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This paper is concerned with the asymptotic expansions of the amplitude ofthe solution of the Helmholtz equation. The original expansions were obtainedusing a pseudo-differential decomposition of the Dirichlet to Neumann operator.This work uses first and second order approximations of this operator to derivenew asymptotic expressions of the normal derivative of the total field. Theresulting expansions can be used to appropriately choose the ansatz in thedesign of high-frequency numerical solvers, such as those based on integralequations, in order to produce more accurate approximation of the solutionsaround the shadow and deep shadow regions than the ones based on the usualansatz.

Asymptotic expansions of the Helmholtz equation solutions using approximations of the Dirichlet to Neumann operator