Frequency Dispersion Relation Research Articles

A mode separation method based on sparse Bayesian learning for explosive sources in a shallow water waveguide

This paper presents a mode separation method based on sparse Bayesian learning (SBL) for explosive sources in a shallow water waveguide. In previous work [Niu et al., JASA, 2021, 4366], the SBL dictionary was constructed by assuming a large number of horizontal wavenumbers and utilized an approximate mode-frequency dispersion relation for low frequencies. Then, modes were separated in the frequency domain by estimating the coefficients of the dictionary atoms. However, challenges inherent to explosive sources, such as bandwidth expansion and the bubble-pulse effect, result in a mismatch in the dictionary matrix built using the approximate mode-frequency dispersion relation for low frequencies, leading to unsuccessful mode separation. To address these issues, this paper builds the SBL dictionary matrix by utilizing the acoustic model (e.g., Kraken) to derive horizontal wavenumbers under various environmental hypotheses and combines it with the secondary bubble-pulse model. By estimating the coefficients of the dictionary atoms, both environmental parameter estimation and mode separation can be achieved simultaneously. Simulation and experimental data results demonstrate the validation of the proposed method.

Nonlocal theory on plane waves in a higher-order thermo-diffusive semiconducting medium under the gravitational field

This article explores the reflection of elastic waves in nonlocally generalized thermo-diffusive semiconducting elastic solids under the influence of gravity. The problem is framed within the higher-order fractional derivative heat equation. The frequency dispersion relation reveals the presence of five coupled waves. The wave speeds are graphed against angular frequency for the nonlocal parameter under the influence of gravity, and the cut-off frequency of the waves is visually represented. The incident wave at the free surface of the solid is the coupled longitudinal P-wave, and its reflection coefficients are computed. The study investigates the impact of gravity, fractional order, and nonlocal parameters on propagation speed and amplitude ratios, revealing their significant influence. The findings are substantiated in the context of energy conservation. This research provides valuable insights for scientists engaged in geophysics and various branches of mechanics.

Propagation and reflection of thermoelastic wave in a rotating nonlocal fractional order porous medium under Hall current influence

This investigation relates to the research on Hall current on propagation and reflection of elastic waves through non-local fractional-order thermoelastic rotating medium with voids. The system is split up into longitudinal and transverse components using the Helmholtz vector rule. It is observed that, through the frequency dispersion relation four coupled quasi-waves exist in the medium. The rotating solid modifies the nature of purely longitudinal and transverse waves toward the quasi-type waves. All the propagating waves are dispersive as they depend upon angular frequency. The quasi-longitudinal wave qP and quasi-transverse wave qSV faces cut-off frequencies. The nonlocal parameter affect all the waves except the quasi void wave. Analytically, the reflection coefficients of the wave are computed using suitable boundary conditions. MATLAB software is used to perform numerical computations for a chosen solid material. The amplitude ratios and the speed of propagation of the wave are plotted graphically for rotational frequency, nonlocal, fractional order, and Hall current parameter. The significant effect of the physical parameters on the computed results has been observed. The cut-off frequency of the waves is also presented graphically. The energy conservation law is proved in the form of energy ratios. The earlier findings in the literature are obtained as special case in the absence of rotation, Hall current parameter and porous voids.

Nonlocal theory of propagation and reflection of plane waves in higher order thermo diffusive semiconducting medium

The present article is related to propagation and reflection of elastic wave through a nonlocal generalized thermo-diffusive semiconducting elastic solid. The non-local theory is employed to study the wave behavior. Three phase lag model with higher order fractional order derivative is incorporated to discuss heat propagation through the medium, in addition with two phase lags diffusion equation. The Helmholz vector rule is applied to decompose the system into longitudinal and transverse components. The frequency dispersion relation indicates the presence of four coupled longitudinal and one un-coupled shear vertical wave. The speed of the waves is plotted against angular frequency for local and nonlocal medium. The cutoff frequency of the waves is also depicted graphically. The longitudinal P-wave is taken to be an incident wave at the free surface of the solid to compute the reflection coefficients. The influences of fractional order and nonlocal parameters on amplitude ratios are also studied. The effect of these parameters is found to be significant. The results are proved in the context of energy conservation. The results obtained from the current investigation are very useful for scientists working on problems of geophysics and various fields of mechanics.

Dispersive landslide

Considering the non-hydrostatic mass flow model (Pudasaini, 2022 [1]), here, I derive a novel dispersive wave equation for landslide. The new dispersive wave for landslide recovers the classical dispersive water waves as a special case. I show that the frequency dispersion relation for landslide is inherently different than the classical frequency dispersion for water waves. The wave frequency with dispersion increases non-linearly as a function of the wave number. For dispersive landslide, the wave frequency without dispersion appears to heavily overestimate the dispersive wave frequency for higher wave number. Due to the dispersion term emerging from the non-hydrostatic contribution for landslide, the phase velocity becomes a function of the wave number. This gives rise to the group velocity that is significantly different from the phase velocity, characterizing the dispersive mass flow. The dispersive phase velocity and group velocity decrease non-linearly with the wave number. Yet, the group velocity is substantially lower than the phase velocity. I analytically derive a dispersion number as the ratio between the phase velocity and the group velocity, which measures the deviation of the group velocity from the phase velocity, provides a dynamic scaling between them and summarizes the overall effect of dispersion in the mass flow. The dispersion number for landslide increases rapidly with the wave number, which is in contrast to the dispersion in water waves. With the definition of the effective dispersive lateral stress, I prove the existence of an anti-restoring force in landslide. I reveal the fact that due to the anti-restoring force, landslides are more dispersive than the piano strings. So, the wave dispersion in landslide is fundamentally different than the wave dispersion in the piano string. My model constitutes a foundation for the wave phenomenon in dispersive mass flows.

Experimental investigations of linear and nonlinear periodic travelling waves in a viscous fluid conduit

Conduits generated by the buoyant dynamics between two miscible Stokes fluids with high viscosity contrast, a type of core–annular flow, exhibit a rich nonlinear wave dynamics. However, little is known about the fundamental wave dispersion properties of the medium. In the present work, a pump is used to inject a time-periodic flow that results in the excitation of propagating small- and large-amplitude periodic travelling waves along the conduit interface. This wavemaker problem is used as a means to measure the linear and nonlinear dispersion relations and corresponding periodic travelling wave profiles. Measurements are favourably compared with predictions from a fully nonlinear, long-wave model (the conduit equation) and the analytically computed linear dispersion relation for two-Stokes flow. A critical frequency is observed, marking the threshold between propagating and non-propagating (spatially decaying) waves. Measurements of wave profiles and the wavenumber–frequency dispersion relation quantitatively agree with wave solutions of the conduit equation. An upshift from the conduit equation's predicted critical frequency is observed and is explained by incorporating a weak recirculating flow into the full two-Stokes flow model. When the boundary condition corresponds to the temporal profile of a nonlinear periodic travelling wave solution of the conduit equation, weakly nonlinear and strongly nonlinear, cnoidal-type waves are observed that quantitatively agree with the conduit nonlinear dispersion relation and wave profiles. This wavemaker problem is an important precursor to the experimental investigation of more general boundary value problems in viscous fluid conduit nonlinear wave dynamics.

Clusters of Bloch waves in three-dimensional periodic media

We consider acoustic wave propagation through a periodic array of the inclusions of arbitrary shape. The inclusion size is much smaller than the array period while the wavelength is fixed. We derive and rigorously justify the dispersion relation for general frequencies and show that there are exceptional frequencies for which the solution is a cluster of waves propagating in different directions with different frequencies so that the dispersion relation cannot be defined uniquely. Examples are provided for the spherical inclusions.

Investigation of the Physical Mechanism of Acoustic Attenuation in Viscous Isotropic Solids

The traditional acoustic attenuation coefficient is derived from an analogy of the attenuation of an electromagnetic wave propagating inside a non-ideal medium, featuring only the attenuation of wave propagation. Nonetheless, the particles inside viscous solids have mass, vibrating energy, viscosity, and inertia of motion, and they go through transient and damping attenuation processes. Based on the long-wavelength approximation, in this paper, we use the energy conservation law to analyze the effect of the viscosity of the medium on acoustic attenuation. We derive the acoustic attenuation coefficient by combinations of the dynamical equation of a solid in an acoustic field with conventional longitudinal wave propagation under a spring oscillator model. Considering the attenuation of propagating waves and the damping attenuation of particle vibration, we develop a frequency dispersion relation of phase velocity for the longitudinal wave propagating inside viscous solid media. We find that the acoustic impulse response and vibrational system function depends on the physical properties of the viscous solid media and their internal structure. Combined with system function, the impulse response can be an excellent tool to invert the physical properties of solids and their internal structures. We select a well-known rock sample for analysis, calculate the impulse response and vibrational system function, and reveal new physical insight into creating acoustic attenuation and frequency dispersion of phase velocity. The results showed that the newly developed acoustic attenuation coefficients enjoy a substantial improvement over the conventional acoustic attenuation coefficients reported in the literature, which is essential for industrial applications; so are the dispersion characteristics.

Depth-integrated wave–current models. Part 2. Current with an arbitrary profile

The depth-integrated wave–current models developed by Yang & Liu (J. Fluid Mech., vol. 883, 2020, A4) are extended to investigate currents with an arbitrary vertical profile in the water column. In the present models, horizontal velocities are decomposed into two components. The first part deduces the prescribed current velocity when waves are absent. The second part is approximated in a polynomial form. The resulting depth-integrated wave–current models are obtained by applying the weighted residual method. In the absence of currents, the present models are identical to those in Yang & Liu (J. Fluid Mech., vol. 883, 2020, A4) and are validated with several three-dimensional (3D) benchmark laboratory experiments. A theoretical analysis is conducted to study the frequency dispersion relation of linear waves on currents with an exponential vertical profile and the results are compared with numerical solutions of the Rayleigh equation. Using the new models, validations and investigations are then conducted for periodic waves and solitary waves on currents with an arbitrary profile in one-dimensional horizontal (1DH) space. Furthermore, the new models are applied to wave–current interactions in two-dimensional horizontal (2DH) space. Two scenarios are considered: (1) wave propagation over a vortex-ring-like current and (2) obliquely incident wave propagation over a 3D sheared current on a varying bathymetry. The vertical and horizontal shear of the current have significant effects on modifying various wave properties, which are well captured by the present models. However, the time-averaged velocity under wave–current interaction shows small differences with the prescribed current velocity, except in the region between the wave trough and crest.

A semi-analytical theory for water waves interacting with a submerged/suspended circular cylinder patch

Semi-analytical solutions are proposed to investigate the effects of a submerged/suspended circular patch on water waves scattering. The patch is simplified to periodic cylindrical lattices with a strong contrast between the cylinder spacing and the typical wavelength. The homogenization theory is employed to derive the effective equations on the macro-scale (wavelength), in which the constitutive coefficients are obtained by numerically solving the micro-scale (cylinder spacing) problem in a unit cell. To simulate the turbulence in the circular cylinder patch, a bulk eddy viscosity model is applied through balancing the time-averaged energy over a wave period. Eigenfunction expansions method is used to solve the macro-scale problem, where a complex frequency dispersion relation is solved by the multiple successive approximation technique. Comparisons between the laboratory measurements and the waves/submerged circular patch model results show acceptable agreements for small amplitude waves. The waves/suspended circular patch model is not checked experimentally due to the lack of data. However, this suspended patch model is shown to reproduce the results of wave/emergent circle forest problem. A comprehensive comparison between the submerged, suspended and emergent circular patches on wave scattering is also presented.

On Stokes wave solutions

Using a perturbation method with the aid of MATLAB®Symbolic Toolbox, a new set of Stokes wave solutions, up to the fifth order, are derived. The new solutions are expressed in terms of free surface profile, velocity potential and wave celerity (or frequency dispersion relation). The solutions in the deep water limit are also deduced and discussed. The new solutions are compared with the existing solutions up to the fifth order. Differences appear among the existing and the present solutions, starting at the third order. The causes for the differences are identified. The characteristic differences were highlighted in the figures for high order solutions of Stokes waves. Numerical examples are provided to quantify the differences between the new fifth-order solutions and Fenton’s, and Skjelbreia and Hendrickson’s solutions.

Reflection of plane wave at free boundary of micro-polar nonlocal semiconductor medium

This work presents the reflection phenomena of thermo-elastic waves in nonlocal micro-polar semi-conducting solid. The thermo-elasticity theory of factional-order time derivative three phase lag thermo-elastic model is applied to study the thermal effect on the waves. Governing equations of the problem are decomposed by using the Helmholtz decomposition. Two frequency dispersion relations corresponding to couple longitudinal and transverse waves are obtained. It is observed that three coupled longitudinal and two coupled transverse waves are propagating through the elastic solid. Relations for amplitude ratios, Energy flux balance are obtained analytically. All the results obtained mathematically are represented graphically for the selected material. The silicon crystal has been selected for computational purpose to check the variation of amplitude ratios against incident angle. The dependence of reflection coefficients upon nonlocal and fractional order parameters is been represented graphically.

Toward real-time optical estimation of ocean waves’ space-time fields

Stereo 3D reconstruction is continuously increasing its popularity in the study of mid-to small-scale sea waves. In the recent past, different approaches have been proposed to reconstruct the space-time sea surface elevation field from synchronized stereo frames. Usually, the reconstruction is performed by first recovering a dense and sparse 3D point cloud from stereo pairs and then by interpolating it into a regular grid. Even considering state-of-the-art methods, with typical image resolutions, it's unlikely to perform the first step in less than dozens of seconds (without paralellization), that can easily doubled if we include the subsequent interpolation step. This will limit the applicability of stereo based wave analysis for all the approaches (like monitoring or data assimilation) in which the time between the raw acquisition and the output of processed data is critical.In this paper, we propose a new method to directly estimate the sea surface spectrum over time from a sequence of stereo frames. We exploit the frequency dispersion relation of gravity waves and the (Non-uniform) Fast Fourier Transform to continuously update the 3D surface and 2D wavenumber spectrum estimations given a sparse set of matching features between the two frames. This effectively combines the two aforementioned steps with a performance gain of more than an order of magnitude while keeping a reconstruction accuracy that is extremely close to the current state-of-the-art approaches. Code is available at https://gitlab.com/fibe/wassfast.

Periodic water waves through suspended canopy

Small-amplitude water waves passing through a suspended canopy is studied. The area of suspended canopy is modeled by an array of vertical rigid cylinders with periodic spacing. Assuming that the diameter of cylinders and their spacing are much smaller than the typical incident wavelength, the homogenization theory (method of multiple-scale perturbation) is applied to create coupled micro-scale (cylinder spacing) and macro-scale (wavelength) problems. The micro-scale problem describes turbulent flows within a unit cell of the cylinder array, being driven by the macro-scale pressure gradients. Employing the concept of averaged energy balance over a wave period, the micro-scale flows determine the eddy viscosity, which damps the waves in the macro-scale flows. Eigenfunction expansions method is used to solve the macro-scale problem, in which a complex frequency dispersion relation is solved numerically by a multiple successive approximation technique. The potential decomposition method, which can avoid solving a complex frequency dispersion relation, is also employed to validate the accuracy of the eigenfunction expansions method. Both methods yield accurate solutions. However, the eigenfunction expansions method is relatively straightforward and can unify the solutions for suspended canopy and emergent vegetation. A new set of flume experiments of waves through suspended canopy is conducted and experimental data are used to check present solutions. Very good agreement has been observed. Finally, the effectiveness of suspended canopy, submerged and emergent vegetation on wave attenuation is discussed.

Super-Resolution Ultrasound Lamb Wave NDE Imaging of Anisotropic Airplane Laminates via Deconvolutional Neural Network

A deconvolutional neural network (DNN) is integrated into the ultrasound (UT) pulse-echo Lamb wave nondestructive evaluation (NDE) imaging technique to achieve subwavelength super-resolution defect images for the application of anisotropic composite airplane-laminated structures. First, numerical simulation has been performed to simulate the multilayer velocity–frequency dispersion relation of the symmetric and antisymmetric Lamb wave modes. Then, a super-resolution DNN is developed with all Lamb wave modes as the convolutional kernels to obtain the subwavelength defects image of the laminated structures. After that, the effectiveness of the pulse-echo Lamb wave NDE imaging technique is verified by the experimental test of a standard aluminum metal plate with three calibration holes of 2/3/5 mm diameters at the UT center frequency of 2 MHz and a line defect. Finally, the experiment of the pulse-echo Lamb wave NDE imaging technique is carried out with an L-joint coupon sample made of anisotropic graphite–epoxy airplane materials. The comparison between the experimental result and the conventional pitch-catch C-scan shows that the Lamb wave NDE technique can reveal more details of the defects, indicating its promising application in anisotropic layers’ structures defects inspection.

Higgs and Goldstone Modes in Crystalline Solids

In crystalline solids, the acoustic phonon can be described either as a Goldstone or as a non‐Abelian gauge boson. However, the non‐Abelianity of the related gauge group apparently makes the acoustic phonon a frequency‐gapped mode, in contradiction with the other description. In a different perspective, overcoming this contradiction, both acoustic and optical phonon—the latter never appearing following the other two approaches—emerge, respectively, as the gapless Goldstone (phase) and the gapped Higgs (amplitude) fluctuation mode of an order parameter arising from the spontaneous breaking of a global symmetry, without invoking the gauge principle. In addition, the Higgs mechanism describes all the phonon–phonon interactions, including a possible perturbation of the acoustic phonon's frequency dispersion relation induced by the eventual optical phonon, a peculiar behavior that is able to produce mini‐gaps inside the phonon Brillouin zone.

Derivation of gravity wave intrinsic parameters and vertical wavelength using a single scanning OH(3-1) airglow spectrometer

Abstract. For the first time, we present an approach to derive zonal, meridional, and vertical wavelengths as well as periods of gravity waves based on only one OH* spectrometer, addressing one vibrational-rotational transition. Knowledge of these parameters is a precondition for the calculation of further information, such as the wave group velocity vector. OH(3-1) spectrometer measurements allow the analysis of gravity wave ground-based periods but spatial information cannot necessarily be deduced. We use a scanning spectrometer and harmonic analysis to derive horizontal wavelengths at the mesopause altitude above Oberpfaffenhofen (48.09∘ N, 11.28∘ E), Germany for 22 nights in 2015. Based on the approximation of the dispersion relation for gravity waves of low and medium frequencies and additional horizontal wind information, we calculate vertical wavelengths. The mesopause wind measurements nearest to Oberpfaffenhofen are conducted at Collm (51.30∘ N, 13.02∘ E), Germany, ca. 380 km northeast of Oberpfaffenhofen, by a meteor radar. In order to compare our results, vertical temperature profiles of TIMED-SABER (thermosphere ionosphere mesosphere energetics dynamics, sounding of the atmosphere using broadband emission radiometry) overpasses are analysed with respect to the dominating vertical wavelength.

Collective magnetic skyrmion gyrotropic modes in a dot chain

We investigate the collective gyrotropic modes in a linear chain of ferromagnetic nanodots in a skyrmion magnetization state. Individual skyrmions in the dots (with thickness about of 1 nm and radius about of 100 nm) are assumed to be stabilized by interplay of isotropic exchange and Dzyaloshinskii-Moriya exchange interactions, magnetic anisotropy and magnetostatic interactions. We consider the dipolar and quadrupolar interactions between moving skyrmions in the dots as prevailing mechanism responsible for dynamic interdot magnetostatic coupling. The frequency dispersion relations of the collective skyrmion gyrotropic excitations are calculated for Bloch and Neel skyrmion configurations and related differences in the frequencies are discussed. The accounting of magnetostatic interactions between the skyrmions removes phase degeneracy of the gyrotropic frequency existing for isolated skyrmions and allows distinguishing the Bloch and Neel skyrmionic configurations by their dynamic response. The interdot magnetostatic coupling and frequency bandwidth are maximal for the large-radius Bloch skyrmions in the relatively thick dots.

Galloping Reduction of Transmission Lines by Using Phononic Crystal

Considering the combination of the transmission lines and phononic crystals (PCs), we propose a new method to solve the problem of the galloping of overhead transmission lines. The method has two key points: attaching the suitable mass-spring system on each spacer, and periodically arranging the modified spacers along a transmission line. Based on the Bloch’s theorem, the PC transmission lines could generate vibration band gaps (BGs), which would reduce galloping. In order to implement our point, we establish the two-dimensional model of the PC transmission lines and derive the transfer matrix method to calculate the frequency dispersion relation of the vertical transverse vibration. Then, the extremely low frequency BG, in the range of galloping frequency, is obtained and verified based on an example of single conductor. To widen the BG range, we also study the effects of the spacer and the attached mass-spring system on the BG. The wide BG, which even covers the range of 0.338–0.909 Hz, could be given just by using the suitable setting of the spacer and mass-spring system.

Seismic isolation of buildings on two dimensional phononic crystal foundation

In order to realize the seismic isolation of buildings, we establish the two dimensional phononic crystal (PC) foundation which has the cell with the size close to the regular concrete test specimens, and is composed of the concrete base, rubber coating and lead cylindrical core. We study the in-plane band gap (BG) characteristics in it, through the analysis of the frequency dispersion relation and frequency response result. To lower the start BG frequency to the seismic frequency range, we also study the influences of material parameters (the elastic modulus of coating and density of cylindrical core) and geometry parameters (the thickness of coating, radius of cylindrical core and lattice constant) on BG ranges. The study could help to design the PC foundation for seismic isolation of building.