Klarinet Archive - Posting 000112.txt from 2010/08

From: Diego Casadei <casadei.diego@-----.com>
Subj: Re: [kl] Fundamental error on
Date: Sat, 14 Aug 2010 15:22:23 -0400

Dear Jennifer,

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 there is nothing really
fundamental on it.

I commented in my previous message about the possibility of having one
additional node in between, and concluded that it is unlikely that there
is another one.

For what I know (which is not very much), it seems that the position of
the main node is a feature of the clarined, mostly tied to the shape of
the mouthpiece and the choice of a single reed.

BTW, thanks for the spell checking: "ear" is indeed the correct noun.

Instead, the sentence "A wave is a process which depends both on time
and position." is not restricted to a sine wave, though it is possible
to decompose any general wave into sines and cosines. A wave can be a
rather strange function... the only constraint is that is must be a
function of the distance x and time t of the form f(x-ct) or g(x+ct),
where c=sound speed.

Cheers,
Diego

Jennifer Jones wrote:
> Dear Diego,
>
> Your description of the misplacement of the node in the diagram on
> page four of the Bore design article is really neat and the
> calculations are quite instructive. It is very interesting that the
> node is located 8 cm down from the reed. There is something that
> puzzles me though,
>
> It appears that the length of the first antinode to node (reed to
> upper barrel) interval described is less than one quarter of the
> length of the clarinet (mouthpiece tip to 1 cm below the end of the
> bell). Is my interpretation correct and, if so, why would this be?
> Is it truly a maximum of pressure at the reed?
>
> Thank you for your kind attention.
>
> Sincerely,
>
> Jennifer Jones
>
> On Fri, Aug 13, 2010 at 1:58 PM, Diego Casadei<casadei.diego@-----.com> wrote:
>> Dear Tony,
>>
>> if you forgive me for a simplified (but non so short... I've spent ~50
>> min to write it!) acoustic lesson, I can try to explain why it should be
>> trivial to understand that the amplitude cannot be zero at the
>> generator. Perhaps other readers could find it useful, given how easy
>> it is to misinterpret diagrams etc.
>>
>
>
>> A real sound can be decomposed into "harmonics", which are sinusoidal
>> waves with frequencies that are integer multiples of a number, called
>> the "fundamental frequency". The first harmonic is the octave (double
>> frequency), the second harmonic is the twelfth (triple frequency), the
>> third harmonic is the second octave, and so on. The decomposition
>> itself is called Fourier analysis or spectrum analysis.
>>
>
>
> Ear?
>> The hear does make a spectrum analysis: we perceive as different timbers
>> acoustic waves with a different power (=energy/time) distribution over
>> the different harmonics. When we can recognize and sing a pitch, it is
>> because most of the power goes to the fundamental frequency, which is
>> the case for most musical instruments.
>>
> SINE WAVE
>> A wave is a process which depends both on time and position. Let's fix
>> the position (for example, the position of the hear relative to the
>> sound source), so that only the time dependence remains. A single
>> harmonic can be written as a simple sinusoidal wave, in terms of the
>> sine (as in http://en.wikipedia.org/wiki/Sine_wave) or cosine function,
>> and has 3 parameters: the amplitude (A on wikipedia), the frequency
>> (=2*pi*omega, where omega multiplies the time t on wikipedia), and the
>> phase (phi on wikipedia). [One can change the sine into a cosine by
>> changing the phase by pi/2 = 90 degrees.]
>>
>> A simple sinusoidal wave carries some energy. Hence, a source of waves
>> must provide the power (=energy/time) to create and maintain the sound.
>> The wave energy is proportional to the square of its amplitude.
>>
>> NB: here, with "amplitude" I exactly mean the parameter A on wikipedia.
>> The actual value of the wave [which is unfortunately also called
>> amplitude by most people] is the product of A with the sine function,
>> which oscillates between -1 and +1. This means that the value of the
>> wave is oscillating between -A and +A, as function of time and position.
>>
>> What is called "node" is the point (in space) at which the wave is
>> always zero, i.e. at which the amplitude A=0. For the guitar string,
>> the extrema are by definition nodes (they are rigidly fixed by
>> construction). Depending on the frequency of the vibration, there could
>> be other nodes. For example, harmonics are obtained by grazing the
>> string with a finger to force a node without dumping the oscillations
>> too much. Grazing the mid point makes a jump of one octave (and
>> automatically dumps all harmonics which have an odd multiple of the
>> fundamental frequency).
>>
>> In general, the amplitude A itself will change (in space and time). For
>> example, acoustic waves usually start as spherical waves: at each time
>> the energy is spread over the surface of a sphere centered on the
>> source. Hence, the energy density per unit surface (think about the
>> hear aperture, for example) decreases with the inverse of the square of
>> the distance from the source. [The amplitude is inversely proportional
>> to the distance: A ~ A_s / d where A_s is the amplitude at the source.]
>>
>> On the other side, the stationary waves propagating in a tube behaves
>> more or less as parallel waves. The amplitude A still decreases, but
>> not because of geometry: energy is dissipated as heat and the
>> temperature of the air (and pipe) is increased. [This effect is also
>> present for spherical waves, but for them the geometrical reduction is
>> dominant and I neglected thermal dissipation in the previous paragraph.]
>>
>> Given that the wave energy decreases with time and distance, it is easy
>> to guess that it is maximum at the sound generator. Hence, also the
>> amplitude is maximum at the source, at least until the latter continues
>> emitting power in the form of acoustic waves.
>>
>> In the guitar, the position in which the finger first plucks the string
>> is the instantaneous source: at the very beginning, it is one antinode.
>> Later, the string is left free to vibrate accordingly to its natural
>> modes, and the energy is redistributed across all harmonics, so that the
>> very same point could find itself no more in the position of the maximum
>> amplitude. Similar things are valid for the piano, the xilophone, and
>> all percussive instruments.
>>
>> For the violin, the player continues providing energy into the same
>> position, which is forced to remain one antinode. Playing in different
>> positions changes the timber, because it forces the antinode to
>> different positions.
>>
>> For the winds, a continuous tone is sustained by a continuous power: the
>> vibrating reed (or reeds, or lips) feeds energy into the pipe, being the
>> point with the maximum amplitude. Energy is then lost as heat inside
>> the pipe and in the form of outgoing acoustic waves far from the instrument.
>>
>> In the clarinet, there is no reflection at the reed, in the sense of a
>> closed pipe. Indeed, as soon as we stop blowing the sound is cut. What
>> happens is that the sound wave is produced in such a way that the
>> reflected front from the other side of the pipe is in sync with the
>> incoming front from the reed (luckily, the reed does not vibrate
>> randomly: it must be synchronized with this mechanism).
>>
>> On the other hand, it is true that the clarinet almost behaves as a
>> closed pipe, because of the almost fixed position of the first node.
>> This is the position which I measured yesterday. Ideally, I should have
>> found the exact frequency of a lot of different pitches, to find their
>> wavelengths. Then I should have measured the node position starting
>> from the tone hole used to generated each pitch. But I had no
>> instrumentation (apart from a simple tuner) and the purpose was simpler.
>> I don't remember the exact citations, but people have already done
>> this and found that the node is indeed quite stable across the range.
>>
>> Finally, the diagrams showing a closed pipe are good to explain several
>> things, but they are _approximations_ and we should be aware of this.
>> When this is forgotten, we can make mistakes like the drawing which was
>> at the origin of this thread. For example, a diagram showing a pipe
>> closed on the left and some amplitude should also remind people that
>> some source is needed on the right, at the open end.
>>
>> Cheers,
>> Diego
>>
>>
>>
>> Tony Pay wrote:
>>> On 12 Aug 2010, at 20:48, Diego Casadei wrote, in part:
>>>
>>>> I made a measurement, just to cross check what I wrote in my first email.
>>>
>>> Well, what you say certainly does seem to make a lot of sense. I'm checking out the references you give, but I think I've learnt something here -- though I have to say 8 cm seems awfully large.
>>>
>>> I never really went into what counts as received wisdom about the reed behaving like the closed end of the tube, thinking that it was counterintuitive because of some subtlety about how the boundary conditions were applied.
>>>
>>> (Obviously, if you're right, this is the error that Ken Shaw thought the paper was 'full of':-)
>>>
>>> Tony
>>> --
>>> Tony Pay
>>> 79 Southmoor Rd
>>> Oxford OX2 6RE
>>> tel/fax +44 1865 553339
>>> mobile +44 7790 532980
>>> tony.p@-----.org
>>>
>>>
>>>
>>>
>>>
>>> _______________________________________________
>>> Klarinet mailing list
>>> Klarinet@-----.com
>>> To do darn near anything to your subscription, go to:
>>> http://klarinet-list.serve-music.com
>>
>> --
>>
>> Diego Casadei
>> __________________________________________________________
>> Physics Department, CERN
>> New York University bld. 32, S-A19
>> 4 Washington Place 1211 Geneve 23
>> New York, NY 10003 Mailbox J28310
>> USA Switzerland
>> office: +1-212-998-7675 office: +41-22-767-6809
>> mobile: +39-347-1460488 mobile: +41-76-213-5376
>> http://cern.ch/casadei/ Diego.Casadei@-----.ch
>> ----------------------------------------------------------
>> _______________________________________________
>> Klarinet mailing list
>> Klarinet@-----.com
>> To do darn near anything to your subscription, go to:
>> http://klarinet-list.serve-music.com
>>
> _______________________________________________
> Klarinet mailing list
> Klarinet@-----.com
> To do darn near anything to your subscription, go to:
> http://klarinet-list.serve-music.com

--

Diego Casadei
__________________________________________________________
Physics Department, CERN
New York University bld. 32, S-A19
4 Washington Place 1211 Geneve 23
New York, NY 10003 Mailbox J28310
USA Switzerland
office: +1-212-998-7675 office: +41-22-767-6809
mobile: +39-347-1460488 mobile: +41-76-213-5376
http://cern.ch/casadei/ Diego.Casadei@-----.ch
----------------------------------------------------------
_______________________________________________
Klarinet mailing list
Klarinet@-----.com
To do darn near anything to your subscription, go to:
http://klarinet-list.serve-music.com