Klarinet Archive - Posting 000118.txt from 2010/08
From: Diego Casadei <casadei.diego@-----.com>
Subj: [kl] About clarinet acoustics
Date: Sun, 15 Aug 2010 02:45:27 -0400
Dear all,
because I see that the discussion now moves in parallel on two threads
("Fundamental error on..." and "Location of antinodes...") I find it
easier to collect my comments to the different emails here.
Below I try to answer Jennifer and Keith. I must admit that the
discussion is moving toward a quite advanced state and I need to tell
you that I'm a particle physicist who never worked on theoretical nor
experimental acoustics. I don't claim to have any particular authority
on the latter fields (nor in the first, to be honest).
Cheers,
Diego
Jennifer Jones wrote:
> Dear Diego,
>
>> indeed the distance from the vibrating reed to the node is less than a
>> quarter of the fundamental wavelength of the instrument (the "all closed"
>> low E). However, the clarinet is able to play a very wide range of pitches
>> and there will be waves for which it happens that such distance is 1/4 of
>> the wavelength.
>
> but if the clarinet is playing the low E, that note corresponds to a
> certain wavelength. If the distance between the first antinode and
> node is so much smaller than the distance between the node and the
> second antinode, it appears that there are two wavelengths involved.
> Unless we have one wavelength with asymmetrically distributed nodes
> and antinodes, namely, a shortened interval between the reed antinode
> and the barrel node, which seems strange.
Strictly speaking, we don't have an asymmetric wavelength in the sense
you mean. The problem is that we must be aware of the approximations,
when we make use of them.
Now it is true that a stationary wave has a regular pattern of nodes and
maxima. The closest case to the clarinet is the closed cylindrical pipe
of infinite length, in which we assume that the source is infinitely
distant from the closed end.
I hope this sounds a clear ideal case... For the clarinet, the length is
not infinite and the source is located near the supposedly closed end!
The node at ~8 cm from the vibrating reed is a feature of the source,
not of the stationary wave. As a matter of fact, there is a reflected
wave which comes back from the clarinet bell, which implies that the
source cannot freely vibrate, but it is forced to do so in such a way
that we finally get a stationary wave, from the node to the bell.
Between the vibrating reed and the node there is no stationary wave, in
the sense of the approximation made above. It is the interplay of the
source and reflected wave which determines the details in this region,
which (I guess) is highly non-linear and needs a tricky treatment.
> All the diagrams I've been looking at put in mind a series of high and
> low air pressure points in the clarinet that are regularly positioned,
> like the sine wave we learn about in school.
>
> Is the high and low air pressure point distribution more like the wave
> seen using an oscilloscope? If that is the case, then I figure there
> would be more nodes and antinodes of varying extremes of pressure.
>
> Sincerely,
>
> Jennifer
Likely, most diagrams show the approximate case of an infinite pipe with
one close end (see above). Inside the clarinet, there is some
regularity between the vibrating reed and the node, tied to the source
details and to the lower part, and some (different) regularity between
the node and the bell, tied to the tube and tone hole details and to the
source in a different way.
Keith Bowen wrote:
> (...)
>
> I think we all agree that
> 1. A clarinet has a pressure antinode somewhere near the reed end. This
> corresponds to a displacement node, i.e. a tube closed at the top.
I'm not completely sure about it, but it sounds likely, given that we
are very near the source. What I've found is that there is a phase
shift between pressure variations and displacements, but found no
precise statement about the size of the shift. One would need a phase
change of 90 degrees to get the picture described above. This can well
be, but I have nothing to prove or disprove it.
> 2. It has a pressure node somewhere near the bell, corresponding to the
> discontinuity between the wave impedance of the tube and the very low wave
> impedance of the open room, allowing waves to be reflected back into the
> clarinet and build up standing waves in the instrument.
This sounds very strange or at least counter-intuitive to me. The bell
is not only a point in which the reflected wave is bouncing back toward
the mouthpiece, it is also the source of the spherical waves which we
hear. I would need to check the mathematics, but I'm pretty sure that
the situation is that we have refraction at the discontinuity between
the pipe and the external world. This would imply that we have a
pressure maximum at the exit of the bell.
Now, above 1/6 of the wavelength pressure variations and displacements
should be already quite in phase and at the bell we are at least at 1/4
of the wavelength for the lowest possible pitch (for all higher tones,
we are beyond this limit). Hence I would deduce that:
1) we have a relative maximum in the pressure variations at the bell
2) we also have a relative maximum for the displacements at the bell,
for most of the range
3) all diagrams I've seen so far are completely wrong
> I think the points of debate are the distance from the reed tip to the first
> pressure antinode (fundamental vibration), and for my part I also am
> concerned about the effects of the mouthpiece and the bell. I, too, am
> surprised that the distance from the reed tip to the antinode is as much as
> 8 cm.
I was not very surprised. Indeed, the register key is at a fixed
position with respect to this node and it is quite near to it. I would
have been surprised to find the node more distant from the register key,
because this would have implied that a lot of harmonics would have been
impossible to play. So, before making the measurement, I was expecting
to find the node a bit above or below the register hole, and I was even
more relaxed when I found it to be above (this makes very much sense to me).
> (...) Indeed, the method I used was measurement of tone holes, bore
> diameter, wall thickness to calculate the pitch of notes other than the
> bottom note, using Benade's tone hole lattice end-correction. This avoids
> the problem of the bell, for which there isn't a simple calculation; and
> anyway the effective length of the bell is frequency dependent. So I
> regarded measurement of overall tube length as unsatisfactory for
> determining pitch. I have enough data from the Nurnberg museum publications
> to run the calculation for several tone holes for some clarinets, which will
> empirically determine the pressure-antinode position. I do need to do this
> to verify my methodology, but it will take some time.
I know nothing about this approach and will be delighted to know about
it. I hope you'll have time to make this check and let me (us) know.
--
Diego Casadei
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