Embedded Rayleigh–Bloch surface waves along periodic rectangular arrays

In this paper, surface waves in the presence of an infinite periodic array of obstacles of rectangular cross-section are considered. Rayleigh–Bloch surface waves are described by a localised wave motion which does not propagate energy away from the array. The periodicity of the array implies the existence of a cut-off frequency below which Rayleigh–Bloch surface waves may be sought. Such solutions are well established and Rayleigh–Bloch surface waves have been shown to exist for all rectangular cross-section. In the present paper, we generate examples of Rayleigh–Bloch surface waves for the more complicated case of frequencies lying above the first cut-off, such waves correspond mathematically to eigenvalues embedded in the continuous spectrum of the field operator. Numerical results are given for rectangular cross-sections based on an integral equation formulation of the problem. Finally, strong numerical evidence is given for embedded Rayleigh–Bloch waves that exist for a single family of rectangular cross-section above the second cut-off.