Wave equation equivalence

Question

I have to prove that the wave equation $\partial_{tt} \zeta = c^2 \partial_{xx} \zeta$ is equivalent to $$\begin{cases}\partial_tv&=c\partial_xw\\\partial_tw&=c\partial_xv\end{cases}$$ with the variable change $v=\partial_t\zeta$, $w=c\partial_x\zeta$.

If you subsitute the variable change in the wave equation you get the first equivalence, but how about the second? I do not know if I am missing something about the variable change. Any recommendations?

Answer 1

$$\partial_{tt}\zeta = \partial_t(\partial_t\zeta) = \partial_t v = c \partial_x w = c \partial_x (c \partial_x \zeta) = c^2 \partial_{xx} \zeta$$