Wigner-Smith Time Delay Matrix for Acoustic Scattering: Theory and Phenomenology

TL;DR

This paper extends the Wigner-Smith time delay matrix concept from quantum mechanics to acoustic scattering, deriving new formulas and demonstrating how eigenmodes reveal specific scattering time delays.

Contribution

The article introduces a generalized formulation of the WS time delay matrix for acoustic problems, applicable to various geometries and boundary conditions, with derived expressions involving energy density integrals.

Findings

Eigenmodes correspond to distinct scattering phenomena

Formulas valid for different geometries and boundary conditions

Numerical examples confirm the theoretical predictions

Abstract

The Wigner-Smith (WS) time delay matrix relates a lossless system's scattering matrix to its frequency derivative. First proposed in the realm of quantum mechanics to characterize time delays experienced by particles during a collision, this article extends the use of WS time delay techniques to acoustic scattering problems governed by the Helmholtz equation. Expression for the entries of the WS time delay matrix involving renormalized volume integrals of energy densities are derived, and shown to hold true independent of the scatterer's geometry, boundary condition (sound-soft or sound-hard), and excitation. Numerical examples show that the eigenmodes of the WS time delay matrix describe distinct scattering phenomena characterized by well-defined time delays.