[Phys-L] Re: Singing Rod Demo....

[John Denker <jsd@AV8N.COM>]

von Philp wrote in part:
>    Another teacher strongly believed
> that the waves must be transverse because the impressive volume of the sound
> is too great to be achieved longitudinally.

I've been thinking some more about that part of the argument.  We
know that argument leads to the wrong conclusion, but beyond that
there are good scientific reasons -- as well as pedagogical reasons --
for asking _how_ that argument goes wrong.

The argument as quoted above doesn't provide much detail, but one
might assume that the physical basis of the argument goes something
like this:  In a transverse mode, a big piece of the object (on the
order of half a wavelength) will be moving in the same direction,
and that whole piece will have an opportunity to couple coherently
to the sound field.  That's true as far as it goes, but alas it
is not the whole story.

We should ask, what's the _physics_ that couples the object to
the sound field?  An aluminum rod (or a piano string without
sounding board) vibrating transversely will be, to a first
approximation, just like any other cylinder moving through a
fluid.  The ideal flow pattern (neglecting friction) for a
steadily-moving cylinder is well known;  see for instance the
nice animations by Marco Colombini:
   http://www.av8n.com/irro/cilindro1a_e.html

The pressure field that goes with this flow pattern shown at
   http://www.av8n.com/irro/cilindro3_e.html
which requires a little bit of interpretation:  there is high
pressure on the leading edge of the cylinder and an equal amount
of high pressure on the trailing edge (i.e. the 3:00 and 9:00
positions in the diagram).  There is low pressure at the 12:00
and 6:00 positions.

In all cases you can see that Bernoulli's principle is
upheld;  high pressure is associated with low velocity and
vice versa.

As the next step in the argument, we need to consider a cylinder
having an *un*steady motion, and figure out how it couples to
the sound field.

The answer is, not much.  To first order in (object diameter)
over (wavelength of the sound in air), it doesn't couple at all.
The negative pressure regions couple equal-and-oppositely to the
positive pressure regions.  Neglecting friction, the dipole term
doesn't couple either.  The lowest term that actually couples
AFAICT is the quadrupole term, which is pretty terrible.

   Relying on friction for the coupling isn't very nice,
   either.  For one thing, it's nonlinear, so the sound
   would be at a different frequency from the motion of
   the object!

This sheds some light on the most basic physics of stringed
instruments.  A violin is a lot more than four strings: it
consists of four strings a a whoooole lot of wood.  Without
the wood, it would be practically inaudible.


In contrast, the longitudinal mode couples just fine.  Yeah,
only the end of the rod couples to the sound field, but it
couples efficiently.  To a first approximation, locally near
the end of the rod the air thinks there is just suddenly
more metal.  The air is displaced.  It's not quite the ideal
symmetric monopole radiator, but it's close.

   The volume of the rod is not changing nearly as much as you
   might naively think, because while the end is getting longer
   the middle of the rod is getting thinner.  But these two
   regions are separated by more than one wavelength (in air)
   of the sound, so the cancellation argument we used in the
   transverse case does not apply.

Similar physics considerations apply to the design of loudspeakers.
If you move some object (like a speaker cone) to push on the air,
you have to ask where the object moved _from_, and what effect
that had on the air.  That's why (at the most basic level) speakers
consist of a lot more than just the voice-coil and speaker cone.
Without the _enclosure_, the performance would be terrible.

==========

Several people on this list have mentioned the advantages of
having a _theme_ in a physics course.  One could imagine building
an introductory physics course around the physics of musical
instruments.  Scaling laws, energy, force, leverage, mass,
resonance, waves ... there's a lot of good physics there.  Also
I think it's good to remind phyiscs majors (as well as the other
99% of the population) that there's more to physics than quarks
and other esoteric stuff.

A starting point for information on the subject is:
   http://www.catgutacoustical.org/
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