How does one get from $$\frac{d^2f}{dz^2} - c^2 \frac{d^2f}{dt^2} = 0 $$
with $f $ being $f(z,t)$, by performing a coordinate transformation to get $f(r,s)$ with $r=z-ct$ and $s=z+ct$, to $$ \frac{d^2f(r,s)}{dz^2}=\frac{d^2f}{dr^2} +2\frac{d^2f}{drds} + \frac{d^2f}{ds^2}. $$ and $$ \frac{d^2f(r,s)}{dt^2}=c^2(\frac{d^2f}{dr^2} -2\frac{d^2f}{drds} + \frac{d^2f}{ds^2}). $$