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Reed physics - harmonics


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30 years ago, I measured reed resonances (not harmonics ) produced by sympathetic vibration to the sound a variable audio oscillator produced.  All the reeds I tested produced very similar results.  I used a little laser bounced off the reed tip onto the wall to find the different peak amplitudes without affecting the reed by proximity to a magnetic field. (Not really important. I did the same thing using a tiny magnetic pickup ). The resulting resonances were not at all linear, though I seem to recall that each successive resonance was roughly about 2.3 times the previous resonance, not the fundamental.  This is just an approximation of cantilever bar vibration modes.  
   Air driven reeds produce a completely different set of actual harmonics in the 1,2,3,4 etc.X the fundamental.  In my experience, air driven reed waveforms are only slightly sawtooth or triangle wave like.  More an “ish” rather than really resembling the ideal, which has a defined strength for each successive harmonic in the wave form.  The individual harmonics vary greatly from instrument or reed in its environment from one to the next.  I did find most of my forays into reed acoustics bore little of practical use.  Few of my hypotheses bore useful results except to be obviously wrong.  There were basic things that let me improve reed output, but trying out different reed pan thicknesses, wood species, chamber depths and lengths and listening to the results was a much more effective way to adjust the character of the sound and response of reeds in the instrument. 
Dana

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Hi Dana, I know we discussed this before, but now I'm confused.

"Air driven reeds produce a completely different set of actual harmonics in the 1,2,3,4 etc.X the fundamental."

Are you talking about harmonics in the acoustic pressure wave form that we hear, or harmonics in the time dependency of the mechanical tongue vibration as a bar, within it's first transverse mode of cantilever vibration? 


"In my experience, air driven reed waveforms are only slightly sawtooth..."

Again, are you talking about the acoustic pressure waveform that we hear, or the mechanical vibration of the tongue?

There are entirely two different kinds of vibrations, or waveforms going on here, one is the mechanical vibration of the tongue, and the other is the vibration of the air medium, or the acoustic vibrations in the air, what we hear.

 

Likewise, there are two ways the tongue vibrates, one is according to what are called "vibrational modes"; i.e., whether transverse (bending) or torsional (twisting), and the other is the "jerkiness" in the vibration, within each particular mode (e.g., first, second, etc. bending and first, second, etc. twisting).  So there are many vibrational modes, and within each mode, there are many ways the motion can be jerky in time.

 

The complexities in the ways this simple little thing can vibrate is often beyond the comprehension of most people. 

 

Regards,

Tom 
 

Edited by ttonon
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Hi Tom,

I mostly always mean the acoustic wave form that is the end result of the vibration of an air driven reed.  I am aware of the variations and combination of the bar modes for a reed that is excited by other means but see zero evidence of their frequencies in the resulting sound. If you put a magnetic pickup near a reed and drive the reed sympathetically, ( no air flowing by the reed) the bar modes are all pretty strong, so they should at least appear as visible peaks in the spectrum.

Hope you are staying well,

Dana
   

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Hi Dana, I had a hunch that I wasn't getting my point across.  I'll try again.  We agree that the cantilever vibrates in different bending modes, and that its first (transverse, or bending) mode is the one that dominates in the free reed.  Now, with that mode, there are also harmonic frequencies in time.  The fundamental of those harmonics is the chief vibration we observe; i.e., that's the sinusoidal motion (in time) that causes the chopping of the air stream.  However, there are also higher harmonics in that time vibration.  As I said, it's best to describe this as a kind of jerkiness in the motion.  There's no way that you could distinguish these bending harmonics from the acoustic harmonics you hear.  That's why you couldn't measure them. 

 

I attach an article from Acoustics Today that explains this very well, with pictures.  The article explains how a tuning fork vibrates in myriad ways, but since the tuning fork is basically two cantilevers attached together, you can interpret much of this to understand how the free reed tongue vibrates. 

 

The only difference is because the tuning fork is struck and then it vibrates freely, without the continuous forcing that an air driven free reed experiences.  The strike of the tuning fork excites all kinds of modes and harmonics, but in the transient free vibration following that, the higher modes and vibration harmonics dampen quicker than the first mode and its fundamental sinusoidal character.  That's why you get such a pure tone, with little "quality." 

 

The air driven free reed vibrates with a steady periodic motion, long after start transients have damped out, but the aerodynamic forces exciting it are not sinusoidal.  They have essentially step functions in time, yet they are still periodic.  This non sinusoidal forcing function will excite not only the fundamental sinusoidal excitation of the first vibration mode, but higher harmonics of that fundamental.  Also, in many cases, but perhaps not all, even the second bending mode may also be excited by these aerodynamic forces, and this mode can be sometimes measured for all but the lowest bellows pressures.  In fact, during the start transient of the free reed, torsional modes can also be excited.  Although it's a common practice, we have no physical reason to oversimplify the vibratory motion of the free reed.  

 

Read the attached article.  It's very interesting.  In particular, look at Figure 3c.

The-Tuning-Fork-An-Amazing-Acoustics-Apparatus-Daniel-A.-Russell.pdf

Edited by ttonon
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12 hours ago, ttonon said:

There's no way that you could distinguish these bending harmonics from the acoustic harmonics you hear.  That's why you couldn't measure them. 

 

The "bending" harmonics are not generally at integer multiples of the fundamental, so ought to be measurable and visible. Indeed, that's exactly what the text says of Figure 3c in the attached article:

"This spectrum shows nine integer harmonics of the fundamental in addition to the clang tone."

I take the "clang" tone to be the first "bending" harmonic.

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John, again I'm not making myself clear.  So I'll try again.  

 

When I say harmonic I mean harmonic, which is a mathematical term indicating whole number ratios.  When I say harmonics, I'm not referring to the bending modes of the cantilever, which are not harmonic.  As you probably know, the second bending mode has a frequency roughly 6.2 that of the first bending mode.  

 

The statement, "I take the clang tone to be the first bending harmonic" is erroneous.  The clang tone is the second mode vibration, with it's fundamental roughly 6.2 times the frequency of the first mode fundamental, which is the usual one we observe.  Each mode has it's own fundamental.    

 

"This spectrum shows nine integer harmonics of the fundamental in addition to the clang tone."

 

This is correct.  The "nine integer harmonics" are the same harmonics I refer to.  They are not modes of vibration, but harmonics within the first mode vibration, as the caption says.  Again, they are harmonics in time, you cannot see them with a strobe, which shows only the geometrical shape in space, and these harmonics do not change that shape.  Again, the best way to understand them is to picture the vibration as "jerky" in time, not a smooth sinusoidal motion at the fundamental frequency.  These harmonics vibrate at higher frequencies than the smooth sinusoidal motion.

 

- gasp -

 

Regards,

Tom

 

 

 

 

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