Klarinet Archive - Posting 001403.txt from 1998/10

From: "Mark Charette" <charette@-----.org>
Subj: Re: [kl] re:Altitude pt 2
Date: Sat, 31 Oct 1998 12:19:41 -0500

From: David C. Blumberg <reedman@-----.com>
>I must have not caught the first thread.
>I would not expect the altitude to
>have an effect on intonation, but would
> expect the reed to feel harder at
>higher altitudes. The air column in the
>Clarinet is a different density at
>different altitudes. If the reed vibrates
>at a different frequency, then what
>is the effect of that different frequency?

Warning: simple algebra approaches :)

The formula for determining the allowable
frequencies of an ideal closed end cylinder
is:

f = (n - 1/2) V / 2(l + S/c)

where
n = any integer
V = the local sound velocity
l = length
S = cross sectional area of the tube
c = the acoustic conductivity of the open
end of the tube. The calculation of this
is _way_ beyond the scope of this posting
and really stretches my math knowledge.
Tensors are easier!

V, the local sound velocity, in an ideal gas,
only varies with temperature. For dry air the
formula for V is:

V = V0 sqrt(1 + t/273)

where

V0 @-----.3 M/sec
t = temperature in Celsius

Now, if air were that "ideal dry air", we
wouldn't have to worry about altitude; pressure
doesn't come into the picture. This formula is
used for V all the time for simplification.

In reality, the speed of sound varies with
pressure - it doesn't obey the ideal gas law.
Some approximate numbers (there are formulae
to calculate some very good approximations to
the local speed of sound, but again, the
formulae are beyond the scope of this
posting):

Speed of sound at sea level at 0 C @-----.3 M/s
Speed of sound at 45,000 ft at 0 C @-----.0 M/s

Doing a linear interpolation ( a dangerous
assumption, but who cares at the moment - we
can lip things up or down to make 'em right :),
the speed of sound at 5000 ft would be:

327.27 M/s

Using a ratio (since the other variables will
remain about the same - c changes a tiny, tiny bit):

f1 @-----.30 / 2(l + S/c) (sea level)
f2 @-----.27 / 2(l + S/c) (5000 ft)

f2 will be lower than f1 by a ratio of .962,
therefore (again, this is approximate):

A fingering for Concert A (440 Hz) at sea
level produces 440 * .962, or 423.28 Hz at 5000
ft - just a wee bit flat :)

During all this time we've been at 0 C, so
time to go in and warm up the instrument and
yourself. Which brings back into play the
temperature. I leave the calculations of the
local velocity of sound in "real" air at other
temperatures as an exercise for the readers.
All you have to do is pick up a good textbook
on fluid dynamics, a bit of thermodynamics,
and a good refresher tome on calculus!
----
Mark Charette@-----.org
Webmaster, http://www.sneezy.org/clarinet
All-around good guy and devil-may-care flying fool.
"There can be no freedom without discipline." - Nadia Boulanger

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