by Neil Bauman, Ph.D.

October 29, 2016

A person asked,

How do you calculate the difference in sound intensity in decibels between any two sound intensities. For example, how do you calculate the increase in sound intensity between 0 dB and 15 dB or between 52 and 94 dB?

There is a mathematical relationship between decibels (dB) and sound intensities. It works like this. Each 10 dB increase results in a **10-fold** increase in sound **intensity** which we **perceive** as a **2-fold** increase in sound **volume**.

Thus, from 0 dB to 10 dB there is a 10-fold increase in sound intensity, just as there is from 10 dB to 20 dB or from 34 dB to 44 dB.

Note: Sound **intensity** is the **energy** (power) needed to produce a given level of sound. Don’t confuse sound intensity (the amount of energy needed to produce a given level of sound) with sound **volume** (the level at which we **perceive** the resulting sound.)

The table below shows the increase in sound intensity between 0 dB and each of the values listed.

Decibel

Value Increase in Sound Intensity Perceived Increase in Volume

0 dB

10 dB 10 times the sound intensity 2 times as loud

20 dB 100 (10 x 10) 4 (2 x 2)

30 dB 1,000 (10 x 10 x 10) etc. 8 (2 x 2 x 2) etc.

40 dB 10,000 16

50 dB 100,000 32

60 dB 1,000,000 64

70 dB 10,000,000 128

80 dB 100,000,000 256

90 dB 1,000,000,000 512

100 dB 10,000,000,000 1024

110 dB 100,000,000,000 2048

120 dB 1,000,000,000,000 4096

As you can see, these numbers quickly get large. For example, if you had a 120 dB loss at a certain frequency, in order to hear a sound at that frequency, it would have to be 1 trillion times as intense (it would require 1 trillion times the energy to produce it) as needed for a person who had “perfect” hearing (and thus could hear it at an intensity of 0 dB).

Note this well. Since our ears **perceive** sound logarithmically, we do not perceive a sound of 120 dB as being 1 trillion times louder than a sound of 0 dB. Rather, we perceive it as about 4,000 times louder.

Now that we have a little background, we are ready to proceed with the details of how to calculate the differences in sound intensities and relate them to decibel values.

Unfortunately, far too often people assume that there is a simple linear interpolation between any two decibel values. Thus, since there is a 10-fold increase between 10 dB and 20 dB in sound intensity, they assume the increase at the half-way point (15 dB in this case) is a 5-fold increase.

If you assumed this, you would be wrong. Even hearing health care professionals that should know don’t always get this right.

The reason you can’t just simply interpolate between two decibel values is because we are not working with linear numbers, but with logarithmic numbers. This means there is a logarithmic relationship between such values, not a linear relationship.

The formula for calculating the increase in sound intensity between two decibel values is:

x-fold increase in sound intensity = 10 ^{(ending dB value – starting dB value)/10}

(For the purposes of the above and below equations, always take the larger number as the ending dB value and the smaller number as the starting dB value.)

Therefore, to find the increase in sound intensity between 10 dB and 15 dB, you simply subtract the higher dB value from the lower value and divide the result by 10 to get the exponent. Calculating (15 – 10)/10 gives you an exponent of 0.5. Raising 10 to the 0.5 power (10 ^{0.5}) gives 3.162. Thus, the intensity increase between 10 dB and 15 dB is 3.162-fold.

In like manner, to calculate the difference in sound intensity between 52 dB and 94 dB, just follow the same procedure and use the same formula. (94-52)/10 gives an exponent of 4.2. 10 ^{4.2} = 15,848.9. Thus, the intensity increase between 52 dB and 94 dB is 15,848.9-fold. To put it another way, it takes 15,848.9 times as much energy to produce a sound of 94 dB than to produce a sound of 52 dB.

It’s easy to check your work to be sure you are in the right ball park. You know the difference you are working with is 42 dB. You already know that for a 40 dB increase, the intensity value is 10,000 times higher (10 x 10 x 10 x 10) and that for a 50 db increase, the value would be 100,000 times higher (10 x 10 x 10 x 10 x 10). (See above table.) So your answer must lie somewhere between these two values, and sure enough, it does.

To make things simple, in case you don’t have a fancy calculator*, here is a table to help you.

dB Difference x-fold Multiplier

1 1.259

2 1.584

3 1.995

4 2.512

5 3.162

6 3.981

7 5.011

8 6.309

9 7.943

10 10.000

In order to use this table, just take the multiplier figures for values between 1 and 10 and then move the decimal point to the right one place for each whole 10 dB difference.

Thus, if you want to find the difference in sound intensity between 3 dB and 9 dB, and since the value is less than 10 dB, just read off the value from the table for a 6 dB difference, namely 3.981. Thus for a 6 dB increase, there is a 3.981-fold increase in intensity.

If you want to find the sound intensity increase between 52 and 94 dB, you subtract 52 from the 94 to get 42 dB. Take the units figure (2) and from the table for a 2 dB difference, you see the multiplier is 1.584. Now to get your final answer, move the decimal to the right by the value of the tens figure (4) and you have a 15,840-fold increase in intensity. (If the decibel difference is larger than 100, then use the tens and hundreds figures. Thus if the difference was 124 dB, you’d move the decimal to the right by 12 decimal places.) That’s how simple it is.

And if you ever want to calculate how much louder you **perceive** one sound as compared to another you can do it by using the following formula.

perceived x-fold volume increase = 2^{ (ending dB value – starting dB value)/10}

Therefore, to find the perceived increase in sound volume between 10 dB and 15 dB, you simply subtract the higher dB value from the lower value and divide the result by 10 to get the exponent—(15 – 10)/10 gives you an exponent of 0.5. (So far, everything is the same as for calculating intensity differences. Now comes the change—you use base 2 rather than base 10.) Raising 2 to the 0.5 power (2^{ 0.5}) gives 1.4. Thus you would perceive the sound as being 1.4 times louder.

In like manner, to calculate the difference in perceived sound volume between 52 dB and 94 dB, just follow the same procedure and use the formula. (94-52)/10 gives an exponent of 4.2. 2^{ 4.2} = 18.4 times louder.

Note: Perceived volume varies from person to person so the calculated results may not agree with any given person’s subjective results, but it certainly puts you in the right ball park.

__________________

* Note: if you have an iPhone, you have a fancy built-in calculator. Swipe up from the bottom and you’ll see it there with your flashlight, timer and camera. When you hold your iPhone vertically you have a simple calculator. Turn your phone on its side and it automatically switches to a fancy scientific calculator where you have the 10^{x} and x^{y} functions.

Ron Peters says

how much louder is 51 dB the 49dB?

Neil Bauman, Ph.D. says

Hi Ron:

A 2 dB increase is 1.584 times louder, so 51 dB is 1.584 times as loud as 49 dB.

Cordially,

Neil

Jordan Hewitt says

He was probably asking about the perceived volume, in which case it is 1.148 times as loud, or roughly 15% louder.

James Gardner says

Hi Jordan, that’s really helpful. Can you confirm the formula you used to calculate that the perceived volume is 15% louder?

Many thanks.

Mog says

Taking this the other way, how much better is 55dB sound insulation than 45dB?

Neil Bauman, Ph.D. says

Hi Mog:

That’s easy. The difference is 10 dB which our ears hear as a doubling of sound level.

So sound insulation rated at 55 dB would be twice as good as insulation rated at 45 dB. To our ears, only half as much sound would get through.

Cordially,

Neil

Hope Welsh says

A nearby lawnmower has an intensity of about 10-2 W/m2.

What is this in dB?

If the mower is 2m away, how much sound power is it producing? (in Watts)

Using the hearing chart (the one with phone lines), compare the dB of the mower at 102, 103, and 104 Hz. At which of these three frequencies is the mower perceived as loudest?

How much louder (as a ratio) is 100dB than 70dB in I, intensity?

Neil Bauman, Ph.D. says

Hi Hope:

It sounds like you want me to do your homework for you. That’s not the purpose of this website.

Cordially,

Neil

James says

Hi Niel,

Great info. Thanks.

I’m having issues with my local transport authority regarding train horn noise. They say that horns have to be 85db at 100m. So my house, which is 100m away from the train line, cops 85db to my windows. Internally I measured the sound at 75db (just inside my windows). This occurs at any time of the day (24 hours). However, some horns, which are quiet on the db scale ie measured at 55db, are very piercing. They are at about 1000hz I believe. How does one cater for the frequency in perception of the loudness? A straightforward db measurement isn’t telling the whole story here… any comments or info appreciated. Thanks, Scott

Neil Bauman, Ph.D. says

Hi Scott:

You need to hook a tractor to your house and drag it away from the rail line! LOL Or move to the middle of the outback.

My hearing is so bad that 85 dB wouldn’t be very audible at 100 meters.

Come to think of it, when I was growing up, we lived less than 100 meters from the CPR mainline in Canada. The house would shake–especially the windows–so I’d know a train was going by. And in the early years it was a steam train, so I’d see the smoke or steam rising–our house was perhaps 25 meters above the tracks.

It’s interesting that the most penetrating sound is a 520 Hz.square wave. You can’t easily sleep through that so they are making more and more smoke detectors produce that sound.

Our ears are least sensitive to very low frequency sounds around 25 or 30 Hz and most sensitive to sounds between 2 kHz and 3 kHz.

Cordially,

Neil

Kayla says

how much louder is 50 db than 10 db?

Neil Bauman, Ph.D. says

Hi Kayla:

Obviously it is 40 dB louder. This means it takes 10,000 TIMES the energy to produce a sound at 50 db that one at 10 dB, but to human ears it sounds 16 TIMES louder.

Cordially,

Neil

Michael Smith says

How much louder (perceived by human ear) is 120 decibels and 165 decibels? Thanks.

Neil Bauman, Ph.D. says

Hi Michael:

The difference between the two is 45 dB which would sound roughly 20 times louder as your ear perceive it. Note that everyone’s ears perceive sound intensities slightly differently.

Cordially,

Neil

Nora says

Can you answer this question and please explain the answer for me 🙂 – ‘A 50 dB sound is ____ times louder than a 20 dB sound.’

Neil Bauman, Ph.D. says

Hi Nora:

That’s easy. The answer is that the energy (sound pressure in the air) to increase a sound by 30 dB is 1,000 times more. (Each 10 dB of sound increase is a factor of 10 times–so 50 – 20 dB = 30 dB. which is 10 x 10 x 10 = 1,000 times more.) However, our ears perceive each 10 dB increase as twice as loud (not 10 times as loud) so a 30 dB increase to us sounds 2 x 2 x 2 = 8 times as loud.

Cordially,

Neil

Alex says

Hey, wanted to ask how much louder is 125 DB to 110DB to the human ear ?

Neil Bauman, Ph.D. says

Hi Alex:

The difference is 15 dB, so the intensity difference factor has to lie between 10 and 100. In this case it is a factor of 31.62–so the sound pressure is 31.62 times greater, and you will perceive it as 2.83 times louder.

Cordially,

Neil

Donovan says

How do I find the intensity of 45db?

Neil Bauman, Ph.D. says

Hi Donovan:

I don’t understand your question. Are you meaning you want to know when a sound is at 45 dB intensity? For that you can use a sound meter, or one of the many sound Apps available for cell phones.

Cordially,

Neil

Dean says

Quite confused by all the numbers 😅. A restaurant opened next to my house, pointing to a main street. Our neighborhood was surprisingly quiet despite the proximity to a very busy area.

His vents are pointing backwards at the neighborhood, right at my house, and a tall building from the opposite back is bouncing the noise literally into my door front door!

When its on it moves around 50dB. When it’s off the range is up to 30dB.

I’d say the house is positioned 20-30 meters in each direction (to the vent and to the wall.

Neil Bauman, Ph.D. says

Hi Dean:

I understand, but was there a question you wanted to ask about this situation?

Cordially,

Neil

Harki says

Hi- I was considering some a specific tire and it produces between 65-68 dB compared to another tire which produces 64-66 dB. The rough number I can calculate (from your formulas) is about 15% increase to human hearing.

My question is, is that noticeable difference? For example, a mosquito buzzing 15% louder a few feet away would not be noticeable. A DJ playing music 15% louder would definitely be more noticeable. What other factors are needed to know this? How can I make the right purchase?

I can see that conversations are about 70dBs. Would the “more-noisy” tires be much worse for carrying out conversation in the car?

I guess we can calculate how much louder the conversation is with respect to each tire. Taking the middle-point (66.5dB and 65dB), we can see that less-noisy is about 41.5% louder than conversation, more-noisy is about 27% louder than conversation.

Is that a drastic difference?

Neil Bauman, Ph.D. says

Hi Harki:

Unless you have particularly sensitive ears, you don’t notice a 1 or 2 dB difference in sound intensity.

There are a lot of other factors you want to consider besides the tire dB rating (and that rating will change depending on the speed you are driving). For example, you need to factor in wind noise, motor noise, car speed, etc. to find a practical difference between the tires. Remember, every bit of noise from any source increases the perceived volume of the sounds around you.

Another factor you haven’t considered is whether you have a hearing loss or not. When you have a hearing loss, you need a greater separation between the background sounds and the speaker’s voice than does a person with normal hearing (called the signal to noise ratio). So every little bit you reduce the noise level increases the intelligibility of speech for a hard of hearing person.

Thus, if you want the easiest conversation, choose a quiet car and quiet tires and drive slower, and when/where traffic noise is at a minimum. Quiet tires alone aren’t the main consideration.

Cordially,

Neil

Jeff Hartman says

Hey Neil, you say to subtract the higher decibel value from the lower and divide by 10 (which would always result in a negative number… and incorrect calculation of the exponent), but in your actual calculations you subtract the lower dB value from the higher one. I think you mean to subtract the lower from the higher, right? I’m using this to lodge a complaint to the FCC for commercial volume compared to normal programming, which is consistently 4-6 dB louder on MSNBC during commercials,(law requires average volume to be the same}. Well, nless I’m an idiot…

Neil Bauman, Ph.D. says

Hi Jeff:

Where do I say that? I think you have misunderstood the equations.

The equations shows that you take the ending dB value (the larger or higher number) MINUS the starting dB value (the smaller or lower number), not that you subtract the higher value from the lower value. It’s the higher value minus the lower value.

Cordially,

Neil

Julia says

Hi Neil,

I wonder if you would be good enough to help me please?

I am suing Portsmouth City Council for statutory noise nuisance. Not too surprisingly PCC promptly made the planning consent unavailable (and everything to do with it) which has prevented me instructing anybody to do reports.

I eventually got PCC to take a sound check at midnight of the construction site and the reading is an average sound of 59.10 dB

My question is, is it possible for me to convert that to know how loud the highest sound is please?

I have tried downloading things on my mobile but none of them have been clear in what is an average dB or the loudest dB.

I fully understand if you would prefer not to reply so please don’t worry, I simply wonder if there is a calculation to find the answer.

Thank you so much,

Julia.

Neil Bauman, Ph.D. says

Hi Julia:

Taking a sound level reading is meaningless unless you have some firm parameters that you/they follow.

First, are they using the dBA or dBC rating. If they use the dBA scale, it gives less weight to low frequency sounds, so industrial noise can appear less than it really is.

Second, where are they measuring the sound–where you normally are, or at the work site? And if at the worksite, how far from the noise being produced? For example, the noise at the site might be say 100 dB, but where you are, it may only be 50 dB.

Third, how long do they measure the sound in order to get an average. Was it 10 seconds or 10 hours or what?

In order to answer your question, there is no way if you only have an average dB sound level to determine what the maximum sound levels were. Sound meters can measure the peak levels and the average levels. You want to know both.

Cordially,

Neil

David says

I had my vehicle undercoated with a view to reducing noise. (I have some hearing loss.) I used 2 different cell phone apps to measure the change. (one of them showed a filter of A). It appears the the reduction is from about 73 dB to 70dB on rough pavement and 70 to 67 on smooth pavement. I take it from the chart that this should be a perceived sound reduction of 1.995, which should translate into about a 50% sound reduction. Do I have this correct? How significant is this? Did I waste $325?

Neil Bauman, Ph.D. says

Hi David:

Since you got a 3 dB reduction in the intensity of the sound, this means the sound energy reaching your ears is about 50% as you correctly calculated. However, your ears perceive this new sound level (assuming you have average perception of sound) as being 81% as loud as the original sound, so you essentially achieved a 19% reduction in the perceived sound level.

Was it worth it? That is up to you to decide.

Cordially,

Neil

Alan says

After a lot of research, I chose Brand X generator to be installed at my residence. The tech specs showed 67 db @ 23 feet; but the db meter shows 81.2 db @ 23 feet. How many times louder is it?

City ordinance states that a generator cannot be over 70 db at night.

Neil Bauman, Ph.D. says

Hi Alan:

The increase is 14.2 dB. Thus the sound intensity would be 26.3 times as loud, but human ears would perceive it as being 2.67 times as loud.

At what distance does the city ordinance specify the sound level is to be measured? 23 feet?

Cordially,

Neil

Chris Gore says

Hi there

A new expressway was opened 300m from our home in july, We live in a rural farming area with only fields around us. Direct line of site to expressway, pre construction dbl taken by road ompany at our home 35dbl. Post construction 17dbl increase to 52d bl . Is this a four fold increase.?

We installed aoustic glass but still ca not sleep properly at night when trucks roar past at 110.

Neil Bauman, Ph.D. says

Hi Chris:

To answer your question. The increase in sound intensity in 50.12 times or a 50 fold increase in sound intensity. You would perceive this as 3.25 times louder so you were pretty close to the correct answer.

That is the theoretical answer, but each person, because we are all different, may perceive the increase as a bit more or less than this theoretical calculation.

Cordially,

Neil

Chris Gore says

Thank you

Simon Deakin says

Hi Chris, Just bought a Porsche 911 GT3 Touring on Michelin Cup 2 tyres. Rear tyres are the noisiest & the tyre chart show a dB rating of 74dB. If I swap to Pirelli P zero’s they have a dB rating of 73db. Does this 1 db difference meanIi’ll perceive about a 6.7% reduction in noise? If so i’ll probably not notice the difference but maybe? Thanks

Neil Bauman, Ph.D. says

Hi Simon:

I doubt you’d be able to really hear a difference of only 1 dB, so I wouldn’t buy new tires at this point. But when you finally need to replace the tires, at that point, you can shop around for the quietest tires you can find.

Cordially,

Neil

john says

Dear Dr. Neil,

I’ve come across information stating that prolonged exposure to sound levels of 85db can cause hearing damage. My speculation is that this reference might pertain to band-limited white noise integrated over a frequency bandwidth of 1 to 4kHz. I believe this range is especially susceptible to damage, though I’d appreciate confirmation on this.

Considering the hairs in the inner ear are likely to resonate within a narrow frequency band, if we divide the range into 64 such bands, would the power in each band be 18db less than the integrated power across the entire range?

With this in mind, would it be accurate to suggest that a narrow band tone, spread over 1/64 of 3kHz, could begin to cause damage at sound levels of 67db?

Furthermore, could you shed light on the frequency selectiveness of the ear structures prone to damage?

Thank you for your insights.

Warm regards,

John

PS: I’m a layperson who recently found out that my hearing loss is 80db at 2kHz and 85db at 4kHz. I’m considering if it would be beneficial to set my hearing aid below optimal levels at these frequencies to prevent further damage.

Neil Bauman, Ph.D. says

Hi John:

Prolonged exposure to 85 dB sounds can cause hearing loss, but it typically takes years or decades before it shows up. And since everyone is different, this can vary greatly from person to person. Furthermore, rather than using 85 dB as the starting point like OSHA does (85 dB average over 8 hours for some years can cause hearing loss), the Environmental Protection Agency (EPA) uses a figure of an average of 70 dB over a lifetime and states that it will not cause any hearing loss.

Damage to hair cells typically starts at the base of the cochlea and works towards the tip. Since the high-frequency sounds are processed at the base and low-frequency sounds at the tip, hearing loss typically starts in the highest sounds you can hear–up around 20 Khz and works down. So I don’t agree with your premise that 1 to 4 kHz is where most hearing loss takes place.

Isn’t band limited white noise an oxymoron? White noise is ALL frequencies of sound. Once you cut out certain frequencies, it is technically not white noise anymore.

Here’s a bit of background information for you taken from my book, “Hypersensitive to Sound?” The single row of hair cells in our inner ears are already divided into 24 perceptual units called critical bands. “Each hair cell in your ears vibrates at a different frequency. Rather than working independently, these hair cells work together in their critical bands. Each critical band is about 16% wider than its central frequency. (Thus, for example, the critical band for a tone of 1,000 Hz. is 160 Hz. wide—ranging from 920 Hz. to 1,080 Hz.)”

If you set your hearing aids below optimal levels you won’t hear as well, so the trade off (to your way of thinking) would be not hear well and preserve hearing, or hear as well as possible with possible hearing damage taking place over a number of decades.

In my case, I choose to hear well when people are talking to me, but often the rest of the time I take my aids off and give my ears a rest. Now that I am in a quiet environment most of the time, that isn’t necessary.

Cordially,

Neil