Surface wave interacting with resonating trees

The propagation of seismic waves in a 2D halfspace is a well-known problem in numerical seismology and modeled by solving the P- SV elastic wave equation (P : Primary or compressional wave, SV : vertically polarized shear or Secondary wave), but its coupling with resonating tree-like elements is rather unusual. The accuracy of the method has been thoroughly tested using plate and rods as input model and it has delivered excellent results. The 2D time domain simulations are carried out using SPECFEM2D a code that solves the elastic wave equation using finite difference in time and the spectral element method in space. The parallelization is implemented through domain decomposition with MPI. The mesh is made of quadrilateral elements and it is generated using the commercial software CUBIT. Simulations are then run on a parallel cluster (Froggy at University of Grenoble) on 16 CPUs.

The halfspace is characterized by a homogeneous material with shear velocity vs = 500 m/s and density rg = 1300 kg/m3. For a Poisson ratio typical of soil, the Rayleigh wavespeed vr vs. While these parameters are representative for an average soil, the results are not strictly limited to this wave speed but they can be generalized to very soft soils featuring vs < 300 m/s. We use trees of both constant and random size as well as random spacing between trees to evaluate the effects of the variability that characterizes natural forests ; the heights are drawn from a uniform distribution with mean 14 m varying between + or - 2.5 m.

The above movie confirms the hybridization phenomenon that drives local resonances and bandgaps in this type of metamaterial : the longitudinal resonances of tree-like resonators, excited by the vertical component (uz) of the Rayleigh waves, introduces a phase shift of pi on the incident waves causing a reflection of the wavefield around the resonant frequencies. At anti-resonance the point of attachment between ground and tree (z = 0, the forcing point) is at rest and thus uz = 0. Because the trees are arranged on a subwavelength scale, the effect over a period interferes constructively creating a band gap between resonance and anti-resonance.

– See more information in :
Andrea Colombi, Philippe Roux, Sebastien Guenneau, Philippe Gueguen, and
Richard V. Craster, "Forests as a natural seismic metamaterial : Rayleigh wave bandgaps induced by local resonances", Scientific Reports 6, 19238, 2016.

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Trees with increasing height : the seismic rainbow

Here we bring together critical concepts from these different fields, elasticity, plasmonics and metamaterials to design a seismic ’rainbow’ that uses spatially graded subwavelength resonators, in a wedge shape - the resonant metawedge - atop an elastic substrate to eitehr reflect Rayleigh waves or convert Rayleigh waves into bulk elastic waves.

To illustrate the concepts behind the resonant metawedge as a geophysical metamaterial, we model a graded array of vertical resonators in 2D. The coupling between Rayleigh waves and the longitudinal resonances of the vertical resonators, which is dependent upon the seismic impedance match between substrate and metawedge, creates large bandgaps bounded below by frequencies inversely proportional to the resonator height h. The resonant metawedge is then obtained from a graded array of vertical resonators as f is simply tuned by adjusting the height h.

In the rainbow effect, the incident wavefront approaches the wedge from the left towards the short edge of the metawedge, initially undisturbed as a Rayleigh wave with characteristic elliptical polarization. After travelling a few wavelengths inside the wedge, it slows down until reaching the resonator whose fundamental longitudinal mode matches the input frequency of the signal (50 Hz and 70 Hz in the top and bottom panels of the above movie). At this stage the propagation is characterized by a bandgap which, like a rigid barrier, reflects the energy backward.

– See more information in :
Andrea colombi, Daniel Colquitt, Philippe Roux, Sébastien Guenneau, and Richard Craster, "A seismic metamaterial : The resonant metawedge", Scientific Reports 6, 27717, 2016.

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Trees with decreasing height : a new aspect of the seismic rainbow

The movie below shows the more intriguing case of the inverse wedge. The Rayleigh waves now approach the wedge from the right towards the taller end of the metawedge and the wavefield follows a very different pattern vis-a-vis the classic wedge. At the turning point the wavefront is mode-converted and directed into the interior of the substrate with the Rayleigh waves converted into a S-wave whose motion is polarized in the transverse direction. Confirming this interpretation is geometrical spreading, typical of body waves, that can be clearly observed after the conversion.

Whilst a bandgap still exists for the Rayleigh mode, as in the classic wedge, the elastic energy avoids the reflection by conversion to S-waves that propagate into the substrate. This converted wave propagates at a refraction angle calculated from the beam forming in the numerical simulations and this value matches Snell’s law prediction. The observed turning point in both cases, the classic and inverse wedges, remains approximately the same.

– See more information in :
Andrea colombi, Daniel Colquitt, Philippe Roux, Sébastien Guenneau, and Richard Craster, "A seismic metamaterial : The resonant metawedge", Scientific Reports 6, 27717, 2016.

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The Luneberg Lens applied to Geophysics

By applying ideas from transformation optics we can manipulate and steer Rayleigh surface wave solutions of the vector Navier equations of elastodynamics ; this is unexpected as this vector system is, unlike Maxwell’s electromagnetic equations, not form invariant under transformations. As a paradigm of the conformal geophysics that we are creating, we design a square arrangement of Luneburg lenses to reroute and then refocus Rayleigh waves around a building with the dual aim of protection and minimizing the effect on the wavefront (cloaking) after exiting the lenses.

To show that this is practically realisable we deliberately choose in the above movie 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 that can be directly interpreted as a lens refractive index.

– See more information in :
Andrea colombi, Sébastien Guenneau, Philippe Roux, and Richard Craster, "Transformation seismology : composite soil lenses for steering surface elastic Rayleigh waves", Scientific Reports 6, 25320, 2016.

Mis à jour le 15 octobre 2020