1

concept

## Sound Intensity

7m

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Hey, guys, in this video, we're gonna talk about sound intensity. Intensity is a measure of sound waves. That's very that is related to the energy carried by that sound wave, but is, um or useful measurement to know? Okay, let's get to it. Now. Intensity is the amount of power carried by a wave over the surface area that that wave happens to be spread across. Okay, power is fundamental to that wave. Meaning a source admits way a wave at some power. Okay. And as long as that power isn't dissipated, it's going to remain constant as the wave travels through space. For instance, in this recording studio where I'm making this video, we have sound insulation to protect from outside noises. That means whatever sound is passing through the walls where the insulation is, that installation is supposed to absorb all the energy that sound, absorb all that power. So that sound doesn't pass through into the room that I'm filming in. Okay, now imagine a sound source. Some source of sound is admitting sound in all directions. Okay, I draw a figure here. Okay? Sometimes we call this admitting spiritually, or if you want to get fancy, we would say admitting it s a tropically. Okay, I so meaning the same tropically. I'm assuming it has something to do with direction, but I'm not sure. So I'm not gonna lie to you guys. I just know I see a tropic means the same in all directions. The surface of an area depends upon R squared, right? The surface area. Sorry. The surface area of a sphere depends on r squared. The surface area of a sphere is just four pi r squared. That means the surface area depends on r squared. And so the intensity depends on our square for that very reason that I said assuming there's nothing to absorb the sound than that powers a constant. So the only thing that changes is the sound gets further and further out is the radius of this spear that the sound is spread across. Okay, so the intensity changes the further out you go to intensities of two different distances are related by the following equation. We have I won in the numerator and I two in the denominator. Remember? The way to remember this is it's always the con. It's always the reciprocal. Okay, so our two over our one squared. If I two is in the denominator are two is in the numerator. Obviously, if you reciprocate both sides, the equation still holds. Okay, let's do a quick example. A speaker emits a sound that you measure to have an intensity of 100 watts per meter squared when you are 5 m away from it. What would the intensity be measured over the intensity measured Be if you walk 3 m towards the speaker so we have our one is 5 m an intensity one is 100 m watch per meter squared. That's just the intensity measured at our one and we have our two. We were 5 m away. We walked 3 m towards the source. So now we're 2 m away and I too is our unknown. So I'm gonna write I two over. I won and remember, it's the reciprocal. So this is our one over r two squared and that means I to is our one over r two squared times I won, which is 5/ squared times, 100 watts per meter squared, which is about 625 blots per meter squared So what does that tell us? That the intensity increased the closer you got. Okay. And this is something that we would expect because the closer you got that sphere that contained all that power amended by the source is smaller, so the power density is larger. Okay, The intensity is also related to the maximum pressure in a sound wave. Remember that. It's sound is just oscillating pressure, so it's gonna have a maximum pressure, and it's gonna have a minimum pressure. Okay, The maximum pressure is one. Sorry. The intensity equals one half times the maximum pressure pressure divided by the density of the gas that the sound is propagating in times the speed of sound. Okay, so we can do another quick example here. Air has a density of 1.22 kg per cubic meter. If a sound wave has an intensity of one time center, the seven watts per square meter. What is the maximum pressure off the wave if the air temperature is zero degrees Celsius? Okay, well, what changes? With an air temperature of zero degrees speed, what changes the air temperature of the speed of sound, but the intensity. So it's a little bit of foreshadowing there that we're gonna have to calculate. Speed is P max divided by Roe V. Now, just as a heads up, we are dealing with pressure and power here, both of which are given by capital P No, by context and by your familiarity with these equations that this p here is pressure. This is maximum pressure. This is not maximum power, right? The power doesn't change is the sound travels So rearranging this equation He Max is two i times row times v the speed of the sound. So what is the speed of sound of zero degrees its 331 m per second times one plus the temperature in Celsius which zero over 273. Right? And that is just 331 m per second. All right, so the maximum power is two times that intensity one times 10 to the seven times the density 1.22 That density is given right here already And s I units times the speed which is m per second squared and that is 81 times 10 to the nine Pascal's, which is a huge, huge, huge pressure. Almost 100,000 times atmospheric pressure that would kill you. Definitely. Alright, guys, that wraps up this introduction into sound intensity. Thanks for watching.

2

Problem

A source emits a sound in the shape of a cone, as shown in the figure below. If you measure the intensity to be 100 W/m^{2} at a distance of 0.5 m, what is the power of the source?

A

26 W

B

79 W

C

91 W

D

105 W

3

concept

## Volume And Intensity Level

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Hey, guys, In this video, we're gonna talk about something called intensity level and how it relates to volume. Okay, let's get to it. Volume of the sound is not actually a measure of the intensity of the sound wave, but it's actually a measure of a related quality called the intensity level. Okay, so when I gave the introduction, I said, We're talking about intensity level and how it relates to volume. Well, intensity level is volume. That's how are they related to one another? Really? What we want to talk about is how intensity level and thus volume are related to intensity. Okay, Now, the reason why we use something else than intensity for volume is because volume doesn't really change that much. With small changes in intensity, there are Onley noticeably large changes in volume with incredibly large changes in intensity. Okay. And thus we use a log arrhythmic scale for intensity which we call intensity level. The intensity level, which is given by the Greek letter Beta is 10 d b times The log a rhythm of I over. I'm not where DP is our unit the decibel okay, and I not is called the threshold of hearing and it's one times to the negative 12 watts per meter square. That's the quiet it sound the human ear can hear. Okay, now, because this is a log, a rhythm it on Lee changes with factors of 10. Okay, so when I becomes a factor of 10 larger lage, rhythm becomes one larger or 10 larger because I'm sorry or the intensity level becomes 10 larger because it has a factor of 10 out in front. Okay, so it's those powers of 10. Increases in, um intensity lead to noticeable changes in volume. But if we go from, let's say ah, 100 to 102 watts per meter squared that produces no noticeable change in volume. If we go from 100 to 1000 that produces a noticeable change in volume. Okay, so we use a logarithmic scale because Onley those multiples of 10 and intensity really produce noticeable changes in volume. Alright, I threw on some common volumes for you guys here. The threshold of hearing is by definition, zero decibels. This is because beta right, which is 10 DPS log of I over I Not if we say I'm not over I not. This is one and by definition, the log rhythm of 10 So the threshold of hearing the quietest sound that you can possibly here is zero decibels. A quiet room, about 40 decibels. Conversation about 60 decibels. Probably what I'm speaking at a at about now inside of this room, speakers in a noisy club. I'm from Miami. So we got lost noisy clubs down here. 100 decibels or so threshold of pain. When it starts to hurt your ears is 130 decibels. Just to give you a little bit of context. A jet aircraft, like a 7 47 150 ft away is already above 130 decibels at 140 decibels. Okay, dangerous volumes or above 150 decibels. That's when you start to cause serious damage to your ears in the short term, not the long term, Okay? And theoretically, there are volumes that can kill you, which are usually a theorized to be about above 200 decibels. Okay, but it's obviously never been proven. All right, now the thing is, just because they sound is loud enough it has a volume above zero decibels doesn't mean that a human can hear it. Humans can't hear every sound above zero decibels because sorry, I minimize yourself too quickly because there is a range of frequencies that humans can hear at. Human. Hearing is considered to be between 20 hertz and 20, hertz or 20 kilohertz. Okay, but we can still feel the effects of very, very loud sounds. Even out of the range of hearing for us, for instance, sounds that dangerous volumes at 150 decibels arm or even if they're outside the range of hearing we can feel the pressure coming from the sounds. Okay, that pressure is still of physical. That's still a physical thing, and that pressure is putting a force on our body, even if we can't hear it. All right, let's do an example to close this out. Sound is measured to be 25 decibels loud at a distance of 10 m from the source. If you walked 4 m away from the source of that sound, what would be the volume of the sound? Okay, so beta one is 25 decibels, and this occurs at a distance are one of 10 m. Now Beta two is our unknown and this occurs at a distance of our two. 4 m further away than our one or m. Okay, now the process to solving this problem is a little complicated because, or at least complicated algebraic Lee, because it involves logarithms. But this is pretty much the way you're always going to solve these problems. They're all usually presented in the same way, except instead of maybe walking away, they have you walk towards the sound, but it's pretty much the same problem. The first thing you're going to start with is the relationship between the intensities. The idea is that beta, too, is 10 decibels. Log of I two over I zero. You don't know what I two is and you can't find I too, because you don't know what I one is. But you can express I to, in terms of I won. Once you've expressed I to in terms of I one. Then through some algebraic trickery, we can get 10 decibels log of I won over I not, which is by definition, beta one which we know so we can solve the answer that way. Toe find the relationship between I one and I two. We just need to use our regular old intensity relationship, right? It's the reciprocal. So it's our one over R two, so I to is gonna equal are one which is 10 over 14 squared I one which is about I want Here's where that algebraic trickery comes into play baby too which is decibels log of I two over. I'm not. I'm gonna replace to with the substitution I just found for I won. This is 10 decibels log a rhythm of 051 I won over I not now for a log, a rhythm Any time you have the multiplication of two numbers. In this case, when a multiplying is 20.51 and I won over, I not those are my two multiplication. Any time you have the multiplication oven input for log rhythm, you can split that into two logarithms that air summing together. Okay, so this is equivalent to 10 decibels. Log a rhythm of that first multiplication plus 10 decibels log rhythm of that second multiplication term. Okay, And remember, this right here is just beta one by definition, and we know what beta one is. It's 25 decibels. This right here we can plug into a calculator and we find its negative to nine decibels. All right, I'm gonna get myself just a little bit of room here to finish out this problem. So be the two is 25 decibels minus 29 decibels, which is 22. decibels. All right, And that's pretty much how you're gonna solve any of these problems. They all are pretty much the same. And it comes directly from this log arrhythmic trick that the log of a times B equals the log of a plus the log of B. That's that little log arrhythmic identity that we want to use. If you guys don't remember your log rhythm identities, you're logarithms algebra just for this part. Specifically in this chapter, you should review it just a little, because it will come in handy. Alright, guys, that wraps up our discussion on sound intensity and volume. Thanks for watching

4

Problem

A source emits sound spherically with a power of 2.2 × 10^{4} W. What is the minimum distance away from this sound that would be considered safe (a volume of 150 dB or less)?

A

1.3×10

^{−5}mB

0.13 m

C

1.3 m

D

1300 m

Additional resources for Sound Intensity

PRACTICE PROBLEMS AND ACTIVITIES (8)

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