Decibel Intensity level question

[BirdieGone]

1. A jackhammer produces a sound of 111 dB IL. What is the sound level when 8 jackhammers are operating simultaneously? When 100 jackhammers are operating simultaneously? When 400 jackhammers are operating simultaneously?2. 1 jackhammer = 111 dBIL, 2 jackhammers = 114 dbIL, 4 jackhammers = 117 dbIL. When two sounds are added together, the increase in level is 3 dB. 3. I got 120 dbIL for 8 jackhammers. But I can't seem to figure out how to find 100 jackhammers and 400 jackhammers.

 

[NFuller]

increase in level is 3 dB.

Just to be a little more accurate, the increase will be $10\text{log}_{10}(2)\approx 3.0103$. The rounding may be important when you are adding 100+ sounds together.

3. I got 120 dbIL for 8 jackhammers. But I can't seem to figure out how to find 100 jackhammers and 400 jackhammers.

This is all coming from the definition of the decibel.
$$L=10\text{log}_{10}\left(\frac{P}{P_{0}}\right)\text{dB}$$
When you add two sounds together, the power adds, $P_{T}=P_{1}+P_{2}$. If the sounds are identical $P_{1}=P_{2}$ so $P_{T}=2P_{1}$ i.e. the power doubles. The change in loudness is then
$$\Delta L=L_{P_{1}+P_{2}}-L_{P_{1}}=10\text{log}_{10}\left(\frac{2P_{1}}{P_{0}}\right)\text{dB}-10\text{log}_{10}\left(\frac{P_{1}}{P_{0}}\right)\text{dB}$$
Using the quotient rule for logarithms we get
$$\Delta L=10\text{log}_{10}\left(\frac{2P_{1}/P_{0}}{P_{1}/P_{0}}\right)\text{dB}=10\text{log}_{10}\left(2\right)\text{dB}\approx 3.0103\text{dB}$$
So what would the change in loudness for an arbitrary number $N$ of identical sounds added together be?

 

[J Johnson]

Of more general interest is the addition of an arbitrary number of sources at arbitrary decibel levels. In applied acoustics we convert each decibel level to its power equivalent, add the list up, and convert the sum back into decibel notation.
if your first dB level is, say 95, its power is 10^(95/10). That's a large number you don't have to worry about, 3162277660 units. Just store it to add to the other power equivalents. I'll assume the second one is 98.6 dB, so its converted value is 10^(9.86). (I divided by 10 there, just like the first one.) The total of the two powers is 1.04066...x 10^10 . Taking the base 10 logarithm of this sum yields 10.0173... Multiply this by 10 (remember we divided by 10 before to get the power values, so have to multiply the sum by 10 now to get back to the decibel notation). The sum of of a 95 db and a 98.6 db source is 100.0173 dB.

The history of decibel notation is interesting. The unit "Bel" is in honor of Alexander Graham Bell. The initial logarthmic units were too small to be convenient, so they were upped by a factor of 10, so that the range of human hearing would be represented by a span from approximately 1 up to 100, and a little more. Human factors figure widely in intensity units! Human sensory responses tend to be logarithmic in nature; don't ask me why.

Reference: Acoustics, by Leo Beranek, 1993; ISBN 0-88318-494-X

 

[J Johnson]

Intensity level is one thing, like power level, but to get sound power level at a distance, where a correction needs to be added to account for losses due to wavefront spreading, the relative dimensions of the radiating sound source, reflections from reflecting surfaces also reaching the observer, etc. Doing this within a closed space further complicates the picture, but it's always good, first, to become familiar with the basics, like power to pressure and vice versa. Get into acoustics, and you're down another damned interesting rabbit hole!

 

[collinsmark]

Hello @BirdieGone,

Welcome to Physics Forums (PF)!

1. A jackhammer produces a sound of 111 dB IL. What is the sound level when 8 jackhammers are operating simultaneously? When 100 jackhammers are operating simultaneously? When 400 jackhammers are operating simultaneously?2. 1 jackhammer = 111 dBIL, 2 jackhammers = 114 dbIL, 4 jackhammers = 117 dbIL. When two sounds are added together, the increase in level is 3 dB. 3. I got 120 dbIL for 8 jackhammers. But I can't seem to figure out how to find 100 jackhammers and 400 jackhammers.

Good job on the 120 dB for the 8 jackhammers. That is correct.

As you've surmised, doubling the power is equivalent to adding ~3 dB. But there's one more relationship you should memorize for the other problems: A tenfold increase in power corresponds to what in dB?

• Doubling the power corresponds to adding roughly 3 dB (and halving the power corresponds to subtracting 3 dB).
• A tenfold increase in power corresponds to adding ___ dB (and a tenfold decrease in power corresponds to subtracting ___ dB).

Fill in the blanks and memorize those.

These problems (in the original post) are designed such that they can be done in your head, without a calculator or even scratch paper. Once you fill in the blanks above and memorize those two rules you should have all you need.

 

[davenn]

The initial logarthmic units were too small to be convenient, so they were upped by a factor of 10,

What initial logarithmic units ?

I hope you are not trying to tell us that 1dB is 10 x 1 Bel ?

1 decibel = 1/10 (1 tenth) of a Bel, NOT 1 Bel x 10