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Re: Acoustic impedance



Jeff Marx wrote:
...
Imagine a sound wave traveling inside a tube (treat it as a plane wave).
When the wave reaches an open end of the tube, part of the wave propagates
into the the room and part of the wave reflects back into the pipe.

OK.

How does the longitudinal wave "know" when it
has reached the end of the tube. I feel that a full explanation would
describe the diffraction of the wave at the tube opening, as well.

You've pretty much answered your own question.

The following may make it somewhat easier to visualize.
Consider a set of tubes in series, starting with a very
rigid tube, followed by a somewhat elastic tube, followed
by a virtual tube (i.e. no tube at all).


---------------- - - - - - -

A B C D E F G H

---------------- - - - - - -


Air parcel B has pressure which pushes on air parcel C.
Parcel C has pressure which pushes on parcel D.
But why should parcel D push on parcel E, when it
can just as easily push outwards on the spongy walls
of the tube? And parcel G is even less motivated
to push on parcel H, because G can just expand sideways
into the open air.

So in a hand-wavy kind of way, you can convince yourself
that there is a major impedance change at the end of
the tube. In particular, parcel F will get carried
away by its own momentum, due to the lack of restoring
force from parcel G. Parcel F will overshoot, moving
farther than it would if the tube were endless, and this
overshoot is what radiates the reflected wave.

All of this leads to answering a, perhaps, more interesting question: "Why
does a trumpet, which seems like a closed pipe,

Closed at one end.

produce a nearly harmonic
set of overtones, without any missing harmonics?" The hip-shot response is
because it has a bell (and a mouthpiece), which changes the "effective
length" of the instrument for different frequencies.

That doesn't seem plausible to me.
The length is the length.
There will be a _slight_ change in effective length
depending on frequency, because of weird fringing
fields around the bell, but this is a minor effect.
It will lead to slight "stretching" of the octaves,
nothing more.

A trumpet is basically a cylindrical tube with a
bell "tacked on". According to my information [1],
the result is:
-- To a first approximation, it's just like a
cylindrical tube with slightly augmented length.
-- To a second approximation, the bell increases
the impedance-matching of the higher modes, to
the extent that it wipes out the Q of all modes
above 1500 Hz, more-or-less making those higher
modes non-resonant.

I'm not sure these non-resonant harmonics should
be called "missing". Plenty of sound comes out
at these frequencies, but it results from a standing
wave at lower frequency being up-converted by the
nonlinearity of the lip-popping.

[1] Reference: Arthur H. Benade, "The Physics
of Brasses" in Scientific American, July 1973.