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Tuesday, October 02, 2012

Does More Speakers = More Volume? Pt 2 (Mutual Cou

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In the previous blog, background information was discussed to clarify certain attributes of sound that will play a role in discussing this subject.  Here is a link if you are interested in some background:

http://profile.ultimate-guitar.com/gumbilicious/bl og/110902/

Without further ado, let's dive into the matter.

Calculating Sound Output From a Loudspeaker and Amplifier (1x12 math)


An amplifier’s output is rated in watts (which is a unit of energy conversion).  A speaker’s loudness is called the sensitivity of the speaker, it is rated in decibels and the reference is taken at 1 meter with 1 watt of power supplied to the speaker.  So we will use these calculations to determine nominal output:

Nominal Output = (10*LOG10(P1)) + sensitivity

Where P1 is the amp’s nominal output.  So a 50 watt amp paired with a speaker of 100 dB sensitivity would be plugged in as such:

Nominal Output = (10*LOG10(50)) + 100 = 116.99 dB

This would be the nominal output in a 1x12 cabinet.

Adding Incoherent Acoustic Signals (2x12 math)

Now, these are the calculations that Side B will use to show how their side of the argument works.  When adding acoustic signals from different sources that don’t share attributes we will use the formula:

Incoherent Output = 10*LOG10(10^(S1/10)+10^(S2/10))

Where S1 is first source, S2 is the second source.  You can actually keep extending this formula to include as many sources as you like.

Now, because power is distributed evenly among speakers we have to recalculate our sources (we’ll assume the same setup as before, 50 watt head with 100 dB speakers), but we’ll adjust for how many watts each speaker will ‘see’ (one half of 50 watts is 25 watts).

Nominal Output = (10*LOG10(25)) + 100 = 113.979

So let’s plug this into our function

Incoherent Output = 10*LOG10(10^(113.979/10)+10^(113.979/10) = 116.99 dB

Using this math, a 50 watt head with 100 dB speakers puts out the same amount of sound with a 1x12 cab as it does a 2x12 cab.

http://www.sengpielaudio.com/calculator-spl.htm

There is a problem though.  There is an interaction going on between these two speakers we are not accounting for.  The phenomenon of coherent signal summing is NOT considered in this example.  “When the two signals being summed are of the same frequency… the sum will be the voltage sum of the two signals. This is referred to as coherent summing of the signals, and it causes some unusual effects that are not intuitively obvious.”

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

Adding Coherent Acoustic Signals (more 2x12 math)

Now, by being more rigorous and realizing that both signals are the same and if you apply the concepts of coherent signal summing, then we use another formula for adding the signals and this is where Side A get’s their argument from:

Coherent Output = S1 + 20 log10(N)

Where S1 is the signal output and N is the number of speakers producing the signal.  Plugging in the information for a 50 watt head with a 100 dB speaker we get:

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

Interesting, using this math gives us +3 dB compared to using a single 100 dB speaker alone with a 50 watt head.  If we take this result seriously, then more speakers does equal more volume when completely coherent signals are summed.

http://www.engineeringtoolbox.com/adding-decibel-d _63.html

But there is still 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”;) 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]

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.

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.

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

-    Signal summing will only happen within a certain ratio of wavelength
-    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.

So once again, where does this leave us?  Well, most of the math we ran through earlier gives us no real quantitative idea of how much of a overall dB bump we get when using more speakers because they just don’t apply to mutual coupling, unless I can come across more math we will just have to live with the qualitative results we obtain from the Mutual Coupling section above.   Most likely, 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_mainte nance/cabinets_for_guitars.html

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

Special thanks to Arby911 and Phil Starr for invaluable input and peer editing.
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