Extend the analysis beyond the subwavelength regime

Generalize the band-edge homogenization and sound-soft scattering analysis for finite bubbly crystals beyond the subwavelength regime.

Background

The paper develops a quantitative sound-soft scattering limit for finite periodic arrays of high-contrast bubbly inclusions at frequencies near the upper edge of the first subwavelength Bloch band. The authors explicitly identify extending this framework to frequency regimes beyond the subwavelength setting as an unresolved problem.

References

Several questions remain open. The first is to generalize our analysis beyond the subwavelength regime; see .

— Band-Edge Homogenization and the Sound-Soft Limit of Finite Bubbly Crystals  (2609.27494 - Ammari et al., 23 Sep 2026) in Section: Concluding remarks

Several questions remain open. The first is to generalize our analysis beyond the subwavelength regime; see . Another direction is to consider arrays with several inclusions per cell, in particular, honeycomb structures. Multiple inclusions would naturally lead to matrix-valued Bloch symbols and potentially coupled envelope equations near Dirac degeneracies; see .

— Band-Edge Homogenization and the Sound-Soft Limit of Finite Bubbly Crystals  (2609.27494 - Ammari et al., 23 Sep 2026) in Section: Concluding remarks

A third challenging question is to extend the interior-band analysis to higher dimensions and rigorously derive the effective behavior of scattered fields at a frequency that is strictly inside the first Bloch band of the corresponding infinitely periodic structure. To simplify the problem, we can consider symmetry-plane slabs and attempt to establish the corresponding boundary matching and uniform resolvent estimates.

— Band-Edge Homogenization and the Sound-Soft Limit of Finite Bubbly Crystals  (2609.27494 - Ammari et al., 23 Sep 2026) in Section: Concluding remarks; see also Section 1, Introduction