  Wave propagation and acoustic scattering problems require vast computational
resources to be solved accurately at high frequencies. Asymptotic methods can
make this cost potentially frequency independent by explicitly extracting the
oscillatory properties of the solution. However, the high-frequency wave
pattern becomes very complicated in the presence of multiple scattering
obstacles. We consider a boundary integral equation formulation of the
Helmholtz equation in two dimensions involving several obstacles, for which ray
tracing schemes have been previously proposed. The existing analysis of ray
tracing schemes focuses on periodic orbits between a subset of the obstacles.
One observes that the densities on each of the obstacles converge to an
equilibrium after a few iterations. In this paper we present an asymptotic
approximation of the phases of those densities in equilibrium, in the form of a
Taylor series. The densities represent a full cycle of reflections in a
periodic orbit. We initially exploit symmetry in the case of two circular
scatterers, but also provide an explicit algorithm for an arbitrary number of
general 2D obstacles. The coefficients, as well as the time to compute them,
are independent of the wavenumber and of the incident wave. The results may be
used to accelerate ray tracing schemes after a small number of initial
iterations.
