Visualizing How Different Loudspeaker LF Directivity Patterns Couple to Room Modes

[NTK]

The Python version of the 2D finite difference (FD) code seem to be working! But I haven't thoroughly tested it yet.

Below show the duplicates of what were in post #8 which was simulated using finite element with Mathematica. The 2D room is 8 m X 4 m. I calculated the room modes using both the analytical solution (reference) and by solving for the eigenvalues and eigenvectors of the FD matrix (the discrete Laplace operator matrix).

Room modes from the analytical room mode solution of rectangular rooms.

Solution from solving for the eigenvalues and eigenvectors of the FD matrix — didn't match perfectly, but this method works for non-rectangular geometries too, if you manage to setup the FD matrix.

Below are simulations of the sound wave propagations with monopole and dipole sources to visualize how different the source types and orientations affect how the sound sources couple to the room modes. The sound sources are centered at the pressure null of the 60.6 Hz (2, 1) mode. The excitation is a 6.5 cycle of 60.6 Hz tone burst.

Monopole: Since it is located in the pressure null, the sound radiated dies off quickly.

Dipole, oriented in the direction along the length of the room: Excited the axial (3, 0) room mode at 64.3 Hz but not the (2, 1) mode.

Dipole, oriented perpendicular to the length of the room: Excited the tangential (2, 1) room mode.

The Python Jupyter notebooks used to generate the plots are in the attached ZIP. I haven't started on the explanations of the theories behind the simulation. Will update when I can get it done. Hopefully by next weekend.

[Edit] Finished the Jupyter notebook explaining the theory/method behind the simulation code. Modified the notebooks to also show a plot of the pressure vs time response at a "listener location". Added 3 more demo cases -- double bass array and comparing it against single bass array with full back wall absorption, plus cardioid. The new and updated notebooks are in the attached v20250208 zip archive.

 

[NTK]

There's no proof of that in real life. When all are optimized for their best position, it's very much the same. To my knowledge, there's only been one study that indicated cardiod was a bit less placement sensitive, but it was very minor.

I started this thread primarily as a mathematical exercise to show how different types of low frequency sound sources (monopole, dipole, cardioid) can interact differently with the room. And I think it would be interesting to do something to show it.

Cardioids have the characteristic that they excite both pressure modes and velocity modes. However, the actual benefits to the sound in the room, I agree, are not so clear. Cardioids are also directional, so not only placement is important, but also where they point at. They seem to be in fashion these days, and seem to be gaining followers outside the geographical vicinity where you are located.

I personally have no dog in this race. But I am interest to see how far this concept of constant directivity cardioid speakers will go.

 

[Bjorn]

I started this thread primarily as a mathematical exercise to show how different types of low frequency sound sources (monopole, dipole, cardioid) can interact differently with the room. And I think it would be interesting to do something to show it.

Cardioids have the characteristic that they excite both pressure modes and velocity modes. However, the actual benefits to the sound in the room, I agree, are not so clear. Cardioids are also directional, so not only placement is important, but also where they point at. They seem to be in fashion these days, and seem to be gaining followers outside the geographical vicinity where you are located.

I personally have no dog in this race. But I am interest to see how far this concept of constant directivity cardioid speakers will go.

Directivity is swamped by room modes.
So in other words, below the Schroeder frequency there isn't any control directivity in function any more. Thus they are not "directional" in small rooms, which is the type of rooms all living rooms are.

Cardiod bass therefore only makes sense in large concert venues.

 

[thewas]

Directivity is swamped by room modes.
So in other words, below the Schroeder frequency there isn't any control directivity in function any more. Thus they are not "directional" in small rooms, which is the type of rooms all living rooms are.

Cardiod bass therefore only makes sense in large concert venues.

Is it really like that? I had measured the LP responses of a pair of Neumann KH310 and D&D 8c without EQ and placed as the same positions and the 8c didn't show the typical upper bass SBIR like the KH310.

 

[Bjorn]

Speaker boundary interference is something else and often either above the Schroeder or in the transition zone.

To make it more complex, monopoles may work better in a different place than a cardiod. Thus the best comparison is optimizing placement for both. If one is stuck with one placement, you might find one working better than the other.

 

[neRok]

The Python version of the 2D finite difference (FD) code seem to be working! But I haven't thoroughly tested it yet.

Below show the duplicates of what were in post #8 which was simulated using finite element with Mathematica. The 2D room is 8 m X 4 m. I calculated the room modes using both the analytical solution (reference) and by solving for the eigenvalues and eigenvectors of the FD matrix (the discrete Laplace operator matrix).

Room modes from the analytical room mode solution of rectangular rooms.
View attachment 424942
Solution from solving for the eigenvalues and eigenvectors of the FD matrix — didn't match perfectly, but this method works for non-rectangular geometries too, if you manage to setup the FD matrix.
View attachment 424943

Below are simulations of the sound wave propagations with monopole and dipole sources to visualize how different the source types and orientations affect how the sound sources couple to the room modes. The sound sources are centered at the pressure null of the 60.6 Hz (2, 1) mode. The excitation is a 6.5 cycle of 60.6 Hz tone burst.

Monopole: Since it is located in the pressure null, the sound radiated dies off quickly.
View attachment 424944
Dipole, oriented in the direction along the length of the room: Excited the axial (3, 0) room mode at 64.3 Hz but not the (2, 1) mode.
View attachment 424945
Dipole, oriented perpendicular to the length of the room: Excited the tangential (2, 1) room mode.
View attachment 424946

The Python Jupyter notebooks used to generate the plots are in the attached ZIP. I haven't started on the explanations of the theories behind the simulation. Will update when I can get it done. Hopefully by next weekend.

Nice. I have a couple of thoughts;

1) What parameters have you considered for the "dipole"? Because I believe polarity, delay and distance are all relevant for the various "arrays" that are possible (Edit: and gain I guess). You can get the idea from this online calculator. "Gradient" mode with 2 drivers seems to be cardioid (as 1 driver gets inverted polarity), and ~0.67m gives max effect at 128.3Hz, which is ~2x the (3,0,0) mode from your examples.

2) I presume these are simulating the same "tone burst" from the original post? I wonder if this tone can be "modified" though, in order to do the simulation with EQ? The idea being that it would be nice to see the effects of an "anti-mode" filter on the responses, like for example -10dB and 15Q at 64Hz, to see what happens vs that (3,0,0) mode. Perhaps rather than doing EQ calcs in the code, maybe a user supplied impulse can be supplied and convolved against the tone burst, or perhaps the tone burst is already a WAV that the user can just replace with a custom burst (that is already EQ'd/convolved).

3) And further to #1+#2, how much flexibility can there be total? For example, "gradient"/cardioid is generally Hz*4 :=> spacing :=> delay, but what if the delay is ever so slightly different from the spacing? Or if the EQ is slightly different on different drivers? That would be interesting to see too. (well, seeing it in 3D would definitely be cool, but just seeing the resulting waveform would be good too).

 

[neRok]

monopoles may work better in a different place than a cardiod

"work better" in what way(s)? From the 1st post it can be seen that monopole gives ~70% strength when installed at 2 locations, whereas cardioid gives ~35% strength at any location. So with monopole you get more "efficiency" but less flexibility, but with cardioid you can always just use more speakers to get the same SPL.

Thus they are not "directional" in small rooms

It still sends less energy "backwards" than it does forwards, which would inevitably clean up R60 times (AFAIK...?)


Just thinking about double bass array that have drivers flush on each end wall - is that effectively an "extreme cardioid"?

 

[Bjorn]

"work better" in what way(s)? From the 1st post it can be seen that monopole gives ~70% strength when installed at 2 locations, whereas cardioid gives ~35% strength at any location. So with monopole you get more "efficiency" but less flexibility, but with cardioid you can always just use more speakers to get the same SPL.


It still sends less energy "backwards" than it does forwards, which would inevitably clean up R60 times (AFAIK...?)
View attachment 424980

Just thinking about double bass array that have drivers flush on each end wall - is that effectively an "extreme cardioid"?

Better means less standing waves and a more even frequency response. We need to look at actual measurements in a room. Not simulations.

RT60 is a measurement of the reverberation which required a mixed diffuse field and no Dc. It doesn't exist in rooms we operate in, and is one of the biggest misleading myths within acoustics today.

Lyd & Akustikk | Små rom og etterklangstid

www.lydogakustikk.no

You have on the side standing waves and resonances with decay. This can be seen with a waterfall or wavelet.

Some comparisons one member here did below.

Kii Three vs Devialet Phantom:

Kii Three (blue vs JBL M2 DIY clone (red):

The difference between Phantom Devialet and the Cardioid Kii Three is basically none is the bass in this comparison. They both have the exact same room modes. The higher level with the Phantom is simply tuning of level related.

In the comparison with the JBL M2, the JBL is more even below 50 Hz and avoids the 42 Hz peak for unkown reasons. Above 100 Hz, the Kii Three is more even at certain frequencies but most here is above the Schroeder.

At least in these comparisons we don't see any benefit for the lowest frequencies with cardioid. That's also what I would expect.

 

[MKR]

Better means less standing waves and a more even frequency response. We need to look at actual measurements in a room. Not simulations.

RT60 is a measurement of the reverberation which required a mixed diffuse field and no Dc. It doesn't exist in rooms we operate in, and is one of the biggest misleading myths within acoustics today.

Lyd & Akustikk | Små rom og etterklangstid

www.lydogakustikk.no

You have on the side standing waves and resonances with decay. This can be seen with a waterfall or wavelet.

Some comparisons one member here did below.

Kii Three vs Devialet Phantom:
View attachment 425005

Kii Three (blue vs JBL M2 DIY clone (red):
View attachment 425006

The difference between Phantom Devialet and the Cardioid Kii Three is basically none is the bass in this comparison. They both have the exact same room modes. The higher level with the Phantom is simply tuning of level related.

In the comparison with the JBL M2, the JBL is more even below 50 Hz and avoids the 42 Hz peak for unkown reasons. Above 100 Hz, the Kii Three is more even at certain frequencies but most here is above the Schroeder.

At least in these comparisons we don't see any benefit for the lowest frequencies with cardioid. That's also what I would expect.

No benefit with LF cardioid? I expect Genelec, Geithain, D&D, and Soundfield might disagree with your position … I thing the Beolab 90 also supports LF cardioid, but not 100% certain of that. And note Kii BXT is not LF cardioid, it only reaches 80Hz.

Have you ever had the chance to hear a system with LF cardioid ?

@Thomas Lund Maybe you can share your thoughts on the benefits of LF cardioid?

 

[Bjorn]

No benefit with LF cardioid? I expect Genelec, Geithain, D&D, and Soundfield might disagree with your position … I thing the Beolab 90 also supports LF cardioid, but not 100% certain of that. And note Kii BXT is not LF cardioid, it only reaches 80Hz.

Have you ever had the chance to hear a system with LF cardioid ?

Did you read what I wrote?
I'm saying that below a certain frequency cardioid doesn't offer less room modes in general. Please prove me wrong!

Geithan and Genelec operates in PA where rooms are large enough that directivity works. D&D seems to agree with me from what discussion I had with the CEO some years ago. He basically said sometimes it works, sometimes it doesn't.

Yes, I have heard cardioid bass many times. In a small room and below a certain frequency you basically loose SPL and get higher distortion vs monopole. However, if you're stuck with a certain position you might find one design yielding a more even response vs another. And that can also be the monopole.

 

[NTK]

1) What parameters have you considered for the "dipole"? Because I believe polarity, delay and distance are all relevant for the various "arrays" that are possible (Edit: and gain I guess). You can get the idea from this online calculator. "Gradient" mode with 2 drivers seems to be cardioid (as 1 driver gets inverted polarity), and ~0.67m gives max effect at 128.3Hz, which is ~2x the (3,0,0) mode from your examples.

The dipole is modeled with 2 monopoles, equal in amplitude but in opposite polarities, separated by a distance. You can modify these parameters in the code, e.g. change their amplitudes, add a relative delay, and whatever else you want to try. That's the purpose of open source, so others are free to experiment with it while I can sit back and take it easy

2) I presume these are simulating the same "tone burst" from the original post? I wonder if this tone can be "modified" though, in order to do the simulation with EQ? The idea being that it would be nice to see the effects of an "anti-mode" filter on the responses, like for example -10dB and 15Q at 64Hz, to see what happens vs that (3,0,0) mode. Perhaps rather than doing EQ calcs in the code, maybe a user supplied impulse can be supplied and convolved against the tone burst, or perhaps the tone burst is already a WAV that the user can just replace with a custom burst (that is already EQ'd/convolved).

You can also change the excitation function in the code too.

3) And further to #1+#2, how much flexibility can there be total? For example, "gradient"/cardioid is generally Hz*4 :=> spacing :=> delay, but what if the delay is ever so slightly different from the spacing? Or if the EQ is slightly different on different drivers? That would be interesting to see too. (well, seeing it in 3D would definitely be cool, but just seeing the resulting waveform would be good too).

You can do a lot of that by playing with the code, which also means there will be some work for you
For 3D simulations I would definitely go with canned FEM (finite element method) code, where others have already do the hard work, instead of (half) baking my own. For 3D simulations, you really want something to be able to deal with irregular geometries, and that means much more complicated code. Also, the amount of computer power we need will start to become serious.

 

[thewas]

Speaker boundary interference is something else and often either above the Schroeder or in the transition zone.

To make it more complex, monopoles may work better in a different place than a cardiod. Thus the best comparison is optimizing placement for both. If one is stuck with one placement, you might find one working better than the other.

In the end they were similar sized loudspeakers and the frequency were the difference appeared was quite below Schroeder, although not in the low bass where also the D&D is not a cardioid. Also loudspeaker placement is in many living situation and rooms not a real variable and in such case the upper bass cardioid showed a small but clear advantage.

 

[Purité Audio]

Written this before and not sure how pertinent but D&D once made an 8M, not cardioid, identical drivers and enclosure the 8Cs were clearer, also the 8Cs measurements around the cardioid were much tidier than traditional non cardioid designs.
Keith

 

[Bjorn]

I thing the Beolab 90 also supports LF cardioid, but not 100% certain of that. And note Kii BXT is not LF cardioid, it only reaches 80Hz.

It's difficult to know what they doing as it can be hidden in the DSP. But if you're correct that Kii doesn't use cardioid below 80 Hz, you should ask yourself why. Since the cardioid in the speaker is active, it can easily be done. That's just a matter of some changes in the DSP. So why do you think they have left it out if the benefit of less room modes is obvious?

As for the B&O Beolab 90, it's also something that can be achieved with the active cardioid. Whether they use it below a certain frequency is something I don't know. But if we look at in-room measurement at the review in Stereophile, we see a poor LF response with a lot of peaks and some dips. Obviously they were not done with tuning of the level.
The two graphs represent left and right speaker and with 1/6 octave smoothing.

Source: https://www.stereophile.com/content/bang-olufsen-beolab-90-loudspeaker-measurements

I guess my monopole is far better because it measures flatter in the lows (1/24 oct smoothing):

That's a joke of course because we can't look at one low frequency measurement in one room and compare it to another in a different room. The room and placement is really what dicates the low frequency response from a speaker. If cardioid made a major difference in sub frequencies, we would have known that decades ago. But there's a benefit with controlled directivity down to a certain frequency of course. Even more so with a narrower dispersion than cardioid.

 

[MKR]

It's difficult to know what they doing as it can be hidden in the DSP. But if you're correct that Kii doesn't use cardioid below 80 Hz, you should ask yourself why. Since the cardioid in the speaker is active, it can easily be done. That's just a matter of some changes in the DSP. So why do you think they have left it out if the benefit of less room modes is obvious?

As for the B&O Beolab 90, it's also something that can be achieved with the active cardioid. Whether they use it below a certain frequency is something I don't know. But if we look at in-room measurement at the review in Stereophile, we see a poor LF response with a lot of peaks and some dips. Obviously they were not done with tuning of the level.
The two graphs represent left and right speaker and with 1/6 octave smoothing.

View attachment 425068
Source: https://www.stereophile.com/content/bang-olufsen-beolab-90-loudspeaker-measurements

I guess my monopole is far better because it measures flatter in the lows (1/24 oct smoothing):
View attachment 425069

That's a joke of course because we can't look at one low frequency measurement in one room and compare it to another in a different room. The room and placement is really what dicates the low frequency response from a speaker. If cardioid made a major difference in sub frequencies, we would have known that decades ago. But there's a benefit with controlled directivity down to a certain frequency of course. Even more so with a narrower dispersion than cardioid.

“If cardioid made a major difference in sub frequencies, we would have known that decades ago.”

Huh? Sorry, not following that statement at all.

So I suppose you are telling me Genelec doesn’t know what they are doing with the 371, which absolutely has LF cardioid?

Anyway, you do you and correct your LF room issues with your fancy room treatment. I myself prefer a more modern approach

 

[Bjorn]

“If cardioid made a major difference in sub frequencies, we would have known that decades ago.”

Huh? Sorry, not following that statement at all.

So I suppose you are telling me Genelec doesn’t know what they are doing with the 371, which absolutely has LF cardioid?

Anyway, you do you and correct your LF room issues with your fancy room treatment. I myself prefer a more modern approach

Modern? Cardioid goes way back in time. And the reason why cardioid subwoofers never were aimed to the home marked is for the reasons I've mentioned. There are researchers on the area.

Let us stick to actual measurements and science and not to a try to look for something only to back up what we want to believe in and to defend our choices. Where's the proof that cardioid leads to less room modes in sub frequencies in small rooms on a general basis?

Much of treatment I recommend my customers is DIY and not fancy at all. But it works and the result simply isn't achievable with any speaker design alone.
But I understand well that treatment isn't for everyone. In that case a horn design with much narrower directivity and also more constant than cardioid is generally a better design because of much less early reflections.

With cardioid you're forced to treat the nearest side wall mirror images and ceiling images for a great imaging. Even the wide dispersion CBT speaker is better than cardioid here because it avoids vertical reflections almost completely.

 

[MKR]

Modern? Cardioid goes way back in time. And the reason why cardioid subwoofers never were aimed to the home marked is for the reasons I've mentioned. There are researchers on the area.

Let us stick to actual measurements and science and not to a try to look for something only to back up what we want to believe in and to defend our choices. Where's the proof that cardioid leads to less room modes in sub frequencies in small rooms on a general basis?

Much of treatment I recommend my customers is DIY and not fancy at all. But it works and the result simply isn't achievable with any speaker design alone.
But I understand well that treatment isn't for everyone. In that case a horn design with much narrower directivity and also more constant than cardioid is generally a better design because of much less early reflections.

With cardioid you're forced to treat the nearest side wall mirror images and ceiling images for a great imaging. Even the wide dispersion CBT speaker is better than cardioid here because it avoids vertical reflections almost completely.

So your horn and CBT offerings (and your room treatment) are the only real way to solve the issue, not cardioid?

 

[neRok]

I haven't read all the other responses yet, but...

You can do a lot of that by playing with the code, which also means there will be some work for you

Lol, I was trying to bait you in to doing it. It must be a common problem because I've already got a big list of things to dabble with too

For 3D simulations I would definitely go with canned FEM (finite element method) code, where others have already do the hard work, instead of (half) baking my own. For 3D simulations, you really want something to be able to deal with irregular geometries, and that means much more complicated code. Also, the amount of computer power we need will start to become serious.

Is there anything relevant/useful in this project (code/tech wise) that could be pilfered? - ROOM EIGENMODES SIMULATOR v1.1
It was shared on this forum, which is where I spotted it - https://www.audiosciencereview.com/forum/index.php?threads/room-eigenmodes-calculator.38734/

 

[Bjorn]

Cardioid Dutch & Dutch 8C measured in the room by Erin. Loads of room modes and a generally poor LF response.

If it is correct that D&D uses cardioid all the way down and Kii Three stops at 80 Hz, it's interesting to compare them. Erin has also measured the Kii Three in this room and likely with very similar placement.

Overall the response below 400 Hz is better with the Kii Three. If we track the response below 100 Hz only, we'll notice that the response of both is very similar, thus the room and standing waves dominates the response here.

As mentioned previously; if one is stuck with one position for the speakers one design can work better than another. But the cost of running cardioid and dipole is less SPL and higher distortion in the lows.

 

[Dmitrij_S]

Modern? Cardioid goes way back in time. And the reason why cardioid subwoofers never were aimed to the home marked is for the reasons I've mentioned. There are researchers on the area.

There are many papers that claims otherwise to this your statement. For example, here are some the decay time graphs of room modes ("low frequency reverberation") for monopoles and cardioids (from paper by Ferekidis and Kempe, Controlling the Mode Excitation of Rooms by using Multiple Low Frequency Cardioids in Multichannel Systems). From this investigation follows that the cardioid provides faster modes decay and more even room modes excitation

====================

=======================