Olivier Poncelet

Tunable effective constants of the one-dimensional piezoelectric phononic crystal with internal connected electrodes

One-dimensional propagation of a longitudinal wave through an infinite piezoelectric periodically layered structure is considered. The unit cell consists, in general, of piezoelectric multilayers separated by thin electrodes which are connected through a capacitor with capacity Cj that plays the role of the external electric control providing tunability of the mechanical properties. The main focus of the present study is on the effective properties characterizing the homogenized medium. Due to the electric boundary conditions, the 4×4 transfer matrix M through the period T can be reduced to 2 × 2 dimension governing the mechanical fields only. As a result, the homogenized medium is pure elastic though its effective material parameters depend on the piezoelectric and dielectric coefficients of the actual piezoelectric medium, as well as on Cj. The effective parameters and the impedance in the low-frequency limit ω→0 and at finite frequency are obtained and analyzed.

Spaces of electromagnetic and mechanical constitutive parameters for dissipative media with either positive or negative index

Propagation of electromagnetic and acoustic plane waves in dissipative isotropic homogeneous media is described in terms of the Poynting vector and of the complex-valued wave vector. The negative sign of the refractive index, which is explained by the presence of backward bulk waves, is then directly related to the phase angle of the complex-valued wavenumber. Attention is focused on an alternative description dealing with the complex-valued dynamic material parameters: the relative permittivity ϵ and the relative permeability μ for the electromagnetic wave motion, and the bulk modulus κ and the mass density ρ for the acoustic wave motion. The 2D spaces of material parameters ( ϵ , μ ) and ( κ , ρ ) are found to be split into regions characterized by their abilities both to induce wave attenuation and to exhibit opposite directions between the energy flow and the direction of the plane wave propagation. Finally, the relevance of such representations is illustrated by superimposing experimentally retrieved and simulated constitutive parameters of media supporting both forward and backward wave motions.

Sharp acoustic multipolar-resonances in highly monodisperse emulsions

We report the achievement of highly monodisperse emulsions exhibiting about ten acoustic Mie resonances. Thanks to robotics, the effective acoustic properties of such strongly scattering media can be precisely targeted by means of the production of calibrated (random) liquid-droplets. Ultrasonic experiments are compared, with an excellent quantitative agreement, to theoretical predictions derived within the framework of the independent scattering approximation. The dependence of the sound speed and of the acoustic attenuation on both the size and the volume fraction of droplets is quantitatively examined for dilute and more concentrated emulsions, and is presented in a dimensionless way.

Love waves in two-dimensional phononic crystals with depth-dependent properties

We calculate subsonic spectra of the Love waves, i.e., of the shear horizontal waves in the coated substrate, using developed analytical approach. Coating or substrate or both are two-dimensional heterogeneous in the sagittal plane and uniform along the out-of-plane direction. Slow coating permits multiple subsonic dispersion branches which are folded due to lateral periodicity. It is observed that low-frequency branches may either cross or repulse each other, the latter giving rise to low-frequency band gaps inside the Brillouin zone. Such behavior is likelier when the periodic inclusion occurs within the coating close enough to its free surface.

Impact of polydispersity on multipolar resonant scattering in emulsions

The influence of size polydispersity on the resonant acoustic properties of dilute emulsions, made of fluorinated-oil droplets, is quantitatively investigated. Ultrasound attenuation and dispersion measurements on various samples with controlled size polydispersities, ranging from 1% to 13%, are found to be in excellent agreement with predictions based on the independent scattering approximation. By relating the particle-size distribution of the synthesized emulsions to the quality factor of the predicted multipolar resonances, the number of observable acoustic resonances is shown to be imposed by the sample polydispersity. These results are briefly discussed into the context of metamaterials for which scattering resonances are central to their effective properties.

Spectral properties of a 2D scalar wave equation with 1D periodic coefficients: Application to shear horizontal elastic waves

The paper provides a rigorous analysis of the dispersion spectrum of shear horizontal elastic waves in periodically stratified solids. The problem consists of an ordinary differential wave equation with periodic coefficients, which involves two free parameters ω (the frequency) and k (the wavenumber in the direction orthogonal to the axis of periodicity). Solutions of this equation satisfy a quasi-periodic boundary condition which yields the Floquet parameter K. The resulting dispersion surface ω(K, k) may be characterized through its cuts at constant values of K, k and ω that define the passband (real K) and stopband areas, the Floquet branches and the isofrequency curves, respectively. The paper combines complementary approaches based on eigenvalue problems and on the monodromy matrix M. The pivotal object is the Lyapunov function Δ ( ω 2 , k 2 ) ≡ 1 2 trace M = cos K which is generalized as a function of two variables. Its analytical properties, asymptotics and bounds are examined and an explicit form of its derivatives obtained. Attention is given to the special case of a zero-width stopband. These ingredients are used to analyse the cuts of the surface ω(K, k). The derivatives of the functions ω(k) at fixed K and ω(K) at fixed k and of the function K(k) at fixed ω are described in detail. The curves ω(k) at fixed K are shown to be monotonic for real K while they may be looped for complex K (i.e. in the stopband areas). The convexity of the closed (first) real isofrequency curve K(k) is proved thus ruling out low-frequency caustics of group velocity. The results are relevant to the broad area of applicability of ordinary differential equation for scalar waves in 1D phononic (solid or fluid) and photonic crystals.

Coherent acoustic response of a screen containing a random distribution of scatterers: Comparison between different approaches

Theoretical models underlying the ultrasonic study of suspensions and bubble swarms in liquids often rely on the concept of a coherent wave response, within which the given medium is viewed as an effective homogeneous medium. More specifically, the coherent response can be formulated in a straightforward manner via the effective wave number and the effective impedance. These in turn can be expressed through the actual material properties of the given scattering medium. The derivation of the effective wave number and impedance is the issue of a number of different approaches existing in the multiple-scattering theory. The present work deals with a coherent response of acoustic wave impinging on a screen of cylindrical inclusions randomly distributed in a fluid. The reflection and transmission coefficients have been expressed via the effective wave number and impedance that are provided by three different models due to Foldy, Waterman & Truell, and Linton & Martin. The obtained expressions have been analyzed with a view to illuminate what is the quantitative difference between those approaches as revealed in the coherent response, and how this difference depends on the basic parameters of the problem such as the frequency, the concentration of scatterers and their contrast relatively to the fluid matrix. Another aspect of this work is to compare the above analytical results with the numerical data. It has been obtained by means of a deterministic computational code which delivers the coherent wave field through averaging the outputs for various samplings of the positions of scatterers. Knowing this numerical benchmark allows us to specify the validity domains for each of the three analytical methods under study.

Plane Inhomogeneous Elastic Waves in Transversely Isotropic Media: Explicit Analysis of Different Types of Degeneracy

An intrinsic trait of the plane inhomogeneous waves in elastic media is a particular parametrization duality, stemming from two components characterizing the complex wave vector (bivector) k with respect to the reference frame of orthonormal vectors. Correspondingly, the characteristic condition associated with the equation of motion involves two parameters p and λ, where p is the eigenvalue of the Stroh matrix N (λ) and λ is the eigenvalue of the complex Christoffel matrix λ (p). One of the consequences is a rather elaborate layout of the complex-valued degeneracy types implying repeated p, or λ, or both p and λ. Different types of degeneracy are characterized by dissimilar analytical, algebraic and topological features. The general theory, which has been established for an arbitrary anisotropic medium, is developed in the present paper into an explicit form for a transversely isotropic continuum. By reducing the number of geometrical parameters and enabling factorization of the characteristic polynomial, transverse isotropy fosters explicit resolving of the equations which define different degeneracy types. A particular thrust of the paper is concerned with the algebraic analysis of the conditions, under which the degenerate inhomogeneous modes, having linearly varying or/and circularly polarized amplitude, can be excited on a traction-free boundary by means of reflection or within the Rayleigh wave for, respectively, two- and one-parameter manifolds in the space of orientations of the free surface and propagation direction. Various settings are identified which ensure reflected and surface-localized wave packets comprising a degenerate inhomogeneous mode.

Extension to cuspidal edges of wave surfaces of anisotropic solids: Treatment of near cusp behavior

Extension to the cuspidal edges of wave surfaces in a plane of material symmetry of anisotropic solids is proposed via three different approaches. Two of the possible extensions stem from the definition of wave arrival used in the stationary phase approximation and the Cagniard-de Hoop technique for the calculation of the line source (2D) Green’s function. The third one is based on ray theory generalized to complex values of the group velocity. The stationary phase method yields an extension but only near to the cusp points. Concerning the two others methods, an extension for any angle of propagation can be obtained and it leads to a closed wave surface. The three methods are compared with experimental data and the extension given by the Cagniard technique best matches experimental measurements.

Coherent wave propagation in solids containing spatially varying distributions of finite-size cracks

Multiple scattering results usually concern uniform distributions of scatterers. In this work, we evaluate the propagation of SH coherent waves through a non-uniform distribution of parallel flat cracks. The spatial variation of distribution is taken into account via replacing a heterogeneous medium by a stack of effective homogenous layers. Propagation in each layer is governed by the effective acoustic impedance and the effective wave number, which are derived by considering cracks as finite-size scatterers. On this basis, the reflection and transmission coefficients due to non-uniformly distributed scatterers are calculated by using the transfer matrix method. Effect of the distribution profiles is also investigated.