Andrea Colombi

Forests as a natural seismic metamaterial: Rayleigh wave bandgaps induced by local resonances.

We explore the thesis that resonances in trees result in forests acting as locally resonant metamaterials for Rayleigh surface waves in the geophysics context. A geophysical experiment demonstrates that a Rayleigh wave, propagating in soft sedimentary soil at frequencies lower than 150 Hz, experiences strong attenuation, when interacting with a forest, over two separate large frequency bands. This experiment is interpreted using finite element simulations that demonstrate the observed attenuation is due to bandgaps when the trees are arranged at the sub-wavelength scale with respect to the incident Rayleigh wave. The repetitive bandgaps are generated by the coupling of the successive longitudinal resonances of trees with the vertical component of the Rayleigh wave. For wavelengths down to 5 meters, the resulting bandgaps are remarkably large and strongly attenuating when the acoustic impedance of the trees matches the impedance of the soil. Since longitudinal resonances of a vertical resonator are inversely proportional to its length, a man-made engineered array of resonators that attenuates Rayleigh waves at frequency ≤10 Hz could be designed starting from vertical pillars coupled to the ground with longitudinal resonance ≤10 Hz.

A seismic metamaterial: The resonant metawedge

Critical concepts from three different fields, elasticity, plasmonics and metamaterials, are brought together to design a metasurface at the geophysical scale, the resonant metawedge, to control seismic Rayleigh waves. Made of spatially graded vertical subwavelength resonators on an elastic substrate, the metawedge can either mode convert incident surface Rayleigh waves into bulk elastic shear waves or reflect the Rayleigh waves creating a “seismic rainbow” effect analogous to the optical rainbow for electromagnetic metasurfaces. Time-domain spectral element simulations demonstrate the broadband efficacy of the metawedge in mode conversion while an analytical model is developed to accurately describe and predict the seismic rainbow effect; allowing the metawedge to be designed without the need for extensive parametric studies and simulations. The efficiency of the resonant metawedge shows that large-scale mechanical metamaterials are feasible, will have application, and that the time is ripe for considering many optical devices in the seismic and geophysical context.

Sub-wavelength energy trapping of elastic waves in a metamaterial

Deep sub-wavelength focusing has been demonstrated for locally resonant metamaterials using electromagnetic and acoustic waves. The elastic equivalents of such objects are made of sub-wavelength resonating beams fixed to a two-dimensional plate, as presented here. Independent of a random or regular arrangement of the resonators, the metamaterial shows large bandgaps that are independent of the incident wave direction. Numerical simulations demonstrate that the insertion of a defect in the layout, as a shorter resonator, creates strong amplification of the wave-field on the defect. This energy trapping, which is localized on a spatial scale that is much smaller than the wavelength in the two-dimensional plate, leads to a >1 factor in terms of the local density of energy.

Seismic metasurfaces: Sub-wavelength resonators and Rayleigh wave interaction

We consider the canonical problem of an array of rods, which act as resonators, placed on an elastic substrate; the substrate being either a thin elastic plate or an elastic half-space. In both cases the flexural plate, or Rayleigh surface, waves in the substrate interact with the resonators to create interesting effects such as effective band-gaps for surface waves or filters that transform surface waves into bulk waves; these effects have parallels in the field of optics where such sub-wavelength resonators create metamaterials in the bulk and metasurfaces at the free surfaces. Here we carefully analyse this canonical problem by extracting the dispersion relations analytically thereby examining the influence of both the flexural and compressional resonances on the propagating wave. For an array of resonators atop an elastic half-space we augment the analysis with numerical simulations. Amongst other effects, we demonstrate the striking effect of a dispersion curve which corresponds to a mode that transitions from Rayleigh wave-like to shear wave-like behaviour and the resultant change in the fields from surface to bulk waves.

Green's function retrieval through cross-correlations in a two-dimensional complex reverberating medium

Cross-correlations of ambient noise averaged at two receivers lead to the reconstruction of the two-point Greens function, provided that the wave-field is uniform azimuthally, and also temporally and spatially uncorrelated. This condition depends on the spatial distribution of the sources and the presence of heterogeneities that act as uncorrelated secondary sources. This study aims to evaluate the relative contributions of source distribution and medium complexity in the two-point cross-correlations by means of numerical simulations and laboratory experiments in a finite-size reverberant two-dimensional (2D) plate. The experiments show that the fit between the cross-correlation and the 2D Greens function depends strongly on the nature of the source used to excite the plate. A turbulent air-jet produces a spatially uncorrelated acoustic field that rapidly builds up the Greens function. On the other hand, extracting the Greens function from cross-correlations of point-like sources requires more realizations and long recordings to balance the effect of the most energetic first arrivals. When the Greens function involves other arrivals than the direct wave, numerical simulations confirm the better Greens function reconstruction with a spatially uniform source distribution than the typical contour-like source distribution surrounding the receivers that systematically gives rise to spurious phases.

Transformation seismology: composite soil lenses for steering surface elastic Rayleigh waves

Metamaterials are artificially structured media that exibit properties beyond those usually encountered in nature. Typically they are developed for electromagnetic waves at millimetric down to nanometric scales, or for acoustics, at centimeter scales. By applying ideas from transformation optics we can steer Rayleigh-surface waves that are solutions of the vector Navier equations of elastodynamics. As a paradigm of the conformal geophysics that we are creating, we design a square arrangement of Luneburg lenses to reroute Rayleigh waves around a building with the dual aim of protection and minimizing the effect on the wavefront (cloaking). To show that this is practically realisable we deliberately choose to use material parameters readily available and this metalens consists of a composite soil structured with buried pillars made of softer material. The regular lattice of inclusions is homogenized to give an effective material with a radially varying velocity profile and hence varying the refractive index of the lens. We develop the theory and then use full 3D numerical simulations to conclusively demonstrate, at frequencies of seismological relevance 3–10 Hz, and for low-speed sedimentary soil (vs: 300–500 m/s), that the vibration of a structure is reduced by up to 6 dB at its resonance frequency.

Enhanced sensing and conversion of ultrasonic Rayleigh waves by elastic metasurfaces

Recent years have heralded the introduction of metasurfaces that advantageously combine the vision of sub-wavelength wave manipulation, with the design, fabrication and size advantages associated with surface excitation. An important topic within metasurfaces is the tailored rainbow trapping and selective spatial frequency separation of electromagnetic and acoustic waves using graded metasurfaces. This frequency dependent trapping and spatial frequency segregation has implications for energy concentrators and associated energy harvesting, sensing and wave filtering techniques. Different demonstrations of acoustic and electromagnetic rainbow devices have been performed, however not for deep elastic substrates that support both shear and compressional waves, together with surface Rayleigh waves; these allow not only for Rayleigh wave rainbow effects to exist but also for mode conversion from surface into shear waves. Here we demonstrate experimentally not only elastic Rayleigh wave rainbow trapping, by taking advantage of a stop-band for surface waves, but also selective mode conversion of surface Rayleigh waves to shear waves. These experiments performed at ultrasonic frequencies, in the range of 400–600 kHz, are complemented by time domain numerical simulations. The metasurfaces we design are not limited to guided ultrasonic waves and are a general phenomenon in elastic waves that can be translated across scales.

Resonant metalenses for flexural waves in plates

The dispersion curves of a cluster of closely spaced rods supported by a thin plate are characterised by subwavelength bandgaps and slow group velocities induced by local resonance effects. A recent analytical study [Williams, Roux, Rupin, and Kuperman (2015). Phys. Rev. B 91, 104307], has shown how the slow velocity branch depends, amongst other parameters, on the height of the rods that make up the cluster. Such metamaterial, offering easy-to-tune spatial velocity gradients, is a perfect candidate for building gradient index lenses such as Luneburg, Maxwell, and 90° rotating. Here theoretical results are combined with numerical simulations to design and test metalenses for flexural waves. The lenses are obtained by tuning the height of the cluster of rods such that they provide the required refractive index profile. Snapshots and videos from three-dimensional numerical simulations in a narrow band centered at ∼4 kHz are used to analyse the performances of three types of gradient index metalens (Luneburg, Maxwell, and 90° rotating).

Elastic Wave Control Beyond Band-Gaps: Shaping the Flow of Waves in Plates and Half-Spaces with Subwavelength Resonant Rods

In metamaterial science local resonance and hybridization are key phenomena strongly influencing the dispersion properties; the metasurface discussed in this article created by a cluster of resonators, subwavelength rods, atop an elastic surface being an exemplar with these features. On this metasurface, band-gaps, slow or fast waves, negative refraction and dynamic anisotropy can all be observed by exploring frequencies and wavenumbers from the Floquet-Bloch problem and by using the Brillouin zone. These extreme characteristics, when appropriately engineered, can be used to design and control the propagation of elastic waves along the metasurface. For the exemplar we consider, two parameters are easily tuned: rod height and cluster periodicity. The height is directly related to the band-gap frequency, and hence to the slow and fast waves, while the periodicity is related to the appearance of dynamic anisotropy. Playing with these two parameters generates a gallery of metasurface designs to control the propagation of both flexural waves in plates and surface Rayleigh waves for half-spaces. Scalability with respect to the frequency and wavelength of the governing physical laws allows the application of these concepts in very different fields and over a wide range of lengthscales.

Toward Seismic Metamaterials: The METAFORET Project

We report on a seismic metamaterial experiment in a pine-tree forest environment where the dense collection of trees behaves as subwavelength coupled resonators for surface seismic waves. For the METAFORET experiment, more than 1000 seismic sensors were deployed over a 120 m × 120 m area to study the properties of the ambient and induced seismic wavefield that propagates in the ground and in trees. The goal of the experiment was to establish a link between seismic-relevant scales and microscale and mesoscale studies that pioneered the development of metamaterial physics in optics and acoustics. The first results of the METAFORET experiment show the presence of frequency band gaps for Rayleigh waves associated with compressional and flexural resonances of the trees, which confirms the strong influence that a dense collection of trees can have on the propagation of seismic waves.