Past exam of the theoretical mechanics course of the University of Oxford / 2011 / B / 6 / a / iii / Solution

Write the shock as $x=s(t)$ and let the plus trace be on $x>s(t)$. Locally its discontinuous part can be represented with the Heaviside factor $H(x-s(t))$. Distributional differentiation gives ${\partial }_{t}H=-s^{\prime }(t)\delta (x-s(t))$ and ${\partial }_{x}H=\delta (x-s(t))$. The singular part of the conservation law is thereforeIt must vanish, so the Rankine-Hugoniot conditions areThe smooth parts of the fields still satisfy the classical conservation law on each side. The jump condition ensures that the moving discontinuity creates no extra source of the conserved quantities.