Klarinet Archive - Posting 000339.txt from 1994/05

From: Martin Brown <martinb@-----.AU>
Subj: Re: Some clarinet physics
Date: Fri, 20 May 1994 14:54:09 -0400

I suspect this didn't get posted so here it is again:

Some people may be insterested in some physics of musical instruments
which I picked up from a fourth year maths course I did while studying
electrical engineering concerning overblowing
clarinets/flutes/saxophones/etc, why they are shaped as cylinders and
cones and so on.

1. There are basically two type of wind instruments: those with
"pressure" reeds and those with "velocity" reeds. A clarinet has a
pressure reed because the driving oscillation is the pressure
variation at the mouthpiece. The pressure variation is maximum at
the mouthpiece while the velocity variation is at a minimum.

Instruments that have a hole which you blow across (like flutes)
have a velocity reed. This is because the velocity variation at the
mouth end is a maximum and the pressure variation is a minimum.

2. All wind instruments end up with a hole at the far end. (The bell of
a clarinet for example.) At this point, the pressure variation is
always at a minimum while the velocity variation is at a maximum.
If you lift a finger off the bottom hole of the instrument, that is
now the point at which the velocity variation is maximum.

3. Pressure variation is always inversely proportional to velocity
variation so if you have maximum pressure variation at a point,
that is where you have the minimum (0) velocity variation and
visa-versa.

4. (Heavy maths follows:) These factors form the limitting conditions on
some differential equations which reduce to one of the famous
Bessel equations. There are three solutions to this equation; one
results in an instrument with a cylindrical bore like a flute, one
results in an instrument with a conical bore like a saxophone or
oboe and one results in an instrument with an exponential bore. The
exponential bore is difficult to make for obvious resons and also
creates significant problems when laying out the holes. So
basically, we end up with cylindrical bore and conical bore
instruments.

5. The clarinet is essentially a cylindrical bore instrument ever though
the bore is somewhat tapered. This is because of various other
factors. The saxophone and oboe are conical bore instruments. The
flute is a cylindrical bore instrument.

6. If you draw a cross section of a cylindrical bore, pressure reed
instrument like the clarinet, the pressure variation will start
high at the mouthpiece and gradually decrease to zero at the bell
or the bottom most open hole. This is when you are playing a bottom
register note. So the wavelength of the note you are playing is 4
times the distance between the mouthpiece and the bottom-most open
hole.

When the note is overblown (by opening the speaker key), the
pressure variation starts large at the mouthpiece, decreases to
zero, increases in the opposite sense, then decreases to zero again
at the bottom-most open hole. So the wavelength of the note is 4/3
times the length between the mouthpiece and the bottom-most open
hole.

So the frequency of the overblown note is 3 times its bottom
register counterpart or 12 tones. (Frequency is inversely
proportional to the wavelength.)

7. If you now draw a cross section of a cylindrical bore, velocity reed
instrument like a flute, the bottom register notes have a zero
pressure variation at the "reed" or player's mouth. As you go down
the bore, the pressure variation increases to a maximum then
decreases to zero again at the open end (or the last open hole). So
the wavelength of the note is 2 times the length of the bore (or
the distance between the mouth and the last open hole).

When that note is overblown, the pressure variation increases from
the mouth, decreases, increases in the opposite direction and then
decreases again to zero. So the wavelength of the note is the same
as the length from the mouth to the last open hole.

The frequency of the overblown note is then twice the frequency of
the base register note. Ie. one octave.

8. Consider a saxophone with a conical bore. This is where things get
_really_ confusing because this behaves as a velocity reed
instrument because of the narrowing bore towards the mouthpiece. So
the pressure variation at the mouthpiece is minimum. This is the
same case as the flute and oboe so the saxophone overblows an
octave.

QED.

^~^~^~^~^~^~^~^~^~^~^~^~^~^~^~^~^~^~^~^~^~^~^~^~^~^~^~^~^~^~^~^~^~^~^~^~
Martin Brown, Telectronics Pacing Systems
Sydney, Australia
Ph: (61 2) 413 6973 Email: martinb@-----.au