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We investigate the scattering of scalar plane waves in two dimensions by a heterogeneous layer that is periodic in the direction parallel to its boundary. On describing the layer as a union of periodic laminae, we develop a solution of the scattering problem by merging the concept of propagator matrices and that of Bloch eigenstates featured by the unit cell of each lamina. The featured Bloch eigenstates are obtained by solving the quadratic eigenvalue problem (QEVP) that seeks a complex-valued wavenumber normal to the layer boundary given (i) the excitation frequency, and (ii) real-valued wavenumber parallel to the boundary -- that is preserved throughout the system. Spectral analysis of the QEVP reveals sufficient conditions for discreteness of the eigenvalue spectrum and the fact that all eigenvalues come in complex-conjugate pairs. By deploying the factorization afforded by the propagator matrix approach, we demonstrate that the contribution of individual eigenvalues (and so eigenmodes) to the solution diminishes exponentially with absolute value of their imaginary part, which then forms a rational basis for truncation of the factorized Bloch-wave solution. The proposed methodology caters for the optimal design of rainbow traps, energy harvesters, and metasurfaces, whose potency to manipulate waves is decided not only by the individual dispersion characteristics of the component laminae, but also by ordering and generally fitting of the latter into a composite layer. Using the factorized Bloch approach, evaluation of trial configurations -- as generated by the permutation and window translation/stretching of the component laminae -- can be accelerated by decades.

We consider homogenization of the scalar wave equation in periodic media at finite wavenumbers and frequencies, with the focus on continua characterized by: (a) arbitrary Bravais lattice in $\mathbb{R}^d$, $d\!\geqslant\!2$, and (b) exclusions i.e. "voids" that are subject to homogenous (Neumann or Dirichlet) boundary conditions. Making use of the Bloch wave expansion, we pursue this goal via asymptotic ansatz featuring the "spectral distance" from a given wavenumber-eigenfrequency pair (within the first Brillouin zone) as the perturbation parameter. We then introduce the effective wave motion via projection(s) of the scalar wavefield onto the Bloch eigenfunction(s) for the unit cell of periodicity, evaluated at the origin of a spectral neighborhood. For generality, we account for the presence of the source term in the wave equation and we consider -- at a given wavenumber -- generic cases of isolated, repeated, and nearby eigenvalues. In this way we obtain a palette of effective models, featuring both wave- and Dirac-type behaviors, whose applicability is controlled by the local band structure and eigenfunction basis. In all spectral regimes, we pursue the homogenized description up to at least first order of expansion, featuring asymptotic corrections of the homogenized Bloch-wave operator and the homogenized source term. Inherently, such framework provides a convenient platform for the synthesis of a wide range of wave phenomena in metamaterials and phononic crystals. The proposed homogenization framework is illustrated by approximating asymptotically the dispersion relationships for (i) Kagome lattice featuring hexagonal Neumann exclusions, and (ii) "pinned" square lattice with circular Dirichlet exclusions. We complete the numerical portrayal of analytical developments by studying the response of a Kagome lattice due to a dipole-like source term acting near the edge of a band gap.

In this work, we propose a full-waveform technique for the spatial reconstruction and characterization of (micro-) seismic events via joint source location and moment tensor inversion. The approach is formulated in the frequency domain, and it allows for the simultaneous inversion of multiple point-like events. In the core of the proposed methodology is a grid search for the source locations that encapsulates the optimality condition on the respective moment tensors. The developments cater for compactly supported elastic bodies in $\mathbb{R}^2$; however our framework is directly extendable to inverse (seismic) source problems in $\mathbb{R}^3$ involving both bounded and unbounded elastic domains. A set of numerical results, targeting laboratory applications, is included to illustrate the performance of the inverse solution in situations involving: (i) reconstruction of multiple events, (ii) sparse (pointwise) boundary measurements, (iii) "off-grid" location of the micro-seismic events, and (iv) inexact knowledge of the medium's elastic properties.

The focus of our work is dispersive, second-order effective model describing the low-frequency wave motion in heterogeneous (e.g.~functionally-graded) media endowed with periodic microstructure. For this class of quasi-periodic medium variations, we pursue homogenization of the scalar wave equation in $\mathbb{R}^d$, $d\geqslant 1$ within the framework of multiple scales expansion. When either $d=1$ or $d=2$, this model problem bears direct relevance to the description of (anti-plane) shear waves in elastic solids. By adopting the lengthscale of microscopic medium fluctuations as the perturbation parameter, we synthesize the germane low-frequency behavior via a fourth-order differential equation (with smoothly varying coefficients) governing the mean wave motion in the medium, where the effect of microscopic heterogeneities is upscaled by way of the so-called cell functions. In an effort to demonstrate the relevance of our analysis toward solving boundary value problems (deemed to be the ultimate goal of most homogenization studies), we also develop effective boundary conditions, up to the second order of asymptotic approximation, applicable to one-dimensional (1D) shear wave motion in a macroscopically heterogeneous solid with periodic microstructure. We illustrate the analysis numerically in 1D by considering (i) low-frequency wave dispersion, (ii) mean-field homogenized description of the shear waves propagating in a finite domain, and (iii) full-field homogenized description thereof. In contrast to (i) where the overall wave dispersion appears to be fairly well described by the leading-order model, the results in (ii) and (iii) demonstrate the critical role that higher-order corrections may have in approximating the actual waveforms in quasi-periodic media.

We pursue a low-wavenumber, second-order homogenized solution of the time-harmonic wave equation at both low and high frequency in periodic media with a source term whose frequency resides inside a band gap. Considering the wave motion in an unbounded medium $\mathbb{R}^d$ ($d\geqslant1$), we first use the (Floquet-)Bloch transform to formulate an equivalent variational problem in a bounded domain. By investigating the source term's projection onto certain periodic functions, the second-order model can then be derived via asymptotic expansion of the Bloch eigenfunction and the germane dispersion relationship. We establish the convergence of the second-order homogenized solution, and we include numerical examples to illustrate the convergence result.

This study examines the feasibility of the full-field ultrasonic characterization of fractures in rock. To this end, a slab-like specimen of granite is subjected to in-plane, O(10$^4$Hz) excitation while monitoring the induced 2D wavefield by a Scanning Laser Doppler Vibrometer (SLDV) with sub-centimeter spatial resolution. Upon suitable filtering and interpolation, the observed wavefield is verified to conform with the plane-stress approximation and used to: (i) compute the maps of elastic modulus in the specimen (before and after fracturing) via a rudimentary application of the principle of elastography; (ii) reconstruct the fracture geometry; (iii) expose the fracture's primal (traction-displacement jump) contact behavior, and (iv) identify its profiles of shear and normal specific stiffness. Through the use of full-field ultrasonic data, the approach provides an unobscured, high-resolution insight into the fracture's contact behavior, foreshadowing in-depth laboratory exploration of interdependencies between the fracture geometry, aperture, interphase properties, and its seismic characteristics.

A non-iterative waveform sensing approach is proposed toward (i) geometric reconstruction of penetrable fractures, and (ii) quantitative identification of their heterogeneous contact condition by seismic i.e. elastic waves. To this end, the fracture support $\Gamma$ (which may be non-planar and unconnected) is first recovered without prior knowledge of the interfacial condition by way of the recently established approaches to non-iterative waveform tomography of heterogeneous fractures, e.g. the methods of generalized linear sampling and topological sensitivity. Given suitable approximation $\breve\Gamma$ of the fracture geometry, the jump in the displacement field across $\breve\Gamma$ i.e. the fracture opening displacement (FOD) profile is computed from remote sensory data via a regularized inversion of the boundary integral representation mapping the FOD to remote observations of the scattered field. Thus obtained FOD is then used as input for solving the traction boundary integral equation on $\breve\Gamma$ for the unknown (linearized) contact parameters. In this study, linear and possibly dissipative interactions between the two faces of a fracture are parameterized in terms of a symmetric, complex-valued matrix $\boldsymbol{K}(\boldsymbol{\xi})$ collecting the normal, shear, and mixed-mode coefficients of specific stiffness. To facilitate the high-fidelity inversion for $\boldsymbol{K}(\boldsymbol{\xi})$, a 3-step regularization algorithm is devised to minimize the errors stemming from the inexact geometric reconstruction and FOD recovery. The performance of the inverse solution is illustrated by a set of numerical experiments where a cylindrical fracture, endowed with two example patterns of specific stiffness coefficients, is illuminated by plane waves and reconstructed in terms of its geometry and heterogeneous (dissipative) contact condition.

A theoretical foundation is developed for active seismic reconstruction of fractures endowed with spatially-varying interfacial condition (e.g.~partially-closed fractures, hydraulic fractures). The proposed indicator functional carries a superior localization property with no significant sensitivity to the fracture's contact condition, measurement errors, and illumination frequency. This is accomplished through the paradigm of the $F_\sharp$-factorization technique and the recently developed Generalized Linear Sampling Method (GLSM) applied to elastodynamics. The direct scattering problem is formulated in the frequency domain where the fracture surface is illuminated by a set of incident plane waves, while monitoring the induced scattered field in the form of (elastic) far-field patterns. The analysis of the well-posedness of the forward problem leads to an admissibility condition on the fracture's (linearized) contact parameters. This in turn contributes toward establishing the applicability of the $F_\sharp$-factorization method, and consequently aids the formulation of a convex GLSM cost functional whose minimizer can be computed without iterations. Such minimizer is then used to construct a robust fracture indicator function, whose performance is illustrated through a set of numerical experiments. For completeness, the results of the GLSM reconstruction are compared to those obtained by the classical linear sampling method (LSM).

The concept of topological sensitivity (TS) is extended to enable simultaneous 3D reconstruction of fractures with unknown boundary condition and characterization of their interface by way of elastic waves. Interactions between the two surfaces of a fracture, due to e.g. presence of asperities, fluid, or proppant, are described via the Schoenberg's linear slip model. The proposed TS sensing platform is formulated in the frequency domain, and entails point-wise interrogation of the subsurface volume by infinitesimal fissures endowed with interfacial stiffness. For completeness, the featured elastic polarization tensor - central to the TS formula - is mathematically described in terms of the shear and normal specific stiffness (ks,kn) of a vanishing fracture. Simulations demonstrate that, irrespective of the contact condition between the faces of a hidden fracture, the TS (used as a waveform imaging tool) is capable of reconstructing its geometry and identifying the normal vector to the fracture surface without iterations. On the basis of such geometrical information, it is further shown via asymptotic analysis -- assuming "low frequency" elastic-wave illumination, that by certain choices of (ks,kn) characterizing the trial (infinitesimal) fracture, the ratio between the shear and normal specific stiffness along the surface of a nearly-panar (finite) fracture can be qualitatively identified. This, in turn, provides a valuable insight into the interfacial condition of a fracture at virtually no surcharge -- beyond the computational effort required for its imaging. The proposed developments are integrated into a computational platform based on a regularized boundary integral equation (BIE) method for 3D elastodynamics, and illustrated via a set of numerical experiments.