The Ghost wrote:

What do you mean by "flow velocity?" It's not discussed or defined in any
of my reference books on acoustics, and I have most if not all of them.
Define your terms........Sp'(t), Ra, Vd, etc. If you have a closed form
solution, present it.

I'm not an acoustics specialist, as is obvious, and am not
terribly concerned about satisfying them with terms. What I
mean is the velocity of the acoustic air flow passing a plane
in units of distance/time. If there is a more correct term
I'd be happy to know it.

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
meter/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.

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.

Ra is the acoustic impedence of air in Pascal*sec/meter. In
a tube it is homogenious and isoptropic with an approximate
value of 300 Pascal*sec/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.


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.


Bob
--

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

A. Einstein