Zero Group Velocity Points Research Articles

Evidence of zero group velocity at the lowest dispersion branch through local interactions

Zero group velocity (ZGV) modes can be utilized in many applications in both optics and acoustics. There exist numerous realizations of metamaterials with ZGVs at higher dispersion branches. However, to engineer the lowest dispersion branch to retain ZGVs, non-local metamaterials with couplings beyond the nearest neighbor are usually at play (i.e., roton-like dispersion). To date, there exists no realization of roton-like dispersion for the lowest branch without non-local couplings. Non-locality, while rich in dynamics, can render designs rather complex. Here, we provide the first experimental evidence of a ZGV point for the lowest dispersion branch within the first Brillouin zone with local interactions. We utilize nonlinear magnetic lattices as a platform to sculpt our dispersion. Our findings might enable the realization of exotic metamaterials with simple designs.

Dispersion of axisymmetric longitudinal waves propagating in an inhomogeneous prestressed hollow cylinder immersed in an inviscid compressible fluid

The paper investigates the dispersion of axisymmetric waves in a cylinder with inhomogeneous initial stresses caused by the hydrostatic pressure of the compressible inviscid fluid in which this cylinder is immersed. In the context of this study, the motion of the cylinder is described by the three-dimensional linearized equations for elastic waves in bodies with initial stresses, but the flow of the fluid is described by the linearized Navier-Stokes equations for compressible (barotropic) inviscid fluids. The initial stresses in the cylinder, due to the action of hydrostatic pressure on the outer surface of the cylinder, are determined according to the well-known Lame formulas of classical linear elasticity theory. The corresponding eigenvalue problem for the dispersion of axisymmetric waves propagating in the hydro-elastic system under consideration is solved using the discrete-analytic method. The dispersion equation is obtained and solved numerically. Concrete numerical investigations are carried out for three pairs of materials: Soft Rubber + Glycerin, Lucite + Water and Steel + Water. The dispersion curves obtained for these pairs of materials are presented and discussed for different values of the intensity of the external pressure. In particular, it is found that the inhomogeneous initial stresses in the cylinder cause zero group velocity (ZGV) points to appear on the dispersion curves of the lowest modes. However, the ZGV points (or modes) appear on the dispersion curves of the higher modes as a result of the contact of the hollow cylinder with the fluid.

Measurement on a thin plate of stepped thickness using zero-group velocity effect of an S1 mode Lamb wave induced by a moving laser source.

The desired narrowband mode of a Lamb wave can be generated efficiently, as long as the laser's moving speed matches the mode's phase velocity, which is a typical characteristic of the moving continuous wave (CW) laser excitation method. In this paper, S 1 mode zero group velocity (ZGV) waves can be generated efficiently when the laser's moving speed matches the phase velocity of the S 1 mode at the ZGV point, while the two fundamental Lamb modes, i.e.,A 0 and S 0 waves, can be avoided. Meanwhile, measurements on thin plates of stepped thickness were carried out by using the zero-group velocity effect of the S 1 mode. Those simulation results demonstrated that the S 1-Z G V resonance can be used to measure the thickness of thin plates by tracking the resonance peak as the sample was scanned, and good accuracy can be achieved, since the measurement was reduced to a simple frequency gauging of sample resonances in the ultrasonic domain, which provides a reliable method to measure a thin plate of varying thickness.

Dynamic behavior of a zero-group velocity guided mode in rail structures.

Important characteristics of a zero-group velocity (ZGV) mode in a standard rail are investigated through numerical simulation and experiment. First, the semi-analytical finite element analysis is implemented to compute dispersion curves for the rail structure and the first ZGV point is identified. Backward waves are identified through opposing senses of group and phase velocities. Next, a time-dependent finite element model is used to understand the dynamic response of the rail. Finally, experimental measurements confirm that ZGV modes in rail structures are formed through interferences between two opposite-traveling waves, which is analogous to the S1-S2b ZGV Lamb mode in plate structures.

Lamb waves analysis in high anisotropy alloys material: Propagation and infinite zero group velocity resonances

In the present paper, the dispersion curve of guided modes in a cubic Ni–Mn–Ga thin plate is studied. Exceptional dispersion behavior is reported, with multiple crossings between antisymmetric (An) and symmetric (Sn) Lamb modes. In addition, several dispersion branches contain one or more zero group velocity (ZGV) points. The number of ZGV points per symmetric mode, Sn and per antisymmetric mode An, turns out be 3n-2.

Computing zero-group-velocity points in anisotropic elastic waveguides: Globally and locally convergent methods.

Dispersion curves of elastic waveguides exhibit points where the group velocity vanishes while the wavenumber remains finite. These are the so-called zero-group-velocity (ZGV) points. As the elastodynamic energy at these points remains confined close to the source, they are of practical interest for nondestructive testing and quantitative characterization of structures. These applications rely on the correct prediction of the ZGV points. In this contribution, we first model the ZGV resonances in anisotropic plates based on the appearance of an additional modal solution. The resulting governing equation is interpreted as a two-parameter eigenvalue problem. Subsequently, we present three complementary numerical procedures capable of computing ZGV points in arbitrary nondissipative elastic waveguides in the conventional sense that their axial power flux vanishes. The first method is globally convergent and guarantees to find all ZGV points but can only be used for small problems. The second procedure is a very fast, generally-applicable, Newton-type iteration that is locally convergent and requires initial guesses. The third method combines both kinds of approaches and yields a procedure that is applicable to large problems, does not require initial guesses and is likely to find all ZGV points. The algorithms are implemented in GEW ZGV computation (doi: 10.5281/zenodo.7537442).

Extraction and selective promotion of zero-group velocity and cutoff frequency resonances in bi-dimensional waveguides using the electromechanical impedance method

This study showcases the electromechanical impedance (EMI) technique for extracting and promoting zero-group velocity (ZGV) and cutoff frequency resonances in a waveguide structure. We identify the mechanisms of multiple resonances in the EMI spectra via a wave propagation perspective. Both simulation and experiments reveal the fact that sharp resonances in the conductance spectra are associated with either ZGV or cutoff frequency points. Consequently, we design four test configurations to enhance local resonances by aligning induced motions with considered mode shapes. Reasonable agreement between simulation and experiment results is observed. We evaluate the performance of considered configurations in terms of mode enhancement, and configurations that can selectively promote certain mode families are summarized. This study also shines the light on the EMI technique for quantitative non-destructive evaluation (NDE) by potentially supporting the inverse characterization of mechanical properties of host structures.

On the existence of zero-group velocity modes in free rails: Modeling and experiments

Zero-group velocity (ZGV) modes in rails are studied through simulation and experiments in this work. Local resonances associated with ZGV modes appear as distinct, sharp peaks in the frequency amplitude spectrum, whose resonant frequencies can serve as indicators of the local structural integrity condition of the rail itself, assuming that one can excite, detect, and identify wave mode type with confidence. To better understand these interesting modes, semi-analytical finite element (SAFE) analysis is implemented to compute dispersion curves of a standard rail and to identify potential ZGV points. A fully discretized Finite Element (FDFE) model then simulates responses of a free rail when subjected to impulse-based dynamic testing. Experimental impulse vibration data are collected from a 25-m rail with multiple impact-receiver configurations to understand the detectability and excitability of specific resonances associated with ZGV modes in rails. Spatial sampling of wave disturbance is performed to calculate the dispersion relations experimentally via two-dimensional Fourier Transforms (2D-FFT). The excellent agreement between simulation and experimental results confirms the existence of ZGV modes and cutoff frequency resonances in rails and verifies the feasibility of using impulse-based dynamic tests for the promotion of ZGV modes.

Appearing ZGV point in the first flexural branch of Lamb waves in multilayered plates

Appearing the zero group velocity (ZGV) point belonging to the first quasi flexural branch (A1) of Lamb waves propagating in the multilayered traction-free plates containing homogeneous isotropic layers is observed and analyzed; in contrast to the previously studied homogeneous isotropic plates admitting ZGV points that may belong to the higher flexural modes (A2, A3, …) and higher symmetric modes. The detected ZGV point in a multilayered traction-free plate leads to the transverse quasi resonance occurred at the lower frequency than the previously studied resonance associated with the first symmetric mode (S1) and especially higher symmetric and flexural modes. The analysis is done by the combined method comprising Cauchy sextic formalism and the exponential fundamental matrix method.

Guided acoustic waves at periodically structured edges: Linear modes and nonlinear generation of Lamb and surface waves

Acoustic waves are investigated which are guided at the edge (apex line) of a wedge-shaped elastic body or at the edge of an elastic plate. The edges contain a periodic sequence of modifications, consisting either of indentations or inclusions with a different elastic material which gives rise to high acoustic mismatch. Dispersion relations are computed with the help of the finite element method. They exhibit zero-group velocity points on the dispersion branches of edge-localized acoustic modes. These special points also occur at Bloch-Floquet wavenumbers away from the Brillouin zone boundary. Deep indentations lead to flat branches corresponding to largely non-interacting, Einstein-oscillator like vibrations of the tongues between the grooves of the periodic structure. Due to the nonlinearity of the elastic media, quantified by their third-order elastic constants, an acoustic mode localized at a periodically modified edge generates a second harmonic which partly consists of surface and plate modes propagating into the elastic medium in the direction vertical to the edge. This acoustic radiation at the second-harmonic frequency is investigated for an elastic plate and a truncated sharp-angle wedge with periodic inclusions at their edges. Unlike nonlinear bulk wave generation by surface acoustic waves in an interdigital structure, surface and plate mode radiation by edge-localized modes can be visualized directly in laser-ultrasound experiments.

Multiple zero-group velocity resonances in elastic layered structures

To date, the phenomenon of zero group velocity (ZGV) Lamb wave in a stress-free elastic plate has been well studied and used in non-destructive material characterization, electroacoustic devices and some other applications. This phenomenon is associated with the backward mode appearing in the range limited by the ZGV and cutoff frequencies. The modal frequency response of the structure, primarily the spectrum of the corresponding guided wave (GW) excited in the plate by a force source, is featured by a sharp peak at the ZGV frequency, allowing its experimental detection. Other peaks appearing at the cutoff frequencies indicate thickness resonances. In a layered half-space of infinite thickness, the backward-mode bend of the dispersion curve turns into a double bend that gives rise to three GWs associated with the same branch (one backward and two forward modes). Moreover, several adjacent branches with double bends, and, thus, several peaks, can appear in a narrow range as the number of layers increases, indicating multiple ZGV resonances. Such dispersion curves with double ZGV points were also found in waveguides of finite thickness with anisotropic or contrast material properties; however, their resonance manifestation is poorly studied. In this article, we consider and discuss several kinds of multi-peak ZGV responses using examples of layered samples of various materials. Along with the expected increase in the number of peaks in multilayer structures with contrast layers, it was noted that some ZGV points may or may not appear in peaks, depending on the type of load.

Multiple zero group velocity Lamb modes in an anisotropic plate: propagation along different crystallographic axes

This paper studies the propagation of symmetric and antisymmetric Lamb waves along a 1 mm thick iodic acid plate (HIO3) in the 1–50 MHz frequency range. The Lamb mode propagation along three crystallographic planes was theoretically investigated, for two mutually orthogonal propagation directions. Several frequencies were found that correspond to Lamb modes with zero group velocity (ZGV) and non-null phase velocity values. The first symmetric Lamb mode, S1, was found to possess only one ZGV point, regardless of the propagation direction; higher order symmetric and antisymmetric modes with up to four ZGV points were found, depending on the propagation plane. The dependence of the ZGV frequencies on each elastic constant (c11, c13, c33, c55) of the HIO3 plate material was also investigated by changing the constant values by 5% and 10%. It was found that c33 and c55 affect the number of the ZGV points, while c11 and c13 affect the frequency of the ZGV points. The existence and frequencies of the ZGV points are strongly dependent on the proximity of standing longitudinal and transverse waves at nearby cutoff frequencies.

Anomalous zero-group-velocity photonic bonding states with local chirality

Photonic crystal waveguides (PCWs) are promising candidates for the basic building blocks of quantum information processing because they support circular polarization points that can unbalance the directionality of an integrated quantum emitter (QE). Nevertheless, the Purcell effect at circular polarization points saturates near the band edge, the preferred region for quantum electrodynamics. Consequently, chirality and ultra-strong light–matter interaction are difficult to combine. Here, we detract from the vicinity of the band edge, and couple modes with different parities by breaking the mirror symmetry. Using the three-dimensional finite-difference time-domain method, simulated band structures of the implemented photonic bonding states (PBS) display single-mode anomalous zero-group-velocity (ZGV) points far from the band edge. The electric field patterns of these points feature circular polarization points at high field intensity regions where a QE would acquire uni-directional emission behavior. Fabricated devices in silicon (Si) slabs demonstrate the predicted coupling energy between the modes and the signature of single-mode anomalous ZGV points. This method to engineer PBS in PCWs paves the way for outperforming chiral light–matter experiments on-chip.

Zero group velocity longitudinal modes in an isotropic cylinder

Zero group velocity (ZGV) modes are studied in an isotropic cylinder. The L(0, 2) mode behaves anomalously for the materials with a value of the bulk velocity ratio, \(\kappa \), in the range \(\sqrt{2}<\kappa <2.64\) and normally otherwise. All higher modes, except the first few, have no ZGV point for all isotropic materials. This is explained analytically by finding the slope of phase velocity dispersion curves of modes first when the phase velocity equals \(\kappa \) and then at their initial state.

Zero-group-velocity acoustic waveguides for high-frequency resonators

The propagation of the Lamb-like modes along a silicon-on-insulator (SOI)/AlN thin supported structure was simulated in order to exploit the intrinsic zero group velocity (ZGV) features to design electroacoustic resonators that do not require metal strip gratings or suspended edges to confine the acoustic energy. The ZGV resonant conditions in the SOI/AlN composite plate, i.e. the frequencies where the mode group velocity vanishes while the phase velocity remains finite, were investigated in the frequency range from few hundreds of MHz up to 1900 MHz. Some ZGV points were found that show up mostly in low-order modes. The thermal behaviour of these points was studied in the −30 to 220 °C temperature range and the temperature coefficients of the ZGV resonant frequencies (TCF) were estimated. The behaviour of the ZGV resonators operating as gas sensors was studied under the hypothesis that the surface of the device is covered with a thin polyisobutylene (PIB) film able to selectively adsorb dichloromethane (CH2Cl2), trichloromethane (CHCl3), carbontetrachloride (CCl4), tetrachloroethylene (C2Cl4), and trichloroethylene (C2HCl3), at atmospheric pressure and room temperature. The sensor sensitivity to gas concentration in air was simulated for the first four ZGV points of the inhomogeneous plate. The feasibility of high-frequency, low TCF electroacoustic micro-resonator based on SOI and piezoelectric thin film technology was demonstrated by the present simulation study.

Spatial Laplace transform for complex wavenumber recovery and its application to the analysis of attenuation in acoustic systems

We present a method for the recovery of complex wavenumber information via spatial Laplace transforms of spatiotemporal wave propagation measurements. The method aids in the analysis of acoustic attenuation phenomena and is applied in three different scenarios: (i) Lamb-like modes in air-saturated porous materials in the low kHz regime, where the method enables the recovery of viscoelastic parameters; (ii) Lamb modes in a Duralumin plate in the MHz regime, where the method demonstrates the effect of leakage on the splitting of the forward S1 and backward S2 modes around the Zero-Group Velocity point; and (iii) surface acoustic waves in a two-dimensional microscale granular crystal adhered to a substrate near 100 MHz, where the method reveals the complex wavenumbers for an out-of-plane translational and two in-plane translational-rotational resonances. This method provides physical insight into each system and serves as a unique tool for analyzing spatiotemporal measurements of propagating waves.

Backward waves with double zero-group-velocity points in a liquid-filled pipe.

Hollow cylinders often exhibit backward propagation modes whose group and phase velocities have opposite directions, and these exhibit a minimum possible frequency at which the group velocity vanishes at a nonzero wavenumber. These zero-group-velocity (ZGV) points are associated with resonant conditions in the medium. On the basis of ZGV resonances, a non-contact and laser ultrasound technique has been developed to measure elastic constants of hollow pipes. This paper provides a theoretical and numerical investigation of the influence of the contained liquid on backward waves and associated ZGV modes, in order to explore whether this ZGV technique is suitable for in-service non-destructive evaluations of liquid-filled pipes. Dispersion spectra and excitation properties have been analyzed. It is found that the presence of the liquid causes an increased number of backward modes and ZGVs which are highly excitable by a point source. In addition, several guided modes twice undergo a change of sign in the slopes of their dispersion curves, leading to two ZGV points. This phenomenon of double ZGVs in one backward wave, which is caused by strong mode repulsions, has not been found in isotropic hollow cylinders, but it can be observed in a fluid-filled thin-walled pipe.

Guided modes with multiple zero-group-velocity points in fluid-filled cylindrical pipes

It is known that guided modes in isotropic hollow cylinders exhibit backward wave propagation with negative group velocity. And interferences between backward and forward waves generate zero-group-velocity (ZGV) resonances with a finite wavenumber but vanishing group velocity. These ZGV resonances can be applied for non-destructive evaluation (NDE) of hollow pipes. In this paper, influences of a fluid-loading on ZGV resonances in pipes are studied for the possible application of integrity inspections of oil transportation pipelines. From numerically simulated frequency-wavenumber spectra of axisymmetric guided modes, in addition to the backward mode with a single ZGV point, certain branches change the sign of their slopes for twice (i.e., two ZGV points in one branch). Such multiple ZGV modes might be caused by the strong repulsion between the backward mode with a single ZGV point that is propagating in the hollow pipe and a number of longitudinal modes in the fluid cylinder. It is found that, from wave structure analyses, ZGV points correspond to relatively large displacement amplitudes at the pipe’s inner and outer interfaces. It indicates that guided modes with multiple ZGV points can be sensitive to the surface features of fluid-filled pipes, which is useful for NDE application.

Existence of zero-group velocity modes in an incompressible plate

Zero-group velocity (ZGV) Lamb modes are studied in an incompressible orthotropic plate. The existence of such modes critically depends on the anisotropy parameter a = ( c 11 + c 22 − 2 c 12 − 4 c 66 )/ c 66 . With materials having a > −1, none of the modes possesses any ZGV points and every mode has such a point if a < −1. Several modes have multiple ZGV points.

Two-dimensional discrete granular phononic crystal for shear wave control

The phononic properties of a two-dimensional discrete phononic crystal, made of circular cross-section, infinitely long contacting elastic cylinders arranged on a simple square lattice, are described analytically. The significant interaction between shear and rotational waves is demonstrated in such granular phononic crystals. Controlling these interactions provides an opportunity for shear wave band design in metamaterials. The phononic band structure presents typical features of a square elementary cell of discrete phononic materials and also of continuous (composite) two-dimensional phononic materials with identical elementary cells. The theoretical analysis provides a clear physical explanation for the existence of a zero-group velocity point of the lowest-energy acoustic mode in particular directions of the phononic crystal and demonstrates the birefraction phenomenon.