Local and nonlocal homogenization of wave propagation in time-varying media

Christian Döding; Barbara Verfürth; Christian Döding; Barbara Verfürth

doi:10.3934/nhm.2026040

Networks and Heterogeneous Media

2026, Volume 21, Issue 3: 968-996. doi: 10.3934/nhm.2026040

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Research article

Local and nonlocal homogenization of wave propagation in time-varying media

Christian Döding ^,,
Barbara Verfürth

Institute for Numerical Simulation, University of Bonn, Bonn, Germany

Received: 27 October 2025 Revised: 21 February 2026 Accepted: 06 March 2026 Published: 13 May 2026

Temporal metamaterials are artificially manufactured materials with time-dependent material properties that exhibit interesting phenomena when waves propagate through them. The propagation of electromagnetic waves in such time-varying dielectric media is governed by Maxwell's equations, which lead to wave equations with temporal highly oscillatory coefficients for the electric and magnetic fields. In this study, we analyze the effective behavior of electromagnetic fields in time-varying metamaterials using a formal two-scale asymptotic expansion. We provide a mathematical derivation of the effective equations for the leading-order homogenized solution, as well as for the first- and second-order corrections of the effective solution. While the effective solution and the first-order correction are governed by local material laws, we reveal a nonlocal constitutive relation for the second-order corrections. Special attention is also paid to temporal interface conditions through initial values of the homogenized equations. The results provide a mathematically justified framework for the effective description of wave-type equations of time-varying media, applicable to models in optics, elasticity, and acoustics.
- homogenization,
- asymptotic expansion,
- Maxwell equations,
- wave equation,
- temporal metamaterial,
- time-varying media
Citation: Christian Döding, Barbara Verfürth. Local and nonlocal homogenization of wave propagation in time-varying media[J]. Networks and Heterogeneous Media, 2026, 21(3): 968-996. doi: 10.3934/nhm.2026040

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Abstract

Temporal metamaterials are artificially manufactured materials with time-dependent material properties that exhibit interesting phenomena when waves propagate through them. The propagation of electromagnetic waves in such time-varying dielectric media is governed by Maxwell's equations, which lead to wave equations with temporal highly oscillatory coefficients for the electric and magnetic fields. In this study, we analyze the effective behavior of electromagnetic fields in time-varying metamaterials using a formal two-scale asymptotic expansion. We provide a mathematical derivation of the effective equations for the leading-order homogenized solution, as well as for the first- and second-order corrections of the effective solution. While the effective solution and the first-order correction are governed by local material laws, we reveal a nonlocal constitutive relation for the second-order corrections. Special attention is also paid to temporal interface conditions through initial values of the homogenized equations. The results provide a mathematically justified framework for the effective description of wave-type equations of time-varying media, applicable to models in optics, elasticity, and acoustics.

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