The Ghost wrote:

Bob Cain wrote in message ...

Bob Cain wrote:



snip.....snip


Just realized that you copied only part of what I said in
sci.physics to everywere else this discussion had become
active (except your home group alt.sci.physics.acoustics
oddly enough), so to rectify that the full sci.physics post
follows.

Bob Cain wrote:

For the case of a propegating sinousiodal plane wave (or
more generally oscillating plane waves if at all possible)
where

v(d,t) = Vp cos(w(t-d/c)) or a general v(t-d/c)

describes the planar particle velocity at a position d
relative to some origin, I need a rock solid
proof/derivation of the state equations, position and
velocity, of the particles which are at rest in a plane d
when the field is quiescent. I cannot believe how elusive
this has remained for me and need help.

The answer turns out to be simple. If the velocity and
position of the particles moving about their rest position
at the origin are

Vp(t) = Sp'(t)

then the flow velocity of the particles in the plane of the
origin is

Vf(t) = Vp(t + Sp(t)/c)

and the pressure there is

P(t) = Ra*Vf(t)

This implies that there is a non-linear relationship between
particle velocity about a point and the flow velocity at
that point. In acoustics texts this relationship is swept
away in the approximation that Vf(t) = Vp(t) when motion is
small relative to the wavelength of the sound.

If there is a driver at the origin which imparts its motion
Vd(t) and Sd(t) to the air then

Vp(d,t) = Vd(t-d/c)
Sp(d,t) = Sd(t-d/c)

where d is the distance from the driver's rest position to
the point about which the motion is of interest. The flow
velocity (and proportional pressure) at that point is

Vf(t,d) = Vd(t-(d-Sd(t-d/c))/c)

The Doppler distortion in a plane wave is a consequence of
the non-linear relationship between the signal motion
imparted to the air by the driver and the flow velocity
(pressure) at a point about which that motion is manifest
and measured.

For a two tone Vd(t), one at 40 Hz and another at 2 kHz and
allowing a motion of 2 cm (a reasonable Xmax for a two way
system) the RMS IM distortion sidebands about the 2 kHz
fundamental near the speaker face are on the order of .2% of
that fundamental. While of some signifigance compared to
other portions of an audio chain, it probably isn't for a
loudspeaker given its other distortion mechanisms.


Bob
--

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

A. Einstein