Klarinet Archive - Posting 000206.txt from 1996/10

From: Josias Associates <josassoc@-----.COM>
Subj: Re: CO2 influencing pitch: An Epilog (I Hope)
Date: Tue, 8 Oct 1996 21:24:19 -0400

I suppressed my initial impulse to comment on this lively
discussion, because I was certain that well-informed people would set
matters straight. Since this is exactly what has happened, I limit my
comments to a reinforcement of what has been said by others.

It has been correctly reported that the clarinet reed vibrates at
a frequency determined, in part, by the resonant gas column inside the
clarinet and that the resonance characteristics are determined by the
speed of sound within that gas mixture, which could, for short durations,
be mixtures rather different from air.

A discussion similar to this one occurred in May 1996,
highlighted by a scholarly presentation by Jonathan Cohler (whose
expected response on this CO2 subject arrived as I composed this note).
At that time (May), I was ignorant of the fact that Benade had apparently
used electrical analogs in his acoustical analysis of the clarinet.
Because I was fascinated by the similarities between the clarinet and
electrical transmission lines, which is one of my engineering
specialties, I presented my analog to the list, which explained why the
fundamental wavelength was four times the column length and why the
overtone spectrum was composed of odd harmonics. (If I had gone to the
trouble of reading Benade's work, I probably would not have written my
note.)

What seems important is that the onset of the pressure wave
at the tip opening of the mouthpiece causes a pressure traversal down the
length of the column and back and then down again and back a second time
before the first complete pressure square-wave is formed at the fundamental
frequency. Thus, as others have said in other ways, it is during the
period of four linear acoustic reflections, while the pressure
wave dwells in the gas within the clarinet column, that the wavelength of
the fundamental frequency is established.

In the transmission-line comparison, the analog to the gas medium
is the transmission-line dielectric, and the electrical analog to the
propagated sound pressure wave is a voltage wave.

For those interested in my May 7th posting (or for uninterested
people who are battling with insomnia) that message is included after this
one.

Regards to all,

Connie

Conrad Josias
La Canada, California

>From josassoc@-----.com Tue Oct 8 15:42:29 1996
Date: Tue, 7 May 1996 00:44:29 -0700 (PDT)
From: Josias Associates <josassoc@-----.com>
Cc: Multiple recipients of list KLARINET <KLARINET@-----.BITNET>
Subject: Re: A question about acoustics -- Transmission Line Analog

Jonathan,

I have copied the list members on this message at the risk of its
being a crashing bore to many. However, with their indulgence, I present
the following electronic analog of your acoustic model of the clarinet,
which I had not seen used before in explaining wind-instrument acoustics,
as a way of reinforcing my understanding of your explanation.

In reviewing your excellent explanation to Dan Leeson as to why
the cylinder closed at one end produces a fundamental wavelength at
four times the length of the column and why the harmonic wavelengths
are odd submultiples, I was struck by the similarity of your explanation
to the physics of pulsed electrical transmission lines such as coaxial
cables, parallel pairs, twisted pairs, etc.

During my engineering career I have had frequent occasion to
write about such pulsed lines when used as elements of complex systems I
have designed for customers. Out of the semi-infinite number of
terminations possible in transmission lines (including matched and
complex terminations), the electrical analog of the clarinet, as you
have described it, employs a relatively simple subset of terminations.

The open end is a short circuit (in this case a pressure short
circuit) and the closed end is an open circuit. In my electrical analog,
I use a step-function current source to excite the line, where the current
source simulates the player's air stream into the clarinet. Thus
far, my model does not include a mouthpiece or reed. That comes later.

In the electrical analog, the incident step function of current
induces an input voltage step, the amplitude of which is the current
times the characteristic impedance of the line. The voltage step propagates
down the line to the short circuit, which has a reflection coefficient,
K @-----. The potential (or pressure) at that port
or at any other port is the sum of the incident and reflected
waves, which, at the short-circuit end, is zero.

When the negative step reflects back to the input, which has
been positive until that time, the new incident negative step
brings the input voltage to zero and the new reflected step (K = +1
for an open-circuited line), which is also negative, drives the line
negative by the magnitude of the original pulse, and that step
propagates down the line. Thus far, the voltage waveform at the input,
from the first onset of the step to the first down-and-back time, is one
half a square wave. During the second down-and-back interval, the input
waveform goes negative, thereby completing one square-wave cycle.

So long as the original stimulus remains in place, the
voltage at the input to the line will continue to be a square wave
whose period is twice the down and back times, which is analagous to
your example. In this way, the transmission line behaves like a
resonant square-wave tank circuit. And, as Fourier afficionados know,
the square-wave is comprised of odd harmonics of the fundamental frequency.

The propagation velocity down an electrical transmission line,
which is analagous to the velocity of sound, is closer to the speed
of light and is reduced from that velocity by a factor of the dielectric
constant of the material separating the cable conductors relative to the
dielectric constant of free space.

The mathematical determination of the performance of complex
mechanical systems is often accomplished with the use of electrical
analogs. My first occasion to use an electrical analog in acoustics
occurred about eight years ago when I needed to analyze phase shifts
between the electrical drive signals applied to speakers and their acoustic
output pressure waveforms when loaded by multiresonant chambers. The
application was a feedback control system for acoustic levitation of
solids and liquids in microgravity environments.

Although I suffer from not having read extensively from formal
acoustic textbooks, what little I have read has not employed the medium
of transmission-line analogs to explain clarinet acoustics. I thought it
worthwhile mentioning.

Jonathan, as an afterthought to your frequency equation, which
shows proportionality to sound velocity, I wonder if you'd comment on the
tuning variations of wind instruments at high altitudes, such as at
Denver, Albuquerque, and Mexico City.

Regards,

Connie

Conrad Josias
Engineering Consultant
La Canada, California

   
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