So, we've seen in our Python implementation of the one-dimensional acoustic wave equation, that we can propagate sound waves away from a central source point. So, one of the most important aspects of all numerical implementations of partial differential equations, is the comparison with known analytical solutions. Before we proceed, we now try to find some of the analytical solutions of the acoustic wave equation and go back to the numerical simulation, compare with them and quantify the difference and the error of these simulations. So, let's look again at the acoustic wave equation. On the right hand side, we have the source term S that depends on x and t. Now, there are also solutions of the wave equation without that source term, so, let's start with this situation. The analytical solution to that is based on initial conditions. So, we can formulate an initial condition, the pressure P at t equals zero has a wave form P zero of x, and the first derivative of time of P is zero. So, if we go into the equation with this initial condition, we actually get analytical solution that you see here. So, we have terms P zero of ct minus x and ct plus x multiplied by one-half, which is actually the initial waveform multiplied by a factor of 0.5, propagating away from the central point in both directions. So, that's one analytical solution we can compare our numerical results to. So, what if we have a source term s of xt in the acoustic wave equation? To find analytical solutions for the situation, we have to make use of the concept of Green's functions that we have encountered before. For that, we define a source using Delta functions and let's recall the definition of a Delta function in either space or time, it can be x zero or t zero, and it's defined, for example in space, if Delta of x zero is infinite at point x zero and zero elsewhere. Also, there is a definition that the integral of the Delta function is one, and that's going to be very important later also for numerical implementation. So, provided that we have a source term as a Delta function in space time at x zero and t zero, the solution to that problem is the so called Green's function, and that's defined as G for Green's function, of xt for the source at x zero and t zero, and so we rewrite the wave equation in this form replacing basically with the field p by the Green's function. Now, without deriving the results, the Green's function for this situation is given here. It actually contains the Heaviside function, so, let's define what the Heaviside function is, which is basically the integral of the delta function, the Heaviside is simply a step function that for x starts from minus infinity at zero it goes up to one, so, that's a step function, that's the Heaviside function. So, in other words and that's a very important point for the numerical simulation, actually the solution of the delta function or solution of the wave equation is actually the integral of the source time function, and that's something we will see also later in the numerical solutions.