Travel-time coordinate for a one-dimensional variable-speed wave equation
= 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 is
$$
s(y)=\int^y\frac{dr}{c(r)}.
$$
Its two families of <characteristic curves> are $s(y)-t=\text{constant}$ and $s(y)+t=\text{constant}$ wherever the <wave speed> $c$ is positive.