On the second order geometric optics approximation to fast magnetosonic waves

Abstract

The first order term in the geometric optics approximation to the solution of an hyperbolic differential system is known to satisfy a transport equation along rays which is analogous to the Burgers equation. As such it usually develops shocks. The second order term satisfies a linear transport equation whose coefficients depend on the first order solution; these coefficients are detailed for the case of fast magnetosonic waves in a simple equilibrium state. The problem is that the solutions to this second order equation will blow up as soon as the first order term develops a shock. This fact is analyzed and its relevance to the validity of the asymptotic approximation discussed.