From: pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt)
Subject: Re: RESONANT AIR COLUMN EXPERIMENT and RESONANT MUSICAL INSTRUMENTS
Organization: Computer Science Department,  Stanford University.
Date: Sun, 27 Sep 1992 22:23:05 GMT

In article <ZOWIE.92Sep27133539@daedalus.stanford.edu> zowie@daedalus.stanford.edu (Craig "Powderkeg" DeForest) writes:
>Gordon_Wong@mindlink.bc.ca (Gordon Wong) writes:
>   >Jim Carr writes:
>   > Gordon Wong writes:
>   > > My question: Which (the fundamental, the 1st overtone, or the 2nd ...)
>   > > would yield the more accurate value for the wavelength...
>   >
>   > Good question.  Your experiment should help you answer it.
>   >
>   Experiments show that the results using the 2nd overtone result in a more
>   accurate value for the wavelength and thus the speed of sound. I am just
>   wondering why. 
>
>There are two effects here:  
>
>First, the more nodes there are in the standing wave pattern, the more
>accurately you can resolve the peaks.

(Presumably you mean resonance peak here and not a peak in the standing
wave itself.)   Good point, I'd neglected this in my reply, thinking
this error would be down in the noise compared to the other effect.
But it does bear mentioning.

>Second, competing with that effect are tube-width effects.  The width 
>of the tube changes the resonant frequency, in ways that aren't obvious
>to me without some thought.

This is the effect I was referring to when I said that you should add a
correction term to the cylinder length equal to a multiple of the tube
radius.  I just couldn't remember the multiplier, though for the
fundamental I would expect it to be fairly close to 1.  It is plausible
that the multiplier includes the wavelength as a linear factor, which
would explain why higher overtones give more accurate results (assuming
you don't apply this correction).

You should be able to distinguish these two sources of error
experimentally because the first is a random error while the second is
systematic.  At each overtone, make "enough" measurements of the
resonance peak and get the average and standard deviation for the
measured frequency of that overtone.  Plot these average frequencies
against their corresponding overtone numbers.  If the correction term
does decrease with increasing frequency then this should yield a curve
asymptotically approaching (from below with increasing overtone number)
a line through the origin.  That line is where your averages should
have landed had the correction term been zero.  Estimate the asymptote,
and proceed to estimate the velocity of sound as though your
measurements had landed on the asymptote and there was no correction
term.

You can now read off the source 2 errors as distance from that line, at
each overtone.  These can in turn be compared with the standard
deviations at those overtone, the source 1 errors, to see which of the
two sources of error is the larger, and how their ratio depends on
overtone number.

If the correction term turns out to be independent of frequency then
the averages should lie on a straight line.  You can still proceed as
above if you take the "asymptote" to be the line parallel to this line
through the origin.  In any event the distances from your measurements
to the asymptote gives the source 2 error in the form of a correction
term to the cylinder length, which seems like the most natural
representation of this error.

If you actually do this experiment I'd be extremely interested in
knowing the outcome, textbook treatments notwithstanding.

-- 
=================================================== In short, I am floundering
Vaughan Pratt   pratt@cs.Stanford.EDU  415-494-2545 around and don't take what
=================================================== I say seriously.
Newsgroups: sci.physics