Square Pegs

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When the wave lengths are significantly larger than the acoustic image size of the reflector, the sound transitions into behaving more like laminar flow

Whether the flow is laminar or turbulent depends on the geometry and the flow velocities - both of which affect the Reynolds number (Re) - the main non-dimensional number used to correlate the onset of turbulent flow based on ratio of momentum over viscous forces. Typically, very thin channels or slower flows will be laminar, and large flow ducts and higher velocities will be (or can be turbulent). A Paraline with very thin channel or plate thickness has the advantage of staying laminar over most of its flow regime given flow velocities that are not too high.

Re number is given by Re = velocity x characteristic length / kinematic viscosity.

At ambient room temp air, kinematic viscosity is circa 10^(-5) m^2/sec, for a 0.5cm channel width and say 25 m/s the Re number is 12,500. This is a larger Re in the transitional flow range of laminar-to-turbulent flow.

So, a Paraline when pressed to high SPL's actually has turbulent flow inside its thin little channels.
 
xrk971 - you are talking about static flow here and that is not relevant to dynamic or oscillating flows, i.e. acoustics. You cannot us static flow concepts in the acoustics range of frequencies with a high degree of confidence. Static hydrodynamics is a completely different set of equations and solutions than acoustic wave equations. They approach one another as the frequency falls to DC, but even then they are not comparable because the static flow hydrodynamic equations are nonlinear while the wave equations are linearized. Turbulence is nonlinear.
 
Repeat return to zero perhaps squelches (or perhaps assures) turbulance?
I could see expected behavior perhaps being different for higher frequencies
that it would be for a constant flow. Any turbulance across our thickness axis
would have to be high order modes above HOM cutoff frequency of the guide,
which we have squeezed >20KHz. The other two dimensions that see this turn
as a curved mirror are not sufficiently constrained to assure turbulance and
reflections back down the throat can't set up HOMs we could perhaps hear.

---

As for the turnaround being too large to execute a lossless 180 turn, maybe
we should angle the approach toward and dispatch away from the reflector.
And make the middle layer as knife thin as possible at the turnaround edge to
raise the cutoff frequency of the turn from 3 layers wide to 2. Granted that
knife edge might be turbulant and diffractive, but I'm not sure in the context
of a 180 turn, some sort of disruption is not going to happen anyway...
 
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Repeat return to zero perhaps squelches (or perhaps assures) turbulance?
I could see expected behavior perhaps being different for higher frequencies.

---

As for the turnaround being too large to execute a lossless 180 turn, maybe
we should 45 angle the approach toward and dispatch away from the reflector.
And make the middle layer as knife thin as possible at the turnaround edge to
raise the cutoff frequency of the turn from 3 layers wide to less than 2. Granted
that knife edge might be turbulant and diffractive, but I'm not sure in the context
of a 180 turn, some sort of disruption is not going to happen anyway. I think we
just need to keep it narrow in thickness to avoid setting up modes in thickness.
As for the other two dimensions that see it roughly as a mirror, it is what it is....
At 45 degree bevel, the action would be that of a corner reflector, if we are refering to rays. In this case I would agree with the taper, but around the bend would act more like laminar flow. Thus a smooth rounded bend around the tapered thin edge would be the best approach.
 
Just saying the turnaround don't need to be three layers thick.
And we make the reflection (if still wide enough here for any
such thing to exist) actually help with the turn. Its a theory...

Its not about rays. Except right there at the end where two
or three layers come together, there maybe its wide enough
we need to concern ourselves to also be right about rays.

Its not about HOMs. Except right there at the end where two
or three layers come together, there maybe its wide enough
we need to concern ourselves to also be right about HOMs.

Yes, I wrote exactly the same thing twice.

-----

We could make one smooth turn, but there's cancellations due
to the inside and outside radius not being the same. As drawn,
whats on the inside hopefully flips to the outside and versa...
So, maybe this bends us back to right? I hope but don't know.

All but the highest frequencies of interest will engulf this and
not care what shape it takes. But we want to be right all the
way to the top, if any way possible.
 

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Founder of XSA-Labs
Joined 2012
Paid Member
xrk971 - you are talking about static flow here and that is not relevant to dynamic or oscillating flows, i.e. acoustics. You cannot us static flow concepts in the acoustics range of frequencies with a high degree of confidence. Static hydrodynamics is a completely different set of equations and solutions than acoustic wave equations. They approach one another as the frequency falls to DC, but even then they are not comparable because the static flow hydrodynamic equations are nonlinear while the wave equations are linearized. Turbulence is nonlinear.


The Re number is not dependent on whether or not the flow is steady (I don't like to use the term "static" as that implies no motion) or time-varying (or dynamic flows). In acoustics, the linearized acoustic wave equation is based on the assumption of inviscid flow, small pressure variations, and no-net flow, hence would not contain any reference to a Re number. But this is an idealized approximation. A more accurate form of the acoustic wave equation (which can be solved via DNS) can be obtained where viscosity, momentum, and dissipation are all considered. The Re number is of course, central to this type of analysis which is more accurate than the inviscid. The Re number is a fundamental property of all viscous flows whether steady or time-varying, and a large Re indicates the higher relative magnitude of the momentum forces over the viscous (damping) forces - which leads to turbulence.
 
Hi Ken
The problem is what one needs is not a “named filter” but one that is the “right shape” for the magnitude and phase of the raw response for each range and that given the delay imposed by the front to back / depth separation between the upper and lower range.......

<snip>


Unfortunately, I would guess it’s close to impossible to arrive at this using real drivers responses without a computer program that iterates and that one can assign a phase target as well as amplitude.
Best,
Tom


YESYESYESYES AND YES. This concept is so tough for some to understand. I can't tell you how many times I get "but your crossover looks weird...that's not a normal L-C 2'nd order...."

Agreed, it's not...and it's exactly what I want.


Scott
 
So, given the different characteristics of sound waves, would there be any benefit to creating a paraline for one frequency range, and then building another for the other? It does seem like (although it is probably completely unnecessary) having mid-range and separate high frequency paralines (2-way in essence) that would allow for ease of placement for time aligninment due to their size...
 
This Cassegrain Smith was sized to match a Paraline we had built earlier.
Most obvious difference is the horizontal slot orientation and roundovers.
Less obvious is 90 degree horizontal directivity due to the internal horn.
Yes, the roundovers are undersized in the photo, and we held them in
temporary place with ridiculous duct-tape.

Much to my surprise, performance was almost identical. Both to Paraline
and Paraline restricted to 90 degrees horizontal by adding a board. And
this was measured by Zob, with his RTA and calibrated mic. I'm sure he
will post something when the screenshots are all sorted out.

Not much to see in those shots except that our sloppily built Paraline
is much smoother than expected from experiences others had posted.
Our Folded Cassegrain Smith Horn is only a tiny bit rougher, and has a
better top end. We are talking only a few db, nothing significant...

The slight increase in brightness might be attributable to the internal
shellac, triangular bumps at the driver and exit, and axe-edge grind
we gave to the middle board, with fillets to match on layers 2 (glued
to 1) and 4 (glued to 5). Refer back to the blue doodle in #846.
These extra steps proved time wasted to very little improvement.
I was hoping roundovers and narrowing vertical directivity might
give us some of that Eq, but didn't play much different in reality.

It was just weird walking past that horizontal slot and hearing much
the same dispersion pattern as Paraline with a vertical slot... I don't
think our Paraline has a perfectly narrow vertical due to drunkenness
on the day of its construction, so was similar to Smithagrain with the
"added advantage?" of roundovers. Again I think we wasted our time
with all the extra attention to detail for the Smith, but not knowing
that for sure, till a Smith configuration was properly and truly tried...

Both wanted for (and got via mini-dsp) +6dB per octave CD boost
starting around 2000. And we crossed (24db LR) to woofer at 1000.
Zob's woofer measured way lumpier in RTA than either folded horn.
We used the woofer's RTA measure to equalize below the crossover.
Above crossover, we tried flat and we tried CD, but didn't do allow
any lumpy, shifty filters. Just CD or flat, and CD was slightly better.

Same size slot, same size stack of wood, same compression driver.
You can see the build is radically different from Paraline inside, but
in the RTA measurement they are still almost twins.

This was not a failure. It worked. And every bit as good as Paraline!
Its just that it took us way longer than the Paraline to construct.
And in the end, no special advantage was found. Go figure...

Internal treatments mentioned above are not shown in these
photos, but will be shown next time we disassemble.
 

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Founder of XSA-Labs
Joined 2012
Paid Member
Very nice work Kenpeter and Zobsky. Looks very nicely made - not drunken at all. Given that the performance seems to be similar to completely different internal ray trace "reflection" geometry of Paraline, would you not agree that it is acting more as pressure duct waveguides rather than specular reflection waveguides? Looking forward to seeing posted measurements.
 
This Cassegrain Smith was sized to match a Paraline we had built earlier.
Most obvious difference is the horizontal slot orientation and roundovers.
Less obvious is 90 degree horizontal directivity due to the internal horn.
Yes, the roundovers are undersized in the photo, and we held them in
temporary place with ridiculous duct-tape.

Much to my surprise, performance was almost identical. Both to Paraline
and Paraline restricted to 90 degrees horizontal by adding a board. And
this was measured by Zob, with his RTA and calibrated mic. I'm sure he
will post something when the screenshots are all sorted out.

Not much to see in those shots except that our sloppily built Paraline
is much smoother than expected from experiences others had posted.
Our Folded Cassegrain Smith Horn is only a tiny bit rougher, and has a
better top end. We are talking only a few db, nothing significant...

The slight increase in brightness might be attributable to the internal
shellac, triangular bumps at the driver and exit, and axe-edge grind
we gave to the middle board, with fillets to match on layers 2 (glued
to 1) and 4 (glued to 5). Refer back to the blue doodle in #846.
These extra steps proved time wasted to very little improvement.
I was hoping roundovers and narrowing vertical directivity might
give us some of that Eq, but didn't play much different in reality.

It was just weird walking past that horizontal slot and hearing much
the same dispersion pattern as Paraline with a vertical slot... I don't
think our Paraline has a perfectly narrow vertical due to drunkenness
on the day of its construction, so was similar to Smithagrain with the
"added advantage?" of roundovers. Again I think we wasted our time
with all the extra attention to detail for the Smith, but not knowing
that for sure, till a Smith configuration was properly and truly tried...

Both wanted for (and got via mini-dsp) +6dB per octave CD boost
starting around 2000. And we crossed (24db LR) to woofer at 1000.
Zob's woofer measured way lumpier in RTA than either folded horn.
We used the woofer's RTA measure to equalize below the crossover.
Above crossover, we tried flat and we tried CD, but didn't do allow
any lumpy, shifty filters. Just CD or flat, and CD was slightly better.

Same size slot, same size stack of wood, same compression driver.
You can see the build is radically different from Paraline inside, but
in the RTA measurement they are still almost twins.

This was not a failure. It worked. And every bit as good as Paraline!
Its just that it took us way longer than the Paraline to construct.
And in the end, no special advantage was found. Go figure...

Internal treatments mentioned above are not shown in these
photos, but will be shown next time we disassemble.

Hi Ken
If the curve is correct given the incoming angle at the far left and outgoing angle in the center, this is a valid Paraline shape and will work as intended. We have one much like this with holes in the “bow ties” for mid drivers.
I can’t tell from the picture but IF the angle in the far right incoming passage is equal but flipped to that in 2 and 3, then the slot would be a straight line. If the incoming angle is wider than the exit, the slot is curve, but outward. BUT like any naked Paraline or any horn terminating in a mouth that shape and small, one will have artifacts because of that size and shape.
So, what you have is the back half of a CD horn, now you should make a front horn to terminate the paraline acoustically.

Follow this relationship, to avoid horn pattern flip, if you want the horizontal angle to be say 80 degrees and the Vertical to be 40, then you MUST end up with a horn that is twice as tall as it is wide. If you want 90 by 30, then it has to be 3 times as tall as it is wide and so on.
This IS NOT the simple pyramidal shape as the wider angle forces it’s apex to be in front of that for the narrow angle. Usually that is what you see normally, a slot at the throat with an extension that continues the vertical angle to the apex.
The paraline (yes xrk971, it is a “pressure duct”) is a way to shorten that back part by assigning a new apparent apex or origin where you wish for the exit wave front with a given incoming wave front. The idea of all of this is to drive the horn with a source having a wave front that is “as if” it had originated a greater distance than it is physically and normally at the apex of the desired (narrower) vertical radiation angle
Best,
Tom
 
...

I can’t tell from the picture but IF the angle in the far right incoming passage is equal but flipped to that in 2 and 3, then the slot would be a straight line. If the incoming angle is wider than the exit, the slot is curve, but outward. BUT like any naked Paraline or any horn terminating in a mouth that shape and small, one will have artifacts because of that size and shape.
So, what you have is the back half of a CD horn, now you should make a front horn to terminate the paraline acoustically.
...
Tom

A better view of the raw pieces
 

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On the back of bowtie #2 you can see where we drew a sound bubble
that we expected to project forward from the slot. If we constrain all
paths out to the full diameter of that bubble, would need an extra pair
of plate lips on the front, and exit from a curve just like a Smith horn.

Our purpose in drawing the bubble was to assist in drafting the smiles
that reflect the narrow bowtie horns on the back into the wider horns
on the front. We wanted both the raytrace and the bubble to agree
for all internal dimensions wider than 1/4 wave of highest frequency.
We also wanted to fill the front horns with sound, from edge to edge
without the usual problem of high frequency not curving with the wall.
So definitely, the angle of the outmost corners of the reflector are so
designed to mitigate that expansion.

From the very earliest documents about Smith horns, it has often been
suggested that they truncate to a flat faced slot with very little problem.
And I further hoped that truncation would smear the reflection distance
from slot discontinuity back to driver. That either didn't happen, or wasn't
significant as I had hoped, but that was the thinking behind it...

I also didn't give pattern flip proper consideration. But then again, our
original plan was for roundovers cut from 12" sonotube. Our 4 inch pvc
pipe was a last minute kludge.

You havn't seen the drunken paraline...
 
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Fig 6 of the Smith article illustrates flat truncation as an option.

Also inspired by side to side array-ability of some nexo horns.
Though this pic shows a narrowing parabolic reflector, they do
have a white paper that mentions an elliptical curve the other
way to expand the reflection. I did not know the smile shape
might turn out to be an ellipse. I just winged it by ray traces,
trial and error till I felt confident my remaining error was less
than 1/4 wave. I made no assumptions about the shape, but
glad Nexo properly figured it to be a simple one.

The full paper resides at:
http://nexo-sa.com/public/attachments/products/46/innovation_analysis_tang.pdf

We can take the exit slot to the edges of the cabinet if we wish.
There is no physical obstruction to a seamless horizontal array,
all the way to omni with as few as three is certainly possible.

Though a vertical array of paralines is probably more useful.
 

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Hi Ken
If the curve is correct given the incoming angle at the far left and outgoing angle in the center, this is a valid Paraline shape and will work as intended. We have one much like this with holes in the “bow ties” for mid drivers.
I can’t tell from the picture but IF the angle in the far right incoming passage is equal but flipped to that in 2 and 3, then the slot would be a straight line. If the incoming angle is wider than the exit, the slot is curve, but outward. BUT like any naked Paraline or any horn terminating in a mouth that shape and small, one will have artifacts because of that size and shape.
So, what you have is the back half of a CD horn, now you should make a front horn to terminate the paraline acoustically.

Follow this relationship, to avoid horn pattern flip, if you want the horizontal angle to be say 80 degrees and the Vertical to be 40, then you MUST end up with a horn that is twice as tall as it is wide. If you want 90 by 30, then it has to be 3 times as tall as it is wide and so on.
This IS NOT the simple pyramidal shape as the wider angle forces it’s apex to be in front of that for the narrow angle. Usually that is what you see normally, a slot at the throat with an extension that continues the vertical angle to the apex.
The paraline (yes xrk971, it is a “pressure duct”) is a way to shorten that back part by assigning a new apparent apex or origin where you wish for the exit wave front with a given incoming wave front. The idea of all of this is to drive the horn with a source having a wave front that is “as if” it had originated a greater distance than it is physically and normally at the apex of the desired (narrower) vertical radiation angle
Best,
Tom

I need to draw some diagrams to confirm, but this seems to indicate that asymmetrical horns have the wrong mouth shape.

Hmmm.

For instance, when you have a horn with a coverage angle of 90 degrees by 60 degrees, the 60 degree axis is narrower than the 90 degree axis. But if I'm reading Tom correctly, it should be the opposite. IE, the 60 degree axis would have to be larger than the 90 degree axis.

The only way to do that (which I can think of) would be to change the angle at some point along the horn.