Rayleigh-Bloch waves above the cutoff

Extensions of Rayleigh-Bloch waves above the cutoff frequency are studied via the discrete spectrum of a transfer operator for a channel containing a single cylinder with quasi-periodic side-wall conditions. Above the cutoff, the Rayleigh-Bloch wavenumber becomes complex valued and an additional wavenumber appears. For small- to intermediate-radius values, the extended Rayleigh-Bloch waves are shown to connect the Neumann and Dirichlet trapped modes before embedding in the continuous spectrum. A homotopy method involving an artificial damping term is proposed to identify the discrete spectrum close to the embedding. Moreover, Rayleigh-Bloch waves vanish beyond some frequency but reappear at higher frequencies for small and large cylinders. The existence and properties of the Rayleigh-Bloch waves are connected with finite-array resonances.

Automated summary

This paper investigates the extensions of Rayleigh-Bloch waves above the cutoff frequency using the discrete spectrum of a transfer operator. The study reveals the complex behavior of Rayleigh-Bloch waves at frequencies above the cutoff, showing connections with the Neumann and Dirichlet trapped modes before embedding in the continuous spectrum. A homotopy method with an artificial damping term is proposed to identify the discrete spectrum close to the embedding. The paper also discusses the disappearance and reappearance of Rayleigh-Bloch waves at different frequencies for small and large cylinders, which is connected to finite-array resonances.