Visualizing How Different Loudspeaker LF Directivity Patterns Couple to Room Modes

[NTK]

This opening post is to demonstrate how different loudspeaker low frequency directvity patterns couple to our room modes. If you are unfamiliar with what room modes are, please refer to @DonH56's excellent thread for an in-depth introduction.

Room Modes

Room interactions are often discussed as one of the primary reasons similar (or even dissimilar) systems sound different. If you read the “Reflections and DACs” thread you saw how reflections and standing waves can interact along a transmission line. As it happens, sound waves in a room also...

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In this post we'll look at a simple case of a longitudinal mode. This example ficticious "1D room" is 8 m long. The first (lowest frequency) room mode is when the room length matches 1/2 of the wavelength, λ. The wavelength of a single tone soundwave is the speed of sound (c = 343 m/s at 20 degC) divided by the frequency (f), i.e. λ = c / f. Therefore, for this room, the first mode is: f_mode_1 = c / (2*room_length) = 21.4 Hz. Higher modes occur at integer mulitple of the lowest mode frequency. In this exercise, we will concentrate on the third mode (64.3 Hz).

The test signal is a 6.5 cycle raised cosine windowed tone burst. The driver is represented with an up-and-down oscillating dot.

Monopole
We first start with the most common type by far — the monopole, which radiates omnidirectionally at low frequencies — when wavelengths are much larger than the size of the speakers. We'll look at 3 speaker placement positions, near the left wall, at a null, and at a peak 1/3 of the way into the room. We'll also look at the sound pressures received at 3 listening positions, first one at 5.33 m from the left wall, corresponding to a room mode peak, second is at 6 m and is half way between the peak and the null, and third is at the 6.67 m null.

Monopole located near the left wall

Monopole located at the first null from the left wall

Monopole located at the peak at 1/3 distance into the room

There is no surprise here. When the speaker is located near a peak, it couples strongly to the mode, and the tone reverberates for a long time. Opposite when the speaker is at a null. Notice that the sound as received at the listening position depends on both the speaker position and listener position. If the listener is at a null (the green traces in plots on the right), it doesn't matter if the speaker is located at a peak, the listener will get minimal response for that mode frequency.

Dipole
A dipole can be seen as two monopoles of opposite polarities, separated by a small distance (0.25 m in this case).

Dipole located near the left wall

Dipole located at the first null from the left wall

Dipole located at the peak at 1/3 distance into the room

Dipoles couple to rooms in the opposite way as monopoles. It couples poorly when it is at the room mode peak, and couples strongly at the null. Since you will usually find many room mode peaks near room corners, dipoles would be best placed some distance away. Note that a dipole is directional — its orientation matters. You'll only get the response shown here when the dipole orientation aligns to the room mode direction. Also note that a dipole is a lot less efficient as a bass radiator than a monopole because of the significant cancellations by the opposing sound radiators. In this simulation, two "drivers" each of the same power as the monopole are used, and the resulting responses at the listening positions are less than that of a monopole with half the total output power. (See below)

[Edit - Corrections]
When I wrote the original post, I forgot that you don't need 2 drivers to realize a dipole or cardioid radiator (unlike in my simulations which used point sources). The back waves from the driver diaphragm will serve nicely as the opposite polarity source. Open baffle dipoles are just such an implementation. Cardioids can also be realized by acoustically restricting the back waves to give the delay (passive cardioids). It is still true that dipoles and cardioids are inefficient acoustic radiators compared to monopoles, just not nearly as much as it was suggested in my wordings in the original post.

Cardioid
A cardioid is modeled similarly to a dipole (two monopoles of opposite polarity), but with the rear radiator delayed by the time it takes for the sound waves of the front radiator to travel to the back. The result is that, for the soundwaves radiating towards the back, those waves from the rear cancels those from the front. This give the cardioid its characteristic radiation pattern that it has a much reduced sound output towards the back, which can be seen in the animations.

Cardioid located near the left wall

Cardioid located at the first null from the left wall

Cardioid located at the peak at 1/3 distance into the room (notice that initially there is no sound radiated to the left due to the cardioid directivity pattern)

The responses received at the listening positions are largely independent on whether the cardioid speaker is located at a peak or null or in between. Just like the dipole, cardioid is directional, and its effectiveness is dependent on whether it is aligned to the room mode direction. Again, cardioid has low efficiency.

Cardioids have the benefit of radiating most of its energy towards the front hemisphere, thereby exciting the lateral room modes less. The cardioid radiation pattern is difficult to maintain at the low bass frequencies, as the long wavelengths will require larger separation distance between the drivers. The mighty Genelec W371A transitions to omni at ~50 Hz. The a little less mighty Dutch & Dutch 8C transitions at ~100 Hz.

Summary (TL/DR)
Monopoles and dipoles couple to the room in opposite ways, so a good location for a monopole is probably a bad location for a dipole, and vice versa.
To get the desired effects from dipoles and cardioids, they need to be oriented to align with the room modes.
Cardioids are significantly less affected by room modes.
Cardioids and dipoles are inefficient sound radiators compared to monopoles.

 

[NTK]

What if the excitation frequency does not match that of a room mode? Below is for a monopole with a test signal half way between the mode 2 and mode 3 frequencies. What happens is that we'll no longer get the "standing wave" when the excitation frequency matches that of a room mode.

 

[Scgorg]

Great post as always, NTK!
If anyone reads this and is interested in the behaviour in real, 3D spaces I will heartily recommend this paper by Ferekidis and Kempe: https://www.aes.org/e-lib/browse.cfm?elib=12663

The general conclusions NTK presents for the 1D scenario here also holds quite well in 3D. To quote the authors' conclusion: "A cardioid excites room modes in pressurenodes as well as velocity-nodes. Therefore its coupling to room modes is less position dependent than that of a monopole or a dipole. Experiments prove that the cardioid maintains nearly the same mode excitation pattern even when placed in contrary positions (corner vs. centre). Turning the cardioid can help to adjust for an even mode excitation pattern."

 

[Dmitrij_S]

Thanks for sharing the results !
It seems that your conclusions are very similar to that of Philip Feurtado made in his master thesis

Conclusions
5.1 Discussion
In this thesis frequency response curves were calculated for simulated subwoofers in rectangular rooms to understand how they coupled to the room’s resonances within the typical low frequency operating range of subwoofers. Comparisons were madeby simulating monopole, dipole, and cardioid sound sources at different locations in the room and comparing the acoustic pressure received at the listener location.
The monopole source is the most efficient radiator in the frequency range of interest and thus couples most effectively to the low order modes in the frequency range of interest. The spatial dependence of the modal coupling goes as the shape of the eigenfunction itself. The monopole source couples more effectively as the absolute value of the eigenfunction increases. Thus the monopole is most effective if placed in a corner as it does not excite even order modes if the source is positioned at the midpoint of a wall.
The dipole source is a less effective radiator than the monopole, yet offers additional flexibility due to it’s directional characteristics. The dipole does not couple to modes that are oriented perpendicular to the dipole axis as the source does not radiate in that direction. Thus the dipole may be used to selectively couple to only certain modes and not to others. Spatially the dipole couples to resonance modes most effectively where the eigenfunction changes sign and the dipole axis is parallel to the sign change. Therefore the dipole couples poorly at in the corner of a room as there is never a sign change close to a corner for the eigenfrequenciesof interest in this application. Furthermore, because of the directionality of the dipole source it is possible to obtain a relatively flat low frequency response at a listener location at the center of a room if a dipole is oriented along the shortest axis because the lowest modes will be neither driven nor detected. This does result in a lower level signal, however, as the trade off for flatness with this method is overall SPL.
The cardioid source is a less effective radiator than the monopole but more effective than the dipole. While the modal coupling from the monopole source and the dipole source is highly dependent on the location of the source in relationship to the mode shape, the cardioid source exhibits much less variation, thus allowing for greater flexibility in determining the source location. Additionally, the cardioid shows less variation as a function of the source’s angular orientation as long as the directional source is pointed towards the receiver location. Thus the cardioid offers the most flexibility as a single source coupling to more modes as source location and angle are varied.
For two sources that are closely spaced and are time delayed longer than the propagation time, more interesting results can be obtained. The intrinsic, frequency dependent, source amplitude variation enables the source to radiate in frequency bands in between modes and to selectively excite certain frequency bands, resulting in a response that can compliment other sources well by ”filling in gaps”
left by other sources.

 

[somebodyelse]

I'm going to have to reread the bit with the 'special case' for the dipole sub with listener in the middle of the room. I think I'm missing something about positioning and orientation.

 

[René - Acculution.com]

Very nice. Also note how you do indeed get sound when exciting in a mode null at this mode's eigenfrequency, contrary to what it is sometimes uttered about sound in rooms. (Unless you set up a special excitation but let us not get into that.)

 

[neRok]

Monopole located at the peak at 1/3 distance into the room

Great graphics. I didn't expect the position to play such a role. This one (monopole on a null) seems the most informative. It seems that part of the wave goes towards the left wall (ie the front wall), reflects, and then couples with the portion of wave currently being output. Due to their phase offests (caused by time of the distance travelled), they are therefore cancelling each other, and then they both move towards the other end of the room in unison. Is this interpretation correct?

I came across the "wave packet" whilst reading on wikipedia, and I presume that is what this is? But is that graphic actually for a 64.3Hz tone, which would be ~15.6ms long?

AFAIK (which isn't a lot) cosine waves start at a crest? So a full cosine wave is crest-zero-valley-zero-crest, as opposed to sine wave being zero-crest-zero-valley-zero? Is the full 1 cycle/period represented in that tone burst graphic?

This might get answered as a result of the above answers, but I'm not sure why the dot moves the way it does. I get that the dot in the simulations traces the up and down movement of the tone burst graphic, but is that how the driver actually moves when the signal is a 1 cycle tone, or is that the way the air behaves because of a "single movement" (single in and out)?

What if the excitation frequency does not match that of a room mode? Below is for a monopole with a test signal half way between the mode 2 and mode 3 frequencies. What happens is that we'll no longer get the "standing wave" when the excitation frequency matches that of a room mode.

Could you repeat this test but at 32.2Hz? I believe this would give a null caused by the right/back wall reflection at the 5.33m position. That's what I would like to see - a back wall reflection null that is not a room mode.

 

[NTK]

I'm going to have to reread the bit with the 'special case' for the dipole sub with listener in the middle of the room. I think I'm missing something about positioning and orientation.

Sorry. I missed your post until a few days ago. It doesn't happen in the 1-D case since there is only one orientation (or no orientation). I'll try to explain using a 2-D example. Below show the first 8 room modes of a 8 m X 4 m 2-D rectangular room. Since the room length is 8 m, we still have the 64.3 Hz (3, 0) axial mode.

The dipole sub will be placed where the black dot is.

In the first configuration the sub is oriented pointing at the long axis of the room. The dipole sub excites the 64.3 Hz mode.
(Note: The titles of the plots should say "Excitation Freq = 64.31 Hz" instead of "Mode Freq = 64.31 Hz".)

However, if we reorient it by turning it to align to the short axis, it excites the 60.6 Hz mode instead. So, unlike monopole subs, the room response of dipole subs also depend on their orientations in addition to it placements.

 

[somebodyelse]

Thanks for the response. Rereading my post I should have been clearer - I had just had a look through the Philip Fuertado thesis @Dmitrij_S linked immediately above my reply. Section 4.2 on page 48 entitled "Unique Solution for Particularly Flat Bass Response" is the bit that had me intrigued, but needing to revisit when I had time. I _think_ I've got it now, but it does seem rather impractical!

 

[neRok]

I came across an old thread on another forum where OP showed room modes are like high gain and narrow bandwidth EQ filters in that they ring the same, and that they can be negated by using opposite filter to "ring in reverse". REW creator JohnM says confirms this in a later post;

• Modes have narrow bandwidths, typically 1/10th of an octave or less. Using narrow filters to boost the response creates the same extended ringing problems as room modes.

This might be confirmation bias, but I eventually saw the same thing in the VBA thread.

• But in this strange case, the only reason PK filter is improving the decay is because of the normally "undesirable" post ringing.

With regards to this thread, it would be interesting to see one of the simulations with such an EQ applied (to Monopole located near the left wall for example).

Later in the referenced thread a user suggests;

• ok, so you hear -25db at 60hz for the first 13ms. then the room mode kicks in and things are corrected. seems like the same thing would happen on the way out as well. in other words the bass would be 13ms late arriving at its proper level and 13ms late decaying back down. think about this on something like a kick that is transient. i can tell you little things like 13ms on a kick are noticable.

So he is saying that the early/direct portion of the bass wave will be weaker than it should, but the "overall" "magnitude" will be correct. JohnM says the following about this (emphasis mine);

• The direct sound is affected roughly as you describe, which is why EQ'ing the direct path is rightly frowned upon if not correcting for anomalies in the reproduction of the speaker itself, but you are forgetting the frequency you are considering. At low frequencies we are unable to perceive such short duration effects, our ears detect the total energy integrated over much longer periods than the time it takes sound to travel the length of a domestic room. The transient effects you detect in a kick drum, for example, are in the higher frequency content of the spectrum of that sound, not in the fundamental, and those higher frequencies which determine our perception of the transient are not being altered by the EQ filter. What is being corrected is the total energy which reaches us, which restores our perception of a balanced sound.

So maybe the magnitude is correct, but I still wonder about the affect of the weakened direct wave. If the direct wave is weak, wouldn't that mean any physical "slam" is also weak?

 

[ernestcarl]

So maybe the magnitude is correct, but I still wonder about the affect of the weakened direct wave. If the direct wave is weak, wouldn't that mean any physical "slam" is also weak?

That would depend what frequencies the heard "slam" is mostly coming from (very likely not the low and infra bass). You can sort of get around this issue by basically adjusting certain select bands with EQ. Listen to a set of "good" kick drum test track samples or music with emphasized low bass lines while dynamically modifying the EQ. It helps a lot if you have some room measurements on-hand for easy visualization, and can also see or examine the spectrogram of the kickdrum tracks. Just experiment and listen which EQ profile adj. works better to your ears. Many times, recordings themselves vary and so you could try just settling for the mean or average and leave the rest to general tone controls/saved EQ presets.

*BTW, if the in-room transient response is truly far from perfect and there's significant time "smearing", this will still not fix the loss of "definition" or ability to hear smallest details... like the way you can with equalized closed-back headphones and IEMs.

 

[HappyPantherFan]

Excuse me for my silly question, but why is it that speakers output pressure waves as shown here, instead of velocity waves? I've always thought the speakers in-wall have trouble exciting the modes, because they expect a pressure source (and a firm wall) there, while the cones and domes were inconveniently moving about.

 

[René - Acculution.com]

Excuse me for my silly question, but why is it that speakers output pressure waves as shown here, instead of velocity waves? I've always thought the speakers in-wall have trouble exciting the modes, because they expect a pressure source (and a firm wall) there, while the cones and domes were inconveniently moving about.

The pressure is often what is visualized because that is after all what we hear. When solving for either modes or the resulting pressure under forced (loudspeakers in the room) conditions, it is again the pressure that is typically solved for. There is an associated velocity with that, and the typical boundary condition of the walls is 'zero normal velocity'. There will be some tangential velocity at the walls, dictated by the pressure field. The modes are not really 'expecting' anything. They are excited in different ways by the placement and type of loudspeakers. For a loudspeaker placed in a wall you just have a particular excitation, which may get some modes more or less involved compared to other placements. From a superposition point of view, the wall is still seen as fairly hard for a typical loudspeaker placed in it.

 

[HappyPantherFan]

The pressure is often what is visualized because that is after all what we hear. When solving for either modes or the resulting pressure under forced (loudspeakers in the room) conditions, it is again the pressure that is typically solved for. There is an associated velocity with that, and the typical boundary condition of the walls is 'zero normal velocity'. There will be some tangential velocity at the walls, dictated by the pressure field. The modes are not really 'expecting' anything. They are excited in different ways by the placement and type of loudspeakers. For a loudspeaker placed in a wall you just have a particular excitation, which may get some modes more or less involved compared to other placements. From a superposition point of view, the wall is still seen as fairly hard for a typical loudspeaker placed in it.

I was asking why the speaker in the graph excites the modes near the wall (where the mode has oscillating pressure but no movement), but not when at the velocity node.

I think I understand now. The moving diaphragm alone is a dipole, but a speaker is not.

 

[NTK]

I used Mathematica for the simulations in posts #1 & 2. Mathematica uses the finite elements method (FEM) to solve the underlying partial differential equation. FEM is very flexible in accommodating irregular geometries.

It occurred to me that for the 1-D case, the geometry is very simple and it is regular -- the domain (i.e. the room being simulated) is a line and it is easily discretized into a number of equally spaced nodes. And thus this problem can also be readily solved using the finite difference method.

Attached are a couple of Jupyter notebooks for those who are interested in experimenting with wave equation simulations. Unfortunately, you'll need some knowledge of Python. The "Room Sim 1-D.ipynb" notebook has an explanation of the finite difference solution process. The "Traveling Pulse Simulation.ipynb" is to demonstrate how a different type of problem is setup with finite difference using initial conditions.

When/if I have some more time in the near future I may come up with a 2-D version for rectangular rooms. Finite difference is not very flexible in dealing with irregular geometries, but rectangular rooms should be very doable. After that if I am feeling super ambitious, may be I'll make an attempt at L-shaped rooms.

[Edit] Forgot to include the picture files that are used by the Jupyter notebook. Updated the ZIP file.

 

[Few]

Interesting topic. I’m working to finish a pair of sealed enclosure subwoofers to take over the lowest octave or two from a pair of vertical dipole arrays (six 8 inch woofers, open-back, per side). My intent, to minimize intrusion into my living room, is to stack the dipole arrays on top of the sealed enclosures. It’ll be interesting to see where the crossover needs to be to yield the smoothest response. Maximum excitation of a room mode by the sealed subwoofer will correspond to minimum excitation by the dipole arrays (if the mode aligns conveniently with the dipole axis), and vice versa. I hope I’m not forced to physically separate the subwoofers from the dipole arrays after designing them to stack neatly!

Few