Second term of the asymptotic expansion of the diffracted wave in the shadow

Abstract

In this paper we calculate the second order term of the Neumann operator for the high frequency scattering of a wave by a strictly convex obstacle n using the technijues introduced by G. Lebeau and used in previous works. This second term is used to calculate the scattering by nCR , and the result is compared to previous results of V. Babich and 1. Mo1otkov. We generalize the calculation when n C R3. 1. Introduction and statement of the result In this paper we study the high frequency scattering of a wave Ui solution of the Helmholtz equation in R n by a strictly convex analytic obstacle n. We prove that the techniques we used before in (9) to compute the leading term of the asymptotic expansion in k (wave number) can be used to compute the following term of this expansion. In this paper we prove this additional result for a model operator of Lebeau (see (10)) and we use the lower term found in the model case to calculate the lower order term of the scattered solution of the Helmholtz operator. We use throughout· the text the notation 1 a ik ox = Dx =~. Let us state the problem. We consider an outgoing solution Vi(X, t) in R n x Rt of (~ - O;JVi = ° and we denote by Ui(X, k) its partial Fourier-Bros-Iagolnitzer transform in time (FBI). The Fourier­ Bros-Iagolnitzer transform is roughly a generalization of the Fourier transform which constructs from a eoo(RN) function f a complex valued holomorphic function of the variablez E eN. The precise definitions and properties of the FBI transform can be found in (13). We intend to find Vd(X, t) the solution of