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## Physics library

### Course: Physics library > Unit 8

Lesson 4: Sound# Decibel Scale

Discover the magic of the Decibel Scale, the key to measuring sound intensity. Unpack the equation Beta = 10 log (I/10^-12), and marvel at the sensitivity of human hearing. Explore how logarithms transform large scales into manageable numbers, making the Decibel Scale a powerful tool in physics. Created by David SantoPietro.

## Want to join the conversation?

- What causes me to have ringing in the ear(which is caused by inflammation) sometimes because I have never had an ear infection(which often causes inflammation).

And why is it that my ears get pressurized sometimes even if I don't have an allergic reaction or I am not changing altitude fast enough?(13 votes)- Ringing in your ear is called Tinnitus. google something about it. If it is affecting your quality of life, you should see a doctor(32 votes)

- Why do we even bother to use the Decibel Scale? Instead, why can't we use the Bel Scale? Or does it make your result neater...(4 votes)
- The decibel measures sound pressure or electrical pressure (voltage) levels. It is a logarithmic unit that describes a ratio of two intensities, such as two different sound pressures, two different voltages, and so on. A bel (named after Alexander Graham Bell) is a base-ten logarithm of the ratio between two signals. This means that for every additional bel on the scale, the signal represented is ten times stronger. For example, the sound pressure level of a loud sound can be billions of times stronger than a quiet sound. Written logarithmically, one billion (1,000,000,000 or 109) is simply 9. Decibels make the numbers much easier to work with.

In practice, a bel is a bit too large to use for measuring sound, so a one-tenth unit called the decibel is used instead. The reason for using decibels instead of bels is no different from the reason for measuring shoe size in, say, centimeters instead of meters; it is a more practical unit.(19 votes)

- This is irrelevant but just curious; if it's a decibel shouldn't that be a tenth of a bel and you'd instead be multiplying by 1/10, not multiply by 10? Or is a "bel" equivalent to 100 * log(../...)(8 votes)
- A decibel is one tenth of a bel and the reason it is multiplied by 10 rather than divided is because it is on the right hand side of the equation. The log finds the number of bels then you multiply the number of bels by 10 to get decibels.(5 votes)

- Don't different people have different thresholds of hearing? I mean it seems like some people can hear softer sounds than other people. Why is that?(7 votes)
- Well part of it is how healthy your ears are which is why if you have an ear infection you can't hear as well.

Age is also a factor because old people hear in a shorter range and infants in a wider one respectively.

Genetics plays a role, in particular epigenetics which is the gene expression patterns. This could make your cochlea have more hair cells and thus you would be able to hear better.

However scientists like to work with averages instead of wide ranges and it turns out that 10^-12 W/m^2 is the average threshold of human hearing with all factors taken into consideration.(5 votes)

- why dosen't the intensity change the velocity of the sound wave ?(2 votes)
- Because the velocity of a sound wave is constant for each medium ^^(7 votes)

- how is intensity of sound related to loudness??(2 votes)
- in simple terms, the loudness increases with intensity.

in more specific terms,

a sound gets louder as the amplitude increases

intensity is proportional to the square of the amplitude(5 votes)

- David makes things so easy to understand! Awesome teacher :)(3 votes)
- My reference book says human ear can hear sound wave with a frequency ranging from 20Hz to 20,000Hz, and human ear has the highest sensitivity for sound wave around 3000Hz. What's the link between the frequency of sound wave and the intensity of sound wave? If the base line, lg(10^-12/10^-12) dB or 0dB, corresponds to a frequency of 2997Hz, how a human ear is possible to percept sound wave weaker than that 2997Hz?(2 votes)
- There is no link between frequency and intensity. Intensity is the amplitude of the wave.

However the ear does not respond the same to all frequencies, so the same intensity at one frequency might sound louder than it does at another frequency. So the lowest intensity noise can be heard (according to your data, which I have not checked for accuracy) at 3000 Hz. To hear a noise at another frequency it would have to have greater intensity than that minimum that can be perceived at 3000 Hz.(2 votes)

- what does it imply when you have 20Log instead of 10Log(2 votes)
- Hi Chima,

As a general statement when we see 10Log(some_value) we are talking about power. If you were to talk to an electrical engineer and mention 20 Log(some_value) you would be understood to be talking about current or voltage.

Now for the math - there is a root two relationship between these values:

Power = (voltage squared) / resistance

Power = (current squared) * resistance

If you follow the log rules you will be able to see how an exponent (2 in this case) changes a 10 to 20.

Regards,

APD(2 votes)

- If you have a sound from "one" source the human ear a sound intensity level (SIL). But if the sound comes from two sorces with the same SIL a human ear says that the sound is stronger. How can this be when the SIL is the same on both sorces?(1 vote)
- The sound waves interfere with each other, with the resultant amplitude being the sum of the original waves' amplitude. The intensity I will obey the equation

sqrt(I) = Σsqrt(In) , where In are the intensities of the sources' waves.(2 votes)

## Video transcript

- [Voiceover] All right,
I'm going to tell you about the Decibel Scale. This is the scale that we use to figure out the loudness of a sound. The equation that goes along
with it looks like this. Beta equals 10 log, or
logarithm, base 10 of I divided by 10 to the negative
12 watts per square meter. This looks intimidating. Let's talk about it and break it down. Beta is the number of decibels. So this side gives you
the number of decibels and we abbreviate decibel
with a little d, capital B. This is the number of dBs or decibel. You've probably seen this
number on your stereo. Somewhere reader just in volume because we're going to
measure volume and Decibels. 10, this 10 just denotes
the fact that this is the Deci-bel Scale and
not just the bel scale. If you didn't multiply by
10, you'd have the bel scale but this is multiplied by 10. We like the 10. We're going to call is Decibel. Log, we'll talk about log in a minute. Logarithm here. I is the intensity of the sound wave. So this is the intensity. In Physics, intensity is defined to be the power divided by the area. What this means, think about it this way. You got your ear and a sound wave, say, is coming toward your ear. If you imagine, one. Power's in, what? Power measure going to watts. Area's measure going to square meter. So think about intensity this way. If you have one square meter. Imagine one square meter of area here. This doesn't have to be
an actual physical object. Just imagine a square meter of area. The power that passes through that area will be, how many joules? If you figure you how many joules pass through this one square meter. If you asked how many joules per second pass through that one square meter. How many joulels of sound energy per second pass through the one square meter? That would be the number of watts per meter squared which
would be the intensity. So watts is joules per second. This gives you an idea of how much energy per second pass through
a certain amount of area. This part of the equation is my favorite. This is my all-time favorite right here. This number. This 10 to the negative
12 watts per square meter represents the threshold of human hearing. What that means is this is the softest possible sound you can hear. Any sound with an
intensity less than that. You won't even notice. But if it's anything bigger
than that, a human ear that's healthy should
be able to detect it. Here is why I like this number. This is unbelievably small. This is one trillion of
a watt per meter square. A trillion. What this says is that
even if only one trillionth of a joule per second passes
through the square meter, your ear would still be able
to detect the sound that soft. If that doesn't impress you,
let me put it to you this way. Imagine we did have one watt. Let me put it to you this way. If you have one watt, how big of an area? A watt isn't really that much. A watt is not a lot of power. If you have one watt, how
big could you make this area? How, spread out. How diluted could one watt be spread over? How large of an area could
this one watt be spread over and still be intense enough
for the human ear to hear it? What do you think? Football field? I don't know, a city? No, it turns out. If you do the calculation,
I suggest you do it. It's interesting. You would get, that you can
spread one watt over the entire land area of Germany,
about three times over, and still it's intense enough
for the human ear to hear. That's how unbelievably
sensitive our ears are. It's actually... I told you it's unbelievable. I can hardly believe it myself. Let's come back over to here. So here's our equation. This is the Decibel Scale. Why log? That's what you're thinking. "Why in God's name did the physicist have "to put logarithm in here? "This scare me." This used to scare me too. Well, I'll show you why. Here's the problem. The fact that we can
hear such a soft sound, 10 to the negative 12
watts per meter squared, there's a huge range of human hearing. This means we can hear from 10 to the negative 12 watts per square meter. This is your point zero, zero, say three, four, five,
six, seven, eight, nine, 10, 11, with a one watts per square meter all the way where there's no upper limit. It just blow out your ears. But once you get to about one watt per square meter, this
one will start hurting. This is painful. You're not going to be happy over here. You're just going to start hurting. You'll start getting
hearing losses and not good. So it's a huge range. 12 orders of magnitude. This one watt per square meter is a trillion times bigger than this side. This scale is just way too big. This is awkward. We want to scale that's small or maybe like one to
100 to measure loudness. We don't want to measure from one to a trillion or a trillion to one. That's what log's going to do. Logs are great. This is a trick for this use. This is why I love this trick. Logarithms take really big
or really small numbers and turn them into nice numbers. That's why we're going
to use the logarithm. Let me show you what I mean. Logarithm, if you don't remember, here's what a logarithm does. Log base 10 of a number
equals, here's what it does. I'm going to stick a number in here. Let's stick 100000. What log does, log is a curious guy. Log is always asking a question. Log always wants to know,
okay, if I'm log base 10, log wants to know what
number would I raise 10 to in order to get this number in here. So log looks at this
number in the parentheses. This entire number here
and asks what number should I raise 10 to
in order to get 100000. Well, we know the answer to that. We should raise 10 to the fifth. If I raise 10 to the
fifth, I'll get 100000. So if five is the number
I raised 10 to get 100000, then that's the answer to this that log base 10 of 100000 is five. Look what happened. Log took a huge number, 100000,
and turned it into five. Well, that's outstanding. Log can take huge numbers,
turn them into nice numbers. The logarithm base 10 of
one billion would be... One billion is a big number. That's hard to deal with but log takes 10 and asks what number can I raise 10 to in
order to get a billion. I should raise 10 to the
ninth because I got one, two, three, four, five, six,
seven, eight, nine zeros here. I raised 10 to the ninth
to get this number. So the answer to this question
for the logarithm is nine. Oops, that's not nine. Nine, and that's why logarithms are good. Logarithm took this enormous number of billion and turned it into nine. So logarithms take enormous
scales turn them in nine scales. That's why we like this formula which is our Decibel
Scale because it takes enormous intensities
and small intensities, turns them into nice intensities. Let me show you an example with
this equation really quick. Let's say you're talking to your friend. Maybe you're yelling at your friend. You guys are having a heated exchange. You're yelling. He's next to you. These are the sound waves coming at him. You're yelling with an intensity of, say 10 to the negative fifth. That doesn't sound like a lot but that's actually,
you're pretty upset here. That's pretty loud. I want to know how many decibels is this. How do we figure out the decibels? Well, here's what we do. We use our formula for decibels. Beta, number of decibels, equals 10 log base 10 of the intensity over always 10 to the negative 12 watts per square meter because that's the
softest sound we can hear. What do I get? 10 to the negative fifth is my intensity. So I plug this into here. I'm going to get beta equals
10 times the log base 10 of 10 to the negative fifth,
because that's my intensity, divided by 10 to the negative 12. Now these are both watts per square meter. So let's cancel. Well, what's 10 to the
negative fifth divided by 10 to the negative 12 turns out
that's 10 to the seventh. I end up with 10 log of 10 to the seventh. Now I don't like logs. I'll be honest. They freak me out but
I can even do this one. Log of 10 to the seventh. Remember what log does. It asks what number do I raise 10 to in order to get the
thing in the parentheses. Well, the number I raise
10 to to get the thing in this parentheses, it's
already 10 to the seventh. It's already in this form. So I've raised 10 to the seventh
to get 10 to the seventh. So the answer to log base 10 of 10 to the seventh is just seven. My final answer beta, the loudness, the number of decibels
is going to be 10 times log of 10 to the seventh was just seven because I have to raise 10 to the seventh to get 10 to the seventh. 10 times seven equals 70. I'm yelling at 70 decibels. I need to calm down. My friend's going to
start getting mad at me. That's how you figure out
how loud the sound wave is.