Dispersion Relation For Waves Research Articles

Nonlinear regimes of inertial wave attractors generated by a precessing lid: Zonal flows and Rossby waves

Fluid flow excited by inertial waves in a rotating annulus with a trapezoidal cross section is experimentally investigated. The forcing is introduced by the precession of a flat annular lid bounding the cavity. Such a setup allows to investigate interactions between inertial waves and zonal flows in the presence of equivalent topographic beta-effect, and its dynamic behavior under weak and strong forcing is experimentally studied for the first time. Due to the specific dispersion relation of inertial waves, a cavity shape supports wave attractors that appear only for retrograde lid precession. At a relatively low wave intensity, a zonal flow is generated in the vicinity of a “virtual” axial cylinder, the radius of which coincides with the radius of the focusing reflection from the conical bottom. A succession of non-linear regimes is observed as forcing increases, starting with a clearly identifiable case of triadic resonance. Further, the frequency spectrum is progressively enriched by emergence of additional discrete components, gradually reaching the state of “overheated” wave turbulence with a significant continuous background of the frequency spectrum. Finally, a strong low-frequency component develops in the frequency spectrum, and the continuous background of the spectrum decreases dramatically. The low-frequency component corresponds to azimuthally periodic structures identified as Rossby waves. In the case of strong forcing, a long-term variability of zonal-flow velocity at the timescale of tens of forcing periods is observed, which is likely a consequence of the interaction of azimuthal structures corresponding to a rich wave number set.

ON THE THIRD ORDER SOLUTION OF KdV EQUATION BY USING HOMOTOPY PERTURBATION METHOD

In this research we discussed about the solution of the KdV equation using Homotopy Perturbation method. The KdV equation that describing water wave equation solved by using the mixing method between Homotopy and Perturbation method. Homotopy was built with embedding parameter p∈[0,1] which undergoes a deformation process from linear problems to nonlinear problems and the assumed solution of the KdV equation is expressed in the form of a power series p up to the third order. The result show that in each order solution we obtained resonance term. for handling the condition, we used Lindsteadt-Poincare method.the wave number k2 and dispersion relation can be obtained in the second order solution as the effect of using Lindsteadt-Poincare method.

A new law of Class I Bragg resonance for linear long waves over arrays of parabolic trenches or rectified cosinoidal trenches

Class I Bragg resonances of linear long waves excited by a finite periodic array of widely spaced trenches were studied, where the trenches include two types, i.e., parabolic trench and rectified cosinoidal trench. First, analytical solutions of the reflection coefficient in terms of the associated Legendre functions for wave reflection by parabolic trenches and in terms of Frobenius series for wave reflection by rectified cosinoidal trenches are constructed, respectively. Second, the phenomenon of the phase upshift of each order Bragg resonance is confirmed for linear long waves reflected by the two types of trenches. Third, the dispersion relation of linear long waves over an infinite periodic array of parabolic or rectified cosinoidal trenches is derived. Fourth, based on the dispersion relation, Bloch band structures are obtained and their related periodic gap maps are plotted. Finally, based on the Bloch band theory, a new law for locating the phase of each order Bragg resonance is established, which is much more accurate than the traditional Bragg’s law. By using the new law, the phase upshift phenomenon can be well explained, visually demonstrated and accurately predicted.

Numerical investigation of cavitating tip vortex dynamics and how they influence the acoustic characteristics

The surface waves traveling on tip vortex cavities are one of the main contributors to underwater radiated noise (URN). However, these waves cannot be easily predicted, especially the mode associated with the cavity volume variations, which hinders research into the hydroacoustic mechanism. This paper presents a method for extracting the surface waves from numerical predictions, which successfully captured all the theoretical dispersion relations of these surface waves. This method was then used to analyze the tip vortex cavitation (TVC) dynamics and how they influence the URN with an acoustic analogy. Low- and medium-frequency URN are shown to be mainly emitted by the surface waves related to the cavity volume variations. The significant tones at the dominant frequency and the broadband hump center frequency are further studied with the main noise sources found to be the cavity radial fluctuations and the TVC resonance.

Minimally modified gravity with auxiliary constraints formalism

We investigate the possibility of reducing the number of degrees of freedom (d.o.f.) starting from generic metric theories of gravity by introducing multiple auxiliary constraints (ACs), under the restriction of retaining spatial covariance as a gauge symmetry. Arbitrary numbers of scalar-, vector- and tensor-type ACs are considered a priori, yet we find that no vector- and tensor-type constraints should be introduced, and that scalar-type ACs should be no more than four for the purpose of constructing minimally modified gravity (MMG) theories which propagate only two tensorial d.o.f., like general relativity (GR). Through a detailed Hamiltonian analysis, we exhaust all the possible classifications of ACs and find out the corresponding minimalizing and symmetrizing conditions for obtaining the MMG theories. In particular, no condition is required in the case of four ACs, hence in this case the theory can couple with matter consistently and naturally. To illustrate our formalism, we build a concrete model for this specific case by using the Cayley-Hamilton theorem and derive the dispersion relation of the gravitational waves, which is subject to constraints from the observations.

Explicit derivation of the chiral and generic helical edge states for the Kane-Mele model: Closed expressions for the wave function, dispersion relation, and spin rotation

Although one of the most important and intriguing features of the topological insulators is the presence of edge states, the closed-form expressions for the edge states of some famous topological models are still lacking. Here, we focus on the Kane-Mele model with and without Rashba spin-orbit coupling as a well-known model to describe a two-dimensional version of the ${\mathbb{Z}}_{2}$ topological insulator to study the properties of its edge states analytically. By considering the tight-binding model on a honeycomb lattice with zigzag boundaries and introducing a perturbative approach, we derive explicit expressions for the wave functions, energy dispersion relations, and the spin rotations of the (generic) helical edge states. To this end, we first map the edge states of the ribbon geometry into an effective two-leg ladder model with momentum-dependent energy parameters. Then, we split the Hamiltonian of the system into an unperturbed part and a perturbation. The unperturbed part has a flat-band energy spectrum and can be solved exactly, which allows us to consider the remaining part of the Hamiltonian perturbatively. The resulting energy dispersion relation within the first-order perturbation, surprisingly, is in excellent agreement with the numerical spectra over a very wide range of wave numbers. Our perturbative framework also allows deriving an explicit form for the rotation of the spins of the momentum edge states in the absence of axial spin symmetry due to the Rashba spin-orbit interaction.

Single-point dispersion measurement of surface waves combining translation, rotation and strain in weakly anisotropic media: theory

SUMMARY A new approach is proposed for measuring the dispersion relation of surface waves in weakly anisotropic media using a single station, which consists of translational displacement and rotation or strain. The azimuth-dependent surface wave dispersion curve can be directly obtained by using the amplitude ratio of translational displacement to rotation or strain. Using observations from earthquakes from a variety of azimuths allows us to characterize the anisotropy of subsurface media. The amplitude ratio gives the localized azimuth-dependent dispersion relations of the formation beneath a receiver without requiring knowledge of the source or structure along the path. The theory shows that in anisotropic media the coupled quasi-Rayleigh wave and quasi-Love wave will result in elliptically polarized rotational motions. In addition, rotational motion information allows the determination of backazimuth even in general anisotropic media. We carry out numerical experiments, investigate the effects of noise and degree of anisotropy and demonstrate the potential for field studies.

Study of the Relationship Between Sunspot Number and the Duration of the ≈ 1.6 – 2.2 Year Period in Neutron Monitor Counting Rates

Neutron monitor counting rates show periodicities in the ≈ 1.6 – 2.2-year range. These periodicities have been associated with a solar origin affecting the cosmic ray propagation conditions through the heliosphere. Our hypothesis is that the periodicities in the ≈ 1.6 – 2.2-years range correspond to a single periodicity that changes its duration over time.López-Comazzi and Blanco (Astrophys. J.927(2), 155, 2022) found that the duration of the ≈ 1.6 – 2.2-year period (tau ) is linearly related to the average sunspot number (SSN_{a}) in each solar cycle. The relationship shows that shorter ≈ 1.6 – 2.2-year periods occur during stronger cycles when SSN_{a} is higher. Therefore, the duration of this period varies from one solar cycle to another. This study focuses on this relation. For obtaining this relation, the values of the duration of the ≈ 1.6 – 2.2-year period in global neutron monitor counting rates (a virtual station determined by averaging of the different neutron monitor counting rates along the world) along the Solar Cycles 20 – 24 have been used. We extend the sample by adding the duration of the ≈ 1.6 – 2.2-year period in Huancayo neutron monitor counting rates along Solar Cycle 19 to this linear relationship. Once the linear relationship is extended, tau for the current Solar Cycle 25 is computed giving ≈ 2.24 years. Drawing on this more accurate relationship given by SSN_{a} = (-120 pm 10) : tau + (320 pm 20), we computed tau for the cycles previous to the existence of neutron monitors (Solar Cycles 7 – 18).These ≈ 1.6 – 2.2-year periodicities in neutron monitor counting rates could be produce by variations in the solar magnetic field due to an internal mechanism of the solar dynamo called Rossby waves. Concretely, the harmonic of fast Rossby waves with m=1 and n=8 fit with the detected ≈ 1.6 – 2.2-year periodicity. In addition, the variation of the solar magnetic-field strength from weaker to stronger solar cycles could explain the different periods detected in each cycle. Based on the detected periodicities using the dispersion relation for fast Rossby waves, a solar tachocline magnetic-field strength of ≈ 7 – 25 kG has been estimated.

Topographic Factor and Limit Transitions in the Equations for Subinertial Waves

In this paper, sub-inertial waves propagating on the Kuril shelf and the oceanic trench are considered. Against the background of a historical review of the beginning of the study of topographic waves and the appearance of relevant terms, a description of the features of wave propagation and the derivation of the main dispersion equations are given. We show that all variants of the topographic solutions presented in the article are basically based on the same dispersion relation: this is the dispersion relation for Rossby topographic waves. Two separate classes of localized solutions have been constructed: one is for the shelf, and the second, in fact, is also for the shelf, but which is commonly called trench waves. We demonstrate that the transverse wave number for trench waves is not independent, as for shelf waves, but is a function of the longitudinal wave number. In other words, Rossby topographic waves are two–dimensional waves, while shelf waves are quasi-one-dimensional solutions. The analytical novelty of the work consists of the fact that we can make crosslinking of trench and shelf waves. This fact was not presented in previous articles on this topic.

Phonon transmission through a nonlocal metamaterial slab

Previous theory and experiment has shown that introducing strong (nonlocal) beyond-nearest-neighbor interactions in addition to (local) nearest-neighbor interactions into rationally designed periodic lattices called metamaterials can lead to unusual wave dispersion relations of the lowest band. For roton-like dispersions, this especially includes the possibility of multiple solutions for the wavenumber at a given frequency. Here, we study the one-dimensional frequency-dependent acoustical phonon transmission of a slab of such nonlocal metamaterial in a local surrounding. In addition to the usual Fabry-Perot resonances, we find a series of bound states in the continuum. In their vicinity, sharp Fano-type transmission resonances occur, with sharp zero-transmission minima next to sharp transmission maxima. Our theoretical discussion starts with a discrete mass-and-spring model. We compare these results with solutions of a generalized wave equation for heterogeneous nonlocal effective media. We validate our findings by numerical calculations on three-dimensional metamaterial microstructures for one-dimensional acoustical wave propagation.

Evidence of experimental three-wave resonant interactions between two dispersion branches.

We report the observation of nonlinear three-wave resonant interactions between two different branches of the dispersion relation of hydrodynamic waves, namely the gravity-capillary and sloshing modes. These atypical interactions are investigated within a torus of fluid for which the sloshing mode can be easily excited. A triadic resonance instability is then observed due to this three-wave two-branch interaction mechanism. An exponential growth of the instability and phase locking are evidenced. The efficiency of this interaction is found to be maximal when the gravity-capillary phase velocity matches the group velocity of the sloshing mode. For a stronger forcing, additional waves are generated by a cascade of three-wave interactions populating the wave spectrum. Such a three-wave two-branch interaction mechanism is probably not restricted to hydrodynamics and could be of interest in other systems involving several propagation modes.

Analytical solutions for Bloch waves in resonant phononic crystals: deep-subwavelength energy splitting and mode steering between topologically protected interfacial and edge states

Summary We derive analytical solutions based on singular Green’s functions, which enable efficient computations of scattering simulations or Floquet–Bloch dispersion relations for waves propagating through an elastic plate, whose surface is patterned by periodic arrays of elastic beams. Our methodology is versatile and allows us to solve a range of problems regarding arrangements of multiple beams per primitive cell, over Bragg to deep-subwavelength scales; we cross-verify against finite element numerical simulations to gain further confidence in our approach, which relies upon the hypothesis of Euler–Bernoulli beam theory considerably simplifying continuity conditions such that each beam can be replaced by point forces and moments applied to the neutral plane of the plate. The representations of Green’s functions by Fourier series or Fourier transforms readily follows, yielding rapid and accurate analytical schemes. The accuracy and flexibility of our solutions are demonstrated by engineering topologically non-trivial states, from primitive cells with broken spatial symmetries, following the phononic analogue of the Quantum Valley Hall Effect. Topologically protected states are produced and coexist along: interfaces between adjoining chiral-mirrored bulk media, and edges between one such chiral bulk and the surrounding bare elastic plate, allowing topological circuits to be designed with robust waveguiding. Our topologically protected interfacial states correspond to zero-line modes, and our topological edgestates are produced in accordance with the bulk-edge correspondence. These topologically non-trivial states exist within near flexural resonances of the constituent beams of the phononic crystal and hence can be tuned into a deep-subwavelength regime.

Acoustic-gravity waves in quasi-isothermal atmospheres with a random vertical temperature profile

We study acoustic-gravity waves in a quasi-isothermal atmosphere in the presence of a weak random addition to the vertical temperature profile, which simulates the real atmosphere of the Earth at altitudes greater than ∼200 km. The resulting stochastic equation is closed in the Bourret approximation. The poles of the obtained mean Green’s function determine the generalized dispersion relation for acoustic-gravity waves. Two particular cases are considered: random inhomogeneities in the form of white noise (δ-correlated in space) and the opposite case of a δ-shaped noise spectrum. In both cases, instability of acoustic-gravity waves is predicted and the corresponding instability growth rates are determined.

Relativistic analysis of upper hybrid wave propagation and trapping

We investigate the impact of relativistic effects on upper hybrid (UH) waves in plasmas with thermal electrons, particularly focusing on modifications of the conditions under which UH wave trapping and related low-threshold parametric decay instabilities (PDIs) may occur. A moderately relativistic (MR) dispersion relation for UH waves, valid for electron temperatures up to 25 keV and wave frequencies up to twice the electron cyclotron frequency, is obtained from previous results and shown to reduce to the warm non-relativistic result commonly used for PDI studies at low electron temperatures. The conditions under which MR UH waves propagate are then determined and compared with warm and cold plasma theory, showing a general increase in the electron density and background magnetic field strength at which the UH resonance occurs for finite electron temperatures. We next investigate the impact of the MR corrections on the possibility of UH wave trapping for X-mode electron cyclotron resonance heated (ECRH) plasmas at the ASDEX Upgrade tokamak and scaled versions of the ASDEX Upgrade parameters with core electron temperatures resembling those expected in ITER X-mode ECRH plasmas. The MR UH wave trapping conditions are virtually unchanged for ASDEX Upgrade relative to warm theory, due to the low electron temperatures, while potentially important differences between warm and MR theory exist for ITER-like core electron temperatures; cold theory is found to be insufficient in both cases. Finally, the MR dispersion relation is shown to qualitatively reproduce the PDI thresholds from warm theory for previously studied ASDEX Upgrade cases.

Toward developing a comprehensive algorithm for solving kinetic plasma dispersion relations for parallel propagation with a kappa distribution

In general, it is challenging to numerically solve all the roots of plasma wave dispersion relations. The velocity distributions of multi-component particles in an anisotropic high-energy plasma can be better described by a drift loss-cone bi-Kappa distribution or a mixed drift loss-cone distribution containing bi-Kappa and bi-Maxwellian plasma in space and laboratories. In this work, we have developed a code with a new numerical algorithm to solve all roots of the kinetic dispersion relation for parallel propagation in hot magnetized plasmas with drift loss-cone bi-Kappa distribution. Solving all roots of the rational expansions of the kinetic dispersion relation is equivalent to a matrix eigenvalue problem of a linear system. We have performed detailed numerical solutions for three kinds of plasmas: bi-Maxwellian, bi-Kappa, and cold plasmas. We have also proposed a unified numerical method to solve the mixed dispersion relation based on the bi-Kappa and bi-Maxwellian distributions. The numerical results and benchmark studies demonstrate that the new algorithm is in agreement with the data from previous studies. This is a crucial step toward revealing a full picture of kinetic plasma waves and instabilities.

Excitation of surface waves in 3D ion beam neutralization

Neutralization of beams with 2D and 3D geometries by the electrons emitted from an external source is studied using particle-in-cell simulations. Our work reveals that the high-energy electrons excite Trivelpiece–Gould (TG) surface waves in the beams with 3D axisymmetric geometries. These high-energy electrons are generated because of a large amplitude electrostatic solitary wave (ESW) that forms near the electron source and has an electric potential amplitude more than three times the electron thermal energy. We also find that surface wave excitation only happens when the beam radius is large enough at the ion source to attract enough electrons that could form the large amplitude ESW. A comparison of the 3D TG surface wave dispersion relation with an expression for 2D surface waves reveals that they become excited in 3D axisymmetric but not in 2D planar beam because of a higher phase speed requirement in the latter case.

Collective effect of nonthermal and suprathermal particles on electrostatic waves and instabilities in Vasyliunas-Cairns distributed plasmas

The combined effects of nonthermal and suprathermal particles on electrostatic modes of waves propagating in an electron-ion plasma are investigated. Vasyliunas-Cairns distribution with nonthermal and suprathermal spectral indices is used to describe the electron-ion plasma under consideration. The dispersion relation of Langmuir and ion-acoustic waves are examined using the kinetic theory approach. The population of nonthermal and suprathermal particles is found to have a considerable influence on the real wave frequency and damping rate of both Langmuir and ion-acoustic waves. The suprathermal particle population stabilizes the wave whereas the nonthermal particle population destabilizes the wave.

Rayleigh waves in nonlocal generalized thermoelastic media

PurposeThe purpose of this article is to investigate the propagation characteristics (such as particle motion, attenuation and phase velocity) of a Rayleigh wave in a nonlocal generalized thermoelastic media.Design/methodology/approachThe bulk waves are represented with Helmholtz potentials. The stress-free insulated and isothermal plane surfaces are taken into account. Rayleigh wave dispersion relation has been established and is found to be complex. Due to the presence of radicals, the dispersion equation is continuously computed as a complicated irrational expression. The dispersion equation is then converted into a polynomial equation that can be solved numerically for precise complex roots. The extra zeros in this polynomial equation are eliminated to yield the dispersion equation’s roots. These routes are then filtered for inhomogeneous wave propagation that decays with depth. To perform numerical computations, MATLAB software is used.FindingsIn this medium, only one mode of Rayleigh wave exists at both isothermal and insulated boundaries. The thermal factors of nonlocal generalized thermoelastic materials significantly influence the particle motion, attenuation and phase velocity of the Rayleigh wave.Originality/valueNumerical examples are taken to examine how the thermal characteristics of materials affect the existing Rayleigh wave’s propagation characteristics. Graphical analysis is used to evaluate the behavior of particle motion (such as elliptical) both inside and at the isothermal (or insulated) flat surface of the medium under consideration.

Rossby waves on stellar equatorial β planes: Uniformly rotating radiative stars

Context. Rossby waves arise due to the conservation of total vorticity in rotating fluids and may govern the large-scale dynamics of stellar interiors. Recent space missions have collected a lot of information about the light curves and activity of many stars, which triggered observations of Rossby waves in the stellar surface and interiors. Aims. We aim to study the theoretical properties of Rossby waves in stratified interiors of uniformly rotating radiative stars with a sub-adiabatic vertical temperature gradient. Methods. We used the equatorial β plane approximation and linear vertical gradient of temperature to study the linear dynamics of equatorially trapped Rossby and inertia-gravity waves in interiors of radiative stars. The governing equation was solved by the method of separation of variables in the vertical and latitudinal directions. Results. Vertical and latitudinal solutions of the waves are found to be governed by Bessel functions and Hermite polynomials, respectively. Appropriate boundary conditions at the stellar surface and poles define analytical dispersion relations for Rossby, Rossby-gravity, and inertia-gravity waves. The waves are confined in surface layers of 30–50 H0, where H0 is the surface density scale height, and they are trapped between the latitudes of ±600. Observable frequencies (normalised by the angular frequency of the stellar rotation) of Rossby waves with m = 1 (m = 2), where m is the toroidal wavenumber, are in the interval of 0.65–1 (1.4–2), depending on the stellar rotation, radius, and surface temperature. Conclusions. Rossby-type waves can be systematically observed using light curves of Kepler and TESS (Transiting Exoplanet Survey Satellite) stars. Observations and theory then can be used for the sounding of stellar interiors.

Characterization of Gravity Waves in Three Dimensions in the Daytime Thermosphere Using Combined Optical and Radio Measurements and Estimation of Horizontal Neutral Winds

AbstractGravity waves, which are considered omnipresent in the Earth's upper atmosphere, are generally investigated by monitoring the fluctuations in different atmospheric parameters. Here, we report the propagation characteristics of thermospheric gravity waves both in horizontal and vertical directions obtained using collocated optical and radio measurements from Ahmedabad, India for February 2021. The measurements of OI 630.0 nm dayglow emission rates over zenith are used to derive time periods of gravity waves. Wave number analyses of variations in the emission over a large field‐of‐view have been performed to derive gravity wave scale sizes and propagation characteristics in the horizontal direction. Time periods, horizontal scale sizes, and propagation directions are found to be in the range of 31–125 min, 78–243 km, and 203°–248° from east, respectively. Vertical wavelengths of the gravity waves are obtained from radio measurements and are in the range of 26–247 km. As the gravity wave characteristics are influenced by the ambient neutral winds, the measured gravity wave propagation characteristics in three dimensions have been used as inputs into the gravity wave dispersion relation to estimate the magnitudes of thermospheric horizontal neutral winds. These estimated daytime winds in the direction of wave propagation are found to be in the range of 1–105 ms−1, and they compare well with those measured independently from MIGHTI added HWM14 model‐derived winds. The daytime winds estimated by this approach are possibly the first of their kind as obtained from ground‐based measurements.