Let's talk about wave intensity; intensity is a word that we use to describe how much energy is associated with a periodic wave. Now it's not a direct correlation so let's kind of go through this. So first off the energy carried by a wave can be considered analogous to the energy carried in simple harmonic motion. The energy of a mass in simple harmonic motion is equal to one half times spring constant times the square of the amplitude. So that indicates to us that energy should be proportional to the square of the amplitude of a wave. But with a periodic wave we keep on getting assaulted again and again and again so it doesn't really make sense to talk about how much energy is carried because it keeps on increasing as the wave keeps on coming. So what we really ought is ask about how much power is being carried, how much energy divided by time. So it's the rate of energy flow, the power change in energy over change in time. Now that's measured in watts, 1 watt is 1 joule per second and of course we're all used to that from power.
Alright so that's all fine and good but let's say that I've got a sound source here and it's got a power of 100 watts alright so it's sending out these sound waves. But I'm not hearing all 100 watts, I'm surrounding the speaker taking all the energy, right I'm only hearing part of it. So what I'm really interested in is what part of that power I'm I really absorbing, so what this is associated with is something called intensity. Intensity is power per unit area, so we can see here that a sound wave coming I'm only going to get a little bit of it. So if I want to know how much power I'm absorbing from the sound wave then what I'm going to do is I'm going to take the area of my ear and multiply by the intensity of the sound wave. So then I've got power divided by area times the area that I'm absorbing and it all makes perfect sense. Alright there was a couple of subtleties in these and they're associated with different shapes of waves.
And let's just go ahead and think about sound waves here because I think it's a pretty simple analogy that we can make to rock concerts, so let's say that we go to a rock concert and there's a wall of speakers. So they've got speaker, speaker, speaker just rows of them on top of each other. Now when those speakers send out a sound wave, they send out what we call a plane wave. The wave is a plane and it comes out so equal wave fronts, equal phase comes out in the shape of a plane. Now if I'm twice as far away it's still a plane I get the whole thing I get the exactly the same intensity. So that means that the intensity of a plane wave is constant, it doesn't depend on how far I am from the speakers as long as my distance from the speakers is small compared to how big the plane wave or ray is alright.
What about a cylindrical wave? So in this case we're going to think about a tower of speakers right? So just one on top of the other but not a plane, so now they send out a sound wave and it goes out in cylinders out like that. Now what's the difference, well Geometrically as it goes out in cylinders the surface area of the cylinder gets bigger as I get farther from the tower. And so what happens is, if I'm farther away from the tower I don't get as much intensity because the power is spread out over more area. So that means that power divided by area got to be smaller. It turns out that because the circumference of a circle is 2 pi r , the intensity for a cylindrical wave is proportional to 1 over r, 1 over the distance from that tower. So that means that if I've got a friend that's twice as far from the tower than I am, then he's only going to experience half the intensity because that 2 goes down stairs.
Alright, what about the last one, a spherical wave? So in this case we're thinking about just a single speaker sitting there all by himself. Now he sends out sound waves and they take the shape of spheres. They just go out in all 3 dimensions, so what's the difference here? Well now the sound is spread out over the area of a sphere, but the area a sphere is 4 pi r squared so that means that in this case the intensity is going to drop off like 1 over r squared. So now my friend who's 2 times as far away from the speaker as I am is only going to hear a quarter, 1 over 2 squared of the intensity. If he was 3 times farther away then he'd only hear a ninth of the intensity. Alright so that's intensity and the Geometry of sound waves.