Revisiting imperfect interface laws for two-dimensional elastodynamics.

We study the interaction of in-plane elastic waves with imperfect interfaces composed of a periodic array of voids or cracks. An effective model is derived from high-order asymptotic analysis based on two-scale homogenization and matched asymptotic technique. In two-dimensional elasticity, we obtain jump conditions set on the in-plane displacements and normal stresses; the jumps involve in addition effective parameters provided by static, elementary problems being the equivalents of the cell problems in classical two-scale homogenization. The derivation of the model is conducted in the transient regime and its stability is guarantied by the positiveness of the effective interfacial energy. Spring models are envisioned as particular cases. It is shown that massless-spring models are recovered in the limit of small void thicknesses and collinear cracks. By contrast, the use of mass-spring model is justified at normal incidence, otherwise unjustified. We provide quantitative validations of our model and comparison with spring models by means of comparison with direct numerical calculations in the harmonic regime.

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

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.

Multimodal method for the scattering by an array of plates connected to an elastic half-space.

An accurate and numerically inexpensive method is presented to calculate the reflection of in-plane waves at the free surface of an elastic substrate supporting a periodic array of plates. The method is based on the expansions of the elastic fields on the pseudo-periodic modes in the substrate and on the Lamb modes in the plates. These expansions involve unknown amplitudes which are determined by simple matrix inversion when accounting for the boundary conditions at the surface of the substrate and at the free edges of the plates. Exemplifying results are reported in a wide range of frequency covering two resonances of the plates which are analyzed: a flexural resonance of slender bodies and the quasi-resonance of the so-called edge mode. The convergence of the method is inspected quantitatively; it is shown that it is good in view of the fact that the stress is singular at the corners of the plates in contact with the substrate. In particular, the scattering coefficients converge as 1/Ns 3/2 with Ns the truncation of the expansions.

Modelling resonant arrays of the Helmholtz type in the time domain.

We present a model based on a two-scale asymptotic analysis for resonant arrays of the Helmholtz type, with resonators open at a single extremity (standard resonators) or open at both extremities (double-sided resonators). The effective behaviour of such arrays is that of a homogeneous anisotropic slab replacing the cavity region, associated with transmission, or jump, conditions for the acoustic pressure and for the normal velocity across the region of the necks. The coefficients entering in the effective wave equation are simply related to the fraction of air in the periodic cell of the array. Those entering in the jump conditions are related to near field effects in the vicinity of the necks and they encapsulate the effects of their geometry. The effective problem, which accounts for the coupling of the resonators with the surrounding air, is written in the time domain which allows us to question the equation of energy conservation. This is of practical importance if the numerical implementations of the effective problem in the time domain is sought.

Light scattering by periodic rough surfaces: equivalent jump conditions.

We present an interface model based on two-scale homogenization to predict the coherent scattering of light by a periodic rough interface between air and a dielectric. Contrary to previous approaches where the roughnesses are replaced by a layer filled with an equivalent medium, our modeling yields effective jump conditions applying across the region containing the roughnesses. The validity of the model is inspected by comparison with direct numerics and with experimental measurements on an air/silicium rough interface near the Brewster angle. It is shown that the interface model reproduces accurately the shift in the Brewster phenomenon without any adjustable parameter, which is of practical importance in retrieval methods to get thickness or filling fraction with reliable physical values.

Influence of the neck shape for Helmholtz resonators.

The resonance of a Helmholtz resonator is studied with a focus on the influence of the neck shape. This is done using a homogenization approach developed for an array of resonators, and the resonance of an array is discussed when compared to that of a single resonator. The homogenization makes a parameter B appear which determines unambiguously the resonance frequency of any neck. As expected, this parameter depends on the length and on the minimum opening of the neck, and it is shown to depend also on the surface of air inside the neck. Once these three geometrical parameters are known, B has an additional but weak dependence on the neck shape, with explicit bounds.

Two-scale homogenization to determine effective parameters of thin metallic-structured films.

We present a homogenization method based on matched asymptotic expansion technique to derive effective transmission conditions of thin structured films. The method leads unambiguously to effective parameters of the interface which define jump conditions or boundary conditions at an equivalent zero thickness interface. The homogenized interface model is presented in the context of electromagnetic waves for metallic inclusions associated with Neumann or Dirichlet boundary conditions for transverse electric or transverse magnetic wave polarization. By comparison with full-wave simulations, the model is shown to be valid for thin interfaces up to thicknesses close to the wavelength. We also compare our effective conditions with the two-sided impedance conditions obtained in transmission line theory and to the so-called generalized sheet transition conditions.

Homogenization models for thin rigid structured surfaces and films.

A homogenization method for thin microstructured surfaces and films is presented. In both cases, sound hard materials are considered, associated with Neumann boundary conditions and the wave equation in the time domain is examined. For a structured surface, a boundary condition is obtained on an equivalent flat wall, which links the acoustic velocity to its normal and tangential derivatives (of the Myers type). For a structured film, jump conditions are obtained for the acoustic pressure and the normal velocity across an equivalent interface (of the Ventcels type). This interface homogenization is based on a matched asymptotic expansion technique, and differs slightly from the classical homogenization, which is known to fail for small structuration thicknesses. In order to get insight into what causes this failure, a two-step homogenization is proposed, mixing classical homogenization and matched asymptotic expansion. Results of the two homogenizations are analyzed in light of the associated elementary problems, which correspond to problems of fluid mechanics, namely, potential flows around rigid obstacles.

Improved multimodal method for the acoustic propagation in waveguides with a wall impedance and a uniform flow.

We present an efficient multimodal method to describe the acoustic propagation in the presence of a uniform flow in a waveguide with locally a wall impedance treatment. The method relies on a variational formulation of the problem, which allows to derive a multimodal formulation within a rigorous mathematical framework, notably to properly account for the boundary conditions on the walls (being locally the Myers condition and the Neumann condition otherwise). Also, the method uses an enriched basis with respect to the usual cosine basis, able to absorb the less converging part of the modal series and thus, to improve the convergence of the method. Using the cosine basis, the modal method has a low convergence, 1/N, with N the order of truncation. Using the enriched basis, the improvement in the convergence is shown to depend on the Mach number, from 1/N5 to roughly 1/N1.5 for M=0 to M close to unity. The case of a continuously varying wall impedance is considered, and we discuss the limiting case of piecewise constant impedance, which defines pressure edge conditions at the impedance discontinuities.

Empirical mode decomposition profilometry: small-scale capabilities and comparison to Fourier transform profilometry.

We present the empirical mode decomposition profilometry (EMDP) for the analysis of fringe projection profilometry (FPP) images. It is based on an iterative filter, using empirical mode decomposition, which is free of spatial filtering and adapted for surfaces characterized by a broadband spectrum of deformation. Its performances are compared to Fourier transform profilometry, the benchmark of FPP. We show both numerically and experimentally that using EMDP improves strongly the profilometry small-scale capabilities. Moreover, the height reconstruction distortion is much lower: the reconstructed height field is now both spectrally and statistically accurate. EMDP is thus particularly suited to quantitative experiments.

Modal method for the 2D wave propagation in heterogeneous anisotropic media.

A multimodal method based on a generalization of the admittance matrix is used to analyze wave propagation in heterogeneous two-dimensional anisotropic media. The heterogeneity of the medium can be due to the presence of anisotropic inclusions with arbitrary shapes, to a succession of anisotropic media with complex interfaces between them, or both. Using a modal expansion of the wave field, the problem is reduced to a system of two sets of first-order differential equations for the modal components of the field, similar to the system obtained in the rigorous coupled wave analysis. The system is solved numerically, using the admittance matrix, which leads to a stable numerical method, the basic properties of which are discussed. The convergence of the method is discussed, considering arrays of anisotropic inclusions with complex shapes, which tend to show that Li's rules are not concerned within our approach. The method is validated by comparison with a subwavelength layered structure presenting an effective anisotropy at the wave scale.

Wave propagation in a waveguide containing restrictions with circular arc shape.

A multimodal method is used to analyze the wave propagation in waveguides containing restrictions (or corrugations) with circular arc shapes. This is done using a geometrical transformation which transforms the waveguide with complex geometry in the real space to a straight waveguide in the transformed space, or virtual space. In this virtual space, the Helmholtz equation has a modified structure which encapsulates the complexity of the geometry. It is solved using an improved modal method, which was proposed in a paper by A. Maurel, J.-F. Mercier, and S. Félix [Proc. R. Soc. A 470, 20130743 (2014)], that increases the accuracy and convergence of usual multimodal formulations. Results show the possibility to solve the wave propagation in a waveguide with a high density of circular arc shaped scatterers.

Propagation of elastic waves through textured polycrystals: application to ice.

The propagation of elastic waves in polycrystals is revisited, with an emphasis on configurations relevant to the study of ice. Randomly oriented hexagonal single crystals are considered with specific, non-uniform, probability distributions for their major axis. Three typical textures or fabrics (i.e. preferred grain orientations) are studied in detail: one cluster fabric and two girdle fabrics, as found in ice recovered from deep ice cores. After computing the averaged elasticity tensor for the considered textures, wave propagation is studied using a wave equation with elastic constants c=〈c〉+δc that are equal to an average plus deviations, presumed small, from that average. This allows for the use of the Voigt average in the wave equation, and velocities are obtained solving the appropriate Christoffel equation. The velocity for vertical propagation, as appropriate to interpret sonic logging measurements, is analysed in more details. Our formulae are shown to be accurate at the 0.5% level and they provide a rationale for previous empirical fits to wave propagation velocities with a quantitative agreement at the 0.07-0.7% level. We conclude that, within the formalism presented here, it is appropriate to use, with confidence, velocity measurements to characterize ice fabrics.

Local transformation leading to an efficient Fourier modal method for perfectly conducting gratings.

We present an efficient Fourier modal method for wave scattering by perfectly conducting gratings (in the two polarizations). The method uses a geometrical transformation, similar to the one used in the C-method, that transforms the grating surface into a flat surface, thus avoiding to question the Rayleigh hypothesis; also, the transformation only affects a bounded inner region that naturally matches the outer region; this allows applying a simple criterion to select the ingoing and outgoing waves. The method is shown to satisfy reciprocity and energy conservation, and it has an exponential rate of convergence for regular groove shapes. Besides, it is shown that the size of the inner region, where the solution is computed, can be reduced to the groove depth, that is, to the minimal computation domain.

Extraordinary transmission through subwavelength dielectric gratings in the microwave range.

We address the problem of the transmission through subwavelength dielectric gratings. Following Maurel et al. [Phys. Rev. B 88, 115416 (2013)], the problem is reduced to the transmission by an homogeneous slab, either anisotropic (for transverse magnetic waves, TM) or isotropic (for transverse electric waves, TE), and an explicit expression of the transmission coefficient is derived. The optimum angle realizing perfect impedance matching (Brewster angle) is shown to depend on the contrasts of the dielectric layers with respect to the air. Besides, we show that the Fabry-Perot resonances may be dependent on the incident angle, in addition to the dependence on the frequency. These facts depart from the case of metallic gratings usually considered; they are confirmed experimentally both for TE and TM waves in the microwave regime.

Propagation in waveguides with varying cross section and curvature: a new light on the role of supplementary modes in multi-modal methods.

We present an efficient multi-modal method to describe the acoustic propagation in waveguides with varying curvature and cross section. A key feature is the use of a flexible geometrical transformation to a virtual space in which the waveguide is straight and has unitary cross section. In this new space, the pressure field has to satisfy a modified wave equation and associated modified boundary conditions. These boundary conditions are in general not satisfied by the Neumann modes, used for the series representation of the field. Following previous work, an improved modal method (MM) is presented, by means of the use of two supplementary modes. Resulting increased convergences are exemplified by comparison with the classical MM. Next, the following question is addressed: when the boundary conditions are verified by the Neumann modes, does the use of supplementary modes improve or degrade the convergence of the computed solution? Surprisingly, although the supplementary modes degrade the behaviour of the solution at the walls, they improve the convergence of the wavefield and of the scattering coefficients. This sheds a new light on the role of the supplementary modes and opens the way for their use in a wide range of scattering problems.

Improved multimodal admittance method in varying cross section waveguides.

An improved version of the multimodal admittance method in acoustic waveguides with varying cross sections is presented. This method aims at a better convergence with respect to the number of transverse modes that are taken into account. It is based on an enriched modal expansion of the pressure: the N first modes are the local transverse modes and a supplementary (N+1)th mode, called boundary mode, is a well-chosen transverse function orthogonal to the N first modes. This expansion leads to the classical form of the coupled mode equations where the component of the boundary mode is of evanescent character. Under this form, the multimodal admittance method based on the Riccati equation on the admittance matrix (the Dirichlet-to-Neumann operator) is straightforwardly implemented. With this supplementary mode, in addition to the improvement of the convergence of the pressure field, results show a superconvergence of the scattered field outside of the varying cross sections region.

Wave propagation through penetrable scatterers in a waveguide and through a penetrable grating.

A multimodal method based on the admittance matrix is used to analyze wave propagation through scatterers of arbitrary shape. Two cases are considered: a waveguide containing scatterers, and the scattering of a plane wave at oblique incidence to an infinite periodic row of scatterers. In both cases, the problem reduces to a system of two sets of first-order differential equations for the modal components of the wavefield, similar to the system obtained in the rigorous coupled wave analysis. The system can be solved numerically using the admittance matrix, which leads to a stable numerical method, the basic properties of which are discussed (convergence, reciprocity, energy conservation). Alternatively, the admittance matrix can be used to get analytical results in the weak scattering approximation. This is done using the plane wave approximation, leading to a generalized version of the Webster equation and using a perturbative method to analyze the Wood anomalies and Fano resonances.

Propagation of guided waves through weak penetrable scatterers.

The scattering of a scalar wave propagating in a waveguide containing weak penetrable scatterers is inspected in the Born approximation. The scatterers are of arbitrary shape and present a contrast both in density and in wavespeed (or bulk modulus), a situation that can be translated in the context of SH waves, water waves, or transverse electric/transverse magnetic polarized electromagnetic waves. For small size inclusions compared to the waveguide height, analytical expressions of the transmission and reflection coefficients are derived, and compared to results of direct numerical simulations. The cases of periodically and randomly distributed inclusions are considered in more detail, and compared with unbounded propagation through inclusions. Comparisons with previous results valid in the low frequency regime are proposed.

Number of propagating modes of a diffusive periodic waveguide in the semiclassical limit.

We study the number of propagating Bloch modes N(B) of an infinite periodic billiard chain. The asymptotic semiclassical behavior of this quantity depends on the phase-space dynamics of the unit cell, growing linearly with the wave number k in systems with a non-null measure of ballistic trajectories and going like ∼square root of k in diffusive systems. We have calculated numerically N(B) for a waveguide with cosine-shaped walls exhibiting strongly diffusive dynamics. The semiclassical prediction for diffusive systems is verified to good accuracy and a connection between this result and the universality of the parametric variation of energy levels is presented.