The mathematics of amplifiers

[Greenman72]

Forgive me if this is a stupid question. I'm a CPA/CFP, not a sound engineer.

I was talking to somebody far smarter than me over the weekend. She said that 2x10 is better (louder) than 1x15.

I assume that bigger = louder, at least to some degree. And the formula for the area of a circle is Pi x R-squared. Since Pi is constant, let's ignore it and just compare R-squared.

1/2 of 15 is 7.5. 7.5 x 7.5 = 56.25 square inches for the 15-inch speaker.

1/2 of 10 = 5. 5 x 5 = 25. So a 10-inch speaker has 25 square inches, which logically means that two of them has 50 square inches.

56.25 > 50, I think.

So why is 2x10 louder than 1x15?

 

[agedhorse]

It's not necessarily louder, that's the first error it the logic.

There's WAY more to this than just driver size.

Also, using the nominal diameter for area calculations is misleading, only the moving part of the diaphragm (which is smaller) matters. A 10" driver has a diaphragm diameter of about 8.5" and a 15" driver is about 13.5".

 

[gtirard]

Forgive me if this is a stupid question. I'm a CPA/CFP, not a sound engineer.

I was talking to somebody far smarter than me over the weekend. She said that 2x10 is better (louder) than 1x15.

I assume that bigger = louder, at least to some degree. And the formula for the area of a circle is Pi x R-squared. Since Pi is constant, let's ignore it and just compare R-squared.

1/2 of 15 is 7.5. 7.5 x 7.5 = 56.25 square inches for the 15-inch speaker.

1/2 of 10 = 5. 5 x 5 = 25. So 2 x 25 = 50.

56.25 > 50.

So why is 2x10 louder than 1x15?

The title of your post calls for an answer from @Wasnex

But before he brings the light, the loudness of a cab depends on its ability to move air. It depends on multiple factors, and the surface of the speakers is one of them, but it's not the only one (I know about "excursion", there are probably others).

So, the answer is probably "it depends" (on other parameters). Some 1x15 cabs are more efficient than others, same for 2x10.

 

[Zooberwerx]

I'll make it simpler: a 2x10 is "5" better. Now, if you want to add the host enclosure into the equation, we'll have quite a different discussion.

Riis

 

[lomo]

Yes there is more than 1 variable to this equation. That said, IF excursion of the drivers was equal......cone area is key, and proportional to the radius squared. 25 + 25 is a tad less than 56.25. There is absolutely less than no reason to assume a 210 will be louder than a 115 unless all other parameters are known and accounted for. OP, she may be smarter than you in some areas, but her blanket conclusion is neither smart nor complete . Perhaps she, like soooooooo many people, knows of a specific case in which the 210 is indeed louder, and has decided based on a tiny sample that the broad conclusion is "truth". Here we are

 

[Greenman72]

Well, the Fender Rumble 500 is a 2x10, and the Rumble 200 is 1x15.
Again, based on nothing but the math, why does a 500 take 2.5x as much power as a 200, when it actually has less volume? (Volume = square inches, not "loudness" in this case.)

It seems to me (the ignorant and uneducated) that higher volume (size) would equal higher volume (sound).

 

[ardgedee]

I assume that bigger = louder

There are other factors, for example...

• Speaker (the driver itself) design -- one might be optimized for low-bass, or high efficiency, for lower power levels, is only efficient at a very narrow or suboptimal frequency range, or otherwise is not intended for how it's being used (for example using cheap car subwoofers for a bass guitar amp is a popular way to make things sound bad).
• Excursion (the distance the cone travels) -- the longer the excursion the more air it can move.
• Cabinet design -- something appropriate for the speaker and what it's being used for can make a speaker more efficient, extend lower, or affect its characteristics in other ways. An ideal cabinet is designed around the speaker's specifications. Cabinet volume has to be optimized, bigger is not necessarily better since the air in the cab acts as a spring behind the cone and too much air can weaken that spring, so to speak.
• Other factors -- for example some of the horn-loaded speakers of the 70s could be reasonably efficient BUT the audio fidelity was pretty terrible. Not necessarily a unilaterally bad thing since guitarists and bassists often want a particular kind of bandwidth limiting or distortion as part of their tone, but these days the appeal is pretty niche. Also they tended to be ludicrously heavy.

 

[KohanMike]

I went with a combo 1x15 because it's more compact than a 2x10, especially my Carvin MB15. I have to cart it around and with nerve damage to my neck from past radiation treatments, the smaller and lighter, the better. And it also sounds very good.

 

[agedhorse]

Well, the Fender Rumble 500 is a 2x10, and the Rumble 200 is 1x15.
Again, based on nothing but the math, why does a 500 take 2.5x as much power as a 200, when it actually has less volume? (Volume = square inches, not "loudness" in this case.)

It seems to me (the ignorant and uneducated) that higher volume (size) would equal higher volume (sound).

Because the 2x 10" drivers used in the Rumble 500 have a little more power handling AND the 500 drives the 210 as an 8 ohm load at about 300 watts PLUS an extension speaker output that can drive an additional 8 ohm load 250 watts to each speaker section when driving an extension speaker.

You need to open your mind to the fact that there are a lot more things going on here.

 

[S-Bigbottom]

Play an Ampeg B15, Peavey 15, MarkBass 15, Messa 15, GK Neo 15 and a Rumble 15 and you will see that the number 15 does not make these cabs equal.

 

[Wasnex]

Diameter alone does not really tell you which is louder. Loudness come from usable power and the sensitivity.

Usable power is the lower value between how much power the amp can deliver and how much power the speaker can handle. Use RMS or IEC power ratings. Ignore program power and peak power.

Sensitivity can be expressed a couple of different ways. The most common is to state the SPL measured at 1 meter when 1W is applied. For example Sensitivity = 97dB 1W/1m.

Some specs state the SPL measured at 1m when 2.83 Volts is applied. The trick here is to know that 2.83V applied to 8 ohms will result in 1W. So 97dB 2.83V/1m at 8 ohms is the same as 97dB 1W/1m. 2.83V at 4 ohms results in 2W. To convert to a 1W Sensitivity rating simply subtract 3dB. Sensitivity of 100dB 2.83V/1m at 4 ohms is the same as 97dB 1W/1m

Once you have the usable power and sensitivity stated at 1W/1m you can use this formula.

• Calculated SPL = 10Log(Usable Power) + SPL

Lets assume usable power is 350W and the Sensitivity of the cab is 97dB 1W/1m

Calculated SPL = 10Log(350) + 97 = ~122.4dB


The first part of the equation tells you the decibel change that occurs from raising the usable power from 1W to 350W.

Decibel Change 1- 350 = 10Log(350) = ~25.4dB.


So if you are comparing multiple speakers and the usable power stays the same, the cab with the higher sensitivity rating will be louder.

The specs will not always accurately compare cabs because there is no standard for how the specs are developed. They should be accurate when comparing two cabs from the same product line. They may not always be accurate when comparing cabs from different product lines, even if both lines are from the same company.


Example of 210 Vs 115 comparison.

Aguilar DB115

• Power Handling = 350W
• Impedance = 8 ohms
• Sensitivity = 96dB 1W/1m
• Usable Response 32Hz
• Calculated SPL at rated power = ~121.4dB

Aguilar DB210

• Power Handling = 350W
• Impedance = 8 ohms
• Sensitivity = 98dB 1W/1m
• Usable Response = 41hz
• Calculated SPL at rated power = ~123.4dB

So in this example the 210 should play noticeably louder but the 115 plays noticeably lower.

 

[ctmullins]

Forgive me if this is a stupid question. I'm a CPA/CFP, not a sound engineer.

I was talking to somebody far smarter than me over the weekend. She said that 2x10 is better (louder) than 1x15.

I assume that bigger = louder, at least to some degree. And the formula for the area of a circle is Pi x R-squared. Since Pi is constant, let's ignore it and just compare R-squared.

1/2 of 15 is 7.5. 7.5 x 7.5 = 56.25 square inches for the 15-inch speaker.

1/2 of 10 = 5. 5 x 5 = 25. So a 10-inch speaker has 25 square inches, which logically means that two of them has 50 square inches.

56.25 > 50, I think.

So why is 2x10 louder than 1x15?

Apparently this person is not "far smarter than [you]" after all.

 

[logdrum]

Calculus and algebra would be helpful math tools for predicting behaviors of sound. That's a start.,

 

[logdrum]

It's not necessarily louder, that's the first error it the logic.

There's WAY more to this than just driver size.

Also, using the nominal diameter for area calculations is misleading, only the moving part of the diaphragm (which is smaller) matters. A 10" driver has a diaphragm diameter of about 8.5" and a 15" driver is about 13.5".

Hey Andy, a driver is also partly a cone correct,, so it is not just calculating the area of a flat circle correct? Aand isn't part of the center which the dust cap covers also deducted from the area calculation?

 

[MCS4]

I gigged for years with a Carvin 2x10 and 1x15 cab from the same series (Red Line) and also spent some time using two 1x15s from that series instead. At least for those specific cabs and the amps I used at the time, there didn't seem to be any meaningful difference in volume between the 2x10 and 1x15. This is obviously purely anecdotal and might not represent any type of overall trend, though.

 

[Kindness]

Play an Ampeg B15, Peavey 15, MarkBass 15, Messa 15, GK Neo 15 and a Rumble 15

This guy knows how to party.

 

[monsterthompson]

A lot of "rules of thumb" seem to have originated from an era when the available products didn't include the variety we see today. There was probably a time when only a few cab options existed and maybe one kind was "louder" than another and that led to assumptions by consumers.

Now we have more driver types, and the cabinet designs have evolved. Excursion and efficiency and tuning and a host of other variables impact output, which has already been pointed out in this thread. Generalizing is risky business. A 10" driver is probably going into a smaller cab than a 15" driver, and the 10" is likely to not be as loud and/or have as much low end extension. A comparison to a 12" might become a closer race. Having 2x10" drivers adds more variables. Unless you know the specifics of the cabs and drivers being compared it can be challenging to know what to expect. It's not surprising people fall back on broad general assumptions.

 

[agedhorse]

Hey Andy, a driver is also partly a cone correct,, so it is not just calculating the area of a flat circle correct? Aand isn't part of the center which the dust cap covers also deducted from the area calculation?

It's the part of the cone that operates as a piston (for low frequency calculations) and the radiating surface affects higher frequencies (response + polar pattern). The cone is shaped like that for mechanical/structural reasons mostly but affects other performance parameters too.

 

[Hurricane Jimmie]

I was talking to somebody far smarter than me over the weekend. She said that 2x10 is better (louder) than 1x15.

So Anyway, when someone makes a general statement like that, I don't put much stock into it. An old Black Widow 15 vs a neo 2x10? There are so many manufactures, cabinet, amp, watt, impedance, PU signal, etc. considerations that you'll rarely get an apples-to-apples comparison.

I currently have a 1x15" combo, a 1x12 combo, a 2x10 combo, a 2x4 combo, a 15 cabinet and 12 cabinet.
They all have their place, and if one setup was obviously better for me, I wouldn't own so much stuff.

 

[SharpCat]

Play an Ampeg B15, Peavey 15, MarkBass 15, Messa 15, GK Neo 15 and a Rumble 15 and you will see that the number 15 does not make these cabs equal.

I dunno about all, but most of these are 15" speaker size, but the Rumble 15 is 15W and clearly not louder. Unless the 6" speaker in it is REALLY efficient.

Rumble 115 cab, different story.