Now let's solve that. And this is the world's simplest differential equation. It's a second derivative of E equal to constant times E and that, of course, has a complex exponential solutions. If we include the time dependence which we often don't, we get a J omega T. Really, we're also interested in the space dependents here which is k x x, plus k y y plus k z z. I remember because the differential equation had three space dimensions. So again, you should know how to solve that equation and come up with a solution, if not, go take a quick look at that. This is really important. And we're going to get an enormous amount of intuition and quantitative understanding of how light propagates from this plane wave solution. And not formally, that this is eigensolution of the Scalar Wave Equation in Cartesian coordinates. And eigensolutions are important. The Eigan here just means it goes forward with only changing something about the phase. So let's start making a new function. If you don't know what it is, you make some plots for it. So let's make some plots for this. We'll fix time. Let's say, T equals zero. So we're not going to worry about that everything's flying around at the speed of light. So a fixed time. We'll throw out one-space dimension because I don't know of a three-dimensional screen to show you things on, so we'll say we're at the plain z equals zero. And so now, we'll be looking at functions just to help x and y or any of the two-space dimension which uses spatial variables if you'd like. What we're gonna find is if you pick some k vector, this wave vector which describes really where this going. Is this essentially our ray direction right here? And it has two components k_x and k_y, that then, if you think about changing your special coordinate x, y, which we have buried here in our. Since there's a dot product here, any time you move orthogonal to k, nothing happens because that's what dot products do, is they don't care about things that are orthogonal. Some are, which is orthogonal to k, is zero. So the only variation that you get of this plane wave is along k. So the phase fronts are normal to k, which again supports the idea that k is a lot like the ray. Also interesting, is that if you break this into terms, k x x plus k y y, like we've drawn over here, that we could run just along the x axis, and we'd find that there's some period along the x axis, which must be two pi over k_x. And if we did the same thing just along y, we'd find that every time this quantity goes through two y, we get one change in period of this periodic function, and so we'd find that there's some period, length of y, that's related to k_y, through whatever two pi. So not only is the wavelength as we propagate along k equal to two pi over the magnitude of k, as I've written here. It's also true in each dimension. There's a wavelength along just the x direction. And if I just slice along there, that's related to k by two pi over lambda x. And if I walk along y, there's a wavelength in y that's related to the y component of k. And these are important little concepts that we're about to use. And finally, remember, when we do have polarizations, the polarizations have to be an isotropic media perpendicular to k. So we have two possible polarizations in a three-dimensional space because we've thrown out one to two in waves where the electric field oscillates along the direction of propagation and they get that basic electromagnetics.