[MUSIC] So now let's leave the static case and go to the dynamic one. The wave equation, the one-dimensional wave equation where things happen in time. So let's again make use of the one-dimensional elastic wave equation, as you see here. Now we have on the left-hand side the density multiplying the second derivative in time of the displacement. And on the right-hand side, we have the space derivative with respect to x times mu times the derivative with respect x of the displacement field plus a force term. Now we have the exact same procedure as we did in the static case to develop the finite-element solution scheme for this one-dimensional elastic wave equation. The only difference is now we have on the left-hand side our time dependent term. So what are the most important steps? Well, first, we integrate over the whole domain D and we multiply by a test function that we now know, same as in the static case, we denote by phi j. So we have j basis function with which we multiply, and then integrate. The next step is we do an integration by parts of the first term on the right-hand side, as you know very well how this works. We do this in the exercises. And we obtain the following formulation for this particular integral. And we see we have an antiderivative. And inside that antiderivative, we have the term mu multiplying the first derivative of the displacement for x, and that's the stress. Now the antiderivative is evaluated at both sides of the physical boundaries. In that case, it's to the left and to the right inside of our one-dimensional domain. And if the stress is zero, we call this the free-surface or zero-stress condition. This is actually implicitly fulfilled exactly. And for seismology, for example, and for other problems, that's really great news. Because, for example, with finite difference methods, this is actually quite tricky to implement accurately. Here we get it for free. Now, this is really great. So we can actually scrub out this term, assuming that we have this so called free-surface boundary condition. And then we end up with the weak form of the elastic wave equation, as written here. Now this is the equation that we're now going to solve, again, basically doing the same steps as we did in the static case, and that's going to be the next few minutes. So again, the next step is to go into the discrete space. And again, we now describe our unknown function u by a sum over some basis functions, okay, like we did before. We call that approximation u-bar, and it's now a sum from 1 to N of, and here comes the big difference now we have coefficients that are time dependent. So u becomes ui(t) multiplying the basis functions phi i. So with this, we go into that equation. So we replace the unknown u by the approximation, u-bar. And u-bar is, of course, now a sum, so we have an integration over sum. This can be interchanged. And you can see already by the indices, just like in the static case, there is a matrix vector system developing. So we actually now have a linear system of equations in j. And the next step will be to actually identify the matrix and vectors in this 1D wave equation in its weak form. And then find a solution to this matrix vector system. So what vectors and matrices can we identify in that equation where we used index notation? Well, let's start with a displacement, the unknown displacement field, ui, which is now time dependent. So these are coefficients that we multiply the basis functions with. We also have a second time derivative, and we'll see how we deal with this second time derivative later. On the left-hand side, we have integrals over density multiplying the basis functions phi i, phi j, integrated over space. This we denote, we call the mass matrix because we have density inside this integral there, and it's called Mij. On the right-hand side on the first term, this is familiar to us. We have integrals over the space derivatives of the basis functions phi i, phi j multiplying the shear modulus mu. And that's the well-known stiffness matrix. We also have integrals of the force terms multiplying the basis functions integrated over space. So these are the system matrices and vectors that we have to evaluate, which is going to be the next step. And we will put all those together. Write this equation now in matrix vector form, which is the next step towards finding solutions to the weak form of the wave equation. One more point. You see here in the equations for each of those matrices and vectors, these are integrals over the entire domain D. From a practical point of view, that's not very good. From an analytical point of view, that's not very good. We're going to break this again down to the local element level, which we've also done in the static case, and that's also going to be one of the next steps.
