Hey, guys. So in a previous video, we talked about the intensity of a wave. The wave intensity equation was given by this equation over here. Remember that this equation actually works for all different kinds of waves, whether they are transverse waves or longitudinal. But we're going to take this equation, and we're going to talk about a related idea that has to do only specifically for sound, called the sound intensity level. So let's go ahead and check this out here. Basically, the idea is that humans can actually hear over a huge range of intensities. Now you don't need to memorize any of these numbers. I'm just using them to make this point here. But if you're walking through the park, the rustling of leaves has an intensity on average of about \(1 \times 10^{-11}\) watts per meter squared. If you're having a normal conversation with your friend, that's \(1 \times 10^{-6}\). And if you're standing close to a jet engine, that's actually \(1 \times 10^{1}\). Notice how the numbers are getting smaller as they go down, but they're actually negative exponents, so the number is actually getting bigger. They're more intense sounds. Now these numbers are pretty impractical to use in everyday language. So instead of using watts per meter squared, we're going to use a logarithmic scale when we talk about sound intensity. So this logarithmic scale is going to make our numbers a lot simpler. So here's the equation the textbooks are going to give you for sound intensity level. They're going to use the letter beta, the Greek letter beta, and it's going to be \(10 \times \log_{10} \left(\frac{I}{I_0}\right)\), where \(I\) is the intensity of the sound source, and \(I_0\) is a constant that has to do with the lower threshold of what humans can on average hear. It's just a number and it's always going to be \(1 \times 10^{-12}\). Alright. So it's just those two letters there. Really, there's only one variable because this is a constant and the units for this are called decibels. So this is where we get our decibel system from, which is a representation of how loud a sound is, and the units for this are going to be in dB. So basically, we can use this once we use this logarithmic scale, we can take these numbers that are pretty impractical and long, and we can turn them into much nicer numbers, like for instance, 10 decibels or 60 or 130. So that's all there is to it, guys. Let's go ahead and take a look at an example. So we have a siren that is producing a power of 9 milliwatts. So that's power \(P\), which equals 9 milliwatts, which equals 0.009. Now what happens is this siren is continuously producing a sound in 3 dimensions in all directions. We're going to calculate the sound level in decibels at a distance of 3 meters. So basically our \(r\) is equal to 3. Alright? So if we want to calculate the sound level, we're just going to use our new equation here, which is beta. So this is \(\beta = 10 \times \log_{10} \left(\frac{I}{I_0}\right)\). The great thing about this equation is that there's actually only just one unknown that you could possibly have, which is this intensity. So how do we actually find this out? Well, remember this equation for intensity is really just this, \(I = \frac{P}{4 \pi r^2}\). So this is going to be \(I = \frac{P}{4\pi r^2}\), and what you're going to get here is you're going to solve for \(0.009 \div 4\pi \times 3^2\). If you go ahead and work this out, what you're going to get is you'll get \(7.96 \times 10^{-5}\), which are watts per meters squared. So now what we do is we take this number here, and we're going to plug it inside for the intensity in our logarithm equation. Alright. So to finish things off, we're going to have that the beta is going to be \(10 \times \log_{10} \left(\frac{7.96 \times 10^{-5}}{I_0}\right)\), and we're going to just divide it by our \(I_0\). \(I_0\) again is just that lower threshold of human hearing and it's kind of like a reference point or a reference level, and basically this is going to be \(1 \times 10^{-12}\) like this. So if you go ahead and work this out, what you're going to get is exactly 79 decibels. And that's the answer. So that's it for this one, guys, and let me know if you have any questions.
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Sound Intensity
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