Klarinet Archive - Posting 000111.txt from 2010/08

From: Diego Casadei <casadei.diego@-----.com>
Subj: Re: [kl] Location of antinodes of vibration in an air column
Date: Sat, 14 Aug 2010 15:14:34 -0400

Dear Richard,

please find my comments below.

Cheers,
Diego

Richard Sankovich wrote:
>
>> 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.
> Consider, as you did, that bowed violin string. The fundamental mode of vibration will be excited, as well as many higher modes. The fundamental mode always has its maximum vibration displacement at a point that is halfway between the two fixed ends, in other words, at the middle of the string. You can even see this happen if you bow the G string so as to produce a loud sound. This fundamental mode antinode will always be located at that same halfway point *regardless of where the bow (energy source) is placed along the string*. Normal bowing is done close to one of the fixed ends of the string, near the bridge, where there is a vibration *node*. So energy is fed into the string close to one of its nodes. Definitely not at the halfway point antinode.

I spoke about "one antinode", not the fundamental antinode, and about
the timber, not the pitch. The point in which the string is bowed
cannot be a node but it is forced to be one antinode. This cuts away
all harmonics which need a node at that point or very near that point,
and this is why the timber changes.

String player are used to see indications about the bowing position.
For example, in Rossini's music this is often done at the beginning of
the "crescendo rossiniano", when the composer asks to play near the
bridge. The nearer the bridge one excites the string, the fewer
harmonics are dumped. The result is a strange "metallic" sound which is
used by Rossini in ridiculous situations.

The fundamental wave always has an antinode at the center of the string,
and this can only be avoided by explicitly grazing the string with a
finger, to get the first harmonic (and suppress all odd multiples of the
fundamental frequency).

In general, the energy is provided to the string by bowing it into some
position. Though such position will be near a node (which is true for
many positions, but we need to quantify "near"; see below), the point at
which you bow the string must have a local maximum in the amplitude,
compatible with the constraints. If the point is very near the bridge,
where a node is (*), then the amplitude will be less than when bowing
near the center of the string. Still, it will be a local maximum of
amplitude, because this is the point in which you excite the string.

(*) Strictly speaking, at the bridge the string is not still. Indeed,
the bridge "bridges" the sound to the structure, which works as an
amplifier. However, given that the stiffness of the bridge is much
larger than for the string, considering a node of the string at the
bridge is an excellent approximation.

> What happens in a clarinet air column is analogous to the vibrating
> violin string. Energy is fed into the system close to one of the
> vibration *nodes*. In other words, the tip of the vibrating reed is
> close to a *node* of vibration, not to a point of maximum amplitude
> (antinode) as you concluded. When we play the lowest note on our
> clarinet, there will be a vibration node close to the mouthpiece and a
> vibration antinode close to the bell. The distance between the node and
> antinode is always one-fourth of a wavelength of the sound that is being
> generated.

As I said above, you will find nodes near many points. Still, "close"
means few centimeters for the clarinet. When playing the lowest
possible pitch (a real D in a Bflat clarinet), one quarter of wavelength
is of the order of 58-59 cm and the single node corresponding to this
funcamental pitch is at ~8 cm from the vibrating reed.

Given that the vibrating reed is one antinode and that there is a node
after 8 cm, the first harmonic which has a node between the two points
is the one which has a wavelength equal two 2/3 of such distance, i.e.
2.6-2.7 cm. This is obtained when the fundamental wavelength is divided
by 88 or 89, which corresponds to a frequency of ~13 kHz.

Given that we are playing the low E, whose frequency is 445/3=148.3 Hz,
and that we are taking the 88-th or 89-th harmonics at ~13 kHz, in
excellent approximation you can assume that they are not excited at all.
This means that in practice there is no intermediate node between the
vibrating reed and the node which is at ~8 cm from it. Hence, the
nearest node to the vibrating reed is the one at ~8 cm from it.

This is surely valid for most of the range but, when playing
"sovra-acutes" [my free adaptation from the Italian term... sorry] it
could be that some frequency as high as 13 kHz is indeed powerful enough
to make a real node in between the vibrating reed and the main node.
But my guess is that this is quite unlikely. Starting from the central
A at 445 Hz, we normally can play up to 2 octaves higher, reaching the
"sovra-acute" A at 1780 Hz. This tone will have its own harmonics, with
frequencies 3.56 kHz (octave), 5.34 kHz (twelfth), 7.12 kHz (2nd
octave), etc. until we reach the 3rd octave at 14.2 kHz. However, the
power of such high frequencies is very small, if not completely
negligible. The problem is that the normal reed is not able to vibrate
at such high frequencies: it would sound "metallic" if it could. Hence
my feeling is that there is really no other node between the vibrating
reed and the main node.

> One more point while I'm at it: antinodes of vibration of the air
> molecules are not the same as antinodes of pressure. (Same can be said
> of nodes.) Vibration antinodes and pressure antinodes do not occur at
> the same places in an air column, in fact, a point where there is
> vibration antinode will also be the location of a pressure node, and
> vice versa. I point out this distinction because much confusion can
> result when people do not make it clear which kind of node or antinode,
> vibration or pressure, is being discussed.

This apparently simple question made me crazy... I'm not an expert
hence I preferred to browse "The Physics of Musical Instruments" by
Fletcher and Rossing. It took me almost the full day, but it was very
instructive :-) though I'm unsure to have the fully correct answer.

On section 6.1 the plane acoustic waves in air are treated and it is
shown that the variations in pressure and particle velocity are in
phase. Plane waves are a good approximation for any real acoustic wave
in a homogeneous medium far away from the source (more precisely, at
distances of many wavelengths or greater). However, this approximation
is not valid inside a clarinet.

On section 6.2 I found that the spherical waves very near the source (no
more than 1/6 of the wavelength from it) have a phase difference between
the variations of the pressure and the particle velocity. But this is
for the free space and could be not necessarily valid also inside the
clarinet.

[Incidentally, in chapter 8 the impedance of cylindrical pipes is
addressed and it is shown that a straight cylinder with an open end has
an effecting length of L+0.6a where L is the physical length and a is
the radius. It is interesting that for the clarinet this 0.6a is about
half a centimeter, as I've found by myself, despite from the shape of
the clarinet bell, which is not cylindrical. I'm happy to see that real
experts confirm my "poor-man" measurement :-) ]

On section 15.1 I found that the mode frequencies for Bessel horns (of
which the cylinder pipe is a particular case) correspond to impedance
maxima at the mouthpiece, hence I conclude that the amplitude for the
pressure variations is maximum at the vibrating reed.

My feeling -- I'm not 100% sure -- is that this is the very same
situation as for the other end of the pipe: maxima in the pressure
variations both at the vibrating reed and at the exit from the bell.

I'm still reading that big book, hence I can find other interesting
things which I can report. But not very soon...

Sorry for the incomplete answer,
Diego

> ---Richard Sankovich
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--

Diego Casadei
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