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

TL;DR

This paper investigates the behavior of nonlinear sound waves near a star's surface, establishing conditions for shock formation and demonstrating the regularity of velocity and sound speed in smooth solutions.

Contribution

It generalizes shallow water wave methods to stellar surface waves and provides criteria for shock formation in nonlinear sound wave evolution.

Findings

Shock formation criterion near the stellar surface.

Velocity and sound speed remain regular in smooth solutions.

Extension of shallow water wave methods to stellar physics.

Abstract

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.