 Travel-time coordinate for a one-dimensional variable-speed wave equation

For a one-dimensional wave equation whose principal symbol is ${\xi }_t^2-c(y{)}^2{\xi }_y^2$, a travel-time coordinate isIts two families of characteristic curves are $s(y)-t=\text{constant}$ and $s(y)+t=\text{constant}$ wherever the wave speed $c$ is positive.