Generic behaviour of nonlinear sound waves near the surface of a star: smooth solutions

We are interested in the generic behaviour of nonlinear sound waves as they approach the surface of a star, here assumed to have the polytropic equation of state $P=K\rho^\Gamma$. Restricting to spherical symmetry, and considering only the region near the surface, we generalise the methods of Carrier and Greenspan (1958) for the shallow water equations on a sloping beach to this problem. We give a semi-quantitative criterion for a shock to form near the surface during the evolution of generic initial data with support away from the surface. We show that in smooth solutions the velocity and the square of the sound speed remain regular functions of Eulerian radius at the surface.