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Improve Your Soundstage for $2

289K views 644 replies 200 participants last post by  seafish  
#1 ·
Some people asked me to post this info, so here goes:

How to improve your soundstage for two bucks.

If you've ever listened to a set of audiophile mini-monitors, you may have noticed that they throw a really amazing soundstage. The reason that they do this is that the enclosure is very small.

But we can replicate this effect with a much larger enclosure by reducing diffraction. Here's how this works:

When the sound from your speaker hits *anything*, it creates a reflection. These reflections are perceived by your ears as a secondary sound source. So for instance, if you take a set of mini monitors and place them six inches away from a wall, the reflection off that wall will muddy the image. That's why speakers sound better when you pull them away from a wall.

What's interesting about this phenomenon is that it's VERY time dependent. A reflection that occurs an inch or two from your speaker is 10x worse than a reflection which occurs a foot away.

Having established that reflections are a BAD thing, how do we address that? Here's a few options:

  • Use a waveguide to 'funnel' the sound in one direction. I do this in most of my projects. It's a LOT of work :(
  • Absorb the sound. You can do this with sound absorbing pads, like they use in a studio. Not really practical in a car.
  • Diffuse the sound. This works really well. Despite all the work I've investigated in waveguides, I'll admit that it works nearly as well, and it's a hell of a lot less work.

With sound diffusion, we're basically taking that reflection, and spreading it waaaaaaaay out in frequency in time. So instead of a big ol' nasty reflection at one spot and one frequency, we're spreading it out over an octave or two, over a few milliseconds.

Here's a couple pics of some speakers which use diffusion to improve their soundstaging abilities.

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Subjectively, the speaker becomes more difficult to localize when it's in a spherical enclosure. The soundstage becomes more diffuse, and seems to float "around" the speakers instead of being tied to them. The sound is also sweeter and less fatiguing.

And this is only common sense of course. Imagine a sound wave striking a sharp edge, diffracting violently:

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Now imagine the same sound wave in a spherical enclosure. The energy is the same, but the sound is diffused gently across the smooth surface. The improvement in the polar response is measurable, and the subjective improvement is obvious.

In part two I'll describe how to do it...
 
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#2 ·
In part one I described why reflections are a BAD thing, some options for controlling them, and how this will improve your soundstage.

Now the practical part, how to build it.

Let's say we wanted to diffuse sound down to 100hz. The speed of sound is 13500in/second. So to diffuse sound down to 100hz, we'd need a spherical enclosure that's 42.98" in diameter. Here's the equation:

required sphere radius = 13500 / lowest frequency / 2 / pi
required sphere radius = 13500 / 100 / 2 / 3.14159

Hmmm that's not gonna work is it? :confused:

Psychoacoustic studies have shown that matching frequency response above 1khz is more important than *below* 1khz.

Based on that we could reduce our sphere from almost four feet to a little over four inches. At that size, it starts to get practical to do this in the car.

My local craft store sells some clear plastic spheres, I have no idea what they're for, but they work for spherical enclosures. They're 8cm in diameter and they're less than a buck. An 8cm sphere will control diffraction down to 1365hz.

Here's a picture of a vifa ring radiator in one of these 99 cent spheres:

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So there's the math above, and they're readily available at craft stores. In the pic I'm using half a sphere. You'd want to take the entire sphere and cut it down to a point where it blends seamlessing with the diaphragm. The way that B&W does it is just about ideal:

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The teardrop shape is better than a sphere; it flattens out the frequency response and reduces diffraction. My local craft store has egg shapes, but they're not big enough for this application. YMMV
 
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#3 ·


I found a better pic of what diffraction looks like:

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click on the link to watch the animated version:

http://www.silcom.com/~aludwig/images/diffdem.gif

Note how the sound diffracts off the tweeters edge and off the baffles edge. That's why tweeters are carefully recessed.

If you look at the math above, you can see that it's practical to reduce diffraction from a tweeter down to 1365hz for a buck. Better yet, put that spehere about six inches away from the dash and the windshield, and you'll eliminate or reduce ALL reflections down to 470hz. If you take it that far, your tweeters will literally disappear. No amount of electronic manipulation or DSP can do this. There isn't a magic amplifier that can do this. Replacing your head unit won't do this.

But a 99 cent plastic sphere can.

 
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#4 ·


Here's some articles on diffraction from guys that are a lot smarter than me:

Diffraction from baffle edges

Baffle Step Compensation

I post a lot of measurements of my car online. The first thing that people notice is that the response isn't particularly flat. That's because they're accustomed to response graphs from manufacturers, who measure speakers on a big ol' flat baffle. To give you an idea of why that's misleading, here's what happens to a speaker when you put it in a box:

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and in a sphere:

Image

 
#6 ·
And I saved the best for last.

Have you ever noticed that your speakers sound good at moderate levels, but grow fatiguing at high volumes? You crank them up for a minute or three, but quickly you want to reduce the volume. This is particularly common with people using horn loaded compression drivers in the car. They sound great for a few minutes, but grow fatiguing quickly.

According to research by Earl Geddes, the perception of diffraction is level dependent. That's one of the reasons you're speakers sound fatiguing at high levels. You could use the most expensive speakers from Dynaudio or Focal, with vanishingly low distortion, along with the most expensive electronics. But if you ignore the diffraction issue, they'll still be fatiguing.

Here's the quote:

"The paper that Lidia and I just did on the perception of very small time delayed signals, such as would occur for diffraction in a waveguide or off a cabinet edge, is also enlightening for its proof that these effects are strongly dependent on the playback sound pressure level. The ear appears to mask these effects at low levels, while they rapidly become perceptible at higher SPLs. This basically puts much of the previous work on the perception of diffraction and very early reflections into a questionable light since playback level was seldom a controlled or control variable."

http://www.audioxpress.com/magsdirx/voxcoil/addenda/media/mowry1008.pdf

 
#7 ·
Very interesting. I will keep this in mind when i build my home theater speakers. I may even rebuild my tweeter pods in my car.
 
#9 ·
Thanks! I've been doing horns and waveguides for almost two decades now, and the improvement that I've seen from adressing diffraction (almost) has me considering "conventional" components again. The improvement in soundstaging is just ridiculous.

I first noticed it with the Anthony Gallo speakers a few years back at CES. They use ridiculously cheap drivers, and they're basically a glorifed Bose system, yet they sound a million times better than they should. Considering the junky components you'd think they'd sound like crap, but they sounded better than speakers that cost ten times as much.

Really made me think about how they sound so good.

Image
 
#10 ·
I've been thinking about sticking my seas neos (which are small format tweeters) in kind of a softball-sized sphere in the kicks to deal with the diffraction issue. As a bonus it would really blend in with the interior theme. I would just need to vinyl up a foam ball.

mini interior

either that or put the tweets on the A-pillars like I have been considering...
 
#83 ·
I've been thinking about sticking my seas neos (which are small format tweeters) in kind of a softball-sized sphere in the kicks to deal with the diffraction issue. As a bonus it would really blend in with the interior theme. I would just need to vinyl up a foam ball.

mini interior
Black Vinyl with Red stitching... DO IT!
 
#11 · (Edited)
Would you not want the tweeters in a sphere that's somewhat larger than the diameter of the faceplate so you have a smooth curved sides round the outside of the faceplate rather than a sharp 90 degree transition at the edge as you would get with a small hemisphere as you show in the photo?

I think one of the other reasons for using a spherical enclosure is to try to reduce standing waves to a minimum.

there are some good tools for simulating baffle diffraction on the FRD consortium site IIRC.

Another technique that helps reduce baffle diffraction is to surround the drivers with fairly thick felt, I remember reading some studies on it although it didn't always look that neat in the examples.

You can buy fairly large polystyrene eggs and spheres, maybe it would be possible to use one as a template for a fibreglass enclosure and then dissolve out the polystyrene.

<<edit http://www.speakerdesign.net/felt_amelioration/feltssenter.html stuff about using felt>>
 
#17 ·
Would you not want the tweeters in a sphere that's somewhat larger than the diameter of the faceplate so you have a smooth curved sides round the outside of the faceplate rather than a sharp 90 degree transition at the edge as you would get with a small hemisphere as you show in the photo?

I think one of the other reasons for using a spherical enclosure is to try to reduce standing waves to a minimum.

there are some good tools for simulating baffle diffraction on the FRD consortium site IIRC.

Another technique that helps reduce baffle diffraction is to surround the drivers with fairly thick felt, I remember reading some studies on it although it didn't always look that neat in the examples.

You can buy fairly large polystyrene eggs and spheres, maybe it would be possible to use one as a template for a fibreglass enclosure and then dissolve out the polystyrene.

<<edit David Ralph's Speaker Pages - Felt Effects on Baffle Diffraction stuff about using felt>>
The problem with felt is it doesn't operate very low; below 15khz it doesn't make much of a difference.

A roundover operates down to 1/4 wavelength of the roundover's radius.

That's another thing too; it doesn't *have* to be a sphere. Anything that's done to reduce diffraction will yield an audible and measurable improvement.

You could look at this as a series of stages, and each stage is cumulative. IE, do some of them, or all of them for full effect.

  • Use a roundover with a radius that's equivalent to one quarter of the frequency you need to cover. For instance, to go down to 2000hz you'd use a roundover with a radius of 1.6875".
  • Go all the way and use a sphere. A sphere is basically a roundover on all sides.
  • Add a waveguide to the face, and round THAT over.

If you look at the profile of the B&W, you can see the woofer's cone itself forms a waveguide. For a tweeter you'd need to build one. Even *without* a waveguide on the tweeter, the enclosures shape reduces diffraction.

And reducing diffraction makes the speakers disappear.
 
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#12 · (Edited)
I noticed this with a friend's car. He had a bunch of Apple iMac sphere speakers like are shown in the link below, and we were basically trying to use them as car speakers.

Shown here: iMac G4 Sphere Speakers

I was immediately surprised by how good they sounded for the size and how they seemed to be constructed. One might even be able to remove the original tweeter inside and mount something else in it. I would be interested to see an RTA plot on one of those.

edit: better pics here: http://www.cubeowner.com/forums/index.php?showtopic=13516
 
#18 ·
Subwoofers are easy to localize, but the mechanism that we use to localize subs is different than the one we use to localize high frequencies. At low frequencies the pathlength is paramount. That's due to the length of the soundwaves. A 500hz soundwave is twenty seven inches long. Low frequencies are so long, we're unable to detect a difference from one ear to the other. (google interaural time delay.)

The reason that subwoofers are localized is generally due to being too far or too close, and localization via the 2nd and 3rd order harmonic distortion which they produce.
 
#16 · (Edited)
Subscribed.. FINALLY.. The idea of sphericals anywhere near the letters DIY.
I've been contemplating ideas on enclosure construction for a better-than driver in something that hopefully ends up about the same size and shape as an Orb or a Gallo.

I was thinking layered sandwiched MDF maybe, or some craft material, etc.. A couple small bowls molded together with a baffle attached/cut-out, etc. But keeping in mind the important thing of smooth and angled all the way to the edge of the driver.

Hoping for some good DIY's of sphericals with some good wide-banders. Aura's, tangs, etc.
 
#21 ·
Diffraction distortion happens on the outside of baffle/enclosure.
Image


The internal space around the driver (inside of the front baffle, etc) would mainly affect driver ventilation and early reflections but I think the main issue to be resolved here is on the exterior surface.
 
#23 ·
Patrick, great work applying this to car audio! It's a tough thing to accomplish but I think it's a great idea. I've known about this diffraction idea for a while but I've never thought about applying it to car audio. Ironically, this is probably the BEST place it could be applied since objects that cause reflections are much closer to the drivers.

This idea was why I spent so long doing the baffles on my frugal horns. I initially thought I could cut a circle, do a roundover, and I'd be good to go. Then Dave at Planet 10HiFi explained to me the reason it was so important. That's when I went the extra mile (4 hours) of sanding. As a result, these things image like CRAZY! I've been told that the ideal shape is that of a tear drop much like what B&W has on the tower you posted above.

Image

Image
 
#24 ·
This idea was why I spent so long doing the baffles on my frugal horns. I initially thought I could cut a circle, do a roundover, and I'd be good to go. Then Dave at Planet 10HiFi explained to me the reason it was so important. That's when I went the extra mile (4 hours) of sanding. As a result, these things image like CRAZY! I've been told that the ideal shape is that of a tear drop much like what B&W has on the tower you posted above.
Thanks! I basically discovered it the same way. Here are *my* home speakers, note the ridiculous roundover:

Image


(the horn on top is just to demonstrate the scale)

One afternoon I was waiting for some stuff to dry on my current car project, and added a roundover to an old set of HLCDs that I had laying around, and measured them. Took literally fifteen minutes. And then I listened to them, when the measurements were WAY improved, and was startled by the transformation. Never heard them sound so good. It was actually kind of depressing LOL, since the speakers that I applied the roundover on were never competitive with my own DIY designs. But once the roundover was applied, they sounded 10x better.
 
#26 ·
With those ikea bowls, they probably aren't fully round, but flat on bottom so they stay standing. Would that defeat the idea of diffusion?

I'm a bit confused on how to pick the size of the cup/ hemisphere, If I'm using a 3.5 FR88 full range driver, what specs would I use to calculate what i would need?
 
#28 ·
With those ikea bowls, they probably aren't fully round, but flat on bottom so they stay standing. Would that defeat the idea of diffusion?
Your gonna have to make an edge somewhere to mount it. I think the main idea is to focus around the front part of the enclosure. The back part isnt as critical as the front edge.
 
#31 ·
You I was reading about this about a month or so ago. It now makes more sense. Great info and I had already been thinking of ways to make it work in my up coming build, now I have a better understand on how it works and I have a pretty good idea of how I'm going to do it now.
 
#38 ·
Question, would the objects around a typical install location (ie dash, defrost vents, pillar angles, glass, ect.) affect the response the same way as baffle shape would? If so then couldn't the changes that a less then optimum baffle bring actually correct the overall response once the surrounding objects own changes are summed in?

In other words, what if my pillar creates a dip where the stock baffle steps it back up to where it sounds better?
 
#42 ·
Yes, other objects will contribute to the frequency response but there are two important things to keep in consideration:
1. the closer the sharp edge is to the driver, the more it matters, especially at high frequencies.
2. diffraction doesn't just screw up the frequency response, it does it differently for each listening angle. That is why you can't effectively EQ out diffraction effects. If you EQ to make it better in one position, it makes it worse in another.

It is useful to think of this like water flowing in a stream or ripples in a pond. A smooth and large rock will have the water flow smoothly around it. One with lots of edges creates all kinds of turbulence. If you put a bunch of jagged rocks in the stream you may be able to create an area where the flow is smooth, but you can't do that without making it rough in other places.
 
#39 ·
Also I think B&W used Tapering Tubes more for the internal benefits then the added bonus of it having the same external shape.

"The sound of silence. Not all sound generated by speaker drive units is good sound. The kind that emerges from the back of a working driver, into a conventional box cabinet, can bounce around and make a mess of the good sound coming out of the front. B&W’s trailblazing Nautilus™ speaker found a way around boxes. Tapering tubes filled with absorbent wadding soaked up the wayward sound energy and reduced resonances to an insignificant minimum.

Nautilus™ Tapering Tubes are fitted to nearly all B&W speakers, even when they’re not visible to the eye. Sound is channeled through a hollow pole magnet, away from the diaphragm, and disappears into the tail.. So all the sound you hear is good sound."

"When a driver is loaded by a tube of a similar diameter to the diaphragm, sound propagates down the tube as a series of simple plane waves. When the sound reaches the other end of the tube it is reflected back up the tube towards the driver. If it reaches the driver it causes delayed radiation that time-smears the original signal, blurring the clarity of the sound. If, however, you fill the tube with absorbent material and the tube is long enough, you can dissipate the energy before it reaches the end of the tube. The sound from the driver then remains clean and true to the input signal. Tapering the tube enables you to make it shorter for the same level of absorption. It acts like a horn in reverse - reducing the sound level instead of increasing it.

The limit of this type of loading is reached when the wavelength gets small enough to be comparable with the diameter of the tube. Above a certain frequency, the sound ceases to propagate as a simple plane wave and a series of cross-mode resonances are set up which can re-radiate through the driver diaphragm. To maintain the effectiveness of tube loading, you must restrict the bandwidth of each driver. This is one reason why the Nautilus loudspeaker is divided into a 4-way system. A more complex type of loading is required to cover a wider bandwidth and the sphere/tube enclosure was developed for the Nautilus™ 800 Series."
 
#41 ·
I've been thinking a lot about how I want to shape the a-pillars for my closed back Dyn mids and F#1 tweets (they don't need enclosures). I think this thread has given me food for thought. I am thinking that since the tweets are crossed over from 4K up that they would be fine in a sphere a little over 1" and the mids would work in 4" sphere. somebody let me know if I am missing something. Is diffraction going to be as big an issue with the mids?
 
#43 ·
Diffraction is an issue at all frequencies, but it's particularly audible above 1khz. But listen for yourself; go to the craft store, buy some spheres for a buck, and cram a sock in there to seal it off. I think you'll be surprised how quickly the speakers "disappear."

If you want to go the extra mile, buy some rope caulk from Home Depot. It's $5. It's used to seal plumbing, it's in the insulation section. That will seal the sphere completely, and it's removable.

So you can experiment with the spheres, and see if they make an audible improvement.

I think you'll be amazed...
 
#44 ·
I remember reading about how Biggs from Jbl used a diffraction ring on the mids in his kicks. is it just the exterior surrounding the driver that is most important with a particular angle on the edge or the actual sphere shape behind the speaker, or both?
 
#45 ·
is it just the exterior surrounding the driver that is most important with a particular angle on the edge or the actual sphere shape behind the speaker, or both?
I'm guessing here but I think it has more to do with the area closest to the edge of the speaker. Reason being is that those edges will get more sound bouncing off them than the rear of the enclosures.
 
#48 ·
Good read here folks!

I have wanted to do some type of spherical or egg shaped enclosure for awhile now. My thinking is....a spherical enclosure along with a mat might be about the best way to get a small mid on the dash.....as it has to be one of the toughest places in the car to put drivers....reflections everywhere. However, when you have a car with a wide and long dash it is worth a try to put speakers there as there are benefits to dash mounting as well.

I was looking at some of the Gallo and Orb sets to play with. However, I ended up picking up a set of the KEF KHT1000 "egg" speakers in the Picofote1 IPOD dock. Mated with a sub, these are some nice sounding small speakers. The enclosure and the Uni-Q driver do a nice job in these....even more so in the larger models.

They may look cute - but they sound anything but. Experiencing a movie or favorite album through KHT1005.2 is like seeing an old friend after a makeover: fresh, involving, more intense.The centre and satellite speakers all have KEF’s stunning 75mm (3in.) Uni-Q driver - the smallest ever made, its 15mm (0.6in.) metal dome tweeter and double neodymium magnets deliver the extended bandwidth you normally get with much larger units.

The curvy die-cast aluminium enclosures eliminate the diffractions and internal resonances that so often blur the output of conventional small speakers, so the sound is cleaner and crisper.
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I hope to do some type of sphere or egg shape enclosure for some 3" fullranges to be mounted on the dash...will see...
 
#547 ·
Good read here folks!

I have wanted to do some type of spherical or egg shaped enclosure for awhile now. My thinking is....a spherical enclosure along with a mat might be about the best way to get a small mid on the dash.....as it has to be one of the toughest places in the car to put drivers....reflections everywhere. However, when you have a car with a wide and long dash it is worth a try to put speakers there as there are benefits to dash mounting as well.

I was looking at some of the Gallo and Orb sets to play with. However, I ended up picking up a set of the KEF KHT1000 "egg" speakers in the Picofote1 IPOD dock. Mated with a sub, these are some nice sounding small speakers. The enclosure and the Uni-Q driver do a nice job in these....even more so in the larger models.

I hope to do some type of sphere or egg shape enclosure for some 3" fullranges to be mounted on the dash...will see...

I wrote the following post about the Kef, over on diyaudio. Thought I'd repost it here, since it definitely fits the bill of a 'low diffrraction wide bandwidth' loudspeaker.

And, yes, I did buy ten of em :)

I know there's a lot of people reading this thread that don't have the time to build a synergy horn, or the space that's required to get it to play down to 300hz. Here's an alternative you might consider:

Image

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This is the satellite from a KEF KHT1005. Here's a list of reasons you might consider it in lieu of a Synergy horn:

1) after screwing around with Unity horns for over half a decade, I'm convinced that their fundamental goodness has little to do with their efficiency, and a LOT to do with the near-perfect transition from midrange to tweeter. Basically a Unity/Synergy horn fools our brain into thinking it's one wide-bandwidth driver.
2) Once you hear this done properly, conventional two-ways will never sound the same; I can't listen to a conventional two-way without *immediately* knowing that something is wrong. Literally the second I walk into a room.
3) The KEF UNI-Q isn't perfect, but it has a LOT of the Unity goodness. I think this is due to (you guessed it) the near-perfect transition from midrange to tweeter. This is especially notable on vocals and percussion.

Image

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4) One way to determine if a loudspeaker has some of that Unity goodness is to look at the polar response. The graph above shows the polar response of two loudspeakers. The first is a conventional two-way (Focal Chorus 807V) and the second is a Kef UNI-Q (Q900).

Note how the transition from the Kef's midrange to tweeter is nearly perfect? To the microphone, it seems to be one driver. There's a little 'wiggle' at 5khz where the midrange hands off to the tweeter. But that 'wiggle' isn't as severe as the noticeable widening of directivity that occurs with the Focal two-way. (Note how the Focal directivity gets much broader at 4khz? That's because the tweeter is much smaller than the woofer, and so has wider directivity. In the Kef, the midrange cone acts like a waveguide for the tweeter, so the directivity change is verrrrrrrry slooooooow.


I couldn't find any directivity plots of the KHT1005 online, but I'm willing to bet it's comparable to the Q900 that was measured by Stereophile.

Image

Here's a frequency response measurement of the KHT1005 satellite, from Home Theater magazine.


Last but not least, I think that one of the most important reasons that Synergy horns don't sound like regular speakers is that they're symmetrical. For instance, a regular two-way speaker is generally optimized for good horizontal directivity (if it's optimized at all.) It's hard to get good vertical directivity from a two-way, due to the fact that the pathlength differences vary greatly depending on angle. The Synergy horns don't have this problem; if you flip an SH-50 on it's side, the directivity is the same.

The same is true with a Kef UNI-Q, at least down to 300hz or so. (Synergy SH-50 is admittedly symmetrical down to a much lower frequency.)


If all of this has piqued your curiosity, the final icing on the cake is that the UNI-Q can be had for about $50 per satellite for the next 24 hours. Newegg has them on their Black Friday flyer. Even if you DIY, it's pretty easy to spend $50 per channel on a Synergy horn *crossover*, and that doesn't even factor in the cost of drivers, labor, failed experiments, and anti-depressants when your midranges don't work ;)

I'm putting my money where my mouth is, and buying ten :)

Newegg.com - KEF KHT1005.2SE Home Audio Speaker System w/ White Satellite Speakers

promo code is EMCJJHF34 and it's good for 24 hours.

Happy Thanksgiving!"
 
#51 ·
Patrick,

A couple of quick points to make sure that I am on the same page:

Better yet, put that spehere about six inches away from the dash and the windshield, and you'll eliminate or reduce ALL reflections down to 470hz.
The case stated above eliminates or reduces diffraction down to 470 hz as the boundary surfaces are not now acoustically part of the baffle. However, reflections still occur (particularly in the omnidirectional radiation frequencies). I would agree that they are of secondary importance to diffraction but they do occur and color the sound (hence the reason room treatments are still important).

Lowest frequency = 13500 / 2 / pi / radius of roundover

For instance, you're standing at Home Depot, and trying to figure out if 2" PVC pipe will do the job. First, you have to take that pipe and quarter it, so that it give you a roundover with a radius of 1". A PVC pipe with a diameter of 2", chopped into four pieces, will give you a roundover with a radius of 1" and a perimeter of 1.57. ( pi / radius / 2)
Plugging those numbers into our equation, we find that a 2" PVC pipe, quartered, will reduce diffraction down to 1368hz. (13500 inches per second / 2 / 3.14159 / 1.57"
In this section you use "radius" and "perimeter" interchangably. I believe that perimeter is correct but I haven't been able to find an authoratative source to confirm.

As many have noted this is great stuff. I just wanted to make sure that I and others are reading the post are on the same page.
 
#53 ·
Patrick,

A couple of quick points to make sure that I am on the same page:



The case stated above eliminates or reduces diffraction down to 470 hz as the boundary surfaces are not now acoustically part of the baffle. However, reflections still occur (particularly in the omnidirectional radiation frequencies). I would agree that they are of secondary importance to diffraction but they do occur and color the sound (hence the reason room treatments are still important).



In this section you use "radius" and "perimeter" interchangably. I believe that perimeter is correct but I haven't been able to find an authoratative source to confirm.

As many have noted this is great stuff. I just wanted to make sure that I and others are reading the post are on the same page.
Yes, you are correct. I was going to edit my original post last night, then realized I'm supposed to be on vacation :D

(Typing this from an airport in Nevada...)

Also, I had the same issue, trying to find a formula for this online was a p.i.t.a. The forumula that I posted is based on a single wavelength.

My measurements seem to demonstrate that a roundover is partially effective at even a fractional wavelength. In other words, in the real world, a roundover has an effectiveness that appears to increase with frequency. Yet it's even effective at frequencies where the formula indicates it shouldn't be, albeit not as effective as it is at higher frequencies.

Then again, we need it the most at high frequencies, as we're naturally more sensitive to polar response above 1khz then below.