Re: Doppler Distortion - Fact or Fiction?

From: Edward Green (spamspamspam3_at_netzero.com)
Date: 08/21/04 

Date: 21 Aug 2004 10:13:36 -0700

Bob Cain <arcane@arcanemethods.com> wrote in message news:<cg67f10jk2@enews3.newsguy.com>...
> Richard Herring wrote:
>
> > In message <cg1ibl0266t@enews4.newsguy.com>, Bob Cain
> > <arcane@arcanemethods.com> writes
> >
> > [big snip]
> >
> >>
> >> Whatcha think?
> >>
> > I think you missed my point about the pressure being a linear function
> > of piston displacement _evaluated at a retarded time which depends on
> > the piston's position_.
>
> Nope, I attempted to describe why it isn't correct in the
> case of a piston in a tube.

In which case, as I just commented, you would have to explain why the
net velocity of sound adjusts itself for a high frequency ping
superimposed on a low frequency oscillation according to the piston's
position, to just compensate for this lead/lag. I'm not saying it
might not so adjust in air confined to a tube, based on some factors I
don't understand -- but you have to explain how this can happen!

One possibility is that the velocity of the high frequency ping
depends on factors which vary over a cycle of low frequency sound, and
in the pings' passing through (n + x) wavelengths of the low frequency
sound (i.e. integer + fractional part), x depends on displacement of
the piston just so to null out the discrepancy in travel time to the
auditor.

See? I've handed you an argument on a platter: you merely need to
fill in the details, to see if it might work out in detail. ;-)

> Another simple arguement for
> the lack of any kind of Doppler distortion in that
> situation, assuming SPL's are low enough to keep the air in
> its linear regime,

The air may well stay locally in its linear regime -- meaning say that
it behaves like a linear elastic solid -- but the overall process have
non-linear input/output pairs. Richard Herring alluded to retarded
times, and I originally said we had a non-linearity attributable to
finite motion, which is consistent with Herring's POV -- only
infinitesimal displacements, no finite leads or lags, no finite piston
velocity, no effect.

> is that the radiation impedence seen by
> anything driving the piston is identically the
> characteristic impedence of the air, real and frequency
> independant. Given that, where is there room for any error
> in the process?

Perhaps the assumptions which went into this simple model of radiation
impedence are exactly those which have thrown out the effect at the
onset -- effectively that all displacements are infinitesimal? I
allowed myself to be temporarily dazzled by yours and Franz Heymann's
bandying of this exciting and important concept :-), but now it
suddenly comes to me that I know exactly what this formalism means
after all; and it may serve to conceal the assumptions which went into
it.

I would concentrate on more direct physical arguments.

> > All the physical processes are individually 100% linear, but the outcome
> > is not.
>
> Odd, that. I just don't understand how that could possibly be.

Stated to be maximally perplexing, like Schroedinger's cat. Bad
Richard! ;-)

Like the tuning fork I mentioned earlier, the answer (if you want a
compact oolie-ish answer illustrating some general principle) is that
finite displacements do not behave linearily. In general, if A and B
are finite displacements, and "A + B" means "A followed by B", then A
+ B /= B + A. As I also suggested, it's a mistake to be too fast and
loose with "linear" and "non-linear", so if Joe Blow says "it's a
non-linear system" at the bar, we all nod sagely as if _we_ knew all
about those infernal non-linear systems, puff on our pipes and sip our
whiskey.

Individual relationships may be linear, or fail to be, according to a
simple definition. Complex systems may exhibit both forms of behavior
depending how we hold them up to the light.