# decibels, loudness, and energy **URL:** \ **Category:** Factual Questions\ **Created:** [March 10, 2008, 10:01pm UTC](https://boards.straightdope.com/t/decibels-loudness-and-energy/440893 "2008-03-10T22:01:29Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![Cardinal](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/cardinal/32/4000_2.png) [@Cardinal](https://boards.straightdope.com/u/Cardinal)\ **Post date:** [March 10, 2008, 10:01pm UTC](https://boards.straightdope.com/t/decibels-loudness-and-energy/440893/1 "2008-03-10T22:01:29Z")
I can’t figure this all out, and we’re becoming convinced that the book we’re using has a misprint. What I think I need is a table. Can you complete this table? Reference level \_\_\_ bel 1 bel \_\_\_ x louder \_\_\_\_ x more energy 2 bel \_\_\_ x louder \_\_\_\_ x more energy 3 bel \_\_\_ x louder \_\_\_\_ x more energy 4 bel \_\_\_ x louder \_\_\_\_ x more energy Just for further info/checking: 1 db \_\_\_ x louder \_\_\_\_ x more energy 2 db \_\_\_ x louder \_\_\_\_ x more energy 3 db \_\_\_ x louder \_\_\_\_ x more energy 4 db \_\_\_ x louder \_\_\_\_ x more energy ---
**Author:** ![commasense](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/commasense/32/3017_2.png) [@commasense](https://boards.straightdope.com/u/commasense)\ **Post date:** [March 11, 2008, 12:31am UTC](https://boards.straightdope.com/t/decibels-loudness-and-energy/440893/2 "2008-03-11T00:31:05Z")
How does your source compare to the [Wikipedia](http://en.wikipedia.org/wiki/Decibel) entry: > [@Wikipedia](#): > > The decibel (dB) is a logarithmic unit of measurement that expresses the magnitude of a physical quantity (usually power or intensity) relative to a specified or implied reference level. Since it expresses a ratio of two (same unit) quantities, it is a dimensionless unit. A decibel is one tenth of a bel (B). > > [snip] > > It is seen that there is a 10 dB increase (decrease) for each factor 10 increase (decrease) in the ratio of the two power levels, and approximately a 3 dB increase (decrease) for every factor 2 increase (decrease). In exact terms, the factor is 103/10, or 1.9953, about 0.24% different from exactly 2. Similarly, an increase of 3 dB implies an increase in voltage by a factor of approximately √2, or about 1.41, an increase of 6 dB corresponds to approximately four times the power and twice the voltage, and so on. (In exact terms the power factor is 106/10, or about 3.9811, a relative error of about 0.5%.) ---
**Author:** ![Q.E.D](https://avatars.discourse-cdn.com/v4/letter/q/51bf81/32.png) [@Q.E.D](https://boards.straightdope.com/u/Q.E.D)\ **Post date:** [March 11, 2008, 2:19am UTC](https://boards.straightdope.com/t/decibels-loudness-and-energy/440893/3 "2008-03-11T02:19:03Z")
It’s important to understand that dB isn’t a measure of loudness, which is a largely subjective quantity. As it applies to sound, dB is normally used to express sound pressure level, although there are several weighting schemes which attempt to approximate sound level by compensating for the frequency response of the human ear. The reference level for 0 dB, the baseline level, is normally the threshold of hearing for the average human ear, which is a SPL of 20 microPascals, a quantity that Wiki states is about the same as the sound of a mosquito from 3 feet distant. Every 3 dB represents a doubling of this level; every 10 dB is an increase of one order of magnitude. ---
**Author:** ![Q.E.D](https://avatars.discourse-cdn.com/v4/letter/q/51bf81/32.png) [@Q.E.D](https://boards.straightdope.com/u/Q.E.D)\ **Post date:** [March 11, 2008, 2:25am UTC](https://boards.straightdope.com/t/decibels-loudness-and-energy/440893/4 "2008-03-11T02:25:57Z")
[QUOTE=Q.E.D.] …about the same as the sound of a mosquito from 3 feet distant. [/QUOTE] Er, make that 3 meters. Yeah, I can read, really. ---
**Author:** ![Cardinal](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/cardinal/32/4000_2.png) [@Cardinal](https://boards.straightdope.com/u/Cardinal)\ **Post date:** [March 11, 2008, 5:39am UTC](https://boards.straightdope.com/t/decibels-loudness-and-energy/440893/5 "2008-03-11T05:39:19Z")
[QUOTE=Barron’s Physics the Easy Way] If something sounds twice as loud, it has 10 times the intensity. A sound whose intensity is twice as much as the reference level has a loudness of 2 bels; if you can barely hear a murmuring brook, another that has a loudness of 3 bels is producing 1000 times as much energy. [/QUOTE] But everything else I’ve read says that a 2 bel increase is a 10 fold increase in intensity, while “sounding” twice as loud. How then does a 3 bel increase have 1000 times the original energy? I can’t reconcile this, and neither could my MIT engineer dad. > [@Wikipedia](#): > > It is seen that there is a 10 dB increase (decrease) for each factor 10 increase (decrease) in the ratio of the two power levels This also seems to back up my understanding. If the factor of 1000 is really supposed to be 100, then it’s about the fifth error we’ve found in this, and things start to make sense. ---
**Author:** ![treis](https://avatars.discourse-cdn.com/v4/letter/t/bc79bd/32.png) [@treis](https://boards.straightdope.com/u/treis)\ **Post date:** [March 11, 2008, 5:56am UTC](https://boards.straightdope.com/t/decibels-loudness-and-energy/440893/6 "2008-03-11T05:56:07Z")
> [@Cardinal](#): > > How then does a 3 bel increase have 1000 times the original energy? I can’t reconcile this, and neither could my MIT engineer dad. A 2 bel increase should be a 100x increase in intensity. bel=log(X/Xo) bel=log(Xo/Xo)=log(1)=0 bel=log(10_Xo/Xo)=log(10)=1 bel=log(100_Xo/Xo)=log(100)=2 bel=log(1000\*Xo/Xo)=log(1000)=3 ---
**Author:** ![scr4](https://avatars.discourse-cdn.com/v4/letter/s/59ef9b/32.png) [@scr4](https://boards.straightdope.com/u/scr4)\ **Post date:** [March 11, 2008, 6:01am UTC](https://boards.straightdope.com/t/decibels-loudness-and-energy/440893/7 "2008-03-11T06:01:19Z")
[QUOTE=Cardinal] But everything else I’ve read says that a 2 bel increase is a 10 fold increase in intensity, while “sounding” twice as loud. How then does a 3 bel increase have 1000 times the original energy? [/quote] Maybe you’re confusing amplitude and energy? Energy is proportional to square of amplitude. A 1-bel increase is a 10-fold increase in energy, which means a sqrt(10)=3.1 fold increase in amplitude. A 2-bel increase is a 100-fold increase in energy, or 10-fold increase in amplitude. A 3-bel increase is a 1000-fold increase in energy, or sqrt(1000)=31 fold increase in amplitude. ---
**Author:** ![Cardinal](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/cardinal/32/4000_2.png) [@Cardinal](https://boards.straightdope.com/u/Cardinal)\ **Post date:** [March 11, 2008, 5:10pm UTC](https://boards.straightdope.com/t/decibels-loudness-and-energy/440893/8 "2008-03-11T17:10:49Z")
[QUOTE=treis] A 2 bel increase should be a 100x increase in intensity. bel=log(X/Xo) bel=log(Xo/Xo)=log(1)=0 bel=log(10_Xo/Xo)=log(10)=1 bel=log(100_Xo/Xo)=log(100)=2 bel=log(1000\*Xo/Xo)=log(1000)=3 [/QUOTE] This starts to approach the table I asked for. This starts to work. What is still confusing is that 2 bels is supposed to be twice as loud. What then is the perceived loudness increase at 1 bel increase in intensity/energy? And (yes, I’m complaining), I think all this back and forth could have been solved if someone would have just filled in a bit of the table I asked for. Just throwing quotes in from other web pages and so on not only isn’t what was asked for, it’s kind of implying that if I’d just do some extremely simple research, it would be obvious. I think I implied even in the OP that I’m not just running here first because I want someone else to do my thinking for me. ---
**Author:** ![Cardinal](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/cardinal/32/4000_2.png) [@Cardinal](https://boards.straightdope.com/u/Cardinal)\ **Post date:** [March 12, 2008, 12:46am UTC](https://boards.straightdope.com/t/decibels-loudness-and-energy/440893/9 "2008-03-12T00:46:53Z")
OK, after consulting another physics book, this is how I understand it 1 bel 10x more energy 2x as loud 2 bel 100x more energy 4x as loud 3 bel 1000x more energy 8x as loud ---
**Author:** ![Mines\_Mystique](https://avatars.discourse-cdn.com/v4/letter/m/d26b3c/32.png) [@Mines\_Mystique](https://boards.straightdope.com/u/Mines_Mystique)\ **Post date:** [March 12, 2008, 4:05pm UTC](https://boards.straightdope.com/t/decibels-loudness-and-energy/440893/10 "2008-03-12T16:05:46Z")
[QUOTE=Cardinal] OK, after consulting another physics book, this is how I understand it 1 bel 10x more energy 2x as loud 2 bel 100x more energy 4x as loud 3 bel 1000x more energy 8x as loud [/QUOTE] Close, but not quite. Bels and decibels are used to work with power ratios, rather than energy. For example, since decibels are most common in sound, lets use a real world example. The [JBL Vertec V4889](http://www.jblpro.com/vertec1/new_vertec/vt4889.html) is a loudspeaker you will commonly see when you go to see large arena & stadium concerts. It is a 3-way speaker, needing separate amplifiers for low, mid, and high frequencies. The low frequency section has a rating of 99 dB @ 1 W/1 m. What this means is that when you give the low frequency section of the speaker 1 W, you will measure an unweighted SPL of 99 dB 1 meter away from the speaker. Now, I’ll fill in you’re chart starting with this information 1 W=99 dB SPL 2 W=105 dB SPL = 2 x the power = 2 x as loud = +6 dB = +.6 Bel 4 W=111 dB SPL = 4 x the power = 4 x as loud = +12 dB = +1.2 Bel 8 W=117 dB SPL = 8 x the power = 8 x as loud = +18 dB = +1.8 Bel So, you can see that an increase of 1 Bel (10 dB) gets you approximately 4 times the power and 4 times the loudness, while an increase of 2 Bel (20 dB) is approximately 8 times the power and 8 times the loudness. Hope that helps. Mines Mystique ---
**Author:** ![Cardinal](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/cardinal/32/4000_2.png) [@Cardinal](https://boards.straightdope.com/u/Cardinal)\ **Post date:** [March 12, 2008, 7:00pm UTC](https://boards.straightdope.com/t/decibels-loudness-and-energy/440893/11 "2008-03-12T19:00:07Z")
Thanks for chiming in. Can anyone reconcile the table supplied with this? [QUOTE=Giancoli Physics] Producing a sound twice as loud requires 10x the intensity. This is roughly valid at any sound level for frequencies near the middle of the audible range. A sound wave of intensity of 10 to the -2 W/meters squared sounds like it is about twice as loud as one at 10 to the -3 W/meters squared, and 4x as loud as 10 to the -4 W/meters squared. [/QUOTE] ---
**Author:** ![CurtC](https://avatars.discourse-cdn.com/v4/letter/c/ce73a5/32.png) [@CurtC](https://boards.straightdope.com/u/CurtC)\ **Post date:** [March 12, 2008, 7:46pm UTC](https://boards.straightdope.com/t/decibels-loudness-and-energy/440893/12 "2008-03-12T19:46:46Z")
I don’t deal in sound levels too often, but I deal in dB measurements all day every day. A few points: - I _never_ see anyone use bels, never. It’s always dB. - I don’t know what the phrase “sounds like it’s twice as loud” means. I can’t imagine any context in which that would be meaningful. - The ratios are for _power_, not energy. - People here have been throwing around the terms intensity (amplitude) and power - these are different. Power goes up with the square of the amplitude, therefore a 20 dB increase is 10x the amplitude and 100x the power. In your chart, you ask for how many times louder, but what do you mean by that? Are you referring to amplitude? If so, here are your answers: Reference level \_\_\_ bel 1 bel 3 x [del]louder[/del] amplitude \_\_\_\_ 10 x more [del]energy[/del] power 2 bel 10x [del]louder[/del] amplitude\_\_\_\_ 100 x more [del]energy[/del] power 3 bel 30x [del]louder[/del] amplitude\_\_\_\_ 1000 x more [del]energy[/del] power 4 bel 100x [del]louder[/del] amplitude\_\_\_\_ 10000 x more [del]energy[/del] power Just for further info/checking: 1 db 1.12 x [del]louder[/del] amplitude 1.26 x more [del]energy[/del] power 2 db 1.26 x [del]louder[/del] amplitude 1.58 x more [del]energy[/del] power 3 db 1.41 x [del]louder[/del] amplitude 2.00 x more [del]energy[/del] power 4 db 1.58 x [del]louder[/del] amplitude 2.51 x more [del]energy[/del] power ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [March 12, 2008, 11:50pm UTC](https://boards.straightdope.com/t/decibels-loudness-and-energy/440893/13 "2008-03-12T23:50:51Z")
“Louder” probably refers to something with the physiology of human perception of sound. I know that human vision works on an approximately logarithmic response scale, and I’m pretty sure that hearing does, too. That’s why we use decibels to measure loudness in the first place, rather than some linear unit. But of course, it’s very difficult to quantitatively assess human response to stimuli, so any figure you see for how loud something “seems” will be somewhat fuzzy. ---
**Author:** ![Cardinal](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/cardinal/32/4000_2.png) [@Cardinal](https://boards.straightdope.com/u/Cardinal)\ **Post date:** [March 13, 2008, 11:39am UTC](https://boards.straightdope.com/t/decibels-loudness-and-energy/440893/14 "2008-03-13T11:39:07Z")
[QUOTE=CurtC] I don’t deal in sound levels too often, but I deal in dB measurements all day every day. A few points: - I _never_ see anyone use bels, never. It’s always dB. [/QUOTE] I know we usually use dB, but these students have never seen this stuff, so I’m starting at the fundamental level. Plus, the math for dB is kind of confusing, as a 3dB increase is more like a doubling in power. I know dB is based on power, but since the time isn’t changing, it must be the energy component of the Watt that’s changing. I didn’t know that about the amplitude. This book is focusing on sound waves specifically, and so is making points about human perceived loudness. If I can ask for more, can someone explain how the amplitude supplied by CurtC is relating to the power increases? 3x amplitude is 10x power? ---
**Author:** ![CurtC](https://avatars.discourse-cdn.com/v4/letter/c/ce73a5/32.png) [@CurtC](https://boards.straightdope.com/u/CurtC)\ **Post date:** [March 13, 2008, 2:29pm UTC](https://boards.straightdope.com/t/decibels-loudness-and-energy/440893/15 "2008-03-13T14:29:37Z")
[QUOTE=Cardinal] I know dB is based on power, but since the time isn’t changing, it must be the energy component of the Watt that’s changing. [/quote] I have no idea what you’re saying. Getting back to the point of how dBs talk about power ratios, not energy ratios, it would be like asking “how much faster is 40 miles compared to 20 miles?” Those aren’t speeds, they’re distances. > [@](#): > > I didn’t know that about the amplitude. This book is focusing on sound waves specifically, and so is making points about human perceived loudness. My thinking was that if someone put me in a test lab and asked me to say when one sound was “twice as loud” as another, I would be completely baffled about what he wants me to do. No human can do this, and if you ask a set of random people to give you that information, they’ll either be making stuff up or applying some misunderstanding of their perception to pick an incorrect answer. > [@](#): > > If I can ask for more, can someone explain how the amplitude supplied by CurtC is relating to the power increases? 3x amplitude is 10x power? Sound is just a wave of air pressure changes, the molecules move back-and-forth between the source and the listener, propagating the wave. If you start with a bigger stimulus that causes the molecules to move twice as much as they did before, that is what’s referred to as twice the amplitude. If you measure the power transmitted by this, it will be four times as much. Power of a wave is proportional to the square of the amplitude. To answer your question, if a signal has ten times the power of another signal (10 dB more), then the amplitude ratio will be the square root of ten, 3.16 - about three. ---
**Author:** ![Cardinal](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/cardinal/32/4000_2.png) [@Cardinal](https://boards.straightdope.com/u/Cardinal)\ **Post date:** [March 13, 2008, 5:04pm UTC](https://boards.straightdope.com/t/decibels-loudness-and-energy/440893/16 "2008-03-13T17:04:37Z")
[QUOTE=CurtC] I have no idea what you’re saying. Getting back to the point of how dBs talk about power ratios, not energy ratios, it would be like asking “how much faster is 40 miles compared to 20 miles?” Those aren’t speeds, they’re distances. [/QUOTE] Watt = Joule/s Since the seconds aren’t changing, it must be the Joules, the energy that’s changing. ---
**Author:** ![Cardinal](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/cardinal/32/4000_2.png) [@Cardinal](https://boards.straightdope.com/u/Cardinal)\ **Post date:** [March 13, 2008, 5:14pm UTC](https://boards.straightdope.com/t/decibels-loudness-and-energy/440893/17 "2008-03-13T17:14:58Z")
Thanks for the info about the power to amplitude relationship. ---
**Author:** ![CurtC](https://avatars.discourse-cdn.com/v4/letter/c/ce73a5/32.png) [@CurtC](https://boards.straightdope.com/u/CurtC)\ **Post date:** [March 13, 2008, 5:38pm UTC](https://boards.straightdope.com/t/decibels-loudness-and-energy/440893/18 "2008-03-13T17:38:43Z")
[QUOTE=Cardinal] Watt = Joule/s Since the seconds aren’t changing, it must be the Joules, the energy that’s changing. [/QUOTE] What I was saying is that the distance-speed analogy fits this perfectly. If someone asks how much faster is 40 miles compared to 20 miles, what do you say? Maybe to him, he’s assuming you cover those distances in the same amount of time, therefore he’s thinking the answer is “twice as much.” But that’s a pretty sloppy and ambiguous way to express it. You made the assumption that “the seconds aren’t changing.” Why? What is that from? We’re comparing ratios of power levels, not energy. ---
**Author:** ![Cardinal](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/cardinal/32/4000_2.png) [@Cardinal](https://boards.straightdope.com/u/Cardinal)\ **Post date:** [March 15, 2008, 8:06am UTC](https://boards.straightdope.com/t/decibels-loudness-and-energy/440893/19 "2008-03-15T08:06:23Z")
The seconds aren’t changing because in real terms, you can’t slow or speed time. You can alter the energy output, though. That’s all that could realistically be changed. ---
**Author:** ![Mines\_Mystique](https://avatars.discourse-cdn.com/v4/letter/m/d26b3c/32.png) [@Mines\_Mystique](https://boards.straightdope.com/u/Mines_Mystique)\ **Post date:** [March 17, 2008, 2:32am UTC](https://boards.straightdope.com/t/decibels-loudness-and-energy/440893/20 "2008-03-17T02:32:42Z")
[QUOTE=Cardinal] The seconds aren’t changing because in real terms, you can’t slow or speed time. You can alter the energy output, though. That’s all that could realistically be changed. [/QUOTE] And energy output is power, measured in terms of Watts, not Joules.