Bob Cain wrote:


Well, I've asked for help on the general equation for pressure at
a point removed from an ideal piston in an infinite tube as a
function of the force applied to the piston that includes the
effects of Doppler distortion in alt.sci.physicw and on the
moderated group sci.physics.research where the real guns hang
out and there has been no answer.

What I've found is that any attempt to write the expression
from conditions at the interface results in a recursion or
infinite regress unless the term included to account for
the motion of the piston is set to zero. It's really tricky.

So let's look at an argument by reciprocity. Assume an
acoustic pulse of any arbitrary shape running down the tube
with an ideal pistion (no mass, stiff, infinite compliance)
in place.

1. The piston will move exactly in step with the motion of
the air molecules as the pulse passes by it.

Now let's measure and record the velocity of that piston as
the pulse passes by. Next let's mount a voltage to velocity
transducer, again ideal with a zero mechanical impedence, on
the side of the piston from which the pulse came when we
measured it.

2. When we drive that piston so as to reproduce the velocity
that was recorded we will get the identical pulse propegating
off of it as originally measured.

3. Because air is air, the resulting pressure pulse will be
in phase with that velocity and given by p(t) = v(t) * Ra,
where Ra is the characteristic impedence or air, and that
pressure pulse will be identical to the one that the
measured pulse had.

Because this should be true with a pulse of any shape it will
be true of a supposition of any such pulses which implies that
it is true of any signal and is thus a linear transducer with
no distortion of any kind.

The only thing that needs to be said to complete this is
that any arguments relative to velocity can be equally
applied to pressure since pressure and velocity in air are
in phase and related by a constant of proportionality, Ra,
the acoustic impedence of air.

The last nail in the coffin of loudspeaker Doppler
distortion is to find the flaw in the the intuitive, and
highly persuasive, argument that has been used historically
and used a lot as a counter to my argument. The flaw just
came to me today, finally.

Doppler shift is an unquestionable phenomenon, easily proved
from first principles. The thing is that it depends on a
sound source that is moving with respect to, i.e. _within_
the medium (we assume for this that the listener is
stationary with respect to it.) A piston with two
frequencies driving it and stationary with respect to the
medium otherwise will supposedly show those frequencies
modulating each other. The flaw is that the piston is not
in any sense moving within the medium, it is moving the
medium which is fundamentally different.

If you had a little loudspeaker moving back and forth in
space with a low frequency while emitting a high frequency
you would indeed see the "Doppler distortion" phenomenon but
if you move that speaker back and forth with a large plane
to which it is attached, the low and high frequencies will
add linearly with no distortion. In the latter case it is
part of the system moving the air and in the former it is
just moving within the air.

If you move the entire plane with the sum of the two
frequencies, it moves the air the same amount and generates
a pressure wave with only those two spectral components and
no frequency modulation.

Doppler distortion in loudspeakers in indeed a myth
justified by faulty intuitive hand waving.


Bob
--

"Things should be described as simply as possible, but no
simpler."

A. Einstein