We discussed the problem of a cold wind blowing on an immovable obstacle. We derived the steady state, azimuthally symmetric hydrodynamics equations in cylindrical parabolic coordinates. We were able to use the approximation of very large \(\tau\) to reduce these equations into a set of ordinary differential equations in \(\sigma\). We numerically integrated these equations, and compared them to a full hydrodynamical simulation.

Further work is required in order to apply this model to astrophysical problem. For example, planetary bow shocks require magnetohydrodynamics, supernova - cloud interaction involve a time varying wind, and the complete description of the bow shock around Betelgeuse requires radiative cooling \cite{Mohamed_2012}. We note, however, that if cooling can be neglected, the bow shock around Betelgeuse would be a good candidate, due to to high Mach number (of the order of magnitude of 13), and the fact the shock stretches to as far as a parsec from the obstacle.

Steady state oblique shock have recently regained popularity in the context of shock breakout \cite{Matzner_2013}. In that same work, the authors assumed that on very small length scales the shock front may be curved, but on larger scales the shock front has to be self similar, and hence straight. However, the analytical solution for that problem yields a non physical shock angle (such that its cosine is larger than 1). This work hints at an alternative solution. Perhapse the shock is curved on all length scales (as long as the shock can be considered strong).