The Green`s function for an acoustic, half-space impedance problem Part II: Analysis of the slowly varying and the plane wave component
C. Lin, J. M. Melenk, S. Sauter
TL;DR
This work establishes a rigorous geometric-optics–style decomposition for the acoustic half-space Green's function under impedance boundary conditions in arbitrary dimensions: $G_{\text{half}}(\mathbf{x},\mathbf{y})$ can be written as a sum of two terms, each the product of an exponential eikonal factor and a slowly varying function. The authors introduce κ-slowly varying families and prove that the three components $\Theta_{\nu,s}^{\text{illu}}$, $\Theta_{\nu,s}^{\text{refl}}$, and $\Theta_{\nu,s}^{\text{imp}}$ satisfy exponential-convergence type polynomial-approximation bounds on η-admissible blocks, enabling efficient numerical approaches via tensor Chebyshev interpolation and hierarchical/butterfly methods. A key technical toolkit includes a uniform majorant for the Macdonald function $K_{\mu}$, holomorphic-norm extensions for the relevant norms, and careful control of the auxiliary quantities $\tilde{\mu}$ and $t$, which govern the impedance contribution. The results provide both a structural understanding of the Green's function and practical, provable bounds that support high-accuracy, fast computation in applications such as acoustic scattering and ground-impedance problems.
Abstract
We show that the acoustic Green`s function for a half-space impedance problem in arbitrary spatial dimension d can be written as a sum of two terms, each of which is the product of an exponential function with the eikonal in the argument and a slowly varying function. We introduce the notion of families of slowly varying functions to formulate this statement as a theorem and present its proof.