Hey, everyone. So in the last couple of videos, we saw the equation for the wave intensity of a wave, which was this equation over here. Now in some problems, you have to compare the intensity of a wave between different distances. The example we're going to work out down here has a siren that's producing waves in all directions. At some distance, we're given the intensity of this value here. We're going to have to calculate the distance at which the intensity falls to a different value. Now, in order to do this, we're going to use an equation called the inverse square law for intensity. I'm going to show you how it works. It's very straightforward. So, in order to compare distances or to compare the intensity of different distances, we have to understand what happens as waves travel outwards from the source. So I've got this diagram here and let's say there's some power source like a loudspeaker or siren or something like that. What happens as waves travel outwards? Well, the distance outwards from the source is going to increase. So for example, if my power source is here and I have some distance, I'm going to call this r₁. There's a surface area here and I can calculate an intensity like this right here at some surface. So what happens as we travel outwards? Later on, the wave is going to travel some bigger distance. I'm going to call this r₂, this one in blue here. And r₂ is bigger than r₁. So what happens here is that the power of the source is actually going to remain constant. So we have to take a look at what happens in our variables inside of this equation here. The power is going to remain constant because as you travel outwards, as waves travel outwards, the thing that's actually producing the waves doesn't change. If the loudspeaker is producing 100 watts, it always is going to remain as 100 watts. So this PEA remains the same. What happens to the surface area though? Well, hopefully you guys realize that at some greater distance r₂ you're going to have a surface area a₂ that is bigger than a₁ because the sphere, the area that encloses, is going to be much bigger than the one that was enclosed by the red sphere, this r₁ distance. So your surface area is going to increase and therefore, this denominator of this equation is going to go So what do you think happens to the intensity? Hopefully, you guys realize that the intensity is going to decrease because their power remains constant, but your denominator gets bigger. So So that's exactly what happens. The wave intensity is going to decrease here. So I₂ is going to be I<I₁ . Basically, the power is going to get spread out or dissipated over a larger area, so the intensity is going to decrease. Alright. So that's really all there is to it guys. So what we can do is we can actually go ahead and rearrange this equation here, and we want to get it in terms of p. So what happens here is that the intensity times the surface area equals p, but this p here is going to remain the same no matter where you look or no matter how far you look away from that source. So what we can do to compare 2 different distances, we can actually just set these ratios or set a ratio equal to each other. So we can do here is we can say that I₁ times 4πr₁² is equal to I₂ times 4πr₂². I1 I2 = r 2 2 r 1 2 And right because both of these things are actually equal to the power, and if they're both if the power always remains the same, we can just set them equal to each other. Now if we go ahead and rearrange for this, cancel out some variables, we'll end up with this relationship here, which is the I₁ over I₂ is equal to r₂ over r₁ and they're both squared. So this equation right here is sometimes called the inverse square law for intensity. It says that the ratio of the intensities has to do with the ratio of the distances. The most important thing you have to remember is that these letters, the I₁ and the numbers I₂ are gonna be flipped from r₂ and r₁. I₁ over I₂ is r₂ squared over r₁ squared. Alright? That's really all there is to it, guys. Let's take a look at our equation or our problem here. So we have a siren that's producing sound waves radially outwards. Notice that we don't have the power. We have the distance. The first distance is 15. This is r₁, and the intensity at that distance, I₁, is going to be 0.25. So we're going to calculate a distance. This is going to be r₂. So I'm going to calculate this, at which the intensity I₂ falls to 0.01. So this is we're going to calculate here, r₂. Now before we begin, I wanna ask you, what happens what do you think this r₂ is going to be? Do you think it's going to be bigger or smaller than this r₁? Well, hopefully, you guys realize that if the intensity from I₁ to I₂ is getting smaller, is decreasing, if I₂ is less than I₁, then that means that r₂ has to be greater than r₁. We're going farther and so the intensity is decreasing. Let's go ahead and verify this by using our equation. So we have our intensity equation, I, our inverse square law. This is I₁ over I₂ equals r₂ squared over r₁ squared. We wanna figure out this r₂ here. So what I wanna do is I'm gonna rearrange this. I'm gonna move this to the other side. So I've got the r₁ squared times the ratio of I₁ over I₂ is equal to r₂ squared. Now let's just go ahead and plug in some numbers. So I've got my r₁ is going to be 15. So this is going to be 15 squared. My I₁ is 0.25, and my I₂ is 0.01. What you're going to get here is r₂ squared, and this is equal to 5,625. So if you go ahead and take the square root, what you're just going to get is you're going to get 75 meters. So just as we predicted, this r₂ is greater than the r₁. We've gone farther and so, therefore, the intensity is decreased. That's really all there is to it, guys. Let me know if you have any questions.