  An influential result of F. John states that no genuinely non-linear strictly
hyperbolic quasi-linear first order system of partial differential equations in
two variables has a global $C^2$-solution for small enough initial data.
Inspired by recent work of D. Christodoulou, we revisit John's original proof
and extract a more precise description of the behaviour of solutions at the
time of shock. We show that John's singular first order quantity, when
expressed in characteristic coordinates, remains bounded until the final time,
which is then characterised by an inverse density of characteristics tending to
zero in one point. Moreover, we study the derivatives of second order, showing
again their boundedness when expressed in appropriate coordinates. We also
recover John's upper bound for the time of shock formation and complement it
with a lower bound. Finally, we apply these results to electromagnetic plane
waves in a crystal with no magnetic properties and cubic electric non-linearity
in the energy density, assuming no dispersion.
