So using Duhammel's principle you need to find a

Such that with initial conditions

Now by D'lamberts we know that

Now by Duhammel's principle the solution is

Note that the region of integration depends on the x and t varaibles. Also the forcing function is non zero in the rectangle (In red in the diagram). In this diagram the function will be zero in any of the shaded (blue) area's and the wave will travel in the white strips.. I hope this helps.