  Shock waves are supersonic disturbances propagating in a fluid and giving
rise to dissipation and drag. Weak shocks, i.e., those of small amplitude, can
be well described within the hydrodynamic approximation. On the other hand,
strong shocks are discontinuous within hydrodynamics and therefore probe the
microscopics of the theory. In this paper we consider the case of the strongly
coupled N=4 plasma whose microscopic description, applicable for scales smaller
than the inverse temperature, is given in terms of gravity in an asymptotically
$AdS_5$ space. In the gravity approximation, weak and strong shocks should be
described by smooth metrics with no discontinuities. For weak shocks we find
the dual metric in a derivative expansion and for strong shocks we use
linearized gravity to find the exponential tail that determines the width of
the shock. In particular we find that, when the velocity of the fluid relative
to the shock approaches the speed of light $v\to 1$ the penetration depth
$\ell$ scales as $\ell\sim (1-v^2)^{1/4}$. We compare the results with second
order hydrodynamics and the Israel-Stewart approximation. Although they all
agree in the hydrodynamic regime of weak shocks, we show that there is not even
qualitative agreement for strong shocks. For the gravity side, the existence of
shock waves implies that there are disturbances of constant shape propagating
on the horizon of the dual black holes.
