Klarinet Archive - Posting 000084.txt from 1996/05

From: Jonathan Cohler <cohler@-----.NET>
Subj: Physics of flute, oboe and bassoon as opposed to the clarinet
Date: Sun, 5 May 1996 13:59:33 -0400

At 7:09 AM 5/4/96, Dan Leeson: LEESON@-----.edu wrote:
>The explanation was very clear though I still have problems.
>
>What is this business about a flute having a node at both ends.
>Is not a flute, like a clarinet/oboe/bassoon closed at one end and
>open at the other???
>
>The sine wave analogy was very helpful because I am well aware
>that the sine wave reaches it's maximum at pi/4, 5pi/4, 9pi/4,
>etc. and its minimum at 3pi/4, 7pi/4, etc.
>
>But the flute business bothers me.
>
>And finally this: how does the clarinet differ from what happens
>in the oboe. How does the oboe differ so as to produce an overblow
>of an octave?
>
>
>====================================
>Dan Leeson, Los Altos, California
>(leeson@-----.edu)
>====================================

The Flute

On the flute the vibration is driven by an "air reed". Namely, flutists
blow across an opening against a sharp edge. The physics of that process
cause the air stream to alternately bounce inside and outside of the hole.
This is the vibrating "air reed".

When one blows so as to produce a vibration that is at or near the
frequency of one of the natural modes of the tube, one gets a note!

The hole that one blows over, however, is open to the atmosphere (if it was
totally closed by the mouth, you could not produce the air reed effect).
Therefore it serves as a pressure node. The fact that the head joint
extends to the left of where the mouth is blowing is a design feature that
helps to make fine adjustments to the intonation of various harmonics. The
pressure wave that is created by the air reed propagates throughout the
length of the tube (both up to the cork and down to the tone holes, but its
physical behaviour is virtually the same as if the flute were blown from
the top and held more like a clarinet. In fact, the original ancient
"flutes" were made that way. The modern flute has just been moved sideways
(hence the "transverse" appelation) and a little correcting headjoint added
for fine tuning.

So a flute fundamentally has a node at both ends and therefore produces all
harmonics, not just the even or odd ones. By the way, it is possible to
have a closed end flute, where the bottom end is closed, and in this case
one gets only odd harmonics. By the way, as an aside, the myth about the
flute sound being largely a simple sinusoid of the fundamental only, is
pretty much a myth. At any dynamics above the softest, there is a good
amount of 2nd, 3rd and 4th harmonics in the sound.

The Oboe and Bassoon
--------------------

The oboe and bassoon are indeed closed at one end and open at the other,
but they, like the saxophone for that matter, have another major
difference. The bore of these instruments is conical. In other words, it
starts narrow at the blowing end, and gets gradually larger and larger
toward the other end of the instrument. In fact, these instruments are
poly-conical (a sequence of cones of slightly differing angles) to make
fine adjustments to the intonation.

The propagation of pressure waves in a conical enclosure is different from
the propagation of pressure waves in a cylindrical enclosure. Intuitively,
one can see this by imagining a high pressure disturbance at the small end
of the tube. As it propagates to the larger end the pressure will decrease
gradually as the volume of air increases.

I can't think of any simple (i.e.non-mathematical) way of explaining this,
but the simple fact is that the fundamental frequency of a cone is:

f = v/2L (v is the speed of sound, L is the length of the cone)

In other words, a pressure wave travels down and up the cone just once to
complete a full cycle. Therefore, it behaves like a cylindrical tube open
at both ends (or a string held down at both ends), and it has all of the
harmonics 2f, 3f, etc. Hence, the octave key goes up an octave.

Furthermore, the impulse response of a cone is very different from the
impulse response of a cylinder. If you recall, in the cylinder a positive
pulse is reflected negatively at the bottom of the tube. So after an
amount of time equal to 2L/v, we get back a negative pulse, and after a
time equal to 4L/v the pulse comes back positive again. This is what makes
it the reed open and close and vibrate at the frequency v/4L.

In the cone, a positive pulse comes back after a time 2L/v as a brief
negative pulse followed immediately by a broad positive pulse. When
coupled with a double reed (like on the oboe and bassoon), the brief
negative pulse is too short to effect the reed, the broad positive pulse
acts to open the reed again. In other words, a complete cycle has already
occurred, and the fundamental frequency is v/2L.

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