So probably, you are already a little bit hungry for some maths. So let's start looking at the equation that we're later going to use to compare different numerical methods. This is the simplest equation that we're going to use it is the Acoustic wave equation in its second-order form, and it's given here. It's the second time derivative of pressure, that's depending on space and time x and t so, we are restricting ourselves to one dimension here, and on the right-hand side we have c square, c is the propagation Velocity,. Velocity here is the Acoustic Velocity, multiplying in general the Laplacian of the pressure in 1D in the Cartesian space simply the second space derivative with respect to x. Now, we omit any source terms here for the moment, and just the velocity is c, is actually also a function of x of space but it doesn't depend on time, and if we assume that it does not depend on x, so it's constant in our physical domain, so let's say it's c zero, then actually we can find simple solutions to that equation. So, even though we're going to talk about this later in terms of plane waves, let's just assume we can have a trial solution, p of x, t, is equal to p 0 e to the i kx minus Omega t. Now, we'd like you to just take a pencil and paper and see what happens if you go into the equation with this trial solution. Surprise, surprise. Well, you've encountered something that we've seen before, you basically end up simply with the relation between wave number k, angular frequency Omega and the phase velocity. C equals Omega divided by k Omega is two Pi f, where f, is the frequency or two Pi by capital T where T is the period, and k is two Pi by Lambda where Lambda is the wavelength. Again, we will use this kind of relation all the time. Actually, this is an equation called a dispersion relation. A dispersion relation here, is actually relating specifically the velocity, the phase velocity, with wavelength, wave number and frequency. Actually, there is another equation that we are also going to use and solve numerically later, which is an even simpler equation, a partial differential equation of first order form. So, we'll see it here, it's the first derivative with respect to time, again let's take the pressure p, plus c the phase velocity, multiplying the first derivative with respect to space equals zero, we omit any source terms for the moment. So, that raises the question, how can we classify these kind of partial differential equations and there is an elegant way of doing that. You can see this here, a linear combination of basically first and mixed derivatives weighted by some coefficients a, b, c, d, e, f, and actually if you transform such a general partial differential equation using Fourier transforms, you obtain an equation, that you can now look at and investigate using so called discriminants. Now, depending on the coefficients, you can actually distinguish these partial differential equations whether they are elliptic, hyperbolic, parabolic and, the discriminant factor is actually given here, with various conditions. And we'll actually stop here, and again you are allowed to investigate for yourself and calculate what kind of form, what specific form our equation has. I would like you to apply this discriminant to the second order wave equation and then we meet again. So it was simple. By calculating the b squared minus 4ac term, you can actually simply categorize a specific partial differential equation. It turns out that, our equation, our wave equation is of hyperbolic form, and actually there are many many physical phenomena that are based on hyperbolic partial differential equations. Given initial conditions for pressure or actually the time derivative of the pressure, the solution is actually determined for all times and it's going to be probably wave like functions that are propagating away from any source or superimposing each other. By the way, this hyperbolic form also applies to the general three dimensional elastic wave equation, so it's very powerful. Where do these categories come from, elliptical, parabolic and hyperbolic? Well, they actually come from intersections with cones and an example for the hyperbolic form is actually given here in the graph. It's a vertical plane that's intersecting a cone that's leading to a hyperbola, and this is basically intrinsic to the form of the partial differential equation, in the Fourier domain. Now, the next step is we're going to look at the impact of the fact that we have this wave like phenomena, and parallelism in the context of parallel computers.