Uniform Asymptotic Expansion Of The Square-Root Helmholtz Operator And The One-Way Wave Propagator

The Bremmer coupling series solution ofthe wave equation, in generally inhomogeneous media, requires the introduction ofpseudodi#erential operators. Such operators appear in the diagonalization process ofthe acoustic system's matrix ofpartial di#erential operators upon extracting a principal direction of(one-way) propagation. In this paper, in three dimensions, uniform asymptotic expansions of the Schwartz kernels ofthese operators are derived. Also, we derive a uniform asymptotic expansion of the one-way propagator appearing in the series. We focus on designing closed-form representations, valid in the high-frequency limit, taking into account critical scattering, phenomena. The latter phenomena are not dealt with in the standard calculus of pseudodi#erential operators. Our expansion is not limited by propagation angle. In principle, the uniform asymptotic expansion of a kernel follows by matching its asymptotic behaviors away and near its diagonal. Ke words. wave field decomposition, Bremmer series, uniform asymptotics AMS sub je classifications. 35L05, 35C10, 47G30, 35C20, 34E20 PII. S0036139902402300 1