Klarinet Archive - Posting 000070.txt from 1996/05

From: Jonathan Cohler <cohler@-----.NET>
Subj: Re: A question about acoustics
Date: Sat, 4 May 1996 00:38:12 -0400

At 4:15 PM 5/2/96, Dan Leeson: LEESON@-----.edu wrote:
>Would some kind soul out there help me understand why the clarinet
>overblows a 12th while a bassoon, a flute, and an oboe overblow an
>octave.
>
>Please, no shooting from the hip, but serious knowledge about the
>phenomenon. Even better would be an explanation followed by a
>scholarly reference.
>
>Much obliged.
>
>
>====================================
>Dan Leeson, Los Altos, California
>(leeson@-----.edu)
>====================================

Dan,

Others have cited good sources. My favorite is Benade's "Fundamentals of
Musical Acoustics" (Dover). The new edition of this is in paperback and
contains many corrections over the old.

The reason that an ideal cylindrical tube closed at one end and open at the
other supports only odd harmonics is as follows.

The open end is open to the atmosphere and therefore the airpressure at
that point does not vary. In other words, it is a pressure node (always at
0, which represents atmospheric pressure).

The closed end allows for build up and decline of air pressure as the wave
"comes in" and "recedes" (think of a water wave).

Now here's the simplest explanation that I know for why the fundamental
wavelength of such a system (node at one end, antinode at the other) is
four times the length (and not two times the length as with a flute, which
has a node at both ends).

Imagine a pulse (a very short, high pressure spike) traveling down the tube
to the open end. As the pulse "hits" the open end, the pressure must
always add to zero. Therefore, a negative pulse (a low pressure spike) is
reflected back (the negative pulse and the positive pulse sum to zero at
the node).

Now, when the negative pulse "hits" the closed end, it just bounces off the
"wall" and comes back as a negative pulse.

It travels back to the open end, where it is now reflected back as a
positive pulse and finally we have made one complete round trip. We are
back to where we started, and the process starts over again.

This is why the fundamental wavelength is four times the length of the
instrument. This same logic can be used to show why the fundamental
wavelength of a string instrument (node at both ends) or a flute (node at
both ends) is two times the length of the vibrating element.

Now, here's why we get only odd harmonics in the closed-open
(antinode-node) case. Draw a quarter of one cycle of a sine wave on a
piece of paper. The sine wave starts at zero (the node point) and rises to
its maximum (the anti-node point) one quarter of the way through the cycle
(one traversal of the lenght of the instrument).

Continue drawing the sine wave. Where is the next point where the sine
wave reaches a maximum (positive or negative)? It is three quarters of the
cycle. That represents the next possible mode, because all possible modes
must have an antinode at that end.

Keep drawing and you'll see that the next possible mode traverses is 5/4 of
a cycle in one crossing of the length.

State this mathematically and you get that the possible wave lengths are:

4L, 4L/3, 4L/5 etc.

Or, in other words, the possible frequencies are:

v/4L, 3v/4L, 5v/4L, etc. (only odd harmonics!)

Do the same drawing excercise with a node at both ends and you will quickly
discover that the allowed wavelengths and frequencies are respectively:

2L, L, L/2, L/3, etc. (wavelengths)
v/2L, 2v/2L, 3v/2L, etc. (all harmonics)

I hope this helps without getting too technical.

----------------------
Jonathan Cohler
cohler@-----.net