The Ghost wrote:

Bob Cain wrote:
I answered you as to precisely what my symbols mean and how
to get from that relationship to a closed form solution for
two sinusoids.

You answered NOTHING. I asked you to define what you meant by flow
felocity and all you did was to restate your equation for so-called flow
velocity, Vf(t),

I fully explained my terms in a response to you and I gave a
detailed answer to what I mean by flow velocity in response
to Edward Green.

which you seem to have pulled out of thin air. Where did
you get that equation? Did you derive it or did you make it up?
If you derived it, show the derivation

If you will bother to understand my definitions of Vf, Vp
and Xp, then if

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

defined at the origin of a cordinate system is not obvious
to you by inspection there is no hope of your understanding
anything else. I suppose I could draw some pretty pictures
for you if that is the level at which you require
explanation. Having understood that, everything else is
just straithtforward translation to move the origin of the
frame of reference to a driver an arbitrary distance d from
the point where conditions are to be described.

In that translation, there is an assumption I suppose you
could argue with but I don't think you will because it is
correct. That assumption is Vp(d,t) = Vd(t-d/c). I.E. the
motion of particls about position d is the same as that of
the driver face about its rest position at a time earlier by
d/c.

In other places you equate your so-called
flow velocity to pressure. Which is it.

I've explained this. It is the velocity at a point that is
related to the pressure at that point by the acoustic impedence.

Pressure is force per unit area.
Velocity is distance per unit time. They are not the same.

Of course not. The are related by a constant of proprionality
called characteristic impedence or more generally acoustic
impedence which is in units of Pascal.s.m^-1 and your
pretending that you didn't understand my explanation of that
in the definition of terms I gave is nothing but baiting.

Great! We now know that the distortion numbers that you gave resulted from
both an undefined analysis as well as an undefined simulation.

You are simply ignoring the explicit formula I gave you that
I used to generate a discrete time signal and pretending
ignorance of commonly understood methods of discrete time
signal analysis. That is your problem, not mine.

So that you cannot continue to pretend you haven't seen it,
here again, copied from a prior response to you with a bit
more elaboration of what I call flow velocity, is the
definition of the terms I use and the results I obtain.

Definition of terms:

Vp(t) is the velocity of a particle normally at rest at the
origin as an acoustic wave goes through it in MKS units of
meters/second.

Sp(t) is the position of that particle in meters.

Vf(t) = Vp(t+Sp(t)/c) is the flow velocity at the origin
as a function of the particle velocity Vp(t) and position
Sp(t) defined above. This relationship holds everywhere,
not just near a driver. What I call flow velocity is
the velocity of whatever particle happens to be at the
origin at any point in time.

Vd(t) is the velocity of a driver face at rest at the origin.

Sd(t) is it the position of the driver face.

Vp(d,t) = Vd(t-d/c) is the velocity of a particle normally
at rest at the position d as an acoustic wave created by a
driver at the origin goes past it. It is also in MKS units
of meter/second.

Sp(d,t) = Sd(t-d/c) is the positon of that same particle
about position d in units of the meter.

Vf(d,t) = Vd(t-(d-Sp(d,t))/c) is the velocity of particles
passing a plane at d in meter/sec.

P(d,t) = Ra*Vf(d,t) is the pressure at that plane.

Ra is the acoustic impedence of air in Pascal*sec/meter. In
a tube it is homogenious and isoptropic with an approximate
value of 429 Pascal*sec/meter.


On closed forms:

If the driver velocity function of time has a closed form
indefinite integral, the application of my expression for
Vf(d,t) to obtain a closed form for the flow velocity and
pressure should be obvious.

For example if the velocity function of the driver is

Vd(t) = Al*sin(Wl*t) + Ah*sin(Wh*t)

then

Sd(t) = -((Al/Wl)cos(Wl*t) + (Ah/Wh)cos(Wh*t))

and

Sp(d,t) = -((Al/Wl)cos(Wl*(t-d/c)) +
(Ah/Wh)cos(Wh*(t-d/c))

Vp(d,t) = Al*sin(Wl*(t-(d-Sp(d,t))/c)) +
Ah*sin(Wh*(t-(d-Sp(d,t))/c))

I haven't bothered to do the algebraic substitution of the
penultimate equation into the final one. With that, it is
an exact closed form solution for two sinusoids.

Final note; the last two formulas are what I used to
generate a discrete time signal (by, of all things,
calculating it for discrete values of time) for analysis
using standard tools with which I am familiar. There are no
secrets in doing this that even a near novice to discrete
time signal processing would fail to know.


Bob
--

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

A. Einstein