  This paper is concerned with the asymptotic expansions of the amplitude of
the solution of the Helmholtz equation. The original expansions were obtained
using a pseudo-differential decomposition of the Dirichlet to Neumann operator.
This work uses first and second order approximations of this operator to derive
new asymptotic expressions of the normal derivative of the total field. The
resulting expansions can be used to appropriately choose the ansatz in the
design of high-frequency numerical solvers, such as those based on integral
equations, in order to produce more accurate approximation of the solutions
around the shadow and deep shadow regions than the ones based on the usual
ansatz.
