Using D'Alembert Solution to Find Values of Regions

[FAS1998]

How were the values of the the regions found in the grid of this solution? I understand that the value should be 0 in every regions that contains the points x = 0, x=4, etc...

I believe the bottom values can be found from the boundary conditions as well, but what about the others?

 

[Orodruin]

Do you understand why the initial condition looks the way it does when extended to a problem on the real line?

Once you have that and d’Alembert’s solution for the time derivative at t=0 being equal to zero you should be able to understand the figure.