Uniform Asymptotic Expansion of the Square-Root Helmholtz Operator and the One-Way Wave Propagator

The Bremmer coupling series solution of the wave equation, in generally inhomogeneous media, requires the introduction of pseudodifferential operators. Such operators appear in the diagonalization process of the acoustic system's matrix of partial differential operators upon extracting a principal direction of (one-way) propagation. In this paper, in three dimensions, uniform asymptotic expansions of the Schwartz kernels of these 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-angle phenomena. The latter phenomena are not dealt with in the standard calculus of pseudodifferential 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.