Wigner-Smith Time Delay Matrix for Acoustic Scattering: Computational Aspects

The Wigner-Smith (WS) time delay matrix relates an acoustic system's scattering matrix to its wavenumber derivative. The entries of the WS time delay matrix can be expressed in terms of energy density-like volume integrals, which cannot be efficiently evaluated in a boundary element method framework. This paper presents two schemes for efficiently populating the WS time delay matrix. The direct formulation casts the energy density-like volume integrals into integrals of the incident field and the field and/or its normal derivative over the scatterer surface. The indirect formulation computes the system's scattering matrix and its wavenumber derivative, again via surface integration, and then invokes the WS relationship to compute the WS time delay matrix. Both the direct and the indirect formulations yield equivalent results and can be easily integrated into standard boundary element codes.