Closed-form Dispersion Relation Research Articles

Boundary-layer approach to the linear karren instability

The present paper concerns the linear fate of transverse perturbations in a gravity-driven, thin-film flow over a soluble substrate. We propose a reduced-order model, based on a boundary-layer treatment of the solute transport and a depth-integration of the Stokes equations, using two extended lubrication methodologies found in the literature. We obtain a closed-form dispersion relation, which we compare to a previous, fully resolved analytical investigation (Bertagni and Camporeale, J. Fluid Mech., vol. 913, 2021, A34). The results allow us to distil the essential physical mechanisms behind the instability.

Tunable metasurfaces for seismic Love wave manipulation: A theoretical study

Seismic Love waves are horizontally polarized surface waves propagating within an overlying layer, which have been shown to pose potential hazards to infrastructures. In this study, we propose an original analytical formulation to investigate the interaction between seismic Love waves and a metasurface that is composed of a cluster of tunable anti-plane resonators. The formulation utilizes the Green's function of a layered half-space subjected to a surface SH source, to describe both the source-induced incident field and the resonator-excited scattered field. Based on the multiple scattering theory, we obtain the total wavefield of the layered half-space equipped with surface resonators by coupling the incident field and the scattered field. Besides, a closed-form dispersion relation governing the hybrid Love waves is derived by utilizing the effective medium approximation. We demonstrate the capabilities of our analytical formulation by modelling Love waves propagating across a pair and a cluster of resonators, evidenced by some complex phenomena (e.g., surface-to-bulk wave conversion and higher-order hybridized surface modes). The hybridization features of different Love modes and metasurface resonances are clearly revealed. We anticipate the proposed tunable metasurface and the formulation can shed light on versatile control of Love wave propagation.

Flexural waves at the edge of nonlocal elastic plate associated with the Pasternak foundation

In this article, an elastic foundation is used to study the characteristics of flexural edge waves propagating on a piezoelectric structure. The nonlocal elasticity theory is introduced to investigate the microscale effect on edge wave propagation. The traditional Kirchhoff plate hypothesis is used to determine the kinematics of the piezoelectric plate. By considering the sinusoidal waveform, a closed-form dispersion relation can be obtained. In the context of a flexural edge wave on a piezoelectric plate, an analytical and graphical comparison between the two boundary conditions (i.e. short circuit and open circuit) is discussed. In the presence of nonlocal elasticity, the dispersion relation is implied as an implicit function of frequency and wave number. A significant difference occurred between the dispersion curves for piezoelectric plates with different foundations and nonlocal parameters. Due to the Pasternak elastic foundation, an increasing rate of the fundamental mode of frequency is observed at the traction-free edge of a thin, semi-infinite piezoelectric plate. In contrast, the nonlocal elasticity tends to decrease this fundamental frequency mode for a particular wave number value. The corresponding phase velocity decreases rapidly within a short range of wave numbers.

Love-type surface wave propagation due to interior impulsive point source in a homogeneous-coated anisotropic poroelastic layer over a non-homogeneous extended substance

The paper is concerned with the effect of an impulsive point source on the Love-type surface wave propagation in an anisotropic poroelastic medium coated with an isotropic homogeneous thin layer over an infinite extent anisotropic non-homogeneous substance. The material properties of both the layer and extended medium are restricted on linear elasticity. Also, the transmissions of waves are considered as usually incidental transient. Infinitely extended medium is considered highly non-homogeneity with rigidity and density, which varies exponentially with depth. The closed-form equation of motion is formulated in each layered media depending on the conditional property of the surface seismic wave. Due to the presence of an impulsive point source at the interface of the intermediate layer and infinitely extended layer, a force density distribution function in the form of the Dirac-delta function is performed with the equations of motion. Green's function approach is demonstrated to obtain the closed-form dispersion relation of the Love-type wave. Attenuation analysis and numerical error analysis of the dispersive wave are performed for the complex quantity of the wavenumber and error percentage of the main dispersion. The validity of the Love-type dispersive wave is examined with the help of some particular cases. The theoretical prediction of phase and group velocities are illustrated numerically using MATLAB software. A short movie (link) of group velocity is created to show the changes of wave amplitude with porosity. It is observed that the phase/group velocity is highly affected by porosity, non-homogeneity, anisotropy, thickness of the layer, coated layer and attenuation coefficient. The derivation and analysis of the structural model bear the promise of having important applications in the field of seismology and civil engineering, particularly in the studies related to earthquakes and mechanical explosions on the near-surface of the Earth.

Influence of initial stress on shear wave scattering in a functionally graded magneto-visco-elastic orthotropic multi-layered structure

A theoretical model is proposed to analyze the propagation behavior of SH-wave in a functionally graded magneto-visco-elastic orthotropic multi-layered composite structure. Haskell’s matrix technique has been used to derive the closed-form dispersion relation for the assumed model. It is found that under certain conditions, the acquired dispersion relation agrees with the classical Love wave equations. Also, a finite difference technique is adopted to deduce the expression for phase velocity and group velocity. Numerical computation has been done and presented graphically to show the effect of different parameters. Stability analysis for the finite difference scheme is carried out to reduce the growth of errors to make it stable and convergent.

An inverse technique for reconstructing ocean’s density stratification from surface data

In this article, we propose an inverse technique that accurately reconstructs the ocean’s density stratification profile simply from free surface elevation data. Satellite observations suggest that ocean surface contains the signature of internal tides, which are internal gravity waves generated by the barotropic tides. Since internal tides contain the information of ocean’s density stratification, the latter can in principle be reconstructed from the free surface signature. We consider a simple theoretical model that approximates a continuously stratified ocean as discrete layers of constant buoyancy frequency; this facilitates the derivation of a closed-form dispersion relation. First, we numerically simulate internal tide generation for toy ocean scenarios and subsequently perform Space–Time Fourier Transform (STFT) of the free surface, which yields internal tide spectra with wavenumbers corresponding to the tidal frequency. The density profile is reconstructed by substituting these wavenumbers into the dispersion relation. Finally, we consider a more realistic situation with rotation, bottom topography, shear and density profiles representative of the Strait of Gibraltar. Density reconstruction in the presence and absence of shear is respectively found to be 90.2% and 94.2% accurate. The proposed method can be used to reconstruct climatological mean ocean density field of uniform spatial resolution using only surface elevation data obtained via satellite altimetry.

Dispersion-Engineered Guided-Wave Resonators in Anisotropic Single-Crystal Substrates - Part I: Concept and Analytical Design.

This paper presents an analytical approach for implementation of high quality-factor (Q) resonators with arbitrary cross-sectional vibration mode-shapes in anisotropic single-crystal substrates. A closed-form dispersion relation is analytically derived to characterize the dynamics of guided waves in rectangular waveguides. Three categories of waves with propagating, standing-evanescent and propagating-evanescent dynamics are identified and used for energy localization of acoustic excitations with arbitrary cross-sectional vibration patterns. An analytical design procedure is presented for dispersion engineering of waveguides to realize high-Q resonators without the need for geometrical suspension through narrow tethers or rigid anchors. The effectiveness of the dispersion engineering methodology is verified through development of experimental test-vehicles in 20㯌m-thick single-crystal silicon substrate with 500nm aluminum nitride transducers. Various proof-of-concept resonators, representing guided waves with different dispersion types, are presented and compared to highlight the optimum design procedures for Q enhancement and spurious mode suppression. Part I of this paper presents the operation principle of guided wave resonators based on analytical derivation of dispersion relation followed by a systematic resonator design procedure. Numerical and experimental characterization for verification of the proposed design procedure and extensive measurement data on proof-of-concept resonators are presented in Part II.

Control of Love waves by resonant metasurfaces

Metasurfaces of mechanical resonators have been successfully used to control in-plane polarized surface waves for filtering, waveguiding and lensing applications across different length scales. In this work, we extend the concept of metasurfaces to anti-plane surface waves existing in semi-infinite layered media, generally known as Love waves. By means of an effective medium approach, we derive an original closed-form dispersion relation for the metasurface. This relation reveals the possibility to control the Love waves dispersive properties by varying the resonators mechanical parameters. We exploit this capability to manipulate the metasurface refractive index and design two gradient index (GRIN) metalenses, i.e. a Luneburg lens and a Maxwell lens. We confirm the performance of the designed lenses using full 3D finite element simulations. Our work demonstrates the possibility of realizing wave control devices for anti-plane waves.

Love-type wave propagation in a hydrostatic stressed magneto-elastic transversely isotropic strip over an inhomogeneous substrate caused by a disturbance point source

Propagation of Love-type waves emanating due to a disturbance point source in a transversely isotropic layer of finite thickness laid over a semi-infinite half-space is investigated. The layer is assumed under the influence of magnetic field and hydrostatic state of stress, while the half-space is inhomogeneous. The source point is situated at the common interface of the layer and half-space. Maxwell’s equation and generalized Ohm’s law have been taken into account to calculate the Laurent force induced in the layer. Green’s function technique and Fourier transform are used as a powerful tool to calculate the interior deformations of the model; consequently, we obtain a closed-form dispersion relation for the wave. Six numerical examples for the transversely isotropic layer, namely, beryl, magnesium, cadmium, zinc, cobalt, and simply isotropic, have been considered. The role of magneto-elastic coupling parameter, hydrostatic stress, inhomogeneity, the order of the depth variation in inhomogeneity function, and different examples of the layer on the propagation of Love-type wave has been observed by numerical examples and graphical demonstrations.

Study of torsional wave in a poroelastic medium sandwiched between a layer and a half-space of heterogeneous dry sandy media

In this article, the propagation of torsional surface wave in an anisotropic poroelastic layer of finite thickness sandwiched between two heterogeneous dry sandy media is investigated. The first uppermost dry sandy medium is considered as a layer of finite thickness and the second one is considered as a lower half-space. The heterogeneities in both dry sandy media are assumed to arise due to quadratic variations in elastic moduli and mass densities. Whittaker’s functions and variable separable techniques have been taken into the application to calculate the interior deformations inside the assumed model; consequently, we obtain a closed form dispersion relation for the torsional wave using effective boundary conditions. Moreover, casewise dispersion equations for some particular aspects of the problem have been studied, which serve as the focal theme of the study. Some significant observations have been made by detailed numerical calculations and graphical visuals related to the effects of tensile and compressive initial stresses, sandy parameters, heterogeneity parameters, porosity parameter, and thickness ratio parameter on the phase velocity of the torsional wave.

Propagation of in-plane wave in viscoelastic monolayer graphene via nonlocal strain gradient theory

The behaviors of monolayer graphene sheet have attracted increasing attention of many scientists and researchers. In this study, the propagation behaviors of in-plane wave in viscoelastic monolayer graphene are investigated. The constitutive equation and governing equation for in-plane wave propagation is developed by employing Hamilton’s principle and nonlocal strain gradient theory. By solving the governing equation of motion, the closed-form dispersion relation between phase velocity and wave number is derived and an asymptotic phase velocity can be acquired. The effects of wave number, material length scale parameter, nonlocal parameter and damping coefficient on in-plane wave propagation behaviors are discussed in the numerical studies. It is found that, when exciting wavelengths or structural dimensions become comparable to the material length scale parameters and nonlocal parameters, the scaling effects on wave propagation behaviors are significant. For nanoscaled graphene sheet, the effects of nonlocal parameter, material length scale parameter and damping coefficient on phase velocity are tiny at low wave numbers while significant at high wave numbers. The phase velocity would increase with the increase of material length scale parameter or the decrease of nonlocal parameter and damping coefficient. Furthermore, results indicate that the asymptotic phase velocity can be increase by increasing material length scale parameter or decreasing nonlocal parameter.

Combined Rayleigh–Taylor and Kelvin–Helmholtz instabilities on an annular liquid sheet

This paper describes the three-dimensional destabilization characteristics of an annular liquid sheet when subjected to the combined action of Rayleigh–Taylor (RT) and Kelvin–Helmholtz (KH) instability mechanisms. The stability characteristics are studied using temporal linear stability analysis and by assuming that the fluids are incompressible, immiscible and inviscid. Surface tension is also taken into account at both the interfaces. Linearized equations governing the growth of instability amplitude have been derived. These equations involve time-varying coefficients and have been analysed using two approaches – direct numerical time integration and frozen-flow approximation. From the direct numerical time integration, we show that the time-varying coefficients evolve on a slow time scale in comparison with the amplitude growth. Therefore, we justify the use of the frozen-flow approximation and derive a closed-form dispersion relation from the appropriate governing equations and boundary conditions. The effect of flow conditions and fluid properties is investigated by introducing dimensionless numbers such as Bond number ($Bo$), inner and outer Weber numbers ($We_{i}$,$We_{o}$) and inner and outer density ratios ($Q_{i}$,$Q_{o}$). We show that four instability modes are possible – Taylor, sinuous, flute and helical. It is observed that the choice of instability mode is influenced by a combination of both$Bo$as well as$We_{i}$and$We_{o}$. However, the instability length scale calculated from the most unstable wavenumbers is primarily a function of$Bo$. We show a regime map in the$Bo,We_{i},We_{o}$parameter space to identify regions where the system is susceptible to three-dimensional helical modes. Finally, we show an optimal partitioning of a given total energy ($\unicode[STIX]{x1D701}$) into acceleration-induced and shear-induced instability mechanisms in order to achieve a minimum instability length scale (${\mathcal{L}}_{m}^{\ast }$). We show that it is beneficial to introduce at least 90 % of the total energy into acceleration induced RT instability mechanism. In addition, we show that when the RT mechanism is invoked to destabilize an annular liquid sheet,${\mathcal{L}}_{m}^{\ast }\sim \unicode[STIX]{x1D701}^{-3/5}$.

Effects of Anisotropy, Initial Stress, Heterogeneity, and Gravity on Torsional Wave Propagation

AbstractThe present paper provides the results of a study of the propagation of torsional surface waves in an initially stressed self-reinforced layer lying over a dry sandy Gibson half-space in the presence of initial stress and gravity. The closed-form dispersion relation was obtained analytically. The influences of initial stress, reinforcement, inhomogeneity, gravity, and sandy parameter on the torsional wave were investigated by plotting the dispersion curves. It was observed that the rigidities, initial stress, and sandy parameter of the half-space have a favoring effect on the phase velocity, whereas gravity has the reverse effect on the velocity of the torsional surface wave. It also was noted that as reinforcement increased in the medium, the phase velocity also increased significantly.

Diffraction of Torsional Surface Wave in a Voigt Layer Overlying a Prestressed Heterogeneous Substratum

AbstractThis paper discusses the torsional surface wave in a Voigt-type layer placed over an initially stressed heterogeneous semi-infinite substratum. The heterogeneities in the semi-infinite substratum are thought to emerge as a result of exponential variation in rigidity, density, and initial stress. A closed-form dispersion relation for the torsional surface waves under the considered geometry was obtained. The effect of various technical constants, such as internal friction parameter, stress parameter, heterogeneity parameter, dissipation factor, complex wave number, and thickness of the layer, were studied numerically by plotting various graphs between the nondimensional angular frequencies against the nondimensional real wave number for the torsional surface wave propagation. The numerical solution for the limiting case of torsional surface waves is also presented. In the absence of the considered technical constants, the obtained results are in close agreement with the results obtained by other re...

Wave propagation in fluid-conveying viscoelastic carbon nanotubes based on nonlocal strain gradient theory

The governing equation of wave motion of fluid-conveying viscoelastic single-walled carbon nanotubes is formulated on the basis of the nonlocal strain gradient theory and the Kelvin–Voigt viscoelastic model. Based on the formulated equation of wave motion, the closed-form dispersion relation between the wave frequency (or phase velocity) and the wave number is derived. It is found that, the effects of nonlocal parameters and small scale material parameters on the dispersion relation between the phase velocity and the wave number are significant at high wave numbers, however, may be ignored at low wave numbers. The upstream phase velocities decrease as increasing flow velocity, whereas the downstream phase velocities firstly increase as increasing flow velocity and then decrease as increasing flow velocity. The effect of damping coefficient on the phase velocity of both upstream and downstream waves is negligible at low wave numbers, however, is remarkable at high wave numbers.

Wave propagation in viscoelastic single-walled carbon nanotubes with surface effect under magnetic field based on nonlocal strain gradient theory

The governing equation of wave motion of viscoelastic SWCNTs (single-walled carbon nanotubes) with surface effect under magnetic field is formulated on the basis of the nonlocal strain gradient theory. Based on the formulated equation of wave motion, the closed-form dispersion relation between the wave frequency (or phase velocity) and the wave number is derived. It is found that the size-dependent effects on the phase velocity may be ignored at low wave numbers, however, is significant at high wave numbers. Phase velocity can increase by decreasing damping or increasing the intensity of magnetic field. The damping ratio considering surface effect is larger than that without considering surface effect. Damping ratio can increase by increasing damping, increasing wave number, or decreasing the intensity of magnetic field.

On torsional surface wave in dry sandy crust laid over an inhomogeneous half space

An in-depth investigation has been carried out for the propagation of torsional surface waves in a dry sandy layer over an inhomogeneous half space. Both medium are assumed to be under compressive initial stress. The inhomogeneities that arise due to slow variation in rigidity and density in the lower half space is taken as an arbitrary function of depth. Using WKB approximation a closed form dispersion relation of torsional surface wave in the sandy layer has been formulated in terms of the arbitrary bounded functions. This dispersion equation is further employed to study the polynomial and exponential variation in rigidity and density of the half-space. An extensive analysis is carried out through numerical computation to explore the effect of inhomogeneity, initial stress, sandiness and thickness of the layer on the phase velocity of the torsional surface waves. The analysis of dispersion equation manifests the decline of phase velocity in the presence of sandy parameter as well as by increase in thickness of the sandy layer. The results further indicate that the initial stress of both the medium and the inhomogeneity of the half space affect the wave velocity significantly.

Thermoelastic symmetric and antisymmetric wave modes with trigonometric functions in laminated plates

Coupling term in coupled thermoelasticity generally is small for all materials and can be neglected. Neglecting the coupling term simplifies the analysis without noticeable effect on the frequency spectrum. In generalized theory of thermoelasticity, effect of increasing the relaxation time is to lower the speeds of the thermal modes. Since the effect of anisotropy of the material is quite pronounced on waves propagating in different directions along the plate. Thus, it is significant to consider the anisotropy of the material in order to accurately model the propagation characteristics. A theoretical framework is developed for displacements and temperature expression of plane harmonic waves in generalized theory of thermoelasticity using three dimensional thermoelasticity with a thermal relaxation time. Subsequently, the Lamb waves in a single lamina where a compact closed-form dispersion relation is derived by separating thermoelastic symmetric and anti-symmetrical modes using trigonometric functions through the lamina thickness. Propagation of guided thermoelastic waves in a homogeneous, orthotropic, thermally conducting plate is investigated within the framework of the generalized theory of thermoelasticity. The results show that both elastic and thermal modes are attenuated, the thermal modes exhibit much larger attenuation than the elastic modes. The attenuation of the former is quite small. Results obtained are extended to a general laminate with an arbitrary layup. On employing boundary conditions on both mid-plane and top surfaces is developed to decouple the wave modes for symmetric and anti-symmetrical modes for laminates. Propagation of guided thermoelastic waves in a homogeneous, orthotropic, thermally conducting plate is investigated employing generalized theory of thermoelasticity which includes a thermal relaxation time in the heat conduction equation in order to model the finite speed of the thermal wave. Both elastic and thermal modes are attenuated; the thermal modes exhibit much larger attenuation than the elastic modes. The effect of increasing the relaxation time is to lower the speeds of the thermal modes. The effect of anisotropy of the material is quite pronounced on waves propagating in different directions. Thus, it is important to consider the anisotropy of the material in order to accurately model the propagation characteristics.

Analysis and Interpretation of Longitudinal Waves in Periodic Multiphase Rods Using the Method of Reverberation-Ray Matrix Combined With the Floquet-Bloch Theorem

The method of reverberation-ray matrix (MRRM) combined with the Floquet–Bloch theorem, which serves as an alternative method, is presented for accurately analyzing longitudinal waves in general periodic multiphase rods. Closed-form dispersion relation of periodic quaternary rods is derived. Based on this relation, the functions of constituent-rod parameters in the formation of longitudinal-wave band structures are analytically revealed. Numerical examples validate the proposed method and indicate the characteristics/applications of all kinds of dispersion curves that include the frequency-wave number spectra, the frequency-wavelength spectra, the frequency-phase velocity spectra, the wave number-phase velocity spectra and the wavelength-phase velocity spectra. The effect of unit-cell layout on the frequency band properties and the functions of constituent-rod parameters in the band structure formation are also illustrated numerically. The analysis and interpretation of longitudinal waves in periodic multiphase rods given in this paper will push forward the design of periodic structures for longitudinal wave filtering/guiding and vibration isolation/control applications.

Formation of Bending-Wave Band Structures in Bicoupled Beam-Type Phononic Crystals

Beam-type phononic crystals as one kind of periodic material bear frequency bands for bending waves. For the first time, this paper presents formation mechanisms of the phase constant spectra in pass-bands of bending waves (coupled flexural and thickness-shear waves) in bicoupled beam-type phononic crystals based on the model of periodic binary beam with rigidly connected joints. Closed-form dispersion relation of bending waves in the bicoupled periodic binary beam is obtained by our proposed method of reverberation-ray matrix (MRRM), based on which the bending-wave band structures in the bicoupled binary beam phononic crystal are found to be generated from the dispersion curves of the equivalent bending waves in the unit cell due to the zone folding effect, the cut-off characteristic of thickness-shear wave mode, and the wave interference phenomenon. The ratios of band-coefficient products, the characteristic times of the unit cell and the characteristic times of the constituent beams are revealed as the three kinds of essential parameters deciding the formation of bending-wave band structures. The MRRM, the closed-form dispersion relation, the formation mechanisms, and the essential parameters for the bending-wave band structures in bicoupled binary beam phononic crystals are validated by numerical examples, all of which will promote the applications of beam-type phononic crystals for wave filtering/guiding and vibration isolation/control.