Decibels vs Watts Curves

[porcupunctis]

In a week or so, I'm going to be starting lesson in my Algebra II class on Logarithms.  As you might guess, most 16 to 17 year old students have a hard time imagining that something as abstract as a log would have any impact on their lives. 

I would like to present a formula that plots decibels based on wattage for a given sensitivity of speaker.  That is, if I set the constant for a 90db speaker, the plot would show 0 for 0 watts and 90 for 1 watt.  Once I have them hooked, I could have them research the sensitivity of their speakers (car speakers for the most part) and we could demonstrate the effects of various amp/speaker combination.

You should see the looks I get when I say "I have a 2 watt per channel amp and when I crank it up my wife complains".  They think I've lost it.  I had one student insist on coming over to the house to verify.  He was very impressed and I think he learned a lot from the experience. 

I've tried to make a few formulas myself but I must be missing something.  I can't get zero out for zero in unless I just subtract one in the formula and that seems like cheating. 

Any help will be fun, useful, and greatly appreciated.

[VoltSecond]

I think this is what you are after:

[ dB is specified into a constant resistance ]

We need 10 dB more power to sound twice as loud.  10dB more power is 10 times the power. 

20dB more power is four times as loud.  20dB more power is 100 times the power.

For amp power, dB = 10 * log( power). 

10 * log(10:1  ) = 10 * 1 = 10dB
10 * log(100:1) = 10 * 2 = 20dB

This means a 2W amp is 3 dBW, a 20W amp is 13 dBW and a 200W amp is 23 dBW.

We add the amp power to the speaker efficiency to get the sound power.

2W into a 100dB/W speaker gives 103dB of sound power
20W into a 90dB/W speaker gives 103dB of sound power
200W into a 80dB/W speaker gives 103dB of sound power.


The Math:

Log(x) is the "inverse" of 10 raised to the "X" power. Log(10^5) = 5. 
Log of zero doesn't work.  We can't raise 10 to any power and get zero. So there is no dB for zero watts.
1 microwatt would be -60dB

Adding logs is like multiplying.

2W being 3dB 
100dB/W being 10^10 sound power/electrical W

3dB power in + 100dB/W sound power / electrical W = 103 dB of sound power.

2W * 10^10 sound power/ W = 2 * 10^10 of sound power.
Converting this 10 * log( 2 * 10^10) = 103 dB

Good engineers take the easy way. Adding the dB is much easier than multiplying big numbers and 2W into an efficient speaker can rock your world.

Note: I've been told to turn it down with a 250mW headphone amp driving a set of horn speakers. . .

[Paul Joppa]

OK, it's late (i.e. after a martini plus dinner with a nice Syrah...) BUT what the heck....

Decibels are the log of a power RATIO. It's a base-ten logarithm, so a power ratio of 10:1 is 1 bel, or 10 decibels. So 10 log (power ratio) equals decibels. Ten watts is one bel, or 10 decibels, louder than 1 watt.

You can't divide by zero, so you can't ratio power to zero power - that's why speakers are rated in dB referred to 1 watt instead of absolute zero. The smallest audible sound pressure level is nominally zero dB - it's not zero sound, it's the smallest non-zero sound the human ear can hear. (It's also pretty close to the noise made by Brownian motion of the air particles that are adjacent to the eardrum -  there's no biological point in having hearing that is more sensitive than that!)

Power is proportional to voltage squared (it equals voltage squared over resistance) so dB equals 20 log voltage ratio - another teachable moment about logs and exponents.

[Doc B.]

See why I surround myself with these guys? I probably hold the record for getting a physics degree while taking the fewest math classes possible.

[porcupunctis]

Thanks everyone, this will be very helpful.  I can solve log problems all day but setting one up to match a physical phenomenon takes more than just the math knowledge.  I'm always amazed  at what I learn on this site. 

Let me get my head around this and I'll post back later.  Right now I have to go survey and mow a 400 meter track in an unmarked open field.  That's going to keep me busy for a while.

[Laudanum]

See why I surround myself with these guys? I probably hold the record for getting a physics degree while taking the fewest math classes possible.

The lack of requiring a bunch of math classes didnt have anything to do with my chosen major, but it was a fantastic side benefit.  I stink at math, always have.  To this day, I dont know how I passed those two statistics classes that were a part of my major.  I clearly remember to this day, almost 30 years later, how amazed I was that I scored a high B on the final of the toughest of those two classes and yet I remember next to nothing about statistics other than the words ... T score.   :-)    

[porcupunctis]

OK, with the information given, it only took a few minutes to make some really nice graphs using Geogebra.  The function I used looked like this:

f(x) = 10log(10, x) + 97

The log(10, x) is Geogebra's way of doing a base 10 log.  The 97 represents a 97dB efficient speaker.  f(x) will be the sound level in dB and x will represent the given watts.  I can graph several functions using different efficiencies and compare them on the graph. 

So, the next thing to blow their little minds with is the fact that the volume knob isn't linear.  With a 100 watt amp, you don't hit 1 watt at 1/10th the rotation.  Back in the 70s I had a 75 wpc amp that hit 1 watt around 1/3 of the rotation and 10 at around 2/3 of the rotation. 

I was also thinking about the maximum sound a given speaker can produce.  Certainly speakers have limits on the watts they can handle, but ignoring that, the most they can do will be limited by the size of the cone and the amount of excursion.  This is my hunch, am I thinking right?

On one website I noticed them citing the loudest noise possible was 194dB.  This must have something to do with the ability of air to transmit sound. 

This has been very interesting research and I really appreciate all the help everyone has given. 

[matthewmckay]

I think anything over 194db is considered a shock wave instead of a sound wave.

[Paul Joppa]

Yeah, 194dB is unrealistic. It represents an RMS pressure wave of 1.0 atmosphere at sea level - which is a peak sine wave of 2.4 to -0.4 atmosphere. Obviously you can't have negative absolute pressure. This is another teachable moment IMHO - all linear models fail at extreme values, and if you are using a linear model - which is extremely useful since the math is tractable - you do have to make sure it's used in it's nearly-linear region! (And of course even if you use a quadratic model, it may extend the range of linear model, but not by much - very quickly you have to add even higher order terms...)

How do I know this stuff? I had a career in aircraft noise, and the sound levels inside a jet engine can be pretty extreme! I actually did a bunch of measurements of sound inside engines and models of engines. For your students, I got my aerospace job partly through a teenage interest in sound and music. It came with a draft deferment so I didn't have to go to war in Vietnam, which I am very glad of. I have a few Vietnamese friends, they are wonderful people!

[porcupunctis]

Paul, thanks for sharing.  I often tell my kids that even if you pursue your passion, you still have no idea how you may end up using it or what kind of job you may be doing.

So, with the jet engines, were you looking (listening) for sounds that were potentially loud enough to cause stress and damage to the parts?  I often wonder about those that crank over-sized subs in small cars.  I know it damages their ears, but what else in the car is shaking apart? 

I had the pleasure to see Peter Frampton in a small theater here in Springfied, MO last night.  A truly fantastic show and most enjoyable.  However, once in a while I could swear I heard parts of the building resonating/shaking/whatever and making its own sound.  It reminded me of a time when I was still in High School and went to see Blue Oyster Cult play in the Shrine Mosque.  The Shrine was an old building even back then and we ended up standing with our backs to the far back wall.  I could hear that wall making its own noise and it made me fear that it was going to just crumble apart at any moment.

Am I really hearing the walls shake or is this some other sonic phenomenon?

[Paul Joppa]

Oh, sure, you can hear things rattle and shake. But it's hard to excite something acoustically enough to cause structural damage, unless the levels are literally ear-bleeding! Usually the only thing destroyed is the sound source. There's a mythbusters episode about car audio, pretty funny!

Parenthetically, in the early days of air-raid siren development they used wooden horns, which (I have been told) quickly "de-constructed" themselves.

The aircraft work was intended to learn exactly how the sound waves were moving inside the engine, so that we could either design quieter engines, or absorb the sound more efficiently on the engine's interior surfaces.

[Noskipallwd]

I don't watch a whole lot of T.V. But I do like the Mythbusters, they have done some interesting experiments using sound waves. The so called "Brown Note", subs in a car actually setting off firearms. The uber sub they built actually used the driveshaft, with a cam installed to actuate the driver. It began destroying the car in short order. The talk about linear models is interesting to me. The majority of the assays we perform in medical lab are linear in nature, most using spectrophotometry. They all have assay ranges, the point where the reaction is no longer linear.

Cheers
Shawn

[Doc B.]

When it comes to audio stuff the Mythbusters are really lucky in that the guys from Meyer Sound help them out with measurements and such. Meyer has about as much knowledge regarding the construction of great sounding LOUD systems as anyone. I've been given a tour of the factory and it is amazing.