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Description
Rayleigh–Bloch waves are guided modes localized around periodic arrays with unbounded unit cells, and they play an important role in resonant wave scattering by diffraction gratings. In this work, we extend a Riemann-sheet tracking framework for Rayleigh–Bloch waves to two-dimensional acoustic gratings composed of penetrable scatterers. The problem is formulated as a quasi-periodic transmission eigenvalue problem, where the exterior and interior acoustic fields are coupled through continuity conditions on the scatterer boundary. Using quasi-periodic Green’s functions, we derive a boundary-integral formulation that allows selected plane-wave components to become unbounded in the far field, thereby indexing different Riemann sheets of the Bloch wavenumber. The resulting nonlinear eigenvalue problem is solved using the block Sakurai–Sugiura method, enabling robust continuation of Rayleigh–Bloch wavenumbers as the frequency and material contrast vary. Numerical examples demonstrate how penetration into the scatterers modifies the cut-on, cut-off and interaction behaviour of Rayleigh–Bloch waves, including transitions between physical and non-physical sheets. The results reveal qualitative differences from the sound-hard case and provide a systematic way to analyse guided and near-resonant modes in periodic structures with finite material contrast.