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Re: Acoustic impedance



Michael D. Edmiston usually is on track, but I think he missed on this one.

Although the mouthpiece end of a trumpet may seem closed, it is open.
Think
of it this way... the characteristic of an open end is a pressure node...
i.e. constant pressure.

My intuition and several books don't agree with this.


The lips are vibrating in the proper phase
relationship with the reflected wave to maintain a constant pressure at
that
point. When pressure from the reflected wave is not available the lips
open
and the pressure is maintained by the pressure in the players mouth. When
back pressure from the reflected wave is available, the lips close.

The pressure inside the mouth is maintained at a greater pressure than the
inside of the trumpet. When a positive pressure pulse comes back, it
causes the lips to part, releasing a pulse of air in a positive feedback
type situation. Try puckering up like you are about to play a trumpet and
just tap on your lips. Each tap (corresponding to a positive pressure
pulse) releases an additional pulse of air in phase with the original
pulse.

Pressure can vary, hence a pressure antinode, hence a closed-at-one-end
pipe.


The key is the shape of the tube. Consider a clarinet and a saxophone. A
clarinet, with essentially a cylindrical bore, acts like a standard
closed-at-one-end pipe. When you go to the upper register, the frequency
goes up by 3 (1.5 octaves), as expected. A saxophone, which looks similar
but has essentially a conical bore, acts differently. When you go to the
upper register, you only go up be a factor of 2 (1 octave).

I haven't been through the derivation, but a closed-at-one-end, conical
tube produces all the harmonics. (It's probably a bad analogy, but I think
of the small end as "less open" than a cylinder, but the open end as "more
open" than a cylinder, with a net effect of 2.0 "open ends", making it act
about like a cylinder which truly has 2.0 "open ends".)

Try making a "trumpet" out of a piece of PVC pipe and you get (very nearly)
the odd harmonics only. Doesn't matter if you use a clarinet mouthpiece of
a trumpet mouthpiece. Adding a "bell" by adding sections of larger pipe to
the end gets you closer to the "standard" trumpet harmonics. The shape of
the bell and mouthpiece have been chosen to produce the desired set of all
even harmonics.

Tim