Constant Directivity without Horns or Waveguides

I recently made a fascinating accidental discovery: By choosing unconventional crossover frequencies and baffle dimensions, you can achieve Constant Directivity using regular woofers and dome tweeters - without waveguides, horns or dipoles.

I call it Zero Horn Constant Directivity (ZHCD). Jack Regula (@nc535) ran some simulations and helped me refine the model. First I’ll show off the end result, then I’ll explain the theory and model and give you a design guide.

Example #1: 5” 2 way with dome tweeter. Focal 5N402DB 5" woofer + Audax TW025 25mm dome. Front baffle is 8" wide.

focal5N402DB+D25+11liters.jpeg


First I tried a 1700Hz crossover which got me this frequency response at 0, 15, 30, 45, 60, 75, 90 degrees off axis:

focal5N402DB+D25-1700Hz_0-15-30-45-60-75-90.png


Notice some "bunching" around 2500Hz. This bunching comes entirely from the tweeter, as the crossover is 1700. At 45 degrees off axis, the output is +4dB higher than on axis.

BUT guess what… if I set the crossover at 4900Hz I get this:

5N402DB+D25_0-15-30-45-60-75-90-.083oct.png


Theory (what I thought I knew about tweeter dispersion = omnidirectional below 8KHz) said this was impossible. Simple theory would predict the woofer is directional above 1K while the tweeter is not directional until 8-10K, but as you can see here, the tweeter’s directional characteristic matches the woofer’s almost exactly.

I might have just thought it was odd and shrugged my shoulders and went on, but I noticed the same thing with another smaller system – A JVC HSA1014 + Dayton ND16FA. Cabinet is 4.5” wide and 6.5” tall.

JVC-HSA1014+ND16FA+1liter.jpeg


I attempted a 3KHz crossover and got similar results as my first try above (bunching around 4KHz), but then a 7KHz crossover got me this result:

HSA1014+ND16-0-15-30-45-60-75-90.png


This is an almost perfect Constant Directivity result above 1KHz with considerable directionality from a tiny 16mm tweeter… which I would expect to have NO directionality!

This was a big surprise. So I got curious. This reminded me of a phenomenon that I observed with the Live Edge Dipoles, which have a wide 19” live edge wood baffle and an 8” coaxial mid-tweeter. The Live Edge Dipoles have a minor but noticeable “bulge” at +/-45 degrees off axis, at about 1KHz:

live-edge-dipoles-baffle-bulge.png


On the low end, Dipoles start rolling off at λ = 2X cabinet width, and normal box speakers similarly have a -6dB Baffle Step at λ = 2X cabinet width. Lots of speaker designers know about these.

Now there are a couple of other baffle diffraction modes I don’t recall getting much attention. If you know what you’re doing, you can really take advantage of this and achieve Constant Directivity without horns or waveguides.

It turns out this +/-45 degree bulge at 1KHz in a 19” wide baffle corresponds to the 2.5KHz bulge from the 8” wide baffle, which also corresponds to the 4KHz bulge from the 4.5” wide baffle. I call this Baffle Bulge. It occurs across a range of λ = 0.5X to 1X cabinet width. At this frequency the on-axis response is the “normal” and “expected” but the off-axis response has a peak caused by diffraction from the edges of the baffle.

Then at even higher frequencies, where λ = 0.25X to 0.5X cabinet width, you get a Baffle Beam and over this octave, on-axis radiation is strong; but off axis drops off much like if you had a waveguide.

So if I draw these points on the polar plot above, it looks like this:

baffle-step-bulge-beam.jpg


So this means in the 5” 2-way with 8” wide cabinet, most people would cross the tweeter at 2-3KHz fearing that the woofer will get too beamy, and has too much cone breakup. That’s what I did in my first attempt. But what I found was: Even the small 5” woofer begins to beam above 1K and then the tweeter has +4dB lobes at +/-45 degrees at 2.5KHz and the off axis response is very uneven.

Yet if I cross it over at 5K, the woofer’s natural beaming has already kicked in and is not as bad as I feared; also, the woofer’s beaming is not affected much by the front baffle dimensions; and then the tweeter is benefiting from Baffle Beam from 4KHz to 8KHz; and above 8KHz the 1” dome begins beaming naturally because of its dimensions; and viola, I get Constant Directivity behavior from 500Hz up. The combination of woofer beaming and calculated cabinet diffraction gets us almost identical polar patterns and CD behavior from woofer and tweeter alike.

Again:

5N402DB+D25_0-15-30-45-60-75-90-.083oct.png


In the 4.5” 2-way with 4.5” wide cabinet, the woofer’s beaming from 3-7K is matched by the tweeter’s Baffle Beam above 7K and even this tiny system gives me Constant Directivity behavior from 1000Hz up:

HSA1014+ND16-0-15-30-45-60-75-90.png

I’ve been designing speakers for a long time, so I’m conditioned to avoid woofer beaming and breakup and handoff to the tweeter at a low frequency. But it turns out if your woofer has well behaved gradual breakup modes at the upper end of its range, and you design the crossover appropriately (the woofer crossover on both of these was more complex than usual), you get the fantastic imaging that CD designs are known for and both of these systems are well behaved at the upper end of the woofer’s range.

So I asked Jack Regula ( @nc535 ) to run some simulations to shed light on this. All of these simulations are of a 1” tweeter only (there is no woofer in the simulation) mounted on a box of dimensions 8” wide x 13” tall.

So let me step you through the research Jack and I did. First, Jack experimented with rounded vs sharp edges, and found that an 18mm bevel on each edge greatly reduced high frequency ripple. So all the simulations that follow assume that 18mm bevel.

Secondly, asymmetry in diffraction is also desirable. The worst situation is a tweeter in the middle of a round baffle. Second worst is putting it in the middle of a square baffle. Third worst is putting it in the middle (as I admittedly did in my small 2-ways).

When Jack put the tweeter in the center of the baffle, this is the polar heat map he got:

PolarPlot8inchWide_Bevel_SYMMETRICAL.jpg


Notice significant “Baffle Bulge” at 2.5-4KHz which we are avoiding by choosing a 5KHz crossover.

Jack moved it off center and found a really nice spot here. The red dot is the tweeter location:

front baffle tweeter off center.png


At this off-center position, Baffle Beam is working for us from 4-8K and has no nasty peaks or surprises:

PolarPlot8inchWide_Bevel_offcenter.jpg


[Please note that Jack’s models have irregularities above 8K that are a result of not using tons of points in the calculations, to save computer time – artifacts of the simulation.]
 
Jack ran simulations on infinitely wide baffles, where he confirmed at all frequencies below 8K the tweeter has no directivity and the polar plots look like this:

Infinite-Baffle-PolarPlotOmni.jpg



Here are the Horizontal tweeter polar plots at various frequencies, for a tweeter mounted in an 8" wide baffle:

HpolarChart_2060Hz.jpg

Notice Baffle Bulge at 2.06KHz, +/-45 degrees off axis, especially on the upper right (+30 degrees)


HpolarChart_2990Hz.jpg

Notice Baffle Bulge at 2.99KHz

PolarPlot3500Hz.jpg

Notice Baffle Bulge at 3.5KHz


HpolarChart_4005Hz.jpg

Notice baffle Bulge at 4.005KHz

PolarPlot_4350Hz.jpg

Above at 4350Hz, the Baffle Bulge has almost gone away.



PolarPlot_4990Hz.jpg

Above at 4.99K the Baffle Bulge has gone away and is starting to turn into an on-axis Baffle Beam


HpolarChart_5010Hz.jpg

At 5K there's a trace of Baffle Bulge.


HpolarChart_6000Hz.jpg

Slight beaming at 6K

PolarPlot_7060Hz.jpg

Baffle Beam is tilted about 15 degrees to one side, above

PolarPlot_7840Hz.jpg

Above 8KHz the dome's natural dispersion limitations start to kick in (which are NOT modeled in these graphs.) Meanwhile, as you can see, the front baffle drops the +/-90 degree SPL by 5-8dB and we get beaming from the tweeter in the 4-8K range that is similar to what we get from a woofer or waveguide.

So here’s the playbook you can use to create a Zero Horn Constant Directivity (ZHCD) design:

-You set the crossover frequency at Cabinet Width = 2.5X to 3X λ.

Handy-dandy formula in inches: Crossover Frequency = 40,000/cabinet width in inches

Handy-dandy formula in centimeters: Crossover Frequency = 100,000/cabinet width in CM

-Tweeter should be off center. A very good spot according to Jack’s simulations is:

Horizontal Position = .225W

In other words left tweeter is 22.5% of baffle width from left edge; right tweeter is 22.5% of baffle width from right edge.

Vertical distance from top = 0.3W

Tweeter is 30% of Baffle Width from top edge.

front baffle tweeter off center.png


-Left and right speakers should be symmetrical.

Other Design Considerations:

Left speaker should have tweeter left of center; right speaker should have tweeter right of center. This push the lobes towards the inside rather than the outside, which means if the speakers are towed in, when you stand next to the left speaker, you’ll hear less SPL from the left speaker and more from the right speaker. Imaging will then stable all around the room, which is what CD designs are famous for.

This won’t work with just any woofer. You need to choose a woofer that can be EQ’d flat up to this higher crossover frequency, and its off-axis behavior needs to be well-behaved. Some woofers show a nice gradual dropoff as you move off-axis; others get wonky with all sorts of peaks and dips. You can tell from the 30 and 45 degree plots how clean the woofer breakup is.

The woofer crossover will likely be more complicated than you’re used to because the crossover has to deal with Baffle Step, handle top-end irregularities, and match the woofer to the tweeter.

If you account for that you’ll be pleasantly surprised at how well-integrated it sounds. I did not expect such a high crossover frequency to work but both of these designs really sound marvelous with great imaging. As the measurements show, there are no discontinuities and the crossover points are almost seamless across a wide range of angles.

Special thanks to Jack Regula @nc535 for the colorful simulation plots.
 
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You've probably seen these threads, but I think it's on-topic here:

First off, I bought one of Earl Geddes' last set of the original Summas, largely based on curiosity about the waveguide and it's performance, as well as Earl generously listening to all of my noob questions for hours when he premiered them at RMAF in 2005. One of the surprising things about the Summas, was that the cabinet was arguably more innovative than even the waveguide. It's really something else. Turn the lights off, and these speakers that are the size of small refrigerator, they sound like a speaker that's 10% as large.

Second -

I think you can achieve about an octave of "directivity control" by juggling the width and the depth of the loudspeaker enclosure. I believe this happens because the wavefront that is radiated is transitioning from half space to full space but it is neither half space or full space.

It is in that 'transition zone' where the polar response looks similar to what you can accomplish with a waveguide. Tons of speakers leverage this; Revel and Kef in particular.

I believe the reason that we see this "waveguide like" effect is because the wavelengths in that 'transition zone' are larger than the width of the baffle but not so wide that they dwarf the size of the enclosure.

Here's an example:

The Revel M126BE is 20cm wide. The waveguide is approximately 15cm wide. We would expect the waveguide to lose beamwidth control at 2300Hz on an infinite baffle. But the loudspeaker maintains a beamwidth of approximately 100 degrees down to 700Hz, far lower than one would expect. I believe this is due to the phenomenon described above.

Kudos to Erin for the measurements, and to Mabat for writing software that makes it possible for people who can't figure out ABEC (like me) to simulate the enclosure easily.

Something I've wondered, is if it would be possible to push that frequency even lower by repeating the process that works with the M126BE, but adding a woofer below the midbass, similar to the enclosure of the Kef 107.

If any of this sounds interesting, check out this thread: https://www.diyaudio.com/community/threads/loudspeaker-enclosures-are-waveguides.363630/
 

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You've probably seen these threads, but I think it's on-topic here:

First off, I bought one of Earl Geddes' last set of the original Summas, largely based on curiosity about the waveguide and it's performance, as well as Earl generously listening to all of my noob questions for hours when he premiered them at RMAF in 2005. One of the surprising things about the Summas, was that the cabinet was arguably more innovative than even the waveguide. It's really something else. Turn the lights off, and these speakers that are the size of small refrigerator, they sound like a speaker that's 10% as large.

Second -

I think you can achieve about an octave of "directivity control" by juggling the width and the depth of the loudspeaker enclosure. I believe this happens because the wavefront that is radiated is transitioning from half space to full space but it is neither half space or full space.

It is in that 'transition zone' where the polar response looks similar to what you can accomplish with a waveguide. Tons of speakers leverage this; Revel and Kef in particular.

I believe the reason that we see this "waveguide like" effect is because the wavelengths in that 'transition zone' are larger than the width of the baffle but not so wide that they dwarf the size of the enclosure.

Here's an example:

The Revel M126BE is 20cm wide. The waveguide is approximately 15cm wide. We would expect the waveguide to lose beamwidth control at 2300Hz on an infinite baffle. But the loudspeaker maintains a beamwidth of approximately 100 degrees down to 700Hz, far lower than one would expect. I believe this is due to the phenomenon described above.

Kudos to Erin for the measurements, and to Mabat for writing software that makes it possible for people who can't figure out ABEC (like me) to simulate the enclosure easily.

Something I've wondered, is if it would be possible to push that frequency even lower by repeating the process that works with the M126BE, but adding a woofer below the midbass, similar to the enclosure of the Kef 107.

If any of this sounds interesting, check out this thread: https://www.diyaudio.com/community/threads/loudspeaker-enclosures-are-waveguides.363630/
Hi Patrick, YES you're right, and in fact a Summa- or KEF 107-like speaker was a big clue to unraveling this, which I hadn't mentioned. I had a discussion with @PKAudio about some Wavecor Tetoron 30mm domes he was selling right here on the forum. He posted these curves:

TW030WA22 H wavecor pkaudio.png

...in which you see gorgeous CD behavior all the way below 1KHz. But I looked up the Wavecor spec sheet and found this:

wavecor_30mm_spec_sheet.jpeg

Huge difference! In Infinite Baffle the CD behavior doesn't kick in until 2KHz and above, so I asked him about it. I asked him if it was even the same tweeter. He said:

"My measurements are from the real baffle which helps to improve the directivity and minimize diffractions a lot. Additional mild offset of the tweeter to one side would make the responses look ideal without any signs of the diffractions." He posted this photo:

wavecore30-in-rounded-enclosure.jpg


His enclosure looks to be 8-9" wide, similar to mine with the Focal 5" woofers, and with generous round-overs he's basically extended the curvature of the waveguide backwards. @PKAudio made a tweeter with a 4" factory waveguide act like an 8" waveguide, using wide roundovers. Very well behaved, too.

So here's the new way I've come to think of it -

This is a waveguide:
2differentwaveguides1.jpeg


...and this is a waveguide too:
2differentwaveguides2.jpeg

In fact #2 is arguably a better waveguide than #1.

That means we shouldn't just think of a horn / waveguide as a "90 degree" or "120 degree" or even "180 degree" device. We should think of a horn as a 270 degree or even 360 degree device!

To your point about the Revels, it has a more graceful transition from half-space to full-space than waveguide #1. And it's certainly more useful than the small waveguide.

A lot of this has to do with how much work you're willing to put into the roundovers.
 
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Very interesting Perry. One of the most surprising things is how high you have the woofers playing without too many issues.
This comes down to defining "issues" properly. What is the definition of a good hand-off from woofer to tweeter?

I think a HUGE underrated part of designing a crossover is how the hand-off appears at many different angles off-axis, both horizontally and vertically.

There's an old thread "Which shape of cabinet gives a better image?" by @presscot and I say that's not even the right question to start with. A better question is, "Which radiation pattern gives a better image?" which leads to "which cabinet shape gives you that radiation pattern?"

My speaker design results took a quantum leap when I started incorporating radiation pattern into my thinking from beginning to end. Now whenever possible I go for >500Hz Constant Directivity; or even full-band 20-20K Constant Directivity which my own dipole designs aim for. The Flanagangsters, for example, achieve Constant Directivity above 100Hz with a hybrid Open Baffle + Reflex.

Both of these woofers, the 5" Focal and the 4" JVC, are not perfect by any means, but with sufficient attention to the crossover, behave nicely both on- and off-axis to 5K and 7K respectively. By "behave nicely" I mean there are no big surprises off-axis that you wouldn't expect from the on-axis curves. The crossover handles the rest.

Thus, counter to intuition, they sound great. I say they sound great because they measure great both on and off axis.
 
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This is the crossover I designed for the 5" 2 way with the Focal 5N402-DB. The 5N402 has a dip at 2500Hz which in my original attempt seemed to help with the 1700Hz crossover. But after I abandoned the 1700Hz crossover and went for 5KHz instead, I needed the crossover to increase the gain at 2.5K which I achieved with a high-Q rolloff which peaks at 2.5K. The 1MH+6.8uF handle that part.

Here is the VituixCad screen:

BBC#2crossover_Focal5N402DB+D25.png

The 33uF / 0.37MH / 6.2ohm notch filter on the woofer deals with the Baffle Step and corrects a peak that shows up @ 1.2KHz. This is a dual voice coil woofer and one coil has a 5MH inductor in series. This was Focal's recommended approach to Baffle Step with this driver and was the reason for the DVC design.

The driver as modeled here in VituixCad as a composite of Coil 1 and Coil 2 + 5MH inductor, with combined impedance and SPL all rolled together as one driver for this simulation.
 
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This project was an eye opener for me and I'd like to thank Perry for allowing me to participate. I'd seen baffle bulge before but hadn't realized there was anything I could do about it except arrange crossover frequencies to avoid it or compensate the power response in the region of it. Perry is ingenious in using it to improve the overall response.

It turns out that one has a fair amount of control over baffle bulge. Cabinet height, width and even depth affect it as does baffle edge treatment. ABEC/AKABAK will reveal the effect of these on directivity and allow you to optimize them before committing to a prototype. It does extend the design cycle but not excessively, unless you are like me and don't know when to stop simulating and start making sawdust 🙂

The simulations I ran for Perry took a few hours to create the model and a few minutes for each run on a 16-core desktop PC.
 
Here’s another speaker that has arguably fantastic DI and a similar concept in baffle function

IMG_0404.webp

It was discussed many years back here for its performance value


While never investigated as far as I know, I always believed those slot ports on either side of the tweeter had something to do with it…….likely there’s no simulation available though.

The nearly 9” woofer I suppose drags the directivity point needed for the tweeter lower and the steep filter allows for the lower XO point and directivity match?
 
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In fact, it's nothing new. With the selection of the right drivers, the width of the enclosure, crossover point and slope of the filters, you can achieve a pretty good CD.

Simple enclosure, SB23Nbac, SB15Nbac and HDS tweeter.
The phenomenon itself is as old as sound. 13 billion years I suppose. What's new is: I don't recall anywhere in my travels having been exposed to what I call "Baffle Bulge" or "Baffle Beam" with an explanation of where those occur relative to cabinet width, along with radiation patterns and precise design guidelines.

A lot of people have designed great speakers by accident, so I'm interested in doing the right things on purpose.
 
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The baffle is the problem. No baffle = no off-axis bulge! How do you do THAT? When you build an OB and keep the baffle dimensions small WRT the driver diameter you also get constant directivity over a wide frequency band, pretty much all of it. This is why I avoid using baffle when ever possible in my projects. Alas, OB and dipole speakers are not for most people...

But back to the speaker in your OP: this might be why some small mini-monitor speakers sound amazing.
 
Yes Charlie and your work was lingering in the back of my mind while I was working on this. This hugely intersects with your minimal baffle dipole approach.

A lot of people say mini monitors have great imaging because the narrow baffle reduces diffraction but it’s more because the diffraction coincides with the crossover frequency and makes a good radiation pattern.
 
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Horizontal directivity alone can be optimized by expence of vertical. Also slope order plays a role, overlapping range is interesting! Surpisingly small changes will make difference. Just comparing single driver's off-axis responses is not enough to determine ideal xo!

Baffle dimensions and driver location are also important for the tweeter (above 2kHz). Offset will improve axial response at the expense of off-axis and symmetry, but this can be beneficial for listening window average. Some designers prefer 10-15deg off-axis response as priority (before VCAD)

Some quick sims below. VCAD sims with real measurements and different xo types would be fantastic...

tweeter midline  offax edge.jpgtweeter offax edge.jpg
 
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The phenomenon itself is as old as sound. 13 billion years I suppose. What's new is: I don't recall anywhere in my travels having been exposed to what I call "Baffle Bulge" or "Baffle Beam" with an explanation of where those occur relative to cabinet width, along with radiation patterns and precise design guidelines.

A lot of people have designed great speakers by accident, so I'm interested in doing the right things on purpose.
I think that it is thanks to new software and modern computers that today we have more opportunities to analyze old ideas. Especially since these possibilities are available to any amateur. Like with series crossovers, which have been known for a long time, but only with CAD software can they be designed in a sensible way that works.
 
This comes down to defining "issues" properly. What is the definition of a good hand-off from woofer to tweeter?

I think a HUGE underrated part of designing a crossover is how the hand-off appears at many different angles off-axis, both horizontally and vertically.

There's an old thread "Which shape of cabinet gives a better image?" by @presscot and I say that's not even the right question to start with. A better question is, "Which radiation pattern gives a better image?" which leads to "which cabinet shape gives you that radiation pattern?"

YES!

This is exactly what I was getting at.

For instance, after I sold the Summas because they didn’t pass the WAF test, I began to miss their effortless dynamics.

I bought a set of Yamaha DXR-12s.

The Yamahas have a nice waveguide, and they’re a two way like the Summa is.

But they don’t “disappear” like the Summas do.

Around the same time, I met the designer from Revel, Kevin Voecks. He was at CES and he did a demo of the Revel Salon II, then did a demo of Revel’s cheaper stuff. The “cheap” stuff cost about 40% as much.

The DYNAMICS of the cheap stuff was comparable to the Salon II, but the latter “disappeared” better.

And what do the Salon II and the Summa have in common?

A VERY inert cabinet, with a heckuva round over.

I have a hunch that a casual buyer might attribute the superior imaging of the Salon II to electronics or expensive crossover components.

My hunch is that Revel puts the sharp edges on the cheap stuff, because it creates a distinction between the top of the line Salon II, versus everything else in their product line.

Similar to how a Lexus ES and a Toyota Avalon have much in common, but you have to buy the Lexus to get all the bells and whistles. Toyota doesn’t sell a Avalon with all the goodies because that would make the Lexus ES pointless. (Toyota ended up killing the Avalon, but the ES is sold to this day.)