Hybridized Love Waves in a Guiding Layer Supporting an Array of Plates with Decorative Endings

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.

Keywords: elastic energy; elastic metasurface; homogenization; metamaterial; time domain analysis.

Figure 7

Dispersion relations of guided waves in four configurations of plates from direct numerics and from the homogenized solution (20)–(22); the dispersion relations are revealed by large $\left|R\right|$ values. The exact reference dispersion relations of classical and modified Love waves are given in the top panel for comparison. Dotted lines are a guide for the eye showing light lines for Love waves and asymptotes for SPPs. AC1 and AC2 are avoided crossings magnified in Figure 8. Details of homogenized coefficients and geometrical parameters are given in Table 2 and Table 3.

Figure A1

Homogenization in the bulk of the array. In the $\mathbf{x}=(x,z)$ coordinate, the array has a spacing $\ell =\epsilon$. The rescaling in the horizontal $\chi =x/\epsilon \in {\mathsf{Y}}_{\mathrm{P}}$ coordinate is shown in the inset; ${\mathsf{Y}}_{\mathrm{P}}$ is a one-dimensional domain.

Figure A2

Elementary cell at the top of the plate in the $\mathit{\chi }=(\chi ,\zeta )$ coordinate, in the unbounded two-dimensional region $\mathcal{Y}={\mathcal{Y}}_{\mathrm{P}}+{\mathcal{Y}}_{\mathrm{t}}$; the region of the plate is ${\mathcal{Y}}_{\mathrm{P}}={\mathsf{Y}}_{\mathrm{P}}\times (-\infty ,0)$, and the heterogeneity ${\mathcal{Y}}_{\mathrm{t}}$ is bounded in $\mathsf{Y}\times (0,h_{\mathrm{t}})$, with $|{\mathcal{Y}}_{\mathrm{t}}|={\phi }_{\mathrm{t}}{h}_{\mathrm{t}}/\ell$.

Figure A3

Elementary cell at the bottom of a single plate in the $\mathit{\chi }=(\chi ,\zeta )$ coordinate. $\mathcal{Y}={\mathcal{Y}}_{\mathrm{P}}\cup {\mathcal{Y}}_{\mathrm{b}}$ with ${\mathcal{Y}}_{\mathrm{P}}=\{\mathit{\chi }\in {\mathsf{Y}}_{\mathrm{P}}\times (0,+\infty )\}$, and ${\mathcal{Y}}_{\mathrm{b}}=\{\mathit{\chi }\in \mathsf{Y}\times (-\infty ,0)\}$; ${\phi }_{\mathrm{b}}{h}_{\mathrm{b}}/\ell$ is the non-dimensional surface of the heterogeneity of vertical extent $h_{\mathrm{b}}/\ell$.

Figure 1

Periodic array of plates decorated at their endings with spacing $\ell =1$, height $h_{\mathrm{P}}$, and thickness ${\phi }_{\mathrm{P}}\ell$; the substrate occupying a half-space is surmounted by a guiding layer of thickness $h_{\mathrm{L}}$ able to support Love waves. The insets show a zoom on the two endings with heterogeneity surfaces ${\mathcal{S}}_{\mathrm{t}}={\phi }_{\mathrm{t}}{h}_{\mathrm{t}}$ and ${\mathcal{S}}_{\mathrm{b}}={\phi }_{\mathrm{b}}{h}_{\mathrm{b}}$.

Figure 2

Configuration of the effective problem (2) and (3): The region of the plates has been replaced by a homogeneous medium; the effective boundary condition and transmission conditions encapsulate the effects of the heterogeneities at the decorative endings of the plates.

Figure 3

Domain $\mathcal{D}$ where the energy is conserved in the absence of incoming/outcoming fluxes through $\Sigma$. The effective boundary condition on $\gamma$ and jump conditions between ${\Gamma }^{\pm }$ in (3) result in additional effective energies ${\mathcal{E}}_{\mathrm{t},\mathrm{b}}$ in (10).

Figure 4

Configuration of the array. The total thickness $H_{\mathrm{P}}=h_{\mathrm{P}}+h_{\mathrm{b}}=12$ of the array and the total thickness $H_{\mathrm{L}}=h_{\mathrm{L}}+h_{\mathrm{t}}=8$ of the layer are kept constant; $\ell =1$ and ${\phi }_{\mathrm{b}}=1$, ${\phi }_{\mathrm{t}}={\phi }_{\mathrm{P}}=0.5$. When the heterogeneities are considered, $h_{\mathrm{b}}=h_{\mathrm{t}}=1$.

Figure 5

Reference solutions: Dispersion relations of Love waves and modified Love waves (with a bottom layer of thickness $h_{\mathrm{b}}$). In the presence of a thin layer b atop the guiding layer, the dispersion relation is modified (${\Theta }_{\mathrm{b}}$ in (16)) resulting in different shapes of the Love dispersion branches. SPP, Spoof Plasmon Polariton.

Figure 6

Asymptotes of the SPPs (at ${\omega }_n^{\mathrm{SPP}}/\left(2\pi \right)=3,15,25$) and modified SPPs (at ${\omega }_n^{\mathrm{SPP},\mathrm{t}}/\left(2\pi \right)=3.1,12.0,22.4$).

Figure 8

Magnified views of the two avoided crossings AC1 and AC2 from Figure 7 (Case 4).

Figure 9

Displacement fields corresponding to the two branches of guided waves (${\beta }_1$ and ${\beta }_2$) at $\omega /\left(2\pi \right)=24$ for the undecorated plates (top) and decorated plates (bottom). On each panel, the field from direct numerics is plotted for $x<0$, and the homogenized solution from (19) is plotted for $x>0$. In both cases, the displacement at $z=-H_{\mathrm{L}}$ is unitary, which allows for a quantitative comparison without any tuning parameter.

Figure 10

Left: Real part of the reflexion coefficient $R\in (-1,1)$ in colorscale against $\beta$ and $\omega \ge c_{\mathrm{S}}\beta$. The dashed white line corresponds to an incident propagating wave at oblique incidence with $\beta =\frac{\omega }{c_{\mathrm{S}}}sin{45}^{\circ }$. Right: Repartition of the energies in the bottom and top heterogeneities (upper panel) and in the substrate, layer, and plate (normalized with the total energy); see the lower panel. Open symbols are obtained from direct numerics, Equations (24), and plain lines from the homogenized problem, Equations (26) and (27).

Figure 11

Surface densities of energy in the actual problem (computed numerically) and given by the homogenized solution at $\omega /\left(2\pi \right)=19$ (corresponding to the resonance of the Love type with $R=-1$, the energy is stored in the layer and in the bottom layer), at $\omega /\left(2\pi \right)=22.7$ (resonance with $R=-1$ of the SPP type; the energy is stored in the plate), and at $\omega /\left(2\pi \right)$, which is a standard case ($R\simeq 1$, the energy is spread).