Re: Acoustic impedance

[Jeff Marx <jmarx@WMDC.EDU>]

Here are my thoughts on this...

> I'll take a stab at both questions posed here.
>
> (1) Reflection at open end.
>
> In the hypothetical wave on a string with a free end, when the wave gets to
> the free end, the end has a whiplash action in which it travels farther than
> it would if more string were present past that point.  When this reaches its
> limit and snaps back it pulls on the previous section of string more than
> that section would have been pulled if the displaced section were being
> restored to the equilibrium position by string on both sides of it.  So this
> initiates a reflected wave.

No problem here.


> For a sound wave in a tube coming to an open end, the displacement has been
> constrained and becomes less constrained.  It can suddenly propagate in more
> directions.  As the air molecules displace under less constraint, they
> displace more.  When they come back to equilibrium they are coming back and
> interacting with the previous molecules in a different way than the
> interaction that occurred from neighbor to neighbor within the contraints of
> the tube.  That initiates a new wave (the reflection) back down the tube.

Ah. Interesting - less constrained. I'll think about this. How does the
frequency of the wave fit into this picture, though. Why are high-frequency
waves diffracted less than low-frequency waves when they reach the opening?
Looking at it another way: high-frequency waves pass out of the tube with
much less change to their wave shape then a low frequency wave. Why? Also,
if we treat the wave as a plane wave propagating down the length of the
tube, how exactly is it constrained by the walls? A plane wave would not be
reflecting off of the walls, so how would the information about the
presence of a wall be transmitted to the medium at, say, the middle of the
pipe? Is it because there is some sort of drag, or some other, effect at
the wall's surface. I don't see this coming into the calculation for the
magnitude of the reflected pulse.

> If this seems nebulous, just admit that the motions of the air molecules at
> the end of the tube will be different than the motions of the molecules
> within the tube.  Then admit that when the displacments/motions of air
> molecules in a region of space is different than the surrounding areas, that
> means a wave will propagate into the surrounding areas from that region that
> is different.

I was always comfortable admitting that there was something different
between the two situations (inside and outside the pipe); however I
couldn't see how this could be translated into a physical description of
reflection.


> (2) For a trumpet.
>
> 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.  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.  So the
> lips are essentially a pressure-regulator, opening and closing at the right
> frequency to keep the pressure in the mouthpiece constant... hence a
> pressure node... hence an open tube.


This, I think, contradicts what Rossing and Hall have to say on the issue.
If I look at figure 11.8 (p. 232) in Rossings book "The Science of Sound:
Third Edition"  I see pictures of standing pressure waves in a brass
instrument. At one end, the bell, there is a node for pressure - as I would
expect; at the other end, the mouthpiece, there is an antinode for pressure
- also as I would expect - which seems to suggest to me that the
interaction between the lips, mouthpiece, and vibrating air column make the
brass instrument act like a closed pipe. Hall, in his book "Musical
Acoustics: Second Edition" has a similar set of pictures, and a quote that
reads (p. 264) ...

     As with other reed instruments, the blown end of the [brass] tube is
effectively closed."

Later in Hall (p.265)

     The puzzle is much more striking for the trumpet family, for we would
expect by analogy to clarinets to get only the odd-numbered members of a
harmonic series by overblowing. Yet here, too, we get a complete harmonic
series, except for an absent fundamental.

Both authors go on to explain how the presence of the bell and, to a lesser
extent, the mouthpiece effect the length of the vibrating air columns at
different frequencies, thereby creating a spectrum that has no missing
harmonics.

Thanks Michael


> Michael D. Edmiston, Ph.D.              Phone/voice-mail:       419-358-3270
> Professor of Chemistry & Physics        FAX:                    419-358-3323
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