[MUSIC] So we've seen in the simulations, the two dimensional simulations. That in homogeneous medium where we actually expect physically a nicer topic fairly symmetric wavefront, sometimes this is not what we see. Sometimes we see so called anodized tropic behavior. And I promise we'll look whether we can actually understand that from an analytical point of view. And again, we resort to the so-called phenomenon analysis, trying to develop an analytic understanding using a plane wave trial solution. So in two dimensions, a continuous plane wave is given like that, you have p(x,z,t)=e to the i kx- omega t. And you see here, k and x in boldface, so these are both vectors, k is the wave number vector. I'll explain that in a second, and x is actually the position vector. So that's a vector product, and if you write it out, it actually KxX + KzZ. Now what is the wave number rector K? K it has components of the Kx. And Kz is actually a vector that points in the direction of propagation. And basically is a vector that is constant on the plane of the constant phase. And to, with this, actually, we now can go again into the wave equation, and try to solve this analytically. But before we need, again, the discretization, so we move into the discreet world. Because x will be jdx, and z will be kdz. Again, we assume for simplicity that space is discretized with the same space increment in both directions. So dz = dx, and we can replace dz basically by dx. So with this discrete representation of a plane harmonic wave in two dimensions. We now enter the wave equation, and we go through the whole algebra as we did in the one dimensional case which I spare you at this point. But I invite you to actually try this out. We will see this later in the exercises. And I will show the result. Remember that this allowed us basically to determine the velocity C as a function of wave number. And now we can now look at the physical velocity as a function of propagation direction. And the vector k, we can also formulate as with components the modulus of k, multiplying cosine alpha. Where alpha is the angle to the x-axis. And kz which would be the modulus of k, times sine alpha. So basically that allows us to actually calculate the error of the velocity, the physical velocity as a function of propagation direction alpha from 0 to 360 degrees. And the results you see here in this graph, and it shows exactly what we've seen in our simulations. And it's very interesting that you notice that the most accurate direction of propagation is actually 45 degrees to the coordinate axis. That's not necessarily intuitive, but that is the case. And the least accurate directions are actually the directions along the coordinate axis, in that case plus x, minus x, plus z, and minus z. So in conclusion, we say we understand that behavior, that behavior of numerical anisotropy that we've seen in the simulations and our Python implementation. It can be nicely derived here for the scalar acoustic wave equation with the so-called phenomenon analysis. And of course, it's very important to avoid this kind of behavior. Because sometimes, for example, you want to understand physical anisotropy that exist in the earth for example. And by all means you would not want to confuse that with numerical anisotropy. So its very important again as in the Wandy case, to have enough number of grid points per wavelength to be so accurate, as to avoid this effect of numerical anisotropy.