The whispering surface effect

The representation theory of symmetry groups, together with variational and functional-topological methods, are used in a two-dimensional formulation to investigate the waveguide properties of one-dimensionally periodic surfaces (OPS) and interfaces. It is established that all surfaces on which the Neumann condition is satisfied possess the waveguide property—they are open waveguides. This means that there are waves localized in the neighbourhood of the surface which propagate along it without attenuation—waveguide modes. It is shown that for any hard OPS there is always a transmission band of waveguide frequencies, localized in the neighbourhood of zero—the whispering surface effect. Anomalous oscillations localized around OPSs on which the Neumann condition is satisfied are observed and investigated. Examples of surfaces for which anomalous oscillations exist and others for which none exist are presented. It is proved that OPSs on which the Dirichlet condition holds do not have a transmission band for waveguide frequencies in the neighbourhood of zero, and for some frequency bands they do not have waveguide and anomalous properties. It is shown that one-dimensionally periodic interfaces of two media possess waveguide and anomalous properties, provided that the parameters satisfy certain relationships. It is established that if the interface has the waveguide property, then transmission band of frequencies will always exist localized in the neighbourhood of zero—the whispering interface effect. An example is presented in which anomalous oscillations are investigated, dispersion relations are derived and pass and stop bands for waveguide modes are determined.