 OK, we're going to do some geometry now to figure out how much the angle changes when a wave refracts across a boundary. So let's have our boundary. Let's have our wave coming in. And then it's going to change direction. And again, I'll assume it's slowing down just for this case. And so it's changed angle. Now, how much is that angle going to change? The way we normally measure the angle is we measure the angle from the direction the wave is going. And if the wave fronts are looking like this, then it's actually traveling at right angles to that. So it's actually traveling in that direction. And that distance is the wavelength. So that's wavelength 1 for the wave when it's going at speed c1. And when it's going at speed c2, it's going to have a different wavelength, I'm going to have to 2, wavelength 2. If we look at the angles in there, so if we look at this angle here and this angle here, theta 1 and theta 2. Now what we can say is we can say that the sine of theta 1 is this wavelength divided by that length there. That's the definition of sine. So sine theta 1 is the first wavelength divided by this distance. Let's call that distance L. And similarly, sine of theta 2 is just going to be that length divided by that length, which is just lambda 2 divided by L. And if we rearrange for L, and obviously L equals L, and therefore the right-hand sides equal each other. And so if we set the two right-hand sides to be equal to each other, then we get. So what we have here is a relationship between the angle of the wave as it comes in towards the boundary and the angle as it goes away from the boundary after it goes through. And all that depends on is the wavelength as it comes in and the wavelength on the other side. And we calculated this for when the wavelength is getting smaller, but the same formula also works when the wavelength is getting bigger. In the case the angle changes in the other direction. Now if we take both of these sides of the equation and multiply by the frequency. And remember, the frequency can't change on each side, because it's the number of cycles per second. And there's nothing about the speed that's going to change how many cycles we're making the wave to every second. And if we take the frequency times the wavelength, then we get the speed. So unlike mechanical waves, light can travel in a vacuum. And when light travels through materials like glass or water, it actually slows down. And the rate at which it slows down is something called the refractive index. So what happens when a light ray comes to a boundary? Well, if it comes in at one angle, turns out it also bends. And so if it goes from a refractive index of N1 to a refractive index of N2, then that's going to change the speeds. And if you do your geometry, you can see that this angle here is exactly the theta one we had in our other diagram above, and that this angle here is exactly our theta two. And if you look at the relationship between the angles as light goes through a boundary between materials of different refractive index, what you find is that it follows something called Snell's Law. And if you plug in the definition of the refractive index into these two here, then this exactly gives you the Snell's Law relationship there. And so the fact that light obeys Snell's Law is very suggestive that light is actually a wave acting just as all waves do when they pass from one medium where they're going at one speed to another medium where they're going at another speed.