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Just a follow up to say thanks to John M, Curtis O and John D for their
insights.
The animation was particularly helpful, John M., in illustrating the idea
that the mid-point between the speakers is the actual point of reference
for this situation.
Check my thinking here, if you would: If we put the speakers at opposite
ends of a tube we now do have a resonant phenomenon and we will get
essentially 1D standing waves with displacement antinodes at the speakers.
Yes?
Thanks again!
Jeff
On Tue, Mar 25, 2014 at 2:52 PM, John Denker <jsd@av8n.com> wrote:
I agree with what John M. said._______________________________________________
Perhaps I can add a little bit of pedagogical detail.
For simplicity, let's focus attention on one-dimensional
propagation. No spreading. Specifically, consider a long
tube. Imagine to be sturdy but transparent. Inside the
tube is ordinary air. We ignore damping, friction, thermal
conductivity, and dissipation of all kinds.
Start by considering the situation with no speakers.
Imagine that as a far-distant initial condition, there is
a wave packet traveling in the medium. Imagine it is a
simple Gaussian pulse. The air pressure goes up and then
back down again. No pressure nodes. No displacement nodes.
The pulse travels past us and keeps going.
Now imagine that in the tube there is an ultra-thin membrane.
It is so thin that the air doesn't care about it. The air
just pushes on one side of the membrane, and the membrane
pushes on the air on the other side, so it is acoustically
transparent at all frequencies.
Let's call this description #1, and summarize it by saying
the membrane is transparent to the pulse.
I claim such a membrane is a good model of an ideal speaker.
We now switch to description #2. Imagine we can only see one
side of the membrane. We have no idea what is behind it. All
we see is that at a certain time, the membrane moves, and a
pulse emerges from the membrane and propagates away, never
to return.
Note that the membrane is also an ideal microphone. Any
sound incident on the membrane is transmitted (according
to description #1) or simply absorbed (according to
description #2). We can say that the membrane is perfectly
/impedance matched/ to the air.
===============
Let's use description #2 to analyze some scenarios
involving /two/ ideal speakers facing each other.
A) Each speaker emits a pulse every so often. The pulses
sometimes cross and sometimes don't. In any case, the
pulses do not interact when they cross. All we see is
leftward pulses coming from one speaker and rightward
pulses coming from the other speaker.
B) Each speaker puts out a sine wave. Assume the frequencies
and amplitudes are the same. Then there will be standing
waves, with nodes and antinodes. However, the speakers
will not sit at the nodes or antinodes, except by accident,
except on a subset of measure zero within the set of all
possible values for the frequency and relative phase.
C) If the two frequencies are not equal, there will be a
non-stationary pattern of nodes and antinodes that will
march along, sometimes being at or near one speaker and
sometimes at or near the other speaker.
================================
Note that in the real world, there is usually nowhere near
perfect impedance matching between the air and the guts
of the speaker. Note that horns can be used as acoustic
/transformers/ to help improve the impedance matching.
Anyway, when the matching is not perfect, there will be
reflections. Invoking the analogy to electrical transmission
lines (such as coax), air hitting a massive object is like
a short circuit in the displacement field. It guarantees
a displacement node at that point ... and not coincidentally
a pressure antinode. It must be emphasized that this depends
on a particular type of nonideality, you have no chance of
figuring it out from first principles. You have to deal
with it on a scenario-by-scenario basis.
To say the same thing another way, your intuition about the
speaker being a massive obstacle to the wave is probably
right in practice ... but wrong in principle ... which
automatically makes this a tricky problem.
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