# Standing wave in cylindar vs cone

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• Phil Freihofner
In summary: No.In summary, the antinode (driven by the reed) is at the closed end of the cone, while the node (where the tube is open to the atmosphere) is at the open end. The shape of the bore affects the pressure profile, which in turn affects the pitch of the sound.
Phil Freihofner
I have been trying to understand why two woodwind bore shapes behave so differently.

My understanding is that one end of a woodwind is an antinode (driven by the reed of the instrument) and the other end is a node (where the tube is open to the atmosphere).

a - - - - - - - - - - n

In the case of the cylindar (e.g., clarinet) I understand how the length of the bore is 1/4 the total length of the wave that would be heard. I can see also that the next possible mode would be 3 x's shorter in period. There has to be an intervening node and antinode. A good visual equivalent is someone spinning a jump rope that is tied to a post at one end. We can get a half loop, or a loop and a half, or two loops and a half: there's always a half loop for the spinning arm/antinode.

The result, for a clarinet is that the second mode for a given fingering is an octave and fifth above the fundamental.

a - - - n - - - a - - - n

This is also a well known part of the defining timbre for the clarinet: that the odd overtones predominate the low end of the frequency spectrum for a given note.

I'm having trouble understanding why this doesn't also hold for conical bores. With saxophone and oboe and bassoon, all instruments with conical bores, the second mode is a pure octave above, and all the overtones are present in the lower part of the frequency spectrum. This seems to suggest that either the bell end has become an antinode or the reed has become a node, in order to fit a wave with half the period of the fundamental.

NOT POSSIBLE:
a - - - - - n - - - - - a
n - - - - - a - - - - - n

Neither of the above make physical sense, given one end is a reed and the other open to air.

Can anyone point me to a good explanation of what is going on with conical bores? I got a glimmer of an idea looking at a discussion of pitch with a flaring brass instrument bell. It seems like different frequencies experience the "node" as being at a different location in the context of the flare (according to a diagram from a book by Arthur Benade). And a conical bore could maybe be considered analogous to a really long, stretched-out flaring.

Is there some math that can illustrate or model what is going on, e.g., how the location of a node might be a function of the diameter of the bore of the tube?

My understanding is that one end of a woodwind is an antinode (driven by the reed of the instrument) and the other end is a node (where the tube is open to the atmosphere).
... In terms of pressure, this is correct. In terms of motion - the closed end has a node.

Basically - instruments can get complicated. Have you seen:
http://hyperphysics.phy-astr.gsu.edu/hbase/Music/clarinet.html
http://hyperphysics.phy-astr.gsu.edu/hbase/Music/oboe.html
... there are pages for the others. I see they wimp out on the details for the oboe acoustics.

OK - now more detail:
https://newt.phys.unsw.edu.au/jw/woodwind.html
... should help with your question.
tldr: the fact it's a cone changes the pressure profile - yep.

Phil Freihofner
Thank you for the reply! These links certain look like they should help, but they don't seem to, unless I am missing something. The third article you linked contained the following link, "Pipes and Harmonics," saying it discusses the situation in further detail. But I do not see how the discussion there addresses my question, either!
https://newt.phys.unsw.edu.au/jw/pipes.html

There is a diagram "Harmonics and the Instrument Bores" in both this article and your third link, comparing an open cylinder (flute), a closed cylinder (clarinet) and a close cone (oboe). While the diagram organizes the graphs along overtone-series lines, demonstrating the relationship as they exist, I don't see how diagram explains or justifies why the 1.5 pattern on the clarinet sounds the 3rd harmonic on the clarinet and the 1.5 pattern on the oboe sounds the 2nd. Nor do the mathematics that follow show me what is going on, as far as I can tell.

There is a discussion of the increase in bore size being a factor in conical standing wave calculations. It makes sense to me that there is a 1/r^2 component in the calculation that affects the amplitude (loudness of the sound). But how does that effect pitch? An insights? How do we go about describing this mathematically?

I can see determining a pitch on the basis of length and speed-of-sound-in-air. But I don't understand how the shape of the container matters. It is not like the density of the air varies and thus speeds up the wave as it approaches the reed, does it? (Reminder to self, the air pressure inside the oboe ranges both above and below atmospheric air pressure, so increased density is a dubious conjecture at best.) Even so, why would going from 1/2 of a wave to 1.5 waves to 2.5 waves on the oboe constitute a full harmonic series rather than an odd-harmonic series as it is with the clarinet?

In the third link - did you notice that the right-two pressure diagrams follow the same node-antinode pattern?
Yet the conical bore plays some extra notes?

Which of the diagrams correspond to the 2m wavelength?

The (simplified) maths section does actually explain why an oboe and flute both play the full harmonic series but clarinet's don't ... the tldr answer is "because the actual physics is more complicated than secondary school acoustics would have you believe."

You are used to an approximation that assumes the pressure waves are sinusoidal with constant amplitude along the length of the tube ... this is not always the case. The simplified math used on the site is still an approximation (it neglects all but 1st order solutions). A full solution involves bessel functions. How's your maths?
http://www.music.mcgill.ca/~gary/courses/2015/618/week10/node2.html
https://theses.lib.vt.edu/theses/available/etd-11012001-133021/unrestricted/thesis_etd.pdf

(You may get an inkling if you consider that a wave traveling from the open end to the closed and of a conical bore will get reflected off the walls into the middle instead of just off the opposite end as in a straight bore. Do you understand how standing waves happen?)

Most of the reading I gave you was for background ... Penn State also have an animations page for acoustic waves. They also have a page linking to most everything you could want to know about acoustics and waves.

UNSW has animations showing how the pulses propagate along the pipes - it would be nice to be able to show you one of these for a conical bore. That would clarify the maths you didn't find convincing.

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Phil Freihofner
Thank you for the *maths* link. Indeed, these are significantly over my head. I have first year calculus but not a whole lot more beyond what I've picked up as a programmer (ragtag of stuff about vector math and matrices). From where I stand, I assume it would take a multi-year commitment to get up to speed, and that this would be best done in a university context. A road map would be informative, if you are able to sketch one out without too much trouble.

My understanding of the standing wave in the oboe is as follows:

The reed is a valve, allowing air into the oboe in bursts. Not generally known but firmly established, the reed must close during its cycle in order for a note to play.
> When a LOT of air enters (valve being open) the pressure climbs especially near the reed, and the concentrated air pushes outward;
> during the time when the reed is closed, pressurized air continues expands outwards, much like a coiled spring releasing (and the pressure drops);
> this outward motion of the air "overshoots" into the atmosphere at the open bore (again, as a coiled spring will expand beyond its resting position) and a relative vacuum is created as a result;
> this relative vacuum draws air back into the bore;
> the returning pressure wave creates positive feedback, contributing to the next buildup and driving the frequency of the cycle (Benade has a good section on how this can occur).

AFAIK, air bouncing off of the walls of the bore is not really much of a factor (beyond the rigidity needed to sustain the pressure). But it is very important that the bore be without leaks (no bad pads or fingers positioned slightly off of keyholes), else when the pressurized air springs out of the bore, a sufficient vacuum to draw the air back into the instrument will not be created. It took me a long while to understand the pressure/vacuum spring cycle as operating, because I was used to thinking of the bouncing of sound waves in terms of echoes off of a solid wall. However, that is not what is going on at the open end of the bore: there is nothing to bounce off of that would return the sound wave. And if there were, that would imply a pressure antinode and motion node at the open bore, not a pressure node with motion antinode.

The animations by Dan Russell are pretty cool! I made a couple wave animations, myself, using Java, that are similar, a couple years back. I tried to make one of my models include calculations pertaining to a conic section (using the height at each calculation point), but I'm pretty sure my concept of how to do this was faulty.

The USNW animation of pulse waves in a pipe are also neat. It makes me wonder if I should be thinking more about a vacuum wave heading back up towards the reed, rather than an overall vacuum that is drawing air in.

AFAIK, the reed valve "appears" to the standing wave as a stand-in for an additional length of bore. This length of "virtual bore" is significantly longer than the reed itself. Nederveen discusses this in depth (also with math that is over my head) in Acoustical Aspects of Woodwind Instruments from 1969. I seem to recall mention of a high of 35% or 40% of the total "bore length" is accounted for by this virtual bore for notes that are high up on the instrument. The "length" of this virtual-bore is not fixed though. One of my goals is to understand why, when playing oboe, I have to continually change the length of the bore (the virtual component, that is) in order to keep the instrument in tune in different registers. The "virtual bore length" is significantly longer for low notes than for high notes. We progressively do things like raise the air pressure (which raises the Bernoulli forces on the blades) or narrow the opening size (shortening distance shortens required traveling time) to speed up the cycle with which the reed opens and shuts, when going from the lowest notes to the highest notes of the instrument. I enquired here because I was wondering if somehow the conical shape and its modes played a part, and that I might be able to add another piece to the puzzle if I had a better intuitive understanding or visualization of waves in a cone.

Perhaps it will be possible to persuade Dan Russell to create a conical model?

The reed part is the driver ... it produces the initial time-varying pressure by the mouthpiece.
This time variance can be quite chaotic - so it has a mix of a wide range of frequencies in there.
The acoustics of the instrument will promote the required harmonics and damp out the others.
It does not just open and shut though does it - can't you take it out and sound it by itself?

There is a common setup with a wire and a wave-driver - the driver has small amplitude oscillations and there is usually a node very close to it.

... at 0.55 or so there is a standing wave showing a whole wavelength ... but the wave does not go the length of the string notice?
The driver is still moving up and down. You can get the wave to start very close to the driver if the driver hardly moves at all.
You'll notice that you are rewarded by the wind instrument if you blow very very softly?

An open hole forces a pressure node at that location - so it cancels some harmonics, and reinforces others.
This allows different notes to be played. What you are doing is fiddling with the resonant frequencies.

There is a reflection at the open end :)
It seems weird to get a reflection of nothing right - but there is air out there doing something different.
So you get a partial reflection off the opening like you see in the pulse animations.

The reed end can be modeled as an extra bit of tube - that does not mean the air inside sees it that way... it's just the maths.
Similarly, the wave is a description of relationships ... does your marriage distrusts an attractive young woman flirting with you or is it your wife that does?
One is the relationship, the other is the thing.

I don't know what you mean by "change the length of the bore" while playing.
Referring back to the video above - could the driver look a bit like an extra virtual "length" or string which varies depending on the amplitude and frequency? (a bit less than an extra half wavelength right?)

I'm afraid I must disagree with you on an important point here. The confusion of reed as driver versus reed as valve is pernicious and common. The string metaphor does more to confuse the issue than elucidate it in this regard.

A better image (than the driver with string) would be if you turn the spigot back and forth quickly on a garden hose. The tiny mechanism of the valve does not "drive" the water, the pressure from the water behind the valve is the driver.

Arthur Benade and others explicitly declare and describe the valve function of wind instrument reeds. Would you like a citation?

If there was no returning wave, the reed's pitch would be determined solely by the embouchure and breath pressure. That degree of pitch flexibility is not possible on an oboe--the range of pitches for a given note is usually no more than about half-step, if that, with a decent reed. There is more pitch flexibility with a note played via an open tone hole near the reed, less flexibility with a note played near the open bore.

Suppose instead of a reed we had a mini slide, like a trombone, near the mouth piece. it would be perhaps like a telescope rather than a loop like a trombone. The standing wave would be determined by the length of the instrument bore plus the amount of slide. Let's say a bore is two feet long and the slide introduces a variability of 4 inches. Giving or taking the four inches will have a greater effect when the four inches is a larger percentage of the total length of the standing wave. For a note at the bore, we have the standing wave being either 48" or 52" in length. With a note played near the reed, say with a tone hole 1 foot down, the variability is a greater percentage of the total length which can be anywhere from 12" to 16".

What I am contending is that by varying the embouchure and breath pressure, the reed is in effect working like a mini-trombone slide. And, a well known aspect of playing oboe (for those that play the instrument) is that the 'slide' has to be all the way out for low notes and all the way in for high notes. I am curious why this is the case, and am assuming that before one can get to the answer, it would be important to understand more about the modes of a cone, for example, if they get progressively flat for some reason.

## 1. What is a standing wave in a cylinder or cone?

A standing wave is a type of wave that remains in a fixed position in space, rather than traveling through a medium. In a cylinder or cone, a standing wave is created when a wave is reflected back and forth between two boundaries, resulting in a pattern of nodes (points of zero displacement) and antinodes (points of maximum displacement).

## 2. How is a standing wave in a cylinder different from a standing wave in a cone?

A standing wave in a cylinder is created when the wave is reflected between two parallel boundaries, while a standing wave in a cone is created when the wave is reflected between a curved boundary and a flat boundary. This results in a different pattern of nodes and antinodes, and a different frequency of the standing wave.

## 3. What factors affect the formation of standing waves in cylinders and cones?

The formation of standing waves in cylinders and cones is affected by several factors, including the size and shape of the boundaries, the frequency of the wave, and the material properties of the medium. Additionally, the angle of the cone and the type of boundary conditions (fixed or free) can also impact the formation of standing waves.

## 4. What are some real-world applications of standing waves in cylinders and cones?

Standing waves in cylinders and cones have many practical applications. For example, they are used in musical instruments such as flutes and organ pipes to produce specific pitches. They are also utilized in medical imaging techniques like ultrasound, and in industrial processes such as ultrasonic cleaning and non-destructive testing.

## 5. How do standing waves in cylinders and cones relate to resonance?

Standing waves in cylinders and cones are a type of resonance phenomenon. Resonance occurs when an object vibrates at its natural frequency in response to an external force. In the case of standing waves, the natural frequency is determined by the size and shape of the boundaries, and the external force is the wave that is reflected back and forth between the boundaries.

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