Rayleigh–Bloch waves above the cutoff

Abstract

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.

Metamaterial applications of Tmatsolver, an easy-to-use software for simulating multiple wave scattering in two dimensions

“…The resonance occurs at a frequency just below the cut-off, which is indicated by the sudden drop in the response at higher frequencies, and can be confirmed by calculations of the spectrum for the infinite array, similar to [68] (not part of T matsolver ). The profile at the near-resonant frequency (figure 7 b ) is symmetrical with a peak at the centre, which is associated with primary resonances caused by Rayleigh–Bloch waves [59–61,72].

Figure 7Detection of Rayleigh–Bloch waves along a line array of N=21 sound-hard square scatterers forced by a point source.

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“…Rayleigh–Bloch waves propagate along the corresponding infinite array with wavenumber βfalse(kfalse)>k, and decay exponentially away from it [58]. Nevertheless, Rayleigh–Bloch waves can dominate the response along the finite array [59–61]. They can also be used to engineer desired responses along the array [62,63].…”

“…At this frequency just below the cut-off, the Rayleigh–Bloch waves are approximately anti-symmetric with respect to the vertical axes passing through the scatterers, with blue on one side and yellow on the other side of a given scatterer (compare with fig. 3 c in [61]). Although the near-resonant field contains both rightward and leftward propagating Rayleigh–Bloch waves, they are close to standing waves (near-zero group velocity) and have similar shapes.…”

“…In the acoustic/water-wave setting, Rayleigh–Bloch waves have been computed predominantly for circular scatterers [59–61,65–69], with notable exceptions being elliptical scatterers [58], rectangular scatterers [70] and C-shaped resonators [62,71]. For circular cylinders of radius a and centre-to-centre spacing R, Rayleigh–Bloch waves that are symmetric about the x-axis exist for all k≤ kc<π/R, where kcfalse(afalse) is known as the cut-off frequency.…”

A mechanistic evaluation of the local Bloch wave approximation in graded arrays of vertical barriers

“…The following discussion is heavily based on an article by Bennetts & Peter (2022), who studied Rayleigh–Bloch waves above the cutoff in a medium governed by the Helmholtz equation that contains an infinite line arrays of circular cylinders (also see Bennetts, Peter & Montiel 2017; Bennetts et al. 2018, 2019).…”