Characteristic diamond for speed one minus y squared ($|x|+|\mathrm{artanh}y|<S$)

For Cauchy data prescribed on $x=0$ over $|y|<a<1$, put $S=\mathrm{artanh}a$. The maximal region determined solely by those data isIn the travel-time coordinate for a one-dimensional variable-speed wave equation $(x,s)$ this is the ordinary characteristic diamond $|x|+|s|<S$, as follows from finite propagation speed.