  High frequency estimates for the Dirichlet-to-Neumann and
Neumann-to-Dirichlet operators are obtained for the Helmholtz equation in the
exterior of bounded obstacles. These a priori estimates are used to study the
scattering of plane waves by an arbitrary bounded obstacle and to prove that
the total cross section of the scattered wave does not exceed four geometrical
cross sections of the obstacle in the limit as the wave number $k\to \infty$.
This bound of the total cross section is sharp.
