 Characteristic line for the one-dimensional wave equation

These characteristic curves are straight when wave speed $c$ is constant. A right-moving component is constant on $x-ct=s$, and a left-moving component on $x+ct=s$. The full superposition generally is not constant on either family, because the other component can vary along it. Their backwards intersections with the initial line select the endpoints used in D'Alembert's formula.