Re: [Phys-l] music pipes

[John Denker <jsd@av8n.com>]

On 04/17/2007 01:28 PM, Anthony Lapinski wrote:
> ... If ... is a closed pipe, then only odd harmonics can be
> played. Right?

It's not that simple.

Even if we restrict consideration to idealized geometrical shapes, there
are multiple cases that need to be considered:
 1) Cylinder open at both ends.                             [odd+even]
    e.g. flute
 2) Cylinder open at one end and closed at the other end.   [odd only]
    e.g. clarinet
 3) Cylinder closed at both ends.
 4) Cone open at the bell.                                  [odd+even]
    e.g. oboe, saxophone
 5) Cone closed at the bell.


Cases (3) and (5) are uncommon in the context of musical instruments,
but make perfectly good resonators (such as might be discussed in
the context of the speed of sound).

> Can someone clarify/revise this information or know of a reliable
> source/text/web site which lists the instruments in terms of open/closed
> pipes?


Note that a cone is _automatically_ closed at the pointy end.  There
is no air and no sound moving through the pointy end.  This is what
"closed" means!

If you consider only the presence or absence of even harmonics, the
cone harmonics (case 4) are more like the open/open pipe harmonics,
(case 1).  Beware that there is a thriving community of folks
who believe the presence of even harmonics "proves" that a conical
instrument is open at both ends.  This requires
 -- relying on mindless analogies
 -- willful ignorance of conflicting theoretical evidence, namely
  the wave equation.
 -- willful ignorance of conflicting observational evidence, namely
  taking even the briefest look at actual instruments.

I've written to some textbook authors about this.  They basically
said "don't confuse me with the facts".

Here is some good discussion, including sketches of the wave functions:

  http://www.phys.unsw.edu.au/jw/pipes.html#here
  http://www.phys.unsw.edu.au/jw/flutes.v.clarinets.html
  http://www.phys.unsw.edu.au/jw/saxacoustics.html


Keep in mind that real instruments, even when keying the
fundamental, are not perfect cones or perfect cylinders.  Also
the musical cases -- i.e. (1), (2), and (4) -- have significant
"end corrections" at the open end(s).  And with the keyholes
open, things get even trickier.  Real waveforms exhibit
resonances in places far from where the ideal geometry would
suggest.

Solving the wave equation in a cylinder is a standard textbook
problem.

Solving the wave equation for the longitudinal modes in a cone is
straightforward in spherical polar coordinates.  This is hardly
surprising, if you consider the "method of images" i.e. packing
lots of cones together to fill a sphere, packed like the ommatidia
in a bug's eye.  Based on that, you should immediately suspect
that the answers involve spherical harmonics ... which they do.
Practically any E&M book (e.g. Jackson) or mathematical methods
of physics book will have the details.