ElasWave: Diverse Elastic-Wave Methods

• ElasWave is a term applied to distinct elastic-wave constructs for eigenmode computation, inverse elastography, metamaterial filtering, and guided-wave rheometry.
• Its implementations use advanced numerical techniques—such as staggered finite-difference grids, plane-wave decomposition, Bloch analysis, and modal dispersion fitting—to achieve error margins below 1%.
• Collectively, these methods drive innovations in waveguide design, ultrasound elastography, polarization-selective filtering, and non-contact constitutive material identification.

Searching arXiv for “ElasWave” and closely related elastic-wave papers to ground the synthesis in current arXiv records. {"^^^^^^^^4query4 OR \4^^^^^^^^"elastic wave eigenmode solver\"^^^^^^^^4 OR \4^^^^^^^^"Plane Wave Elastography\"^^^^^^^^4 OR \4^^^^^^^^"guided elastic waves\"","max_results":^^^^4ElasWave OR \4query4^^^^,"sort_by":"relevance"} ElasWave is a label attached in the cited literature to several distinct elastic-wave constructs rather than to a single universally standardized object. In the arXiv record, it denotes an elastic-wave eigenmode solver for acoustic waveguides (&&&^^^^4query4^^^^&&&), appears to denote the implementation or solver framework built around Plane Wave Elastography for ultrasound shear wave elastography (&&&^^^^4ElasWave OR \4^^^^&&&), is used as a nickname for an elastic three-dimensional metaframe for selective wave filtering and polarization control (&&&^^^^4 OR \4^^^^&&&), and names a guided-wave mechanical characterization method for soft elastomers (&&&^^^^4 OR \4^^^^&&&). Taken together, these usages place ElasWave at the intersection of waveguide eigenanalysis, inverse elastography, metamaterial filtering, and wave-based constitutive identification.

^^^^4ElasWave OR \4^^^^. Semantic scope and principal usages

In the cited literature, the term is associated with four main technical meanings.

Usage of “ElasWave”	Core technical object	Source
Waveguide solver	Elastic-wave eigenmode solver for acoustic waveguides of inhomogeneous cross-section	(&&&^^^^4query4^^^^&&&)
Elastography framework	Plane-wave-based frequency-domain inverse solver for shear modulus estimation	(&&&^^^^4ElasWave OR \4^^^^&&&)
Metamaterial device	Elastic three-dimensional metaframe with local resonators for selective filtering and polarization control	(&&&^^^^4 OR \4^^^^&&&)
Rheometry method	Guided elastic waves in a pre-stretched elastomer strip for visco-acoustoelastic parameter identification	(&&&^^^^4 OR \4^^^^&&&)

A common misconception is that ElasWave refers to one software package or one experimental platform. The cited record does not support that interpretation. A more accurate reading is that the label recurs across multiple elastic-wave contexts, each centered on a different technical problem: modal computation, imaging inversion, polarization-selective attenuation, or broadband material characterization. This suggests a family resemblance at the level of method and application, not a single canonical implementation.

^^^^4 OR \4^^^^. Waveguide eigenmode computation

In its solver usage, ElasWave is a numerical elastic-wave eigenmode solver for acoustic waveguides with arbitrary inhomogeneous cross sections (&&&^^^^4query4^^^^&&&). The governing model is the source-free linear elastic wave equation

PRESERVED_PLACEHOLDER_^^^^4query4^^^^

with linear, time-invariant, isotropic materials in the main implementation and a structure invariant along the propagation direction PRESERVED_PLACEHOLDER_^^^^4ElasWave OR \4^^^^. The reduction to a ^^^^4 OR \4^^^^+^^^^4ElasWave OR \4^^^^D waveguide problem is effected by PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^, so the solver computes guided modes compatible with a specified propagation constant rather than arbitrary ^^^^4 OR \4^^^^D resonances. The weighted displacement field PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^ leads to the eigenproblem

$$\omega^2 \tilde{\mathbf{u}} = \bar{\bar{H}}\,\tilde{\mathbf{u}}, \qquad \bar{\bar{H}} = \frac{-1}{\sqrt{\rho}} \nabla\cdot \mathbb{C} : \nabla_s \frac{1}{\sqrt{\rho}}.$$

The discretization is a staggered finite-difference grid analogous to the Yee grid in electromagnetics. Normal displacements are sampled on face centers, principal stresses at cell centers, and shear stresses at staggered half-cell positions. Free, fixed, symmetry, and anti-symmetry boundary conditions are implemented, and perfectly matched layers are introduced through complex coordinate stretching to model leaky or radiative modes. In lossless cases without PMLs, the operator can be written in a real symmetric form by redefining the eigenvector as $$\tilde{\mathbf{u}}'=[\tilde{u}_x,\tilde{u}_y,-j\tilde{u}_z]$$, whereas loss or PMLs yield a complex symmetric operator.

Validation is extensive. For a thin free silicon slab, eigenfrequency error scales as $\varepsilon \propto (\Delta y)^2$, confirming second-order accuracy. For a $1\,\mu\text{m}$ thick and $1000\,\mu\text{m}$ wide silicon slab, mode shapes and eigenfrequencies for pure shear, anti-symmetric Lamb, and symmetric Lamb waves matched analytical theory with $<1\%$ error over wavelengths from PRESERVED_PLACEHOLDER_^^^^4ElasWave OR \4query4^^^^ down to PRESERVED_PLACEHOLDER_^^^^4ElasWave OR \4ElasWave OR \4^^^^. For suspended and fixed PRESERVED_PLACEHOLDER_^^^^4ElasWave OR \4 OR \4^^^^ silicon beams, both frequency and mode-shape errors were below PRESERVED_PLACEHOLDER_^^^^4ElasWave OR \4 OR \4^^^^ relative to full ^^^^4 OR \4^^^^D COMSOL simulations for most wavelengths from PRESERVED_PLACEHOLDER_^^^^4ElasWave OR \44^^^^ to PRESERVED_PLACEHOLDER_^^^^4ElasWave OR \4^^^^5 (&&&^^^^4query4^^^^&&&).

Its significance lies in treating elastic-waveguide design in the same computational style as electromagnetic mode solving. The intended workflow is to compute acoustic mode profiles, frequencies, and propagation constants from the cross-section, then combine them with electromagnetic solvers for acousto-optic and optomechanical design. The solver is therefore best understood as an acoustic counterpart of photonic eigenmode tools, not as a general transient elastodynamics code.

^^^^4 OR \4^^^^. Plane-wave elastography framework

In ultrasound elastography, ElasWave appears to denote the implementation or solver framework built around Plane Wave Elastography, a frequency-domain shear wave elastography method that estimates stiffness without first decomposing the measured wavefield into known propagation directions (&&&^^^^4ElasWave OR \4^^^^&&&). Commercial systems such as SSI and CUSE typically rely on time-of-flight analysis and directional filtering. By contrast, PWE models the measured displacement field directly as a superposition of plane-wave solutions of the scalar wave equation in a homogeneous, isotropic, incompressible, linear-elastic window:

PRESERVED_PLACEHOLDER_^^^^4ElasWave OR \4^^^^6

which in the Fourier domain becomes

PRESERVED_PLACEHOLDER_^^^^4ElasWave OR \4^^^^7

Within a homogeneous subdomain, the field is represented as

PRESERVED_PLACEHOLDER_^^^^4ElasWave OR \4^^^^8

with unknown complex coefficients PRESERVED_PLACEHOLDER_^^^^4ElasWave OR \4^^^^9 and candidate directions PRESERVED_PLACEHOLDER_^^^^4 OR \4query4^^^^. The inverse problem is posed as a regularized nonlinear least-squares optimization,

PRESERVED_PLACEHOLDER_^^^^4 OR \4ElasWave OR \4^^^^

and extended to multiple frequencies by summing the same objective over PRESERVED_PLACEHOLDER_^^^^4 OR \4 OR \4^^^^. For fixed PRESERVED_PLACEHOLDER_^^^^4 OR \4 OR \4^^^^, the coefficient update is closed-form, and PRESERVED_PLACEHOLDER_^^^^4 OR \44^^^^ is recovered by a one-dimensional search over PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^5. The local shear modulus then follows from PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^6 (&&&^^^^4ElasWave OR \4^^^^&&&).

The operational consequences are specific. No explicit directional filtering is required, no prior knowledge of propagation direction is needed, no compounding step is required, regularization acts as implicit denoising, and the mean-squared error of the fit serves as a confidence metric. The framework is explicitly presented as highly parallelizable because each frequency and each local window can be solved independently. This distinguishes the PWE-style ElasWave usage from time-of-flight pipelines and from full PDE-constrained inversion.

The reported phantom studies establish competitive performance. On a PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^7 digital phantom with background shear modulus PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^8 kPa and inclusion modulus PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^9 kPa, the noiseless reconstruction gave PRESERVED_PLACEHOLDER_^^^^4 OR \4query4^^^^ kPa, PRESERVED_PLACEHOLDER_^^^^4 OR \4ElasWave OR \4^^^^ kPa, and CNR PRESERVED_PLACEHOLDER_^^^^4 OR \4 OR \4^^^^ dB; under strong added Gaussian noise, the method still produced PRESERVED_PLACEHOLDER_^^^^4 OR \4 OR \4^^^^ kPa, PRESERVED_PLACEHOLDER_^^^^4 OR \44^^^^ kPa, and CNR PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^5 dB (&&&^^^^4ElasWave OR \4^^^^&&&). The main misconception corrected by this work is that complex shear-wave fields necessarily require directional prefiltering before inversion.

4. Metaframe-based filtering and polarization control

A different ElasWave usage names a three-dimensional elastic metaframe embedding local resonators for selective wave filtering and polarization control (&&&^^^^4 OR \4^^^^&&&). The structure is a periodic elastic frame whose nodes are loaded with resonators, each resonator consisting of a thin beam with a spherical tip mass. The key result is a complete low-frequency subwavelength bandgap with strongly polarization-dependent attenuation: the same bandgap window strongly blocks transverse elastic waves while allowing longitudinal waves to remain mostly transmissive.

The physical mechanism is resonance-induced hybridization. A two-degree-of-freedom spring-mass model yields the dispersion relation

PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^6

where PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^7, PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^8, and PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^9. Bandgaps occur when $$\omega^2 \tilde{\mathbf{u}} = \bar{\bar{H}}\,\tilde{\mathbf{u}}, \qquad \bar{\bar{H}} = \frac{-1}{\sqrt{\rho}} \nabla\cdot \mathbb{C} : \nabla_s \frac{1}{\sqrt{\rho}}.$$^^^^4query4^^^^ becomes complex, meaning the cosine lies outside the propagating range. The authors emphasize that the bandgap onset depends not only on the resonator properties but also on the host-frame stiffness; it does not open exactly at $$\omega^2 \tilde{\mathbf{u}} = \bar{\bar{H}}\,\tilde{\mathbf{u}}, \qquad \bar{\bar{H}} = \frac{-1}{\sqrt{\rho}} \nabla\cdot \mathbb{C} : \nabla_s \frac{1}{\sqrt{\rho}}.$$^^^^4ElasWave OR \4^^^^ (&&&^^^^4 OR \4^^^^&&&).

The reported prototype uses Nylon PA^^^^4ElasWave OR \4 OR \4^^^^ with unit-cell size $$\omega^2 \tilde{\mathbf{u}} = \bar{\bar{H}}\,\tilde{\mathbf{u}}, \qquad \bar{\bar{H}} = \frac{-1}{\sqrt{\rho}} \nabla\cdot \mathbb{C} : \nabla_s \frac{1}{\sqrt{\rho}}.$$^^^^4 OR \4^^^^, frame beam cross section $$\omega^2 \tilde{\mathbf{u}} = \bar{\bar{H}}\,\tilde{\mathbf{u}}, \qquad \bar{\bar{H}} = \frac{-1}{\sqrt{\rho}} \nabla\cdot \mathbb{C} : \nabla_s \frac{1}{\sqrt{\rho}}.$$^^^^4 OR \4^^^^, resonator beam width $$\omega^2 \tilde{\mathbf{u}} = \bar{\bar{H}}\,\tilde{\mathbf{u}}, \qquad \bar{\bar{H}} = \frac{-1}{\sqrt{\rho}} \nabla\cdot \mathbb{C} : \nabla_s \frac{1}{\sqrt{\rho}}.$$4, resonator length $$\omega^2 \tilde{\mathbf{u}} = \bar{\bar{H}}\,\tilde{\mathbf{u}}, \qquad \bar{\bar{H}} = \frac{-1}{\sqrt{\rho}} \nabla\cdot \mathbb{C} : \nabla_s \frac{1}{\sqrt{\rho}}.$$5, and sphere radius $$\omega^2 \tilde{\mathbf{u}} = \bar{\bar{H}}\,\tilde{\mathbf{u}}, \qquad \bar{\bar{H}} = \frac{-1}{\sqrt{\rho}} \nabla\cdot \mathbb{C} : \nabla_s \frac{1}{\sqrt{\rho}}.$$6. A finite element Bloch analysis gives a low-frequency bandgap from approximately $$\omega^2 \tilde{\mathbf{u}} = \bar{\bar{H}}\,\tilde{\mathbf{u}}, \qquad \bar{\bar{H}} = \frac{-1}{\sqrt{\rho}} \nabla\cdot \mathbb{C} : \nabla_s \frac{1}{\sqrt{\rho}}.$$7 to $$\omega^2 \tilde{\mathbf{u}} = \bar{\bar{H}}\,\tilde{\mathbf{u}}, \qquad \bar{\bar{H}} = \frac{-1}{\sqrt{\rho}} \nabla\cdot \mathbb{C} : \nabla_s \frac{1}{\sqrt{\rho}}.$$8, with a gap-midgap ratio of about $$\omega^2 \tilde{\mathbf{u}} = \bar{\bar{H}}\,\tilde{\mathbf{u}}, \qquad \bar{\bar{H}} = \frac{-1}{\sqrt{\rho}} \nabla\cdot \mathbb{C} : \nabla_s \frac{1}{\sqrt{\rho}}.$$9. In the Timoshenko-beam-with-lumped-mass interpretation, about $$\tilde{\mathbf{u}}'=[\tilde{u}_x,\tilde{u}_y,-j\tilde{u}_z]$$^^^^4query4^^^^ of the sphere mass participates in the resonant motion, the resonance frequency is $$\tilde{\mathbf{u}}'=[\tilde{u}_x,\tilde{u}_y,-j\tilde{u}_z]$$^^^^4ElasWave OR \4^^^^, and the simplified model agrees with full ^^^^4 OR \4^^^^D finite elements to about $$\tilde{\mathbf{u}}'=[\tilde{u}_x,\tilde{u}_y,-j\tilde{u}_z]$$^^^^4 OR \4^^^^ in the bandgap edge frequencies. The fabricated prototype contains 9 unit cells, forming a cube of about ^^^^4ElasWave OR \4max_results4query4^^^^ mm (&&&^^^^4 OR \4^^^^&&&).

The most important conceptual clarification is that a “complete” bandgap here does not mean equal suppression of all polarizations. The cited analysis states that the imaginary part of the transverse branch is much larger than that of the longitudinal branch, so transverse waves are strongly evanescent in the gap whereas longitudinal waves are only weakly attenuated except near the resonant frequency. This is the basis for the “fluid-like” elasticity analogy: the structure behaves as if it can sustain compression-like motion while suppressing shear-like motion. The experimentally measured transmission spectra, obtained with an LDS v4^^^^4query4^^^^6 electrodynamic shaker, PCB Piezotronics ^^^^4 OR \4max_results4 OR \4^^^^C^^^^4 OR \4 OR \4^^^^ accelerometers, and a 5^^^^4query4^^^^–^^^^4ElasWave OR \4max_results4query4^^^^ Hz swept-frequency cosine excitation, agree well with viscoelastic FEM simulations using a Standard Linear Solid / Maxwell-type model with $$\tilde{\mathbf{u}}'=[\tilde{u}_x,\tilde{u}_y,-j\tilde{u}_z]$$^^^^4 OR \4^^^^ and $$\tilde{\mathbf{u}}'=[\tilde{u}_x,\tilde{u}_y,-j\tilde{u}_z]$$4 (&&&^^^^4 OR \4^^^^&&&).

5. Guided-wave rheometry and constitutive identification

In soft-matter mechanics, ElasWave denotes a wave-based rheometry method in which a thin elastomer strip acts as a waveguide supporting multiple in-plane guided modes (&&&^^^^4 OR \4^^^^&&&). Rather than measuring force in a conventional rheometer, the method sends guided elastic waves through the strip and infers mechanical parameters from complex dispersion relations under controlled static elongation. The samples are thin rectangular strips with typical dimensions $$\tilde{\mathbf{u}}'=[\tilde{u}_x,\tilde{u}_y,-j\tilde{u}_z]$$5 cm, $$\tilde{\mathbf{u}}'=[\tilde{u}_x,\tilde{u}_y,-j\tilde{u}_z]$$6 cm, and $$\tilde{\mathbf{u}}'=[\tilde{u}_x,\tilde{u}_y,-j\tilde{u}_z]$$7 mm; the study uses Ecoflex OO-^^^^4 OR \4query4^^^^, Ecoflex OO-5^^^^4query4^^^^, and SortaClear ^^^^4ElasWave OR \4 OR \4^^^^. A point-like shaker excites monochromatic waves over $$\tilde{\mathbf{u}}'=[\tilde{u}_x,\tilde{u}_y,-j\tilde{u}_z]$$8–$$\tilde{\mathbf{u}}'=[\tilde{u}_x,\tilde{u}_y,-j\tilde{u}_z]$$9 Hz, the source is rotated by $\varepsilon \propto (\Delta y)^2$^^^^4query4^^^^ around $\varepsilon \propto (\Delta y)^2$^^^^4ElasWave OR \4^^^^, and the wavefield is reconstructed by stroboscopic imaging from 6^^^^4query4^^^^ frames per pseudo-period while the strip is stretched to $\varepsilon \propto (\Delta y)^2$^^^^4 OR \4^^^^ (&&&^^^^4 OR \4^^^^&&&).

The data reduction is modal and complex-valued. Digital image correlation yields the in-plane displacements $\varepsilon \propto (\Delta y)^2$^^^^4 OR \4^^^^ and $\varepsilon \propto (\Delta y)^2$4, from which the monochromatic field is reconstructed via

$\varepsilon \propto (\Delta y)^2$5

Symmetric and antisymmetric components are then separated, singular value decomposition is used to extract dominant modal fields, the real part of $\varepsilon \propto (\Delta y)^2$6 comes from a spatial Fourier transform, and the imaginary part from an exponential fit of the spatial decay. The result is a complex dispersion relation

$\varepsilon \propto (\Delta y)^2$7

The constitutive model couples viscoelasticity and acoustoelasticity. The shear modulus is taken as

$\varepsilon \propto (\Delta y)^2$8

and the static deformation is modeled with a compressible Mooney–Rivlin energy. The inverse problem uses five parameters, $\varepsilon \propto (\Delta y)^2$9, $1\,\mu\text{m}$^^^^4query4^^^^, $1\,\mu\text{m}$^^^^4ElasWave OR \4^^^^, $1\,\mu\text{m}$^^^^4 OR \4^^^^, and $1\,\mu\text{m}$^^^^4 OR \4^^^^, with $1\,\mu\text{m}$4 and $1\,\mu\text{m}$5 obtained from tensile tests and $1\,\mu\text{m}$6, $1\,\mu\text{m}$7, and $1\,\mu\text{m}$8 fitted by Matlab’s genetic algorithm using a cost function that combines the real and imaginary parts of the measured wavenumber. The real-part misfit is based on matching theoretical dispersion curves to the experimental intensity map, while the imaginary-part misfit uses a Gaussian-weighted comparison with $1\,\mu\text{m}$9 and weight $1000\,\mu\text{m}$^^^^4query4^^^^ (&&&^^^^4 OR \4^^^^&&&).

The comparison with an Anton-Paar MCR5^^^^4query4ElasWave OR \4^^^^ plate-plate rheometer is central. The overlapping range is roughly $1000\,\mu\text{m}$^^^^4ElasWave OR \4^^^^–1000 μm1000\,\mu\text{m}^^^^4 OR \4^^^^ Hz, whereas the guided-wave method spans approximately $1000\,\mu\text{m}$^^^^4 OR \4^^^^–1000 μm1000\,\mu\text{m}4 Hz. For Ecoflex OO-^^^^4 OR \4query4^^^^ and Ecoflex OO-5^^^^4query4^^^^, the storage and loss moduli inferred from guided waves agree well with rheometer data; the loss-modulus slope and the fractional exponent $1000\,\mu\text{m}$5 are reproduced well, although $1000\,\mu\text{m}$6 can differ by as much as an order of magnitude. The method’s explicit advantages are that no force measurement is required, it is broadband, it is sensitive to pre-stress, and it provides multiple observables through several guided modes (&&&^^^^4 OR \4^^^^&&&).

A closely related guided-wave constitutive-identification program sharpens the interpretation. Measurements of the zero-order modes $1000\,\mu\text{m}$7, $1000\,\mu\text{m}$8, and $1000\,\mu\text{m}$9 in a stretched Ecoflex OO-^^^^4 OR \4query4^^^^ plate show that guided-wave dispersion can lift the degeneracy between Mooney-Rivlin, Gent-Thomas, and Carroll forms that are indistinguishable from static uniaxial stress–strain data alone, while remaining unable to distinguish different generalized neo-Hookean models (&&&^^^^4 OR \4ElasWave OR \4^^^^&&&). In that study, the pseudo-longitudinal mode is especially important because

$<1\%$^^^^4query4^^^^

so it depends on second derivatives of the strain-energy density and therefore probes the derivative of stress with strain (&&&^^^^4 OR \4ElasWave OR \4^^^^&&&). This related work clarifies that the rheometric ElasWave usage is not merely a replacement sensor modality; it is also a constitutive-model discrimination tool.

6. Adjacent lineages and terminological ambiguity

The term also sits near other elastic-wave literatures that are technically related but conceptually distinct. One physically different line reports traveling Elastic Waves in a Hele-Shaw cell with a co-moving Kelvin-Helmholtz-like parallel shear flow at very low Reynolds number, $<1\%$^^^^4ElasWave OR \4^^^^, using co-moving dilute polymeric solutions moving at different speeds in a straight channel. In that setting, the interface between the streams is wavy and unsteady, the differential velocity of the parallel streams is identified as the only possible source of generation, and the frequency and wave speed are found to scale with the Weissenberg number based on the velocity difference between the streams (&&&^^^^4 OR \4 OR \4^^^^&&&). This usage concerns elastic-wave generation in viscoelastic shear flow, not waveguide computation, elastography inversion, or guided-wave rheometry.

The broader computational ecosystem reinforces the non-uniqueness of the label. ElasticMatrix is an open-source MATLAB toolbox for elastic and acoustic wave propagation in isotropic and transverse-isotropic layered media using the partial-wave / global-matrix method (&&&^^^^4 OR \44^^^^&&&). Spectral collocation and spectral element methods have been developed for electroelastic guided-wave dispersion in $<1\%$^^^^4 OR \4^^^^ Y-cut LiNbO$<1\%$^^^^4 OR \4^^^^ plates under open-open, open-shorted, and shorted-shorted electrical boundary conditions, with laser-ultrasound validation (&&&^^^^4 OR \4^^^^5&&&). A separate spectral element method addresses ^^^^4 OR \4^^^^.5-D elastic/acoustic waveguide eigenproblems in anisotropic metamaterials with anisotropic mass density, negative-index parameters, solid-fluid coupling, and Bloch periodic or absorbing boundary conditions (&&&^^^^4 OR \4^^^^6&&&). These neighboring toolchains solve some of the same classes of guided-wave problems addressed by ElasWave-labeled work, but under different modeling assumptions and for different material systems.

The most defensible encyclopedic conclusion is therefore taxonomic rather than singular. “ElasWave” names a cluster of elastic-wave methods and devices unified by their reliance on modal structure, dispersion, resonance, or wavefield fitting, but differentiated by problem class: acoustic-waveguide eigensolving, shear-wave elastography, polarization-selective metamaterial filtering, and visco-acoustoelastic material identification. The label is informative when read locally within each paper, but it should not be presumed to identify one canonical framework across the entire elastic-wave literature.