Welcome back. Last week I promised you that we are going to look into simulations in two dimensions. So far, we've only looked at one dimension. Usually, that is not enough to actually get interesting results for scientific problems. Actually, in two-dimensions, often it's already possible to understand the physical phenomena in more detail, which is why I think that's an important issue. So, to understand some of the peculiarities of two-dimensions, it's actually useful to start with the full three dimensional wave equation. Now, it's written again here. Now, on the left hand side, we have the second time derivative of p. Now, here again, with all dependencies in space x, y, z, and t. On the right hand side, we have the propagation velocity c square that depend also on space, only x, y, and z, multiplying the Laplace operator of the pressure field. That, of course, also depends on space x, y, z, and time. Plus we, again, have the source term s which also depends on space x, y, z, and t. Now, what happens if we want to turn this three-dimensional equation into a two-dimensional problem? Well, what we do is we say all fields that we have here in the wave equation do not depend on one particular direction. So, let's say this is the y direction, so that means all partial derivatives with respect to y and zero. So, let's express that again in mathematical terms. So we have the pressure field that is no longer depending on y. So, it will be only a function of x, z, and t. The propagation velocity c will only depend on x and z. Also, the source term will no longer be dependent on x, y, z and t but now is only a function of x, z, and t. Now, let's try and see what this means geometrically. First, let's look at a coordinate system. Actually, in earth sciences, particularly in exploration we actually, people often take the z coordinate downwards. So, we have here x, y and z. Now, what does it mean if, for example, a wavefield or an earth model is independent of y. Well, this is called then a two-dimensional model and here's an example. Now, let us assume we have a velocity field. This is colored. So, the color corresponds to a propagation velocity, and it only depends on x and z. So, basically, what that means we can prolong it, it's translationally invariant in that third direction, in the y direction. That's expressive of its two-dimensionality. What about the source term? Now, this is very important. Now, if you consider, for example, a point source somewhere inside that medium, this source, this point in the x, z plane is also invariant in y. So what does that mean? It's no longer a point source, it's actually a line source. That actually creates a very particular waveform and very specific effects that are very important to remember. Similar concepts apply to all other simulation types that you do in two dimensions. So, now let's plug in these new dependencies into the wave equation and see how we can proceed to find a finite difference approximation. Let's start again with the whole system. Here is, again, the three-dimensional wave equation. Remember, we require that first all fields do not depend on y. Also that means all partial derivatives with respect to y, are of course zero. So, you can scrap out a lot of things, and we are left with now the two-dimensional wave equation. So, for example, the term with the term in the Laplace operator with the partial derivative with respect to y are dropping out. Finally, we are left with a partial differential equation, the wave equation in two dimensions as it is written here. Now, basically, we're ready, again, to try and formulate this equation using finite differences. But before we do that, we ask ourselves, what is the analytical solution to that equation for a point source. So, we're seeking a solution of the Green's function. So remember, if you would like to find the solution of the partial differential equation for a Delta point source in space and time, we call this the Green's function, and replace basically in the partial differential equation p by g. We will find a solution to that equation. Now, again, like in the one-dimensional case, without showing the derivation, here's the result for the two-dimensional case. So, the two-dimensional Green's function of the acoustic wave equation is actually a scaled Heaviside function. Remember, the Heaviside function is a step function, and it's scaled actually by t. There is the square root of t square minus r square divided by c square. R square by c square is actually also a time. So, the waveform now looks very peculiar. That's no longer or is not an impulsive arrival. If you think of a seismograph or a pressure record that you record at some distance and here's an example. It basically rises, and then it slowly decays and actually never goes to zero. This behavior of the pressure field in terms here as a Green's function never going to zero is actually a direct consequence of the line source, because you can imagine that basically you always get energy from that line forever wherever you are. That's basically what we see here in the analytical expression for a Delta point source. Now, that seems academic, but it's actually in many cases particular if you deal with wave propagation very important to remember, because sometimes you want to compare observations in the three-dimensional world with a simulation. Here's the problem. If you only use two-dimensional simulations it's often quite hard actually to turn the line source into an actual point source solution. But that's really something for a mature science problem. Here, we would like to learn the numerical methods, but it is something at least to remember.
