Variable Wave Speed Research Articles

Nonlinear incompressible shear wave models in hyperelasticity and viscoelasticity frameworks, with applications to Love waves

General equations describing shear displacements in incompressible hyperelastic materials, holding for an arbitrary form of strain energy density function, are presented and applied to the description of nonlinear Love-type waves propagating on an interface between materials with different mechanical properties. The model is valid for a broad class of hyper-viscoelastic materials. For the Murnaghan constitutive model, shear wave equations contain cubic and quintic differential polynomial terms, including viscoelasticity contributions in terms of dispersion terms that include mixed derivatives uxxt of the material displacement. Full (2+1)-dimensional numerical simulations of waves propagating in the bulk of a two-layered solid are undertaken and analysed with respect to the source position and mechanical properties of the layers. Interfacial nonlinear Love waves and free upper surface shear waves are tracked; it is demonstrated that in the fully nonlinear case, the variable wave speed of interface and surface waves generally satisfies the linear Love wave existence condition c1<v<c2, while tending to the larger material wave speed c1 or c2 for large times.

Simulation of gas mixture dynamics in a pipeline network using explicit staggered-grid discretization

We develop an explicit staggered finite difference discretization scheme for simulating the transport of highly heterogeneous gas mixtures through pipeline networks. This study is motivated by the proposed blending of hydrogen into natural gas pipelines to reduce end use carbon emissions while using existing pipeline systems throughout their planned lifetimes. Our computational method accommodates an arbitrary number of constituent gases with very different physical properties that may be injected into a network with significant spatiotemporal variation. In this setting, the gas flow physics are highly location- and time- dependent, so that local composition and nodal mixing must be accounted for. The resulting conservation laws are formulated in terms of pressure, partial densities and flows, and volumetric and mass fractions of the constituents. We include non-ideal equations of state that employ linear approximations of gas compressibility factors, so that the pressure dynamics propagate locally according to a variable wave speed that depends on mixture composition and density. We derive compatibility relationships for network edge boundary values that are more complex than for a homogeneous gas. The simulation method is evaluated on initial boundary value problems for a single pipe and a small network, is cross-validated with a lumped element simulation, and used to demonstrate a local monitoring and control policy for maintaining allowable concentration levels.

Cubic nonlinearity and surface shock waves in soft tissue-like materials

The cubic nonlinearity of shear wave propagation plays a significant role in brain injury biomechanics. However, soft materials, like the brain, also support the propagation of surface waves, which produce a combination of longitudinal and transverse deformation. The order of the nonlinearity of surface waves in soft materials is still unknown. Here, we directly observe nonlinear Scholte waves propagating in an interface formed by an incompressible gelatin tissue-mimicking phantom and a water layer using ultrasound imaging operated as fast as 16667 frames per second. A two-dimensional correlation-based tracking algorithm was utilized to extract movies of the movement produced by the surface wave. Our results show that the initially nearly monochromatic wave becomes progressively distorted with the propagation due to nonlinearity. The distortion of the wave and its frequency spectrum indicate a high content of odd harmonics when compared with even harmonics. Additionally, by fitting our experimental data to a minimalist one-dimensional model based on the wave speed variation as a function of the perturbation amplitude, we found a cubic nonlinear parameter 46 times larger than the quadratic nonlinear parameter. Overall, the wave distortion, the harmonic development, and the dependence of the wave speed with the amplitude prove that cubic nonlinearity is essential to modeling nonlinear Scholte wave propagation.

Adaptive conservative time integration for unsteady compressible flow

Unsteady flow computations typically rely on time integration schemes which employ the same step size everywhere. However, in cases with strong variations of wave speed and/or spatial resolution, local stability criteria and truncation errors would allow for much larger time steps in parts of the domain. To exploit this potential of improving the computational efficiency, adaptive time-stepping methods have been developed. Recently, an adaptive conservative time integration (ACTI) scheme for explicit finite volume methods was devised. Opposed to previous methods, ACTI guarantees exact conservation and periodic synchronization. Both are achieved by splitting a global time step into 2L (L≥0 is a cell dependent level) sub-steps, whereas the order of cell updates is critical. It has been demonstrated for flows in fractured porous media and for the Euler equations that ACTI can reduce the computational cost dramatically while preserving second order accuracy in space and time. In this paper, the ACTI scheme is extended to the compressible Navier-Stokes-Fourier system, and special attention is required for the spatio-temporal discretization of the convective and viscous fluxes. Numerical studies of unsteady flows involving shock boundary layer interaction demonstrate that the same accuracy is achieved with the new compressible ACTI flow solver as with classical time integration, but at much lower cost. Moreover, since the method is explicit, it has the potential for efficient computations on multiple CPUs and GPUs.

Ultrasound-tensiometry: A new method for measuring differential loading within a tendon during movement

BackgroundTendons transmit tensile load from muscles to the skeleton. Differential loading across a tendon can occur, especially when it contains subtendons originating from distinct muscles. Tendon shear wave speed has previously been shown to reflect local tensile stress. Hence, a tool that measures spatial variations in wave speed may reflect differential loading within a tendon during human movement. Research questionDo wave speeds measured via high-framerate ultrasound-based tensiometry correspond with differential loading across a tendon? MethodsUltrasound-tensiometry uses an external mechanical actuator to induce waves and high-framerate plane wave ultrasound imaging (20 kHz) to track tissue displacements arising from wave propagation within a tendon. Local tissue displacements are temporally and spatially filtered to remove high-frequency noise and reflected waves. A Radon transform of the spatio-temporal displacement data is used to compute the shear wave speed across the tendon.We evaluated ultrasound-tensiometry’s ability to measure differential loading across a tendon using in silico, ex vivo and in vivo approaches. The in silico approach used a finite element model to simulate wave propagation along two adjacent subtendons undergoing differential loading. The ex vivo experiment measured wave speed in adjacent porcine flexor subtendons subjected to differential loading. In vivo, we tracked wave speed across the Achilles tendon while a participant performed calf stretches to differentially load the subtendons, and while walking on a treadmill at 1.5 m/s. ResultsWave speeds modulated with local tendon stress under both in silico and ex vivo conditions, with higher wave speeds observed in subtendons subjected to higher loads (6–16 m/s higher at 1.5× load differential). Spatial variations in in vivo Achilles tendon wave speeds were consistent with differential subtendon loading arising from distinct muscle loads (maximum range: 0–137 m/s, resolution: 0.1 mm×0.2 mm, precision: ±0.2 m/s). SignificanceHigh-framerate ultrasound-tensiometry tracks spatial variations in tendon wave speed, which may be useful to investigate local tissue loading and to delineate individual muscle contributions to movement.

A High-Precision Earthquake Catalog for Nevada

Abstract The state of Nevada is home to one of the most seismically active regions in the world, with crustal deformation associated with the Walker Lane transitioning into Basin and Range tectonics as one traverses from west to east across the state. Despite hosting numerous prominent earthquake sequences over the past century and beyond, at present, there exists no unified research-quality earthquake catalog for the state and its surrounding region. Here, we present a newly compiled, high-precision catalog of more than 180,000 earthquakes occurring around Nevada from 2008 to 2023. The data processing workflow to create this catalog includes an absolute location step that accounts for topography and 3D variations in subsurface wavespeed, and a relative relocation step that refines event positions using differential times measured from waveform cross-correlation. We also provide an update to the local magnitude scale that better accounts for the observed distance attenuation of waveform amplitudes as well as local site effects. We describe some fundamental insights that can be derived from the new catalog, including regional variations in event depth distributions and sequence clustering statistics, and publish the catalog to the wider community to facilitate future research efforts.

A Synoptic View of Mantle Plume Shapes Enabled by Virtual Reality

AbstractThe shapes of mantle plumes are sensitive to mantle viscosity, density structure, and flow patterns. Increasingly, global tomographic models reveal broad plume conduits in the lower mantle and highly tilting conduits in the mid and upper mantle. Previous studies mostly relied on 2D slices to analyze plume shapes, but fully investigating the complexity of 3D plume structures requires more effective visualization methods. Here, we use immersive headset‐based virtual reality (VR) to visualize the full‐waveform global tomographic models SEMUCB‐WM1 and GLAD‐M25. We develop criteria for the identification of plume conduits based on the relationship between the plume excess temperature and the VS anomaly (δVS). We trace 20 major plume conduits, measure the offsets of the conduits in azimuth and distance with respect to the hotspots, calculate the tilt angle, and evaluate the δVS along all traced conduits. We compare our traced conduits with the conduits predicted by global mantle convection models and vertical conduits. The wavespeed variations along conduits traced from each tomographic model are slower than modeled or vertical conduits, regardless of which tomographic model they are evaluated in. The shapes of traced conduits tend to differ greatly from modeled conduits. Plume ponding and the emergence of secondary plumes, which could result from a combination of compositional variations, phase transitions, small‐scale convection, and variations in viscosity, can contribute to the complex observed plume shapes. The variation of δVS along the traced conduits and complex plume shapes suggest a thermochemical origin of many plumes.

Analysis of optical solitons propagation in the dual-mode resonant nonlinear Schrödinger dynamical equation with assorted nonlinear interactions

This research explores the dual-mode manifestation within the nonlinear Schrödinger equation, elucidating the amplification or absorption of paired waves. This study delves into the simultaneous generation of two distinct waves associated with the dual-mode phenomenon with three crucial parameters: phase velocity, nonlinearity and dispersive factor. The resulting wave phenomena from these solutions have implications across various fields, including fluid dynamics, water wave mechanics, ocean engineering and scientific inquiry. The study employs the modified Sardar sub-equation method to obtain the optical soliton solutions, encompassing various types such as dark, bright, singular, combo dark–singular, periodic singular and dark–bright solitons. The obtained results highlight the reliability and simplicity of the modified Sardar sub-equation method. Additionally, the paper delves into the parametric conditions crucial for shaping and sustaining these solitons. The research explores the interaction of dual waves and the variation in wave speed. Furthermore, dynamic phenomena are illustrated, and the physical implications of the solutions are interpreted using 3D and 2D plots with different parameter values.

Detonation structure in the presence of mixture stratification using reaction-resolved simulations

Many investigations of detonation-based combustors have identified reactant mixture inhomogeneity as having a leading-order impact on wave dynamics and combustion efficiency. To examine this phenomenon in a simplified context, an array of two- and three-dimensional channel detonation simulations are conducted in the present work. The reactant mixture consists of stratified fuel and air, wherein the randomly distributed equivalence ratio field features a characteristic stratification length scale. Detailed chemical kinetics are implemented in an adaptive mesh refinement solution framework where the region near the shock front is resolved with O(100) cells per representative ZND induction length. The results show that in comparison to baseline cases with uniform reactant mixtures, reactant stratification has a marked impact on the detonation structure. Increasing the stratification length scale increases the size and irregularity of the detonation cells, yielding larger variations in wave speed. Triple point collisions in fuel-rich regions lead to local wave speeds above the notional mean CJ speed, but wave passage through inert regions causes the local wave speed and strength to diminish. Conditional statistics show that increasing the stratification length scale increases the variance in pressure and temperature in the primary reaction zone, as well as the variance in heat release over a range of mixture conditions. In addition to the reactant mixture, the impact of the boundary condition behind the detonation is also investigated. The results show that an inflow boundary condition acts to over-drive the wave, leading to higher peak pressures, smaller detonation cells, and increased reactant consumption. On the other hand, cases with a wall behind the wave exhibit weaker waves with lower peak pressures and heat release rates, as well as greater variance in conditional quantities. Comparisons between complementary two- and three-dimensional simulations show reasonable qualitative agreement in wave structure, speed, and conditional statistics.Novelty and significance statement: The present work considers two- and three-dimensional simulations of detonations propagating through stratified reactant mixtures using detailed chemical kinetics with O(100) cells per representative ZND induction length. To the authors’ knowledge, it is the first study to analyze this type of configuration at this resolution and detail, as well as the first to compare two- and three-dimensional simulations of this configuration. The results have implications for the detonation wave behavior observed in practical propulsion and power-generating applications, such as rotating detonation engines.

On the temperature sensitivity of near-surface seismic wave speeds: application to the Groningen region, the Netherlands

SUMMARY Subsurface temperature measurements play a crucial role, for instance, in optimizing geothermal power plants and monitoring heat-storage systems. Previous studies have demonstrated that time-lapse variations in temperature can be correlated with variations in seismic wave speeds, offering the potential for temperature monitoring via seismic surveys. However, an apparent discrepancy has emerged between field and laboratory experiments. Field studies predominantly report positive correlations between temperature and seismic wave speeds, while laboratory experiments often show anticorrelations. This inconsistency underscores the need for a more comprehensive, physics-based understanding of temperature-induced wave speed changes. In this study, we strive to bridge the gap between field and laboratory findings by examining several mechanisms governing temperature-induced seismic wave speed changes, namely the intrinsic temperature dependency of elastic parameters and thermally induced elasticity. We present a physics-based modelling approach to identify the primary mechanisms responsible for temperature-induced seismic wave speed changes. By considering several end-member models, we find that intrinsic temperature dependency of elastic parameters (negative correlation) compete with thermal pressure effects (positive correlation). The precise initial and boundary conditions and physical parameters of the system under consideration will determine the weight of both effects. Temperature-related dilatation does not seem to play an important role. We apply our approach to loosely consolidated sediments in the shallow subsurface of the Groningen region, where subsurface temperature fluctuations are driven by seasonal atmospheric temperature fluctuations roughly between −5 and $30\, ^\circ$C. For these models, we predict seasonal temperature-induced changes in body-wave speeds of up to 8 per cent in the first few metres of the subsurface, high-frequency (above 2 Hz) surface wave phase velocity variations in the range of 1–2 per cent, and relative changes in site amplification on the order of 4 per cent. These findings contribute to a more comprehensive understanding of the intricate relationship between temperature and near-surface seismic properties, offering insights for applications as subsurface temperature monitoring systems.

Individual Muscle Force Estimation in the Human Forearm Using Multi-Muscle MR Elastography (MM-MRE).

To establish the sensitivity of magnetic resonance elastography (MRE) to active muscle contraction in multiple muscles of the forearm. We combined MRE of forearm muscles with an MRI-compatible device, the MREbot, to simultaneously measure the mechanical properties of tissues in the forearm and the torque applied by the wrist joint during isometric tasks. We measured shear wave speed of thirteen forearm muscles via MRE in a series of contractile states and wrist postures and fit these outputs to a force estimation algorithm based on a musculoskeletal model. Shear wave speed changed significantly upon several factors, including whether the muscle was recruited as an agonist or antagonist (p = 0.0019), torque amplitude (p = <0.0001), and wrist posture (p = 0.0002). Shear wave speed increased significantly during both agonist (p = <0.0001) and antagonist (p = 0.0448) contraction. Additionally, there was a greater increase in shear wave speed at greater levels of loading. The variations due to these factors indicate the sensitivity to functional loading of muscle. Under the assumption of a quadratic relationship between shear wave speed and muscle force, MRE measurements accounted for an average of 70% of the variance in the measured joint torque. This study shows the ability of MM-MRE to capture variations in individual muscle shear wave speed due to muscle activation and presents a method to estimate individual muscle force through MM-MRE derived measurements of shear wave speed. MM-MRE could be used to establish normal and abnormal muscle co-contraction patterns in muscles of the forearm controlling hand and wrist function.

Nonlinear fourth-order elastic characterization of the cornea using torsional wave elastography

Measuring the mechanical nonlinear properties of the cornea remains challenging due to the lack of consensus in the methodology and in the models that effectively predict its behaviour. This study proposed developing a procedure to reconstruct nonlinear fourth-order elastic properties of the cornea based on a mathematical model derived from the theory of Hamilton et al. and using the torsional wave elastography (TWE) technique. In order to validate its diagnostic capability of simulated pathological conditions, two different groups were studied, non-treated cornea samples (n=7), and ammonium hydroxide (NH4OH\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$NH_4OH$$\\end{document}) treated samples (n=7). All the samples were measured in-plane by a torsional wave device by increasing IOP from 5 to 25 mmHg with 5 mmHg steps. The results show a nonlinear variation of the shear wave speed with the IOP, with higher values for higher IOPs. Moreover, the shear wave speed values of the control group were higher than those of the treated group. The study also revealed significant differences between the control and treated groups for the Lamé parameter μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu$$\\end{document} (25.9–6.52 kPa), third-order elastic constant A (215.09–44.85 kPa), and fourth-order elastic constant D (523.5–129.63 kPa), with p-values of 0.010, 0.024, and 0.032, respectively. These findings demonstrate that the proposed procedure can distinguish between healthy and damaged corneas, making it a promising technique for detecting diseases associated with IOP alteration, such as corneal burns, glaucoma, or ocular hypertension.

Vertical dynamic interaction and group efficiency factor for floating pile group in layered soil

AbstractThis paper theoretically estimates the dynamic pile–soil interaction and the group efficiency factor for pile group in layered soil through energy‐based method. The vertical dynamic interaction of partially embedded single piles and their surrounding layered soil is analytically deduced from Hamilton's principle. Combined with a series of numerical simulations, the soil attenuation factor from energy method is modified for adapting to the wave speed variation in various layers. Then, the pile‐to‐pile interaction factor is directly solved with the help of the transfer‐matrix method. The dynamic governing equation of pile group with an elevated rigid cap is established by superposing the pile‐to‐pile interaction factors. Finally, the dynamic impedance of pile group is obtained and derived into a group efficiency factor. Compared with the plane strain method, this present method can produce a more suitable soil attenuation factor and a dynamic interaction factor at low frequency range, which is exploited for practical engineering design. The effects of subsoil layer, subsoil layer, and unembedded pile segment on group efficiency factor are investigated. The results show that the real part of group efficiency factor decreases at high frequency range for a small pile spacing, which may be detriment to the pile group capacity. Besides that, the combined effects of unembedded segment and weak surface soil on group efficiency factor are highlighted.

Simulating gravitational waves passing through the spacetime of a black hole

We investigate how GWs pass through the spacetime of a Schwarzschild black hole using time-domain numerical simulations. Our work is based on the perturbed $3+1$ Einstein's equations up to the linear order. We show explicitly that our perturbation equations are covariant under infinitesimal coordinate transformations. Then we solve a symmetric second-order hyperbolic wave equation with a spatially varying wave speed. As the wave speed in our wave equation vanishes at the horizon, our formalism can naturally avoid boundary conditions at the horizon. Our formalism also does not contain coordinate singularities and, therefore, does not need regularity conditions. Then, based on our code, we simulate both finite and continuous initially plane-fronted wave trains passing through the Schwarzschild black hole. We find that for the finite wave train, the wave zone of GWs is wildly twisted by the black hole. While for the continuous wave train, unlike geometric optics, GWs cannot be sheltered by the black hole. A strong beam and an interference pattern appear behind the black hole along the optical axis. Moreover, we find that the back-scattering due to the interaction between GWs and the background curvature is strongly dependent on the direction of the propagation of the trailing wavefront relative to the black hole. Finally, for a realistic input waveform generated by binary black holes, we find that the lensed waveform in the merger and ringdown phases is much longer than that of the input waveform due to the effect of back-scattering.

Introduction to this special section: Potential fields

This special section of The Leading Edge is dedicated to recent advances in potential field methods of geophysical exploration. In their broadest sense, these methods are rooted in data whose origins are compatible with solutions to the Laplace equation such as gravity, magnetics, and electrostatics, each responding to distinct and complementary material properties of the subsurface such as mass density, magnetic susceptibility, and electrical conductivity. These proxies for the real targets of geologic interest (lithologic boundaries, fluid-saturated zones, regions of mineral alteration, etc.) have proven themselves indispensable, for example, through joint inversion of seismic data and their direct reflection — no pun intended — of subsurface variations in seismic wavespeed. Indeed, joint inversion seismic and gravity data yield a direct path toward inference and constraint of the elastic moduli variations within the subsurface — variations that, themselves, can reveal changes in rock porosity or fracture density.

Adaptive total variation stable local timestepping for conservation laws

This paper proposes a first-order total variation diminishing (TVD) treatment for coarsening and refining of local timestep size in response to dynamic local variations in wave speeds for nonlinear conservation laws. The algorithm is accompanied with a proof of formal correctness showing that given a sufficiently small minimum timestep the algorithm will produce a TVD solution for nonlinear scalar conservation laws. A key feature of the algorithm is its formulation as a discrete event simulation, which allows for easy and efficient parallelization using existing software. Numerical results demonstrate the stability and adaptivity of the method for the shallow water equations. We also introduce a performance model to load balance and explain the observed performance gains. Performance results are presented for a single node on Stampede2's Skylake partition using an optimistic parallel discrete event simulator. Results show the proposed algorithm recovering 59%-77% of the theoretically achievable speed-up with the remainder being attributed to the cost of computing the CFL condition and load imbalance.

A high order compact time/space finite difference scheme for the 2D and 3D wave equation with a damping layer

We consider fourth order accurate compact schemes, in both space and time, for the second order wave equation with a variable speed of sound. For unbounded domains we add a fourth order accurate sponge layer to damp the outgoing waves. We demonstrate that usually this is more efficient than lower order schemes despite being implicit and conditionally stable. Fast time marching of the implicit scheme is accomplished by iterative methods such as multi-grid. Computations confirm the design convergence rate for the in-homogeneous, variable wave speed equation.

Nonlinear Ultrasound Imaging Modeled by a Westervelt Equation

We consider the ultrasound imaging problem governed by a nonlinear wave equation of Westervelt type with variable wave speed. We show that the coefficient of nonlinearity can be recovered uniquely from knowledge of the Dirichlet-to-Neumann map. Our proof is based on a second order linearization and the use of Gaussian beam solutions to reduce the problem to the inversion of a weighted geodesic ray transform. We propose an inversion algorithm and report the results of a numerical implementation to solve the nonlinear ultrasound imaging problem in a transmission setting in the frequency domain.

Subduction of an Oceanic Plateau Across Southcentral Alaska: Scattered‐Wave Imaging

AbstractAn oceanic plateau, the Yakutat terrane, has entered the subduction system across southcentral Alaska. Its down‐dip fate and relationship to overlying volcanism is still debated. Broadband seismometers from the Wrangell Volcanism and Lithospheric Fate (WVLF) temporary experiment were deployed with &lt;20 km spacing across southcentral Alaska to study this region. An array‐based deconvolution procedure is used to isolate the scattered P and S coda of teleseismic P waves for imaging discontinuity structure. This procedure is applied to WVLF and other dense seismic arrays across southcentral Alaska in a manner that accounts for near‐surface wavespeed variations. Two imaging techniques are employed: two‐dimensional migration and three‐dimensional common‐conversion‐point (CCP) stacking. Migrating the scattered phases along WVLF stations shows the ∼18 ± 4 km thick Yakutat crust subducting beneath the Wrangell Volcanic field to the NNE. It is offset from the Alaska‐Aleutian seismic zone laterally by 250 km to the southeast at 100 km depth, and dips more steeply (45°). At depths &lt;45 km, CCP stacking reveals that the Yakutat crust is continuous for over 450 km along strike. This shallow continuity and deeper offset suggest a tear in the subducting Yakutat slab at depths &gt;45 km, around 146°W. CCP stacking also reveals a continuous thin low‐velocity layer atop the underthrust Yakutat crust for &gt;450 km along strike, at all depths &lt;35 km. The uniform low‐velocity thrust zone indicates consistent properties through multiple rupture‐zone segments, showing that low‐velocity channels generally correspond with subduction megathrusts.

Emergence of stegotons in phononic materials with strongly nonlinear rough contacts

The incorporation of nonlinearity in phononic materials enables complex wave interactions in both space and time enriching the dynamic response of the underlying linear media. In this talk, we discuss the strongly nonlinear wave response of continuum phononic material with periodic and discrete nonlinearity. The studied phononic material is a periodic architecture of contact interfaces with rough surfaces, connecting linear elastic layers. These contacts exhibit strong nonlinearity, stemming from variable contact areas under compressive loads and their inability to support the tensile loads. We reveal the evolution of propagating waves using finite element time-domain simulations. The interplay of strong nonlinearity, and dispersion, in the presence of elastic layers, generate traveling localized waves, referred to as “stegotons.” Unlike classical solitons, these stagotons exhibit evolving spatial wave profiles and local variations in wave speed. Moreover, the elastodynamic effects arising from the nonlinearly coupled elastic layers enable strongly nonlinear energy transfer in the frequency domain by activating acoustic resonances of the layers. This study sheds light on the role of the strongly nonlinear coupling of linear media on the formation of traveling localized waves, and could open opportunities for enhanced dynamic response not possible with pure discrete or continuum phononic materials.