Guys, for this video, I want to introduce a new concept and a new variable called the wave intensity, which has to do with the amount of energy the wave produces over a certain distance. Let's go ahead and check this out. So the idea here is that we've seen one-dimensional waves that move to the right, and they also carry some amount of energy. Right? So one-dimensional waves just move along to the right and carry energy from point A to point B. You can also have two and three-dimensional waves as well. If you've ever dropped a rock in a pond, the ripples travel outwards not just in one direction. That's two-dimensional. And in the example that we're going to work out down here, loudspeakers which produce sound waves actually radiate in all directions. That's a three-dimensional wave. So what happens here is that one-dimensional waves carry energy just along the straight line, but two and three-dimensional waves radiate energy in all directions and they spread it out over a surface area, which we use the letter A for. So the idea here is we actually define the wave intensity. The intensity of that wave is just the energy per time, which remember is equal to power. It's going to be the power divided by the surface area.

So the idea here is that if you have three-dimensional waves that travel outwards like the sound source here at the center of our diagram, it's going to radiate energy outwards in all dimensions like this. And at a certain distance, at any distance of r away from that source, you can sort of imagine the sphere, the surface area is \( 4\pi r^2 \). And the intensity of the wave is going to be \( p \) divided by that surface area. So the equation for this is \( \frac{p}{4 \pi r^2} \), and the units for this are going to be in watts per meter squared. Alright. So let's go ahead and take a look at our example here. We have a source that produces 500 watts of power. So our loudspeaker here produces 500 watts of power. We want to calculate what the wave intensity is at a distance of 10 meters. What this means here is that \( r = 10 \). So, basically, at some distance away of 10, we can sort of draw this surface area, which is our sphere, and the intensity at this point is going to be \( I = \frac{p}{a} \). So it's going to be \( p \)divided by so sorry. I'm actually going to use \( 4 \pi r^2 \) here. So \( 4 \pi r^2 \). So we're going to use \( \frac{500}{4 \pi \times 10^2} \). And if you go and work this out, what you're going to get is 0.4, and that's watts per meter squared. So watts per meter squared. Alright. That's that's it for this one, guys. Very straightforward example, and let's go ahead and take a look at some practice problems.