Tracking Rayleigh–Bloch waves swapping between Riemann sheets

Rayleigh–Bloch waves are modes localized to periodic arrays of scatterers with unbounded unit cells. Here, Rayleigh–Bloch waves are studied for line arrays of sound-hard circular scatterers embedded in a two-dimensional acoustic medium, for which it has recently been shown that Rayleigh–Bloch waves exist for higher frequencies than previously thought. Moreover, it was shown that Rayleigh–Bloch waves can cut off (disappear) and cut on (reappear), and additional Rayleigh–Bloch waves can cut on and interact with the existing ones. These complicated behaviours are reconsidered using a family of quasi-periodic Green’s functions that allow particular plane-wave components to become unbounded away from the array. The Green’s function formulation is combined with the block Sakurai–Sugiura method to trace the trajectories of the Rayleigh–Bloch wavenumbers as they swap between Riemann sheets that are categorized according to the unbounded plane wave(s). A detailed analysis is presented for three different scatterer radius values, and contrasting qualitative behaviours are identified. The findings are consistent with those published previously, extend to higher frequencies than allowed by the previous approach, and provide new understanding of Rayleigh–Bloch waves around the critical frequency intervals where they cut on/cut off/interact.