From: zowie@daedalus.stanford.edu (Craig "Powderkeg" DeForest)
Subject: Re: RESONANT AIR COLUMN EXPERIMENT and RESONANT MUSICAL INSTRUMENTS
Summary: Zowie addresses tube question, then diverges & mumbles about flutes
Organization: Stanford Center for Space Science and Astrophysics
Date: 27 Sep 92 13:35:39

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.  This is simple to understand:
the more nodes there are, the narrower each peak region is.  As you
change the frequency, you need to change it less (proportionally) to
go all the way through a peak, so your fractional accuracy goes up.

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. (and, Lord knows, *no-one* bothers to think
on USENET!) Clearly, though, the width will affect the resonance
frequencies of small sound waves more than those of larger ones, ie
higher frequencies will be more affected, since for the long sound
waves transverse motions of air in the column are negligible and the
tube diameter ceases to matter too much.

So, there's clearly some sort of compromise wave number with the best
result.  If I were less lazy (or if I had more used envelopes lying
around to write on), I suppose I could do the math and work out some
of the effect.  But lots of work has been done on tube width resonance
effects -- try looking in American Journal of Physics ( they have
lots of good `how does this ordinary effect work?' type articles ) or 
try Fetter and Walecka's book, `Theoretical Mechanics of Particles and 
Continua' for a good jumping-off point.

For a good practical example, check in The Literature for review
articles about flutes!  I recently came across (but have since lost) a
good review article about flute design.  A flute, like all acoustical
instruments, is a resonant device to pull one note out of random noise
generated by the musician.  In this case, it's a tube with some holes
that can be opened or closed to change the effective length of the
tube, and another hole at the end that can be blown across to generate
white noise and make the flute resonate.

But orchestral music demands very precise tuning, and flutes play over
a range of about three octaves [or more!] -- over this range, small
effects of the tube width, etc. really matter -- lots of thought has
gone into flute design over the years.  Each hole has a little
`chimney' of an exact (but small, and different for each hole) height
sticking out of the flute; the head of the flute is tapered to change
the frequencies of high notes; etc. etc.  There are many small details
that are designed in just so, so that the resonant frequencies come out
right.

Much more enigmatic is the ocarina.  At its simplest, an ocarina is
a small, round resonant chamber -- from the size of a silver dollar up
to about the size of a mango -- with a whistle mouthpiece (the white
noise generator) at one end, and four or five seemingly randomly -
placed holes around the top.  The musician can cover different
combinations of the holes to get a chromatic scale over one or two
octaves (depending on the skill of the musician and the manufacturer).
Here, though, the resonant modes of the chamber are *not* obvious
_a_priori_; it's very difficult to calculate them from first
principles, since the chamber chape is not particularly symmetric in
any easily identifiable way, as in a flute.


--
Craig `No, I *really* *do* play piano in the marching band!' DeForest
                       zowie@daedalus.stanford.edu
Newsgroups: sci.physics,sci.math