Characteristic lines.

  For the wave equation, there are two families of characteristic lines; lines with slope dx/dt = c and lines with slope dx/dt=-c. These are shown in figure 11; the lines along which the singularities of our example propagate are shown thickened. Try following the singularities around.

  

Figure 11: Characteristic lines.

One family of characteristics corresponds to the wave that moves to the right, and the other to the wave that moves toward the left. When the initial condition has a singularity, usually this singularity splits into a part that follows the right moving characteristic at the singularity, and a part that follows the left moving characteristic.