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. I'm darned sure the
result for velocity is the same function but I just can't
get there from here. From the flow at a plane at d, I wish
to get to the state equations for the particles which have a
rest position at d.

This isn't homework and I'm ashamed I have to ask but...


Thanks,

Bob
--

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

A. Einstein

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. I'm darned sure the result for velocity
is the same function but I just can't get there from here. From the
flow at a plane at d, I wish to get to the state equations for the
particles which have a rest position at d.

This isn't homework and I'm ashamed I have to ask but...


Thanks,

Bob

Try:

"Acoustic Fields and Waves in Solids, Vol I & II" by B.A. Auld, 2nd edition
(February 1990), Krieger Publishing Company; ISBN: 089874783X

Sam Wormley wrote:

Bob cain wrote:
From the flow at a plane at d, I wish to get to the state
equations for the particles which have a rest position at d.

Try:

"Acoustic Fields and Waves in Solids, Vol I & II" by B.A. Auld, 2nd
edition
(February 1990), Krieger Publishing Company; ISBN: 089874783X


Thanks, Sam. Are you certain this particular derivation is
there? I was also told that it was in Pierce's "Acoustics -
An Introduction to Its Physical Principles and
Applications", but it isn't (not that I'm at all sorry I
bought Pierce's text.)

I have a heuristic argument but it isn't good enough. FWIW,
it goes like: the wave equation being purely linear, all
state variable functions of particles in the field will be
linear functions of the other field variables, i.e. if the
propegating pressure/velocity wave is sinusoidal then
position, velocity and acceleration of the particles will be
too. This I just cannot prove although it seems obvious.

Thus, particle velocity about its quiescent position will be
purely sinusoidal if the propegating wave is. Since the
peak value at a position d is given as Vp and it is defined
by the phase at d=0 to be a maximum when the particle
passing by d is the one that's there in quiescent
conditions, we know from the extremal values and the fact
that it must be sinusoidal, all there is to know about the
velocity of a particle quiescent at d. I.e., its velocity
function is the same as the velocity function of particles
passing d.


Bob
--

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

A. Einstein

Bob Cain wrote:


Sam Wormley wrote:

Bob cain wrote:

From the flow at a plane at d, I wish to get to the state equations
for the particles which have a rest position at d.

Try:

"Acoustic Fields and Waves in Solids, Vol I & II" by B.A. Auld, 2nd
edition
(February 1990), Krieger Publishing Company; ISBN: 089874783X


Thanks, Sam. Are you certain this particular derivation is there? I
was also told that it was in Pierce's "Acoustics - An Introduction to
Its Physical Principles and Applications", but it isn't (not that I'm at
all sorry I bought Pierce's text.)


Vol. 1, Chapter 1, "Particle Displacement and Strain"
A. Particle Displacement and Displacement Gradient
B. Strain
C. "Local" Rotation
D. Pictorial Representations
E. Transformation Properties
F. Symbolic Notation and Abbreviated Subscripts
Problems
References

Perhaps you can check Auld out of a local library (interlibrary loan)
to check it out. I don't know Pierce's "Acoustics - An Introduction to
Its Physical Principles and Applications".












I have a heuristic argument but it isn't good enough. FWIW, it goes
like: the wave equation being purely linear, all state variable
functions of particles in the field will be linear functions of the
other field variables, i.e. if the propegating pressure/velocity wave is
sinusoidal then position, velocity and acceleration of the particles
will be too. This I just cannot prove although it seems obvious.

Thus, particle velocity about its quiescent position will be purely
sinusoidal if the propegating wave is. Since the peak value at a
position d is given as Vp and it is defined by the phase at d=0 to be a
maximum when the particle passing by d is the one that's there in
quiescent conditions, we know from the extremal values and the fact that
it must be sinusoidal, all there is to know about the velocity of a
particle quiescent at d. I.e., its velocity function is the same as the
velocity function of particles passing d.


Bob

Bob Cain wrote in message ...

Hi, Bob,

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. I'm darned sure the
result for velocity is the same function but I just can't
get there from here. From the flow at a plane at d, I wish
to get to the state equations for the particles which have a
rest position at d.

You can't prove your equation, because in fact the general acoustic
field can be much richer. What you have written down is the equation
for a plane wave of a single frequency. You can have any number of
frequencies, and the waves can be travelling in any number of
directions.

You've got to start off with the equations of motion. I guess you'll
find that in Sam's textbook. I've given them for an isotropic medium
in a thread not too long ago ( Transmission line matrix (TLM) -
with two different velocities??? ). The equations relate the
variations of pressure and of velocity. The velocity is a vector,
while the pressure is a scalar. For isotropic acoustic media, sound
waves are longitudinal, i.e. the velocity variation is parallel to the
direction of propagation of the sound wave. The ratio of the pressure
amplitude to the velocity amplitude is the acoustic impedance.

For the usual acoustic media, the velocity of sound is independent of
frequency. This means that the velocity variation doesn't have to be
sinusoidal. You could have a square-wave variation of velocity, and it
would remain a square wave no matter how far the sound travels. The
reason why all the books talk about sine waves is that this is the
only type of wave which can be guaranteed to preserve its shape no
matter how far it travels, in any uniform, unchanging medium. This is
the case because the assumption that the medium does not change means
that the equations of motion commute with d/dt. Sine waves are
eigenfunctions of (d/dt)^2.

Cheers,

Zigoteau.

zigoteau wrote:


You can't prove your equation, because in fact the general acoustic
field can be much richer. What you have written down is the equation
for a plane wave of a single frequency. You can have any number of
frequencies, and the waves can be travelling in any number of
directions.

Zigoteau,

Thank you very much for the detailed response. I only need
to prove it for the case of a plane wave moving in one
direction normal to it. I absolutely agree that it should
be for a general signal rather than a single frequency, but
a major part of the argument in which I'm involved revolves
around the single frequency case. Having settled that I can
consider the more general case. If however it is just as
easy to prove the general case that would be far better.


You've got to start off with the equations of motion. I guess you'll
find that in Sam's textbook. I've given them for an isotropic medium
in a thread not too long ago ( Transmission line matrix (TLM) -
with two different velocities??? ). The equations relate the
variations of pressure and of velocity. The velocity is a vector,
while the pressure is a scalar. For isotropic acoustic media, sound
waves are longitudinal, i.e. the velocity variation is parallel to the
direction of propagation of the sound wave. The ratio of the pressure
amplitude to the velocity amplitude is the acoustic impedance.

I've got this much under my belt and understand what the
wave equation says (if not how to solve it for all the stuff
I want to show.)

You could have a square-wave variation of velocity, and it
would remain a square wave no matter how far the sound travels.

Absolutely no disagreement from me. Proving it remains
elusive at the level of my mathematical facility.

The
reason why all the books talk about sine waves is that this is the
only type of wave which can be guaranteed to preserve its shape no
matter how far it travels, in any uniform, unchanging medium. This is
the case because the assumption that the medium does not change means
that the equations of motion commute with d/dt. Sine waves are
eigenfunctions of (d/dt)^2.

Yep, I understand that too. The very definition of a linear
system.

What I still can't figure out how to do is, in the case of a
single plane wave, impulsive or oscilating, derive the
velocity function of time of a particle which has an
undisturbed (rest) position d from the velocity function of
time describing the state of whatever particle happens to be
at d. Is my description adequate to describe the simple
problem?

While various considerations make me certain the two
functions are identical, including that it seems intuitively
obvious, I can't for the life of me prove it and the
argument I'm in demands that.


Bob
--

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

A. Einstein

Bob Cain wrote in message ...

Hi, Bob,


Thank you very much for the detailed response. I only need
to prove it for the case of a plane wave moving in one
direction normal to it. I absolutely agree that it should
be for a general signal rather than a single frequency, but
a major part of the argument in which I'm involved revolves
around the single frequency case. Having settled that I can
consider the more general case. If however it is just as
easy to prove the general case that would be far better.

I'm glad that you appreciate my contribution.


You've got to start off with the equations of motion. snip

I've got this much under my belt and understand what the
wave equation says (if not how to solve it for all the stuff
I want to show.)

Right. Once you've got the equations, then all you need to how is how
to analyze and solve them.

You could have a square-wave variation of velocity, and it
would remain a square wave no matter how far the sound travels.

Absolutely no disagreement from me. Proving it remains
elusive at the level of my mathematical facility.

The
reason why all the books talk about sine waves is that this is the
only type of wave which can be guaranteed to preserve its shape no
matter how far it travels, in any uniform, unchanging medium snip

Yep, I understand that too. The very definition of a linear
system.

What I still can't figure out how to do is, in the case of a
single plane wave, impulsive or oscilating, derive the
velocity function of time of a particle which has an
undisturbed (rest) position d from the velocity function of
time describing the state of whatever particle happens to be
at d. Is my description adequate to describe the simple
problem?

It took me some time to analyze your sentence!

Why do you need to know that?

I will give you a few clues about methods of analysis you can use, but
get the feeling that if I gave you a complete solution to your
problem, you would still be none the wiser. What you need a lot more
of is overview.

Firstly the equations I gave in that other thread, and the equations
you will find in Sam's textbook, are linear. They are linear because
it is possible to analyze linear equations. You can express the
general solution in terms of a much smaller set of characteristic
solutions, the eigenfunctions, which often have sinusoidal variation
in all directions. If you have a linear transmission medium, then you
can get distortions of the signal, but they are a relatively mild sort
of distortion, which can often be reversed by using a filter which
cancels them out.

Now, while these equations are a very good approximation to the true
equations for the propagation of low-intensity sound through many
parctical systems, they are nonlinear in the general case. Our ears
extract sound from the air, and nonlinearity starts to become quite
significant for acoustic transmission through air at sound levels of,
say, 120 dB, that is to say 1 W.m^-2. Audio frequencies are high
enough that the compression and expansion of air during one cycle is
adiabatic. Pressure and density are related by P ~ rho^1.4, which is
not linear (1.4 being the value of gamma, ie the ratio C_P/C_V, for an
ideal diatomic gas). Distortion from this mechanism can be a
significant consideration in the design e.g. of horn loudspeakers for
public address systems.

At normal listening levels of 60-70 dB (1-10 W.m^-2), the distortion
is much smaller and is usually neglected.

You can calculate the amplitude and the pressure amplitudes once you
know the acoustic impedance for the medium. For air, the acoustic
impedance is roughly 300 Pa.m^-1.s (if this is important, do your own
calculation or find a reference book). At 60 dB this gives a pressure
fluctuation of ~3e-2 Pa and a velocity fluctuation of ~1e-4 m/s. At
middle C, the amplitude of the displacement of individual air
particles is ~1 µm. This compares to a wavelength of around 1 m.

Hence in the linear regime, the distance between the points whose v
you want to relate is so small that essentially v=v'. If the
displacement amplitude is so large that you are no longer in the
linear regime, then v and v' will start to differ significantly. The
variation of v will no longer be adequately represented by a sine
wave, even if it started off as a sine wave at the source, and in
general you won't be able to analyze the equations to work out how v
does vary.

Secondly, you want to relate the velocities at a fixed point to the
velocity of an individual element of air. Mathematically this problem
is as old as the Navier-Stokes equations for fluid flow. f(x,y,z,t) is
any function defined over all space and time. The particle you want to
follow has x=x(t), y=y(t), z=z(t), so that at the moving particle at
time t the function takes on the value f(x(t),y(t),z(t),t). The master
equation is:

df/dt = &f/&dt + &f/&x.dx/dt + &f/&y.dy/dt + &f/&z.dz/dt

where I am using & for the curly d symbol of partial differentiation.

So: I've given you all the tools you will need. I am not sure what the
sense is of the problem is you are trying to solve, and suspect that
it results from a lack of overview, which I have also tried to
provide. It's now up to you.

Cheers,

Zigoteau.

zigoteau wrote:

Sorry also for not realising earlier that your problem is the same one
you were discussing a month ago. Doh. Now that I understand the
physical basis of your problem, I can suggest more appropriate
approaches.

In fact on 23 August I replied to your last post to the thread you
initiated: " Doppler Distortion - Fact or Fiction?". Did you see
that?

Yes, and it escaped me. I'm afraid, also, that my access to
publication is limited not being at a university or
belonging to any publication societies. My only real source
at the moment for matters acoustic is Pierce's text.

Putting it all together:

P1[(A/a)*sin(a*t) +(B/b)*sin(b*t)-c*t] = Z*[A*cos(a*t) + B*cos(b*t)]

This is a constraint on the arbitrary function P1, and as long as the
maximum velocity |A|+|B| is less than the velocity of sound c, it will
allow just one solution for it, ie just one value of P1(x) for each
value of the argument x.

Are you still with me?

I can scarcely believe it but, yes, every step. (I did need
to get help from Pierce on the details of getting from the
wave equation to the general bi-directional solution through
the domain transformation.)

As I understand the meaning of your final expression from
its explanation, it is a constraint on the pressure at a
point in space oscillating about c*t which will yield a
unique solution for general x. Is that correct?

With that, I'm ready for the finale if you are. I wish I
was ready for that to jump out at me but...


Many Thanks,

Bob
--

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

A. Einstein

Bob Cain wrote in message ...

Hi, Bob,

Sorry I've discouraged you, but I hope I am correct in the assumption
that you want a valid solution, and are willing to spend some time
getting up to speed with the required maths.

Absolutely, I just sorta stepped off the edge of the last
attempt. :-)


Are you still with me?

I'll be back when I can answer that one way or another and I
will put all possible effort into comprehending it first.
Getting a firm grip on this problem has consumed me for some
time. The brief look I've been able to give your response
tonight makes me hopeful again. I'm not a complete dummy
(E.E. having started in physics), just a little out of my
depth at the moment and paddling hard to reach the surface
where the sun shines.

I've looked back through the thread and see that I overlooked syour
expression for the velocity on 2004-09-21 19:50:03 PST, then replaced
it with a different but equivalent form. Sorry.

I was a bit oblique about how to get the function P1 from my last
equation. I've just straightened out my own thoughts on where to go,
so here goes:

The last equation was basically P1[s(t)-c*t] = Z*s'(t) . . . (E1)

where s(t) is the position of the loudspeaker cone at time t, and
s'(t) = ds/dt. The problem is to determine the value P1[x] for a given
value of x.

This comes down to finding the value of t which solves the equation

s(t)-c*t = x . . . (E2)

You could do it graphically. The plot of s(t)-c*t is a wiggly line
with a generally downwards trend. Since s(t) is differentiable, it is
continuous, as is s(t)-c*t. Hence s(t)-c*t takes on every possible
value, so that E2 always has at least one root. By the assumption that
the maximum loudspeaker cone velocity is less than c, the gradient is
always negative, so s(t)-c*t cannot take on the value x more than
once. Hence for any value x, E2 has a unique solution t. The value of
P1(x) is then given by E1.

A graphical solution is not much good to you if you want an analytical
expression for the distortion, but at this stage you can launch into
approximations. The solution to E2 is given approximately by:

t = -x/c . . . (E3)

Substituting t=-x/c+epsilon into E2 gives you:

s(-x/c+epsilon) = c*epsilon . . . (E4)

By expanding the LHS as a Maclaurin-Taylor power series expansion in
epsilon, you can then generate a series of approximate but
increasingly accurate solutions. For your purposes, I think it would
be adequate to stop the power series expansion at the linear term,
from which you get:

epsilon = s(-x/c) /{c-s'(-x/c)} . . . (E5)

which gives you ultimately

P1(x) = Z*s'[-x/c+s(-x/c)/{c-s'(-x/c)}] . . . (E6)

or, at the same level of approximation,

P1(x) = Z*[s'+s*s''/(c-s')] . . . (E7)

where s and its derivatives s' and s'' are evaluated at -x/c. You can
see that this level of approximation gives you a nonlinear term, and
clearly at low sound levels it will be the dominant distortion term.

If you have an ideal pressure transducer at a point d down the tube,
the signal it picks up is P1(d-c*t) = Z*[s'+s*s''/(c-s')] where s and
its derivatives are evaluated at t+d/c .

Cheers,

Zigoteau.

Bob Cain wrote in message ...

Hi, Bob,

I'm glad you're getting somewhere.

Yes, and it escaped me. I'm afraid, also, that my access to
publication is limited not being at a university or
belonging to any publication societies. My only real source
at the moment for matters acoustic is Pierce's text.

Prompted by your postings, I have assembled an electronic library of
about 1.5MB about distortion in loudspeakers, which I imagine is your
ultimate motivation. Would you like me to email them to you? They're
not textbooks, so they will take a bit more digesting than Pierce, but
you are welcome to them. Just say the word.

Cheers,

Zigoteau.