  We consider the two-dimensional high-frequency plane wave scattering problem
in the exterior of a finite collection of disjoint, compact, smooth, strictly
convex obstacles with Neumann boundary conditions. Using integral equation
formulations, we determine the H\"{o}rmander classes and derive high-frequency
asymptotic expansions of the total fields corresponding to multiple scattering
iterations on the boundaries of the scattering obstacles. These asymptotic
expansions are used to obtain sharp wavenumber dependent estimates on the
derivatives of multiple scattering total fields which, in turn, allow for the
optimal design and numerical analysis of Galerkin boundary element methods for
the efficient (frequency independent) approximation of sound hard multiple
scattering returns. Numerical experiments supporting the validity of these
expansions are presented.
