with you that, for all we know, the start transient might contain physical processes not necessarily present in the periodic vibration.

 

Let me ask, can you explain just how it is that the reed can start vibrating, from rest? After thinking about it myself, I still have some questions, and I haven’t yet found any explanation that I consider adequate.

I get the compressibility bit now thanks to your explanation. One question about the jets I might have since I think they have amply demonstrated their shape, is whether they contribute to a reduced pressure under the reed by the same mechanism as a vacuum aspirator. No large scale vortex under the reed is needed for that to occur. In the initial stages of oscillation, the only flow is through the jets since the rest of the area is effectively blocked. As the amplitude increases though the shape if not the nature of the flow under the reed I would expect to be changing greatly throughout the cycle. Perhaps they have dealt with this, but I have't been able to decode it.

This might shed some light on the issue. Take a plate that has a perforation (aperture), the left side of which contains air at constant pressure P1, which is greater than that constant pressure P2 on the right side, which also contains air at the same temperature. Let’s look at the steady state; i.e., long after the motion started. When the air leaves the aperture and enters the right side, it must form a jet. This is not opinion. It must be the case, according to the laws of Physics, and I can go through the verbal proof, if you’re interested. But to save time, I’ll pass on this here in order to explain something a bit more complicated. Let’s say P1 is greater than P2 by 3 inches of water column. This will give a jet velocity of 116 feet/sec, as calculated using the so-called “Bernoulli Equation,” which is valid for incompressible, steady state conditions. Let’s say also that the aperture is a round hole with a diameter of ¼ inch. Now look at a molecule of air as it passes through the hole. (You can imagine yourself trying to get out of a theater in a crowd, through a single set of doors.) It’s velocity isn’t much – it takes you forever - until it gets very close to the hole, say within ¼ inch – the same distance as that which spans the aperture. In a blink, the molecule is accelerated to 116 ft/sec, whips through the aperture, and once it’s another ¼ inch or so downstream of the aperture, it completely forgets that there was an aperture to start with. In a time span of about (1/4*2)/(116*12) = 0.00036 seconds, the molecule completes its experience of the aperture. Now let’s say that, just when the molecule gets within ¼ inch to the left of the hole, you increase P1 to twice it’s value, and you took 0.001 seconds to do it. Will the molecule feel the change? Not much, because it’s long gone before the pressure P1 is significantly increased. If, however, you increased P1 in 0.0002 second, the molecule will definitely experience the change, because information of this change will travel to the molecule at the speed of sound (1030 ft/s), and the increased value of P1 will impart an increased velocity to the molecule during its passage through the aperture. One conclusion from this is that, if you change P1 at a frequency less than about 1/0.001 = 1000 Hz, the air motion can be considered steady, and you can use the Bernoulli Equation for each value of the pressure, even though the pressure is changing. Another conclusion is that, if you change P1 at higher frequencies, the steady flow equations become increasingly inaccurate, and at frequencies above around 1/0.0002 = 5,000 Hz, time changes in the flow field become very important. In these latter cases, the steady flow Bernoulli Equation is not valid, and more complicated unsteady flow equations must be used (e.g., Equation 3 in the paper). These latter equations contain time derivatives that account for the forces needed to time-accelerate fluid. This kind of acceleration should not be confused with the convective acceleration in steady flow. An example of this latter effect can be seen in the example of the theater exit. When you get near the doors, other people push on you, overcoming your inertia, increasing your velocity. You don’t need your own muscles for this. If you were on roller skates, you would be simply convectively accelerated out the door. An example of time acceleration would be a short sprint, accelerating yourself from a standing start.

 

I don’t entirely understand your question involving the vacuum aspirator, but let me simply say that the Bernoulli Equation doesn’t cause suction on the reed. I know you can find this term used in some web explanations on how reeds work, but it’s not accurate. Pressure pushes against things, moves them in the direction of lesser pressure. The lower pressure area doesn’t “pull.” As I understand Ricot’s analysis, he sets the pressure underneath the reed tongue to a constant value, equal to the pressure under the reed (on the back side of the plexiglass). The justification is that, since jets are formed, this pressure boundary condition must be satisfied. An interesting complication here is the difference between the flow going into the reed and that exiting, in the form of jets. In the paper, Ricot treats the exit flow as (quasi)steady, while he writes unsteady equations for the entrance flow. The reasoning here is not obvious to me. First off, you might think that, if the exit region is steady, then why not the entrance region? The fact that Ricot, after finishing the analysis, throws away unsteady terms suggests that the simple question may be valid. This is one aspect of the paper I need to spend more time on.

I'd actually be inclined to mostly discount any Bernoulli effects ( as microscopic an understanding of them I have ) in comparison with the entrainment of the fluid beneath the reed by the stream of fluid running through the reed window. When I did the small experiment using smoke instead of dye, there was no tendency for the smoke to flow under the surface of the reed beyond the edge of the opening. Instead, it simply "poured" through the windowing the same manner you's expect air to flow into the end of an open pipe with a large part of the flow coming perpendicular to the window opening. If enough motion was imparted to the fluid beneath the reed by the jets or larger scale flow at higher amplitudes, the momentum of that "plug" might be significant enough to create a substantial vacuum when the flow is blocked as the reed enters the tight portion of the window and flow is effectively cut off. They seem to address something similar to this by having the downstream side of the reed in free air while the upstream side is still water. While this seems to slam the door shut, I wonder if it doesn't simply introduce it's inverse with the lack of inertia of the lower water mass to counter the momentum of the upper mass.

Are you saying that smoke streamed through the tiny gaps circumventing the tongue? If so, then I presume that your smoke experiment and Ricot’s water experiment agree? The only way I can understand a “plug” effect, causing a vacuum is by invoking some kind of unsteady phenomenon. With the presence of jets, I can’t see a way to incorporate such phenomena.

The action of the mass of air above the reed is confusing to me regardless. I am not sure to what part of the flow they are ascribing bernoulli sorts of interaction. I can easily imagine the following of the water behind the reed on the down stroke as well as some deflecting sideways over the reed to go around through the gaps, but while the pressure easily can force the reed into the opening, since it is equal in all directions in the + side of the chamber, as the reed moves forward, I'd see most of the water entering the gaps coming from the side since it has no direction change to fuss with. When the water changes direction over the reed as shown in their diagram, whatever change in pressure due to the horizontal part of the flow over the surface would seem to be countered by the need to dump the forward momentum into the reed . Visualizations like mine are easily misled by ideas we grow up with that are often wrong, so I'm only musing here. The Sketch in Fig 5 they have here is a bit misleading since the scale factor is way off for the gap by the reed. The fit of a normal accordion reed is very tight as mentioned above and they show the gap in the vertical and horizontal being about equal when they are really about 20:1 almost all of it being in the vertical until the reed rises to an appreciable height during operation. Since they seem to be driving the reed at or near it's starting pressure at very low amplitude, I'm presuming that the gap doesn't much more than double, which doesn't change things much.

I’m having difficulty understanding all you’re saying here. Perhaps my paragraphs above can supply some common terminology for us. In any event, I totally agree with your first sentence here. It seems to me that Ricot finds all his physical mechanisms in the flow above the reed. If I had to make a wild guess from my understanding so far, I’d say that the only term he says is important for the excitation of the reed is the usual V^2 term present in the steady flow Bernoulli equation. If so, the actual situation is much simpler than the way he formulated it. I suppose it’s useful to first make a more general (unsteady) formulation, then trim it down, according to the results you get.

I can see the water flowing around the reed, as it resists the pressure, but as it swings back up on the reverse side of the cycle, the idea of it pushing back a huge mass of water in a plug seems essentially impossible if we are dealing with incompressible fluids, where with air it wouldn't be a big issue. That probably isn't what they are suggesting, but I am confused about what's momentum they are talking about. The Jets obviously have considerable momentum in proportion to their size, but the flow over the reed's surface is pretty darn slow in comparison, and only a small portion of that will actually make it into the jets compared to the amount entering from the side direction extending along the surface of the reed plate a little ways. The inertia of the water above the reed seems like a pretty big thing, and I can see a lot of influence on the reed there ( among which they mention the pitch reduction ), but that doesn't seem to be what they thing is driving things anyway. As far as I can see (and I may be missing their point here) the only place they find something that can drive the oscillation is in the out of phase velocity fluctuations and inertial forces being out of phase

I really can’t make much sensible comment on your points. I also have many intuitive kind of questions. I would like to do my own analysis on this, and this is maybe one reason why I’m motivated to discuss the paper with such interest.

I am beginning to see I think that they are concentrating on the jets which is after all where the flow ends up for the most part. One thing about what they say that troubles me is that they say that because there is no contact with the lower side of the reed by the upstream water that there is no effect of the water (or air ) on the back side of the reed. Are they suggesting some sort of cavitation occurs? I don't' see the water or air on the back side of the reed detaching from the reed, so it will move with it's own inertia, velocity etc.

The fluid on the back side of the tongue does contact the tongue and does have an effect. It transmits its static pressure to the back side. They are not saying that, just because the flow avoids that area, being swept in a jet, there is a vacuum there. The pressure there is equal to the pressure on the bottom side of the plexiglass. (This statement neglects presumably small pressure fluctuations due to turbulence and eddies.)

I also noticed that while they use an accordion reed, they don't use the reed plate ( mounting the reed over a clear plastic window) and their side gaps may in fact be to the scale they have drawn. A normal accordion reed would have to have very high pressure air to operate with that much gap. I also notice that in their picture of the jets, the tip jet is much larger than the side jets even though it represents such a short length in comparison. If a lot more water is moving here, might that be an indication that it might be more important than the side jets and possibly the source of a much larger percentage of the pressure variation readings? But as you say, is it different in kind than the sides? Perhaps, since the physical results are so much different, there is more than one thing at work here.

Although you may be correct, I see no need – without proof - to allude to extra mechanisms for a so-called “real reed.” I can accept that they built a free reed, and I can assume that this reed behaves as other (but maybe not all) free reeds.

I understand your point about the agreement of their model output with measurements, which does suggest to me that they have latched onto something worthwhile here. This doesn't feel like problem solved yet. The starting transient I mention at the beginning ( which may be different for really gappy reeds compared to musically useful reeds), The difference between tip and side jets, and the fact that efficient reeds are tight reeds, and a lot of changes are made at the tip of reeds (arching, twisting, corner cutting etc )to change the reed's motion at the time it is entering the window, makes me think that it should be looked at to see what is going on. Your point probably would be that while the amounts might be different between the sides and the tip area, the mechanism is likely to be essentially the same for the purpose of determining the driving mechanism. Lots to think about, and thanks for your insight and willingness to translate so well the more difficult parts for my more of a practical and less academic education.

I agree that there are many complications that further study can focus on. As usual with early investigations, much credit can be earned if you can put the problem in as simple a way as possible, eliminating all complications that might cloud the fundamentals, then succeed in nailing down something real and not understood before. Not all attempts are successful. This study is recorded in the literature and future researchers with take it apart, piece by piece. If there is a concern about how realistic their choice for side gap clearance, my guess is that the theoretical calculations can be very simply performed by setting this gap to zero. The theoretical calculations don’t have to worry about the tongue sticking in the slot! In fact, maybe they did this calculation, and if they did, I hope to think that they’d report if such a gap has such a prominent role in reed operation. I don’t know, apart from the obvous problem of scraping, is it possible for a free reed to operate with zero clearance?

P.S. the Sheng reed is really cool being made integral with it's reed plate, very flared window walls ( from the cutter used to incise the reed's backside ) zero set, +/- operation and near zero gap. They use a paste of marble dust and copper to fill any gap from the cutting of the reed and break the reed loose creating a close to perfect fit. Another Chinese reed is in the Bawu a reed flute with a reed similarly cut from the plate, but with a set and varying from quite a taper with square tip to a triangular reed with a point at the tip. It only operates in the overblown mode needed to dominate the reed's inherent strength and force it to vibrate in sync with the flute's resonances.

Is the marble/copper mixture applied with the copper in a molten state? How do you mean, “overblown?”

 

 

Best regards,

Tom

Edited November 16, 2010 by ttonon

[Chris Ghent]

I'm not willing to accept Chris's remark about his editing of his reply.

Along with Dana, my experience has led me to understand that Chris has

 

Thank you, but this is not how I experience the guy, and I have to work with him every day. He makes all of the mistakes in the shed and never cleans up. And he acts as if he knows what he is doing when I know for a fact he makes it up as he goes along...

 

Best

 

Chris

[Dana Johnson]

And he acts as if he knows what he is doing when I know for a fact he makes it up as he goes along...

 

Best

 

Chris

Psst... Chris... We all do.

Dana

[Dana Johnson]

Great stuff once again though I feel a bit bad occupying so much of your time. The Reynolds number is clearer to me now having run into it before primarily with respect to flow through openings. This helps me understand better what it refers to in that process. Regarding the choice of characteristic length, I'm a bit confused by the choice of the reed slot width. At the amplitudes they seem to be operating their reed, the water hardly sees the whole window except in the exhaust side, and it isn't going to be the limiting factor. The restriction of the gap variable as it is, would seem to limit the possible flow to vastly less than the window would allow, and if most of the flow is in the jets, what else are we talking about with respect to flow? The reed blocks the rest of the flow throughout most of it's travel at low amplitudes. Once the tip exits the slot at the bottom and the top, a larger more or less rectangular opening grows and then collapses, but at low amplitudes the reed will still cycle ( in air driven reeds at least) barely entering the window. On a well made reed with clearance at the tip and sides at .02mm, this comes pretty close to shutting off the flow except for the air the reed is following until it reopens the gap at the top again. Looking at the reeds from what I can imagine of the flow now, The reed dimensions seem to have a lot to do with the area the pressure operates on, and I wouldn't be surprised if it also made a difference to the amplitude of the sound pressure, but the driving flow seems to be interested in the gap dimensions especially if as they say, there are no vortexes that form under the reed surface. Once the reed is moving with larger amplitude that carries the tip through the window and beyond, the whole slot certainly does begin to open up and I'd agree then about the characteristic length.

 

 

I don't know what is going on during the transient phase, but I doubt if those forces go away. They may end up a small fraction of what drives the steady vibration phase though. Regarding what starts a reed I have had many thoughts, but no proofs. Interestingly, a reed will start if you direct a stream of air into the gap at the tip with a very small tube in a direction parallel to the plane of the reed. ( it doesn't work on the side gaps ) Most of the machinations people go through to improve a reed's starting take place at the tip. Anything from a corner being nipped off, to a bleed hole under the reed tip to a little pocket milled out under the tip. These things seem to be effective. Why? I don't know. The tip seems important because it is the point of greatest motion as well as the point of greatest leverage on the reed.

As far as I can see, some force needs to move the reed downward then disappear to allow the reed to spring back. downward air pressure by itself only closes the reed ( which happens when a reed chokes for instance ) When I mentioned the vacuum aspirator, We used to use them in chemistry class to increase the flow through filter paper and that sort of thing. They use a jet of water to impinge on the air surrounding it and carry it all out the end of the tube. In that example the high speed jet of the reed gaps would be transferring their energy to the water between and beyond them causing it to move away from the reed. While you are obviously right about things not actually "sucking", the plug of water under the reed moving away does get out of the way of the pressure above which then pushes further than it would if the water under did nothing. Even without the Jet pump effect, the reed moving down initially will force the water in front of it forward, The water's momentum will tend to keep it moving forward until water flowed in from the side to take it's place. The initial pulse of water would be in phase wit the reed's motion as caused by the initial pressure, but I'd imagine that the time needed for water to replace the moving " plug" under the reed might shift that phase a bit. What I would see happening is that the reed follows the departing water past the point where the initial top side pressure forced it before the lower plug tried to separate from the bottom of the reed. As flow from the side under the reed replaces the moving plug the pressure is returned to that produced from above the reed only. Now the reed having been deflected farther than would be the case can begin it's return, imparting some reverse momentum to the upper water with a somewhat similar effect. Each time the reed deflects, the momentum of the attached water takes it past where it would have gone were the water in front and behind simply to stop instantly once the forces were in balance. This seems t me to add an out of phase element that would work as a driver. Reading this, it seems that the momentum of the water following and receding from the reed might be enough without any jet pumping added to keep the reed moving. ( by any chance is this a description of what they were saying? It seems looking at it again in section B " Aerodynamic Pressure Force" the bottom of the last paragraph seems to indicate something like it. )

The initial oscillation may need to be started by motion being imparted to the fluid under the reed. Flow through the gaps would do it if followed by enough pressure to shut the reed. This seems more akin to plumbing and "water hammer" in pipes when the flow is cut off too quickly and the momentum of the water in the whole pipe hitting the valve drives the pipe up against a beam with a bang.

 

 

The smoke in my experiment did circumvent the tongue ( or seem to as far as I could tell ) and I felt was a good match to their dye streams though a coarser look at it.

My apologies for my lack of clarity. I am primarily trying to visualize what they are saying piecing together their remarks about the results of the equations, parts of which I can follow but the common terms in fluid dynamics aren't ones I'm familiar with so I spend most of my time trying to guess what the terms in the equations refer to once you get past V and t and P and the more obvious ones. Any experience I had with this sort of math was 40 years ago, and not in this area of physics. Consequently when I try to put things into words, they could be as confusing to someone who understands the math very well.

 

You have a good point about "real" vs experimental reeds I think it is cool they got the thing to work! The problem that I have run into with reed clearance is only the difficulty of keeping the reed from touching the sides which it does at high velocity. On long reeds with tight clearances, minute lack of alignment with the window will cause the reed to touch at the tip. I have made one or two smaller reeds with clearance of .01mm per side and they worked great. I couldn't use them though because the chance of them going even slightly out of alignment in a finished instrument far from my shop was too great. As it is, I have to center the reeds under the microscope. So far, we seem to find that the smaller the clearance the more efficient the reed. The limit seems the harshness of tone that develops when clearances are too close. I mentioned the sheng reed and the stuff they put on them to make them nearly zero clearance. That mix is more like a paste, not molten at all. ( marble decomposes to calcium oxide rather than melting and would be too reactive to last any time at all, soaking up CO2 from the players breath and turning back to calcium carbonate. )

If you play a bawu's reed ( set on the side of the flute where the lip plate of a normal flute is ) blowing at a pressure to set it vibrating , you get a nice musical sound, but the reed is strongly affecting the frequency, and only a few notes are possible. If you blow with much more pressure, the reed is effectively pushed into the choking mode ( where it can't return under it's own force ) and the tube resonance then can control it's action at the tube's resonant frequency allowing a full range of notes. The piper friend of mine I learned about this from uses overblown in their context where the reed is forced to move from one regime of oscillation to a different one.

Thanks for your insights,

Dana

Edited November 17, 2010 by Dana Johnson

[ttonon]

Hi Dans,

 

Thanks for your insightful observations.

 

Great stuff once again though I feel a bit bad occupying so much of your time. The Reynolds number is clearer to me now having run into it before primarily with respect to flow through openings. This helps me understand better what it refers to in that process. Regarding the choice of characteristic length, I'm a bit confused by the choice of the reed slot width. At the amplitudes they seem to be operating their reed, the water hardly sees the whole window except in the exhaust side, and it isn't going to be the limiting factor. The restriction of the gap variable as it is, would seem to limit the possible flow to vastly less than the window would allow, and if most of the flow is in the jets, what else are we talking about with respect to flow? The reed blocks the rest of the flow throughout most of it's travel at low amplitudes. Once the tip exits the slot at the bottom and the top, a larger more or less rectangular opening grows and then collapses, but at low amplitudes the reed will still cycle ( in air driven reeds at least) barely entering the window. On a well made reed with clearance at the tip and sides at .02mm, this comes pretty close to shutting off the flow except for the air the reed is following until it reopens the gap at the top again. Looking at the reeds from what I can imagine of the flow now, The reed dimensions seem to have a lot to do with the area the pressure operates on, and I wouldn't be surprised if it also made a difference to the amplitude of the sound pressure, but the driving flow seems to be interested in the gap dimensions especially if as they say, there are no vortexes that form under the reed surface. Once the reed is moving with larger amplitude that carries the tip through the window and beyond, the whole slot certainly does begin to open up and I'd agree then about the characteristic length.

 

Dana, reading through your descriptions here gives me opportunity to recall how efficient mathematical descriptions are. If a picture can save a thousand words, how many can an equation solve? I’m very pleased to have such an opportunity to think about these things, and in focusing on these details, I’ve finally come up with my own, hopefully realistic, description of how the reed works. I’m not 100% sure it’s correct yet or even that it’s the mechanism Ricot uncovers, but it is simple enough to intuitively understand, once certain physical concepts are understood.

 

But before I go into that explanation, let me comment on your paragraph above. Remember that Ricot invoked the Reynolds Number in order to argue that the water flow field is similar to the air flow field, in reality, for working reeds. I thus would not try to look for a Reynolds Number in his model. For actual operation of actual reeds, as you point out, the width of the slot comes into play for large swings of the tongue. Thus, without any reference to any theoretical model, the choice of slot width as characteristic length, to me, seems reasonable, and I share at least some of your criticism when Ricot throws out the operation of the tip region, which incorporates this dimension. However, as I explained previously, I don’t make an issue about it. In addition, even in Ricot’s model, the slot width IS a key parameter, and it arises because it defines the extent of the flow region above the reed. This parameter, h, appears in their Equation 8, and physically it incorporates the importance of the width of the tongue, which is essentially the width of the slot. The larger the value of h, the more flow is diverted into the variable area gaps. I think this reason alone should somewhat temper your objection here.

 

In addition, I don’t think Ricot’s formulation includes the case where the tongue vibration amplitude is large enough to send the tongue through to the other side of the slot. How can it do so if indeed the tip region is eliminated from the mathematical solution? I think we have to accept that their solution applies only to small amplitude motion.

 

Okay, in order to understand the way I think the reed works, first consider a flow of air striking normally to a wall. Fig. 5 in the paper can suffice for this, but you’d have to first eliminate the gaps and picture a wall of infinite extent. The streamlines of flow will be similar to those drawn near the axis in the figure, with the flow being smoothly diverted, eventually becoming parallel to the wall. The streamline right on the axis is unique. It intersects the wall perpendicularly, and the fluid in this streamline hits the wall and stays there (though it takes an infinite amount of time to get there). The pressure at this normal point of intersection is equal to the “stagnation” pressure in the fluid. This stagnation pressure, Po, is the maximum pressure in the flow field and equal to the pressure that is attained when any part of the flow field is decelerated and brought to rest, without any loss in energy (as from friction). Po is also the pressure that is used to accelerate the fluid to the velocity it has attained anywhere else in the flow field. All this can be expressed by the equation Po = P + rho*V^2/2, where P is the static pressure at any point in the field, rho is fluid density, and V the magnitude of the velocity at the corresponding point. This is the Bernoulli equation for steady, incompressible flow, which is derived from the Euler equation, which in turn is derived from the principles of Conservation of Mass and Momentum. Since we are not including heat and temperature effects, this equation is also the Energy Equation for the motion. It reflects the law of Conservation of Mechanical Energy, with P representing the local potential energy, and with the V-squared term representing local kinetic energy. Po is the maximum pressure in the system (analogous to the largest height or most extended spring position). It is a conservative system (no losses) and at all points in the system, the sum of total energy (potential + kinetic) is constant (and equal to Po). Thus, the maximum velocity in the system must be Vo = (2*Po/rho)^0.5. Thus, everywhere in the system, P is less than or, at most, equal to Po.

 

Now go back to Figure 5, as it’s drawn in the paper, with the gaps. Using the above concepts, we see immediately that the average pressure on the top of the reed tongue is less than Po. Only the stagnation point has Po, and everywhere else has less pressure. Furthermore, as the flow velocity increases above the tongue, the pressure of the air that touches the top of the tongue becomes lower. This is a direct result of Bernoulli (above) – the higher the velocity, the lower the static pressure, P. Keep in mind also that Po is the pressure in the bellows, away from any air motion. We call this simply bellows pressure, or driving pressure.

 

A second important concept arises from the ASSUMPTION that the pressure on the bottom side of the tongue is everywhere constant and equal to atmospheric pressure (outside the accordion).

 

With these two concepts, we might understand how the reed works. Assume periodic motion and imagine the tongue moving downward, approaching the slot. At any moment, there’s an average pressure, P1, on the top side of the tongue and atmospheric pressure Pa on the bottom, with net pressure (P1 – Pa) pushing the tongue towards the slot. As the tongue gets closer to the slot, the gap circumventing the tongue gets smaller, causing the average air velocity above the tongue to decrease. From Bernoulli, this causes an INCREASE of the average pressure, P2, acting on the top side of the tongue. Thus, (P2 – Pa) > (P1 – Pa), which means that the force pushing down the tongue increases as the tongue approaches the entrance to the slot. Conversely, as the tongue moves upward away from the slot, it experiences a lessening force. If the tongue/slot fit were perfect, when the tongue enters the slot, all air flow stops, in which case the pressure forces on the tongue are the maximum possible, and equal to (Po – Pa). (Dana, keep in mind that steady state – actually quasi-steady – is assumed. With such sudden stoppage, there would be some kind of “water hammer” effect, but apparently, Ricot’s work showed them to be negligible, compared to other forces at play.)

 

Let’s step back now and look at a simple property of all vibrating systems, consisting of a spring and a mass, being excited by a periodic force. When the mass is near the center of its motion, it’s displacement is near zero, which means its spring force is near zero, and its velocity is near max. Importantly, at resonance, the periodic force causing the motion is also at a maximum. Said simply, the periodic force causing motion is IN PHASE with the Velocity. This means, and it can be shown mathematically, that the work done by the pressure forces goes directly into maintaining the motion against all friction and losses in the system.

 

We have thus see the mechanism by which the pressure forces acting on the tongue are made to time their action so that these forces are in phase with the oscillatory tongue velocity. For me, at least for the moment, I think I understand this, and furthermore, I perhaps also understand Ricot’s conclusions – but I’m still not 100% sure, and here’s why. First of all, it looks to me that his excitation force arises simply from the steady flow Bernoulli equation and that his more general, unsteady, analysis for the flow field above the tongue simply produced the numbers he needed to justify throwing out unsteady effects. I’m still confused, however, by his explanation in the paragraph after Equation 8. He says that the “pressure fluctuations associated with volume flux fluctuations due to the reed aperture variations” do not transfer energy to reed motion, because they are in phase with the displacement (and out of phase with the velocity). I guess I don’t know what he means by the effects of “reed aperture variations.” In my reasoning above, it looks to me like these variations do indeed produce a force out of phase with displacement and in phase with velocity.

 

Lastly, I’m not sure if the mechanism I propose can explain the start transient. Perhaps for this, we would need to incorporate unsteady effects, such as a “water hammer” or some other inertial effect.

 

I don't know what is going on during the transient phase, but I doubt if those forces go away. They may end up a small fraction of what drives the steady vibration phase though. Regarding what starts a reed I have had many thoughts, but no proofs. Interestingly, a reed will start if you direct a stream of air into the gap at the tip with a very small tube in a direction parallel to the plane of the reed. ( it doesn't work on the side gaps ) Most of the machinations people go through to improve a reed's starting take place at the tip. Anything from a corner being nipped off, to a bleed hole under the reed tip to a little pocket milled out under the tip. These things seem to be effective. Why? I don't know. The tip seems important because it is the point of greatest motion as well as the point of greatest leverage on the reed.

As far as I can see, some force needs to move the reed downward then disappear to allow the reed to spring back. downward air pressure by itself only closes the reed ( which happens when a reed chokes for instance ) When I mentioned the vacuum aspirator, We used to use them in chemistry class to increase the flow through filter paper and that sort of thing. They use a jet of water to impinge on the air surrounding it and carry it all out the end of the tube. In that example the high speed jet of the reed gaps would be transferring their energy to the water between and beyond them causing it to move away from the reed. While you are obviously right about things not actually "sucking", the plug of water under the reed moving away does get out of the way of the pressure above which then pushes further than it would if the water under did nothing. Even without the Jet pump effect, the reed moving down initially will force the water in front of it forward, The water's momentum will tend to keep it moving forward until water flowed in from the side to take it's place. The initial pulse of water would be in phase wit the reed's motion as caused by the initial pressure, but I'd imagine that the time needed for water to replace the moving " plug" under the reed might shift that phase a bit. What I would see happening is that the reed follows the departing water past the point where the initial top side pressure forced it before the lower plug tried to separate from the bottom of the reed. As flow from the side under the reed replaces the moving plug the pressure is returned to that produced from above the reed only. Now the reed having been deflected farther than would be the case can begin it's return, imparting some reverse momentum to the upper water with a somewhat similar effect. Each time the reed deflects, the momentum of the attached water takes it past where it would have gone were the water in front and behind simply to stop instantly once the forces were in balance. This seems t me to add an out of phase element that would work as a driver. Reading this, it seems that the momentum of the water following and receding from the reed might be enough without any jet pumping added to keep the reed moving. ( by any chance is this a description of what they were saying? It seems looking at it again in section B " Aerodynamic Pressure Force" the bottom of the last paragraph seems to indicate something like it. )

The initial oscillation may need to be started by motion being imparted to the fluid under the reed. Flow through the gaps would do it if followed by enough pressure to shut the reed. This seems more akin to plumbing and "water hammer" in pipes when the flow is cut off too quickly and the momentum of the water in the whole pipe hitting the valve drives the pipe up against a beam with a bang.

 

I think we take on too much difficulty by mixing the start transient and periodic motion into the same physical explanation for how the reed works. I do indeed suspect that some unsteady effects would be necessary to explain the start transient, but apparently, these effects are not, and do not account for the periodic motion. One interesting experiment would be to very closely observe the tongue tip during start up, with some kind of flow visualization to see if any jets are formed, or if there’s any flow separation from a boundary. If not, then my guess would be that unsteady effects are important. I would guess this is the case, simply because, during start up, velocities are very small, and particle stay times (as I described in my previous post) may indeed be comparable to the period of oscillation – a dead giv-a-way for unsteady effects.

 

The smoke in my experiment did circumvent the tongue ( or seem to as far as I could tell ) and I felt was a good match to their dye streams though a coarser look at it.

My apologies for my lack of clarity. I am primarily trying to visualize what they are saying piecing together their remarks about the results of the equations, parts of which I can follow but the common terms in fluid dynamics aren't ones I'm familiar with so I spend most of my time trying to guess what the terms in the equations refer to once you get past V and t and P and the more obvious ones. Any experience I had with this sort of math was 40 years ago, and not in this area of physics. Consequently when I try to put things into words, they could be as confusing to someone who understands the math very well.

 

You have a good point about "real" vs experimental reeds I think it is cool they got the thing to work! The problem that I have run into with reed clearance is only the difficulty of keeping the reed from touching the sides which it does at high velocity. On long reeds with tight clearances, minute lack of alignment with the window will cause the reed to touch at the tip. I have made one or two smaller reeds with clearance of .01mm per side and they worked great. I couldn't use them though because the chance of them going even slightly out of alignment in a finished instrument far from my shop was too great. As it is, I have to center the reeds under the microscope. So far, we seem to find that the smaller the clearance the more efficient the reed. The limit seems the harshness of tone that develops when clearances are too close. I mentioned the sheng reed and the stuff they put on them to make them nearly zero clearance. That mix is more like a paste, not molten at all. ( marble decomposes to calcium oxide rather than melting and would be too reactive to last any time at all, soaking up CO2 from the players breath and turning back to calcium carbonate.

 

I certainly believe that you can build reeds with 0.01 mm (0.0004 inch) clearance, at least in some areas of the gap. I have a Baldoni accordion with “handmade” reeds, and on the East Coast, in Winter, I often have to slowly aspirate air through the instrument in order to warm up the reeds. Until I do, they are often so cold from the night before that most of them buzz, presumably because the tongue hits the sides of the gap, and this is a reasonable assumption because Aluminum plates have such a higher coefficient of thermal expansion than steel tongues. Calculations certainly bear this out, when you are talking about clearances near the tip end that are less than a thousandth of an inch.

 

If you play a bawu's reed ( set on the side of the flute where the lip plate of a normal flute is ) blowing at a pressure to set it vibrating , you get a nice musical sound, but the reed is strongly affecting the frequency, and only a few notes are possible. If you blow with much more pressure, the reed is effectively pushed into the choking mode ( where it can't return under it's own force ) and the tube resonance then can control it's action at the tube's resonant frequency allowing a full range of notes. The piper friend of mine I learned about this from uses overblown in their context where the reed is forced to move from one regime of oscillation to a different one.

Thanks for your insights,

 

As I mentioned before, I love the sound of the bawu. You can find some good examples on YouTube. In the literature, and according to Fletcher’s notation, the Western free reed is classified as a closing/opening, or (-,+) reed. In this notation, the first symbol denotes how air flow changes with increasing pressure from the front side of the reed, and the second symbol from the back of the reed. The clarinet (beating) reed is (-,+). The trumpet (lip) reed is either (-,-), or (+,-). The Asian free reed is (+,+). It’s not surprising that there could be two different operating regimes here, but in most normal use, as you say, the Asian tongue is blown entirely downstream of the slot. As you know, a resonating air column is required for this mode of operation. Interestingly enough, the Western free reed can, and does, also operate in this configuration, though not with accordions and concertinas. With harmonicas, however, the vocal tract of the musician does provide the required resonator, and indeed, Steve Levy became famous for his master of his “overblowing” technique. Incidentally, Ricot is aware of all this issues, and this explains his statement at the beginning of the paper, “However, in the instrument, the accordion reed only operates in its blown-closed configuration…” I point this out only to answer one of your original comments, calling into question whether Ricot realizes that the Western free reed can “only” be operated this way. “Only” isn’t true, as in the case of the harmonica, and Ricot merely accounted for that possibility.

 

Dana, I have enjoyed very much this discussion, and I know that there are at least a few other areas where we could find much interest in sharing experience on musical instruments. In these days, I’ve had extra time on my hands, as I’m between jobs – rather, I hope I’m between jobs. I’m not sure why I have such interest in free reeds. It’s true my grandfather, father, uncles, etc. played accordion, but my own siblings think I’m a bit weird for it. It does give me both musical, social, and intellectual outlet. Since studying them, I’ve learned about other musical instruments, and it’s interesting to see how different features among different kinds of instruments vary. No instrument is “perfect” from any point of view, and most of the difficult features simply provide challenges for the makers, many of whom are extremely skilled and competent individuals, who have done much to elucidate and even extend the fundamental principles by which the instruments work. I certainly concur with your aside to Chris.

 

Best regards,

Tom

[Dana Johnson]

Tom, Thanks for the very clear explanation. It makes eminent sense to me. Especially as it functions without being upset by the presence or absence of fluid of similar density on the back side of the reed. One question I have is how does the momentum of the reed feed into this? ( or is it one of those throw aways? ) I'm presuming most of the momentum of the water is directed perpendicular to the reed. It effects timing if nothing else. Perhaps this will be a good jumping off point for someone working on the amplifying effect of the chamber, but it seems as though things will get complicated fast. Thanks for making clear what the +/- referred to, The Bawu reed is more or less like the western reed in that it has a set above the window that has to be just right to function. and doesn't need the resonator to work unlike the sheng reed. In order to take advantage of the resonator it has to be blown above a certain pressure or the reed simply sounds approximately it's natural free frequency Like the sheng and I guess harmonica though I've never been able to manage it it will work in reverse on the draw, but only at pressures that can dominate it's natural tendency. It is also very thin in comparison to normal western reeds and so quite weak and easily forced beyond it's normal tendency.

Thanks again for the thought provoking discussion.

Dana

[Chris Ghent]

I don't want to hijack this thread; I'd be happy if it continued for years, but I note an air of conclusion in your last remarks Tom, and your reference to other possible areas of exploration. I am interested if you have any light to shine on the effect of composition, size and shape of chambers on tone.

 

I love that people are researching reeds and that one day I might be able to get hold of the results in a more simple and accurate formula than I use now. And I'd love to make an contribution to that moment. But while reeds are about 70% of making a concertina, concertinamakers seem have a better practical grasp on reedmaking than on the vagaries of wood, the species and the piece, its thickness/density and the shape/size of chambers, etc.

 

My current thoughts/questions are around whether producing the lowest formant possible is an issue only to do with the size of the chamber or whether the overall dimensions of the outer casing are involved.

 

I know experiments are good and I work on them slowly, but as you acknowledge they are time consuming and difficult. I apologise for being so practically geared, trouble is the 24 hour day...

 

Best Wishes

 

Chris

[ttonon]

One question I have is how does the momentum of the reed feed into this? ( or is it one of those throw aways? ) I'm presuming most of the momentum of the water is directed perpendicular to the reed. It effects timing if nothing else.

Dana, the reed tongue vibration is the primary system for vibration. All these air flow considerations we have made are done simply in order to understand the forcing (exciting) mechanism that causes reed vibration. The momentum of the reed itself is very large, compared to any momentum present in the air stream, and it certainly is not a throw away. In the paper, the reed equation of motion is Equation 10. All our discussion involving pressure is represented by the term F1(t) on the right hand side of this equation. The first term of this equation is the force required to accelerate the tongue (Newton’s law), the second term is the force required to generate the acoustic sound field that radiates sound to the surroundings (friction is neglected), and the third term is the restoring force due to the springiness of the reed. This is pretty much the classical vibration equation of motion for a cantilever beam.

 

If my own analysis of the excitory mechanism is correct, and if Ricot is correct in throwing away unsteady terms because they are negligible, you can forget about momentum, because it’s an unsteady effect inherent to fluid motion. Recall that, in the suggestion I made in the previous post concerning the excitory mechanism, I invoke only the steady state Bernoulli equation, and the conclusions were made without any consideration of fluid momentum, as it affects motion outside the steady state Bernoulli equation. I reason that, as the tongue moves upward away from the slot, the pressure force decreases, and I didn’t bother to add in a force that would be necessary to push the fluid out of the way. Because the motion is so (relatively) slow, the fluid is really not rushed enough to produce appreciable force, compared to the forces invoked by the Bernoulli equation. It’s as though, as you are exiting the theater, people are very gentle, and you on your roller skates are able to move along. There’s no bully accelerating himself and pushing everyone else out of the way. I agree with you that the water reed may have peculiarities not present in the air reed, but here, I’ll rely on Ricot’s argument using the Reynold’s number, which appraises the level of inertial forces, mainly because I find myself a little too lazy to go elsewhere.

 

The Bawu reed is more or less like the western reed in that it has a set above the window that has to be just right to function. and doesn't need the resonator to work unlike the sheng reed.

 

How well does it work without the resonator? Can it produce a sound as loud as the Western free reed? It seems that, in this case, what we have is basically a Western free reed with a relatively thin plate. I’d be surprised if it can provide a very loud tone, since the mean, equilibrium, position of the tongue would move to the other side of the thin slot with a relatively low blowing pressure, and you cannot develop a high blowing pressure without choking the reed. But, as always, I may be wrong.

 

In order to take advantage of the resonator it has to be blown above a certain pressure or the reed simply sounds approximately it's natural free frequency Like the sheng

 

I’d guess the reason for the minimum blowing pressure is so that enough energy can be transmitted to the air column within the tube, enabling a vibration mode in the air column to become established. These sound sources (reeds and edge tones), from an energy point of view, are very inefficient.

 

It is also very thin in comparison to normal western reeds and so quite weak and easily forced beyond it's normal tendency.

My guess here is that it must be fairly thin in order for the air column to control its motion, or timing, so that the reed’s vibration can be made to line up with how the air column wants to vibrate. We have a co-operative system here, much like that in a reed organ pipe, the sounding frequency of which is slightly above the natural vibration frequency of the reed. The reed in this case is a beating reed and not a free reed, but I think the physical principles have similarities. As an aside, the beating reed in clarinets is very supple, and the musical tone, coinciding with vibrational modes of the air column, is well below the natural vibrational frequency of the reed itself. The reed can couple with the air column in this case because the player’s lips introduce a lot of damping, which greatly widens the resonance response curve of the thin reed. With brass instruments, the players lips vibrate at frequencies close to the musical tone, and here, the damping is relatively small. I guess this is a bit off topic, but I find it enormously interesting.

 

Do you have a sheng, or a bawu? I’ve been toying with the idea of getting some of these instruments.

[ttonon]

I don't want to hijack this thread; I'd be happy if it continued for years, but I note an air of conclusion in your last remarks Tom, and your reference to other possible areas of exploration. I am interested if you have any light to shine on the effect of composition, size and shape of chambers on tone.

 

For years? We get a moment now and then when the universe seems to click in our favor, and this, for me, is one of these moments, when I feel community among fellow human beings is at a premium. I do echo your wishes, at least the spirit of them, and I say, lets give it a try.

 

Chris, I’ve been thinking about free reeds for some time now, and one thing I’ve isolated is the possibility that the apparatus involving their construction can be better suited to provide a more predictable and desired sound. I’m also aware that I’m not the only one on this quest, and right now, I think the most competent people for such an undertaking are the makers. Of course, you are welcome to study my PICA article, though I’m sure you understand that this article concerns cavity resonance, and it’s my experience that cavity resonance is detrimental to free reed operation. So it’s a paper on what not to do, rather than what you’d want to do. Nevertheless, the concepts there are fundamental. Concerning cavity designs that can be optimized for musical tone, I cannot offer anything significant from a theoretical point of view, and as I say, I think the makers have a far better grasp on this topic than I do, in a practical sense. I do however have some new ideas and theories that I’ve been playing with, in my daydreams, though these require experimentation for verification in the real world (drat!). I think that this, for me, is indeed a years’ long project, since there are many things on my plate now, though when I get through them, and IF I arrive at worthwhile conclusions, I’ll be sure to let you know. In the meantime, I think interesting topics will pop up now and then on this very interesting discussion group.

 

Best regards,

Tom

[Dana Johnson]

Hi Tom,

I do have a Bawu, and a Dizi. Both what you could call student models, a step above tourist grade, but not great. Bawu's aren't easy to find in China except in the south. Oddly in Bejing there were a lot of people playing the three pipe in a gourd relative. I have a friend who has a few shengs. one of which is a orchestral grade instrument . He sent me some reeds to look at ( on short term loan ) which were remarkable for the craftsmanship in them. The sheng reeds were cut from relatively thick plates while the Bawu reeds are from sheet I'd estimate at about .25mm. The Sheng reeds were essentially tuned to the pitch of the closed resonator and would play off pitch if out of tune even with the controlling influence of the resonator. In this respect the were still free reeds I'd think since they weren't dominated by the resonator, only supported enough to allow either direction operation. The Bawu reeds as you say are weak enough to vibrate in accordance to the demands of the resonator though the reed's natural frequency is very much in the range of the notes being played. When blowing hard enough to play properly, oddly not much more air is used and the volume produced is only slightly greater than when the reed is in control. My bamboo is old enough now I may try to make some this year. The one I bought is some nice tropical hardwood, but the ones my friend sent me to measure were bamboo and ranged from a very simple small one to a keyed two piece with ferrule joint that was a nice instrument. Unfortunately someone had fussed with the reed and it wasn't functional anymore.

Dana