Fig. 2: Dispersion curves and Rayleigh–Bloch (RB) modes.

a, b Finite element (FE) dispersion curves for differing aspect ratio (radius to unit cell width) r/a = 0.35, 0.15, respectively. The sound line is shown in dashed blue with non-radiative regime highlighted; dotted and dashed black lines correspond to standing-wave and band-edge frequencies, respectively, in Fig. 1b, d. Colourscale indicates localisation to the array (Finite Element Methods section). Mode-shapes show RB and generalised RB modes numbered in (b). Neumann and Dirichlet trapped modes (purely real eigenfrequencies) are labelled 1 & 3, respectively. c–f Representations of dispersion spectra as complex eigensolutions μ (black points) for r/a = 0.35 (c, e) and 0.15 (d, f). Coloured arrows show direction around the continuous spectra along the unit circle, corresponding to a fixed frequency and altering RB wavenumber β in (a, b)—the real parts of the eigensolutions are highlighted with corresponding colours in (a, b). Below the cut-off, the eigensolutions lead the unit circle (black arrow), as in (c, d). The first generalised RB mode above the cut-off appears as complex solutions off the unit circle for large aspect ratio (f) that are not present in (e). Dashed boxes in (d, f) show a zoom near the cut-offs.