I Using D'Alembert Solution to Find Values of Regions

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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

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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.
 

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