#1
The purpose of this thread is to shed some light on the topic of what is roughly happening when you play an amplifier with more than one speaker connected to it. This thread is for those who desire more description than the simple “Double the speakers and you get a +3 dB bump” description. Having a better idea of what is going on when extra speakers are added to an amp can be a useful tool when deciding on how many and what kind of speakers will best suit your purposes.

In order to understand what is going on here, we are going to have to get a bit technical. It may help to read the Basics of Sound and Calculating Sound Intensities threads as they were originally made to give proper backstory to this topic.

Adding Coherent Acoustic Signals (One amp with two speakers)

We will first briefly revisit the math dealing with adding coherent audio signals together.

Coherent Output = S1 + 20 log10(N)

Where S1 is the signal output and N is the number of speakers producing the signal. Remember, sound power is divided evenly among the speakers so each speaker should be ‘seeing’ 25 watts so S1 will get calculated with 25 watts. Plugging in the information for a 50 watt head with a 100 dB speaker we get:

S1 = (10*LOG10(25)) + 100 = 113.979 dB

Coherent Output = 113.979 + 20 log(2) = 120 dB

http://www.engineeringtoolbox.com/adding-decibel-d_63.html

But there is a problem. There are stipulations to coherent signal summing, one of the stipulations states that coherent signals are signals that are completely ‘in-phase’ with each other. What does that mean?

Phasing

Two signals that are ‘in phase’ are perfectly synced up, their crests will add up to a larger crest, their troughs will add up to a deeper trough. This is considered constructive interference, and this is what summing a coherent signal assumes.

But what if the two signals are not perfectly synced up? Well, you get a phenomenon known as ‘destructive interference’, this is where the crests and the troughs of the waves are not perfectly lined up and can actually start canceling one another when summed together and the result is a final wave that is actually less intense than the two input waves.

http://dunagan.biz/tutor/cohoadd.htm

http://en.wikipedia.org/wiki/Interference_%28wave_propagation%29

The Real World

So, in the real world it would be incredibly hard for two sources to be perfectly in phase with each other for these reasons:

- Individual sound sources must be displaced some distance away from one another
- Sounds have frequencies that have inherent wavelength
- Most real sounds are made of a mix of these wavelengths
- Most musical sounds will produce notes of different frequencies (musicians will play different notes)

When we consider these points we come to a conclusion: in order for two signals from two sources to be perfectly in phase, the distance between the sources must be a constant multiple of the wavelength of the note being produced. In other words, if the distance from output source A to output source B is not equal to the wavelength of the note being produced then the signals won’t sum perfectly.

That is not all, since the distance between the speakers in a 2x12 is fixed (and by necessity they can only get so close together, so they will be displaced by a minimum of ~12&rdquo signals will only be perfectly in phase for frequencies of particular wavelengths. Since real sounds (like notes from a guitar) are made of many wavelengths, this makes it impossible for pure coherent signal summing to occur. Beyond that, while certain frequencies may coherently sum to sound louder, other frequencies will destructively sum to and cancel each other out.

There is also another problem with coherent signal summing, we have not taken into consideration the location of the listener. All of the formulas for coherent summing assume the listener is directly in line, in front and on axis with the signal sources; in other words we have been assuming the most ideal position for signal summing. Once the listener displaces himself from the ideal listening position more phasing issues arise because sound from the closer source will arrive at the listener before sounds from the further source. So the +3 dB bump is an actual phenomenon, you just have to keep in mind that the listener must be in an optimal location in order to perceive it and it is only active in particular frequency ranges.

So coherent signal summing only works for certain frequencies (so simple sounds would work best) and only if the listener is in the optimal position. Where does that leave us?

http://www.recordingeq.com/EQ/req1001/mmi.htm
* the concepts in this section are equivocal to the microphone placing in the above link. Most of the concepts in this section were brought up in sound engineering books I have read as well. For example: Sound and Recording - Francis Rumsey, Tim McCormick [0240519965]
punk isn't dead, it's always smelled that way.

"A perfection of means, and confusion of aims, seems to be our main problem."
-ae
Last edited by gumbilicious at May 14, 2013,
#2
Mutual Coupling

So with all this complicated stuff going on, what is the result? The result is mutual coupling; I will quote another source:

Let's say you have a 12" speaker that produces 100db/w. If all specs stay the same, but you double the surface area of the cone, the speaker will now have a 103 dB/w sensitivity. Each time you double the cone surface area you get a 3 dB increase in SPL, which is a mechanical/acoustical transfer of air.

When adding another speaker to the 12" equation, you are trying to merge both speakers into one. Essentially adding the surface area together for the 3db gain. The problem here is that the cones of the speakers are physically not capable of being close enough to reproduce the 20hz-20khz frequency range as a single system.

The problem introduced here is that as the speaker centers separate, the benefit of speaker coupling tapers off from the high end. Since two 12" speakers can only be physically 12" in distance from each other - cone center to center - , this presents a limitation of a wavelength at a certain frequency. The 3db effect starts at a 1/2 wave distance, and is pretty much full at a 1/4 wave. So for 12", you get a 550hz half wave, to 275hz quarter wave. This is the best performance you will get from two 12" speakers in proximity in relation to speaker coupling 3db gain. As the speakers gain distance you will start lowering the frequency of this effect.

To reiterate, the speakers need to be within 1/4 to 1/2 wavelength for that frequency range to get any gain. This means, if a wavelength of 200 hz is 67 inches, by four (quarter wave) and two (half wave) gives you 17 to 34 inches respectively, which is the distance both speakers need to be from each other (cone center to center) to get a gain at that frequency. So once you get to 400hz, that gap is half the distance. So the speaker cone centers would need to be 8 to 16" from each other. Get to 800Hz and you need the cone centers to be 4 to 8" close, which is physically impossible with 12" speakers. So we can safely say that doubling speakers only has a benefit in the low end frequencies.

Now let's take the same math to a 4x12, since the speaker centers have to match for all 4 speakers, this means that the furthest centers are to be used for that calculation. So for a 4x12, we are talking about 300Hz or less being a realistic region for speaker coupling gain.

Adding another 4x12 stacked on top, would mean that the very top left speaker and very bottom right speaker is now the new length to use for the formula to work, which means roughly 50-60inches in distance. At this point the only frequency range getting advantage is ~100hz and below. So there is still an advantage, but the window closes quickly as you add distance to the speakers.

Now let's say you have two 4x12 cabs, and you position them angled in, the more you angle the cabs the less you will get that gain. The speakers have to be directed in the same way and be even, so you can't have one closer than the other. This means an angled cab won't have as much gain in the low end as a straight cab.


-diaz (From: The Gear Page)

A concept brought up in the above quote, but not addressed so far in this blog, is the fact that signals don’t need to be 100% coherent in order to constructively interfere. Signals that are closely in phase can start to constructively sum when the signals are as far apart as half a wavelength and they can sum to almost a full +3 dB by the time the signals are a quarter wavelength apart.

TLDR: Mutual coupling is the result of all the imperfections mentioned before. To summarize

- Signal summing will only happen within a certain ratio of wavelengths
- Most of the signal summing will happen below a low end threshold
- This threshold frequency is determined by furthest distance between drivers in the cabinet
- Frequencies higher than this threshold frequency will have destructive interference introduced in the form of comb filtering
- The frequencies where signal summing occurs are frequencies that our ear is not most sensitive, while the frequencies that our ear is most sensitive to is subject to destructive interference
- The amount of summing perceived will depends on the location of the listener

http://www.zainea.com/mutualcoupling.htm

http://www.rane.com/par-c.html

The Real World… Again

So what does this mean to us? It means if you add more speakers, you should get more low end at the expense of some (most likely not overly noticeable) distortions in your higher frequencies. The more speakers you add, the lower the threshold bump becomes effective (reducing the range of frequencies that benefit from the bump) and the more phase cancellation you introduce in the higher frequencies. It is also worth noting that this ‘low end bump’ you get from mutual coupling deals with frequency ranges that are quite pertinent for guitar playing as long as you don’t introduce too many speakers.

Another interesting tidbit: mutual coupling benefits closed back cabs much more than open back cabs. Open back cabs suffer from phase cancellation of low notes (due to it’s limited baffling)*, so extra speakers in a open back cab won’t get as much ‘low end bump’ because mutual coupling especially effects the low end. It should also be mentioned that just because open back cabinets don’t get as much benefit from mutual coupling doesn’t mean they are inherently quieter, open back cabs allow sound to propagate out the front and back of the cabinet and proper placement of the open back cab can allow for even more perceived volume than a closed back cabinet.

Most of the math discussed in Calculating Sound Intensities gives us an ideal quantitative idea of how much of a overall dB bump we get when using more speakers. Mutual Coupling explains what is happening in a more qualitative way and can help answer questions like:

-How many speakers should I run
-Should I use open or close back cabs
-Where should I expect increased frequency response

Adding speakers just for the volume boost is not the most efficient way to go about making a louder rig though. Depending on where you are standing and what phasing issues apply to you, you may notice anything from a +0 dB boost to a +3 dB boost in certain frequency ranges (keep in mind, obtaining a full 3 dB in practice is difficult to do as it represents the max increase that can be observed).

One thing to note: even in the most ideal situations all we are getting from doubling the speakers is a +3 dB boost overall. When you introduce an amp and speaker into any sort of room (you know, the opposite of a near field) and the sound starts interacting with that environment, you can get MUCH more dramatic comb filtering, standing waves, and increased SPL’s that just swallow any type of boost using an extra speaker may enable**.

* http://www.ultimate-guitar.com/columns/gear_maintenance/cabinets_for_guitars.html

** http://www.zainea.com/mutualcoupling.htm
punk isn't dead, it's always smelled that way.

"A perfection of means, and confusion of aims, seems to be our main problem."
-ae
Last edited by gumbilicious at May 14, 2013,
#3
I'm thinking a valid mutual coupling model across for given array size range would require you to calculate the coupling from a given speaker to every other speaker in the array for each speaker in the array, then an integration of all the results, no?

So a full stack requires an integration of 28 calculations?

Or am I missing something?
“Ignorance more frequently begets confidence than does knowledge.”
Charles Darwin
#4
Quote by Arby911
I'm thinking a valid mutual coupling model across for given array size range would require you to calculate the coupling from a given speaker to every other speaker in the array for each speaker in the array, then an integration of all the results, no?

So a full stack requires an integration of 28 calculations?

Or am I missing something?


i am not sure. i noticed the math for mutual coupling was much more complex than simple algebra and i tried to stay more on the qualitative side rather than quantitative. i have also noticed that as more speakers get introduced then you are introducing more complexities, i have tripped across articles on speaker arrays and how to efficiently arrange multi speaker arrays and that stuff is mind numbing.

the main thing i would like to convey with this article is that you are going to get a bump in low end when adding more speakers, and that this bump in low end narrows as more speakers are added (and more comb filtering is added). also it may be good for people to know that 'more speakers = more volume' but it is not a totally even volume boost.

some of the references that are linked to above actually have some great information about the more technical math involved with mutual coupling.

you are not missing anything, you are noticing me distancing myself from the math on this topic
punk isn't dead, it's always smelled that way.

"A perfection of means, and confusion of aims, seems to be our main problem."
-ae
Last edited by gumbilicious at May 14, 2013,