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Decibel Scale

Teacher/Instructor Jonathan Osbourne
Jonathan Osbourne

PhD., University of Maryland
Published author

Jonathan is a published author and recently completed a book on physics and applied mathematics.

The decibel scale is used to measure the intensity of sound waves. A wave with an intensity of 0 decibels is impossible to hear. Normal speech, on the other hand, has an intensity of 60 decibels.

Alright let's talk about the decibel system, the decibel system is a way that we use to measure sound intensity. Now you might say why don't we just measure sound intensity? Well the issue is that sound intensity varies extremely broadly, the quietest noises that we as human beings can hear have intensities of around 10 to the minus 12 watts per square meter. Remember that intensity is power divided by area so its unit will be watts per square meter. The loudest intensity levels that we can hear without really feeling at least too much pain are on the order of 1 watt per square meter. So we want to somehow divide the region between 10 to the minus 12 which is essentially 0 and 1 in some sort of useful way where we can point to it and say, you know that sound is 20 times as loud as that other sound. And that's not really easy to do when we're just working at, with the region between 0 and 1. So when we've got a situation like this, it's usually convenient to instead use a logarithmic scale. So this is also used when we talk about the Richter scale and how much energy is released in an earthquake because we got the same problem now, so we'll just solve it in the same way.

Alright, so the decibel level is defined as 10 times the log base 10 of the intensity divided by some base intensity. Now this base intensity can be arbitrary chosen but it's been chosen as the so called threshold of hearing, which 10 to the minus 12 watts per square meter, okay the threshold of hearing. Now the other side of it when the intensity is 1 if we do 1 divided by 10 to the minus 12 we get 10 to the 12, log base 10 of that is 12 multiplied by 10 and we get a decibel level of 120 decibels. Now that's called the threshold of pain, which is kind of the opposite of the threshold of hearing. Now these things, really if I want to say I'm I actually going to feel pain at this decibel level, I'm I actually going to be able to hear at that decibel level, that's really frequency dependent. If you have a frequency that's less than about 20 hertz you can't hear it. Instead what you'll, the way that you'll experience that sound is vibration. So if you've ever been sitting next to somebody on the freeway, whose got their bass booming a lot of times you can't really hear it as much as you can feel it right? And that's kind of that transition from sonic to infra sonic, doesn't really come across as a sound wave anymore it more comes across as a vibration.

Now on the other hand when the frequency is greater than about 20,000 hertz we get ultrasonic and that of course is dog whistles and those types of things come in. And we don't really experience those types of sound waves at all because they vibrate too quickly for our ear drums to catch up, so they just kind of are a wash and we don't even experience them alright and in between the actual threshold of hearing and the actual threshold of pain will depend on the frequency but these are fairly standard basic numbers that we can use. Alright so let's look and see how this decibel scale works. If I have an increase of 10 decibels, then that means that I've got a factor of 10 I multiply the intensity by 10 it was 10 times louder. So let's just look here at some characteristic values, 40 decibels is fairly quiet it's like leaves rustling in the wind something like that, maybe a mosquito buzzing not too close to my ear 55 decibels is ordinary conversation so I can hear it fine but it's not loud.

70 decibels so as a vacuum cleaner alright so that's loud it's something that I can't really filter out very easily and 90 decibels or so is going to be a rock concert. We go up farther and maybe like 100, 120 decibels that's like right next to a plane taking off. Alright not again as threshold of pain, you get too much louder than that your eardrums will bleed. Alright so let's go ahead and do a problem with this. Suppose that you're at a rock concert and you'll hear a decibel level of 90 decibels and I want to know if your friend is 5 times farther away from the speakers what's the decibel level that he hears. And we're going to do this in 3 different cases, plane wave, cylindrical wave and spherical wave. Remember plane wave is associated with a wall of speakers sending out a wave that just goes. Cylindrical wave is associated with a tower of speakers sending out a wave that expands in a cylinder. And spherical wave is associated with a single speaker just sitting there sending out a wave that goes out in a sphere now as you remember a plane wave has an intensity that doesn't depend on how far away you are. So if the intensity is the same then the decibel level is the same. So for a we'll just have 90 decibels no difference it doesn't matter because it's a plane wave.

Alright what about b? Well for b, we have a cylindrical wave now as you remember cylindrical waves intensity goes down with the distance. So if I'm 5 times farther away I only have one fifth of the intensity. Now the way that logs work, the log of a ratio is equal to the difference of the logs. So if I have log 10 times log base 10 of intensity over 5 that's what my friend is hearing my intensity divided by 5 it's not 15 that's the bottom of my data and then we'll divide by i node now I can make this real, real simple if I just say look it's just 10 log base 10 of i over i node this was i over i node divided by 5 but the log property said that when I divide I just subtract 10 log base 10 of 5. Now what's nice about this is that I already know this value, it's 90 because that's my decibel reading. I can calculate this in a calculator and what I end up with is 83 decibels alright so my friend hears a much smaller decibel level because he's 5 times farther away almost 10 times smaller but not quite right it can't be 10 times because would have to be factor of 10.

Alright now what about c? Well c is the same thing except instead of dividing by 5 I'm going to divide by 5 squared, because now I've got a spherical wave and spherical intensities go down like the square of the radius. So if it was 5 squared there again log properties, the 2 comes down and multiplies by the 10 so what we'll end up with is 90 minus 2 times 10 9s 20, 20 log base 10 of 5 and when we put that in the calculator we'll see that, that's 76 decibels alright that's the decibel scale and some of its uses.