Abstract
This study follows from Maurel et al., Phys. Rev. B 98, 134311 (2018), where we reported on direct numerical observations of out-of-plane shear surface waves propagating along an array of plates atop a guiding layer, as a model for a forest of trees. We derived closed form dispersion relations using the homogenization procedure and investigated the effect of heterogeneities at the top of the plates (the foliage of trees). Here, we extend the study to the derivation of a homogenized model accounting for heterogeneities at both endings of the plates. The derivation is presented in the time domain, which allows for an energetic analysis of the effective problem. The effect of these heterogeneous endings on the properties of the surface waves is inspected for hard heterogeneities. It is shown that top heterogeneities affect the resonances of the plates, hence modifying the cut-off frequencies of a wave mathematically similar to the so-called Spoof Plasmon Polariton (SPP) wave, while the bottom heterogeneities affect the behavior of the layer, hence modifying the dispersion relation of the Love waves. The complete system simply mixes these two ingredients, resulting in hybrid surface waves accurately described by our model.
Highlights
The problem of waves propagating in an elastic half-space supporting an array of beams or plates is well known in seismology, where the site–city interaction aims at understanding the interaction of seismic waves with a set of buildings
Arrays of trees with a gradient in their heights [16,17,18]. This configuration is the elastic analog of a corrugated interface able to support surface waves, studied in acoustics [19] and in electromagnetism [20,21], where they are known as Spoof Plasmon Polaritons (SPPs)
Simple dispersion relations have been obtained from the effective model for the resulting spoof Love waves, so-called because of the characteristics they share with classical Love waves and SPPs
Summary
The problem of waves propagating in an elastic half-space supporting an array of beams or plates is well known in seismology, where the site–city interaction aims at understanding the interaction of seismic waves with a set of buildings. Simple dispersion relations have been obtained from the effective model for the resulting spoof Love waves, so-called because of the characteristics they share with classical Love waves (surface waves supported by the layer on its own) and SPPs. to account for the presence of heterogeneities (a foliage) at the top of the plates (trees), a hybrid model was used where the homogenization was performed locally (near the top of the plates) at the second order. The present study generalizes and complements this study following two ways: (i) from a physical point of view, we include the effect of heterogeneities at the bottom of the plates (Figure 1), and (ii) from a technical point of view, we derive the full model at second order.
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