Klarinet Archive - Posting 000089.txt from 2010/08

From: Diego Casadei <casadei.diego@-----.com>
Subj: Re: [kl] Fundamental error on
Date: Fri, 13 Aug 2010 16:58:47 -0400

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.

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.

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
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--

Diego Casadei
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Physics Department, CERN
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