Re: Acoustic impedance

["John S. Denker" <jsd@MONMOUTH.COM>]

Jeff Marx wrote:
...
> Why does a trumpet, which seems like a closed pipe,
> produce a nearly harmonic
> set of overtones, without any missing harmonics?

That is a verrrrry interesting question.
It was hiding in among some other questions
and I missed it the first time around.

The point is that a tube with one end closed should
only produce odd harmonics.  That's an elementary
wave-mechanics problem.  But in a real trumpet, all
modes are observed, even and odd, so there is quite
a puzzle here.

> The hip-shot response is
> because it has a bell (and a mouthpiece), which changes the "effective
> length" of the instrument for different frequencies.

Not a chance.  The bell does some impedance matching
at the open end, but only at higher frequencies, and
it doesn't have any effect on the "closedness" of the
"closed" end.

===========================

I am not an expert and we are entering into very
tricky territory, so some of what follows will
be hypothetical.  I don't have a trumpet to
experiment on, or even look at, I can't be sure
my ideas apply to real trumpets.  But they might.

We will have to look very closely at the "closed"
end.  I keep saying "closed" in quotes because
the assumption of "closedness" is very much in
doubt.

For starters, I hypothesize that if you _really_
closed the tube with a cork or some such, the
high-school physics prediction would come true:  you
would observe resonances with odd-integer spacing;
half the modes would be missing.

So what can we say about the lips?  Lips obviously
have mass.  They also have some springiness.  And
the trumpeter has arranged it so that this mass-on-
a-spring system is essentially "on resonance" for
whatever note is being played.

General non-hypothetical physics advice:  When you
have a resonant system, expect the impedance to do
fancy things!

Let me start with an electrical analogy, because
some people are more familiar with the laws for
series/parallel electrical impedances.  The
correspondences are as follows:

    trumpet                 circuit
  length of tube     <--> length of coax
  reflection at end  <--> reflection at end
  mass               <--> inductance
  spring             <--> capacitance

Suppose we have a piece of coax that terminates
in a discrete capacitor.  At high frequencies, it is
a dead short, and we get the upside-down reflection you
would expect.  At low frequencies, it is wide open,
and we get the right-side-up reflection you would
expect.  At an intermediate frequency, roughly at
the point where one wavelength's worth of coax
has the same capacitance as the discrete capacitor,
you get a reflection that is +90 degrees out of phase.
For all frequencies, you can plot the phase and
amplitude of the reflection in polar coordinates
(parameterized by frequency) and it's all very
understandable.

Now suppose that the coax is terminated in a
discrete inductor instead.  You get the mirror-
image story:  dead short at DC, wide open at
high frequencies, -90 degrees out of phase at
some intermediate frequency.

Now (drum roll, please) terminate the coax with
a capacitor _and_ an inductor in parallel.
Obviously it's a dead short at DC because of
the inductor.  Obviously it's a dead short at
high frequency because of the capacitor.  But
what about the intermediate frequency?  If
these were resistors in parallel, the parallel
combination would necessarily have _less_
impedance than either component separately.  But
for these reactive components, if you keep
track of the phases properly, you will find
that they combine to have _more_ impedance.
(Work it out.  It's just the parallel-circuit
formula with the obvious complex impedances.)
Right at the LC resonance the impedance of
the parallel combination goes to infinity.
It becomes equivalent to an open circuit,
even though you might have expected (based
on the off-resonance behavior) something
pretty close to a closed circuit.

So, returning from circuit-land to acoustics-
land, there is a pretty obvious hypothesis:
The lips _look_ like they produce a closed
end, and they probably do produce a closed
end under all conditions EXCEPT the conditions
that prevail when the trumpet is played!
The on-resonance lips look like an open end.

Everything I said about circuits is pretty
much true;  whether the analogy works for
trumpets is hypothetical at the moment.

This hypothesis leads to some interesting
predictions.  For one thing, if we think
the lips are on-resonance for the fundamental,
they will be off-resonance for all the
harmonics.  This will affect the timbre
of the instrument, because only odd partials
will appear.  I'm not at all confident of
this prediction, because the lip-popping
is so nonlinear that some of the partials
might find something other than a closed
end.  This is getting pretty complicated.

Another semi-prediction is that you might
be able to play two types of notes:  open-end
notes and closed-end notes, depending on
just exactly how you hold your lips.

The existence of a closed-end and/or open-end
notes can be easily verified just by measuring
the length of the trumpet tube and comparing it
to thewavelength of the lowest playable notes.
If it's a half-wavelength, it's an open-end
note.  If it's a quarter-wavelength, it's
a closed-end note.

There's a lot more one could say.  But I'll
stop here and let some others have some fun
with it.....