  Extensions of Rayleigh-Bloch waves above the cut-off frequency are studied
via the discrete spectrum of a transfer operator for a generalised channel
containing a single cylinder. Their wavenumbers are shown to become
complex-valued and an additional pair of wavenumbers to appear. For small to
intermediate radius values, the extended Rayleigh-Bloch waves are shown connect
the Neumann and Dirichlet trapped modes, then embed in the continuous spectrum.
Rayleigh-Bloch waves vanish as frequency increases but reappear at high
frequencies for small and large cylinders. The existence and properties of the
Rayleigh-Bloch waves are connected with finite-array resonances.
