Dispersion Relation For Linear Waves Research Articles

Asymptotic integrability of nonlinear wave equations.

We introduce the notion of asymptotic integrability into the theory of nonlinear wave equations. It means that the Hamiltonian structure of equations describing propagation of high-frequency wave packets is preserved by hydrodynamic evolution of the large-scale background wave so that these equations have an additional integral of motion. This condition is expressed mathematically as a system of equations for the carrier wave number as a function of the background variables. We show that a solution of this system for a given dispersion relation of linear waves is related to the quasiclassical limit of the Lax pair for the completely integrable equation having the corresponding dispersionless and linear dispersive behavior. We illustrate the theory with several examples.

Whitham modulation theory and Riemann problem for the Kundu–Eckhaus equation

In this paper, the Riemann problem for the defocusing Kundu–Eckhaus equation is investigated by Whitham modulation theory. First, we study the dispersion relation for linear waves. Then, the zero-phase and one-phase periodic solutions of the Kundu–Eckhaus equation along with the corresponding Whitham modulation equations are derived by the finite-gap integration method. Further, employing the Whitham equations parametrized by the Riemann invariants, the main fundamental wave structures induced by the discontinuous initial data are found. Analytical and graphic methods are utilized to provide the wave structures of rarefaction waves and dispersive shock waves, and thus for a complete classification of solutions under general step-like conditions of initial discontinuity.

On a stabilization of the Ingard-Myers impedance boundary condition and its time domain implementation

It has been well-known that the Ingard-Myers impedance condition, while simple to apply, is subject to the hydrodynamic Kelvin-Helmholtz-type instability due to its use of a vortex sheet in modeling the flow at the liner boundary. Recently, in the development of a time domain boundary element method for acoustic scattering by treated surfaces, it was found that by neglecting a certain second-order spatial derivative term in the Ingard-Myers formulation, the hydrodynamic instability can be avoided. The present paper aims to provide further analysis of this modified condition, hereby referred to as the Truncated Ingard-Myers Impedance Boundary Condition (TIMIBC). It will be shown, based on the dispersion relations of linear waves, that the instability intrinsic to the Ingard-Myers condition is eliminated in the proposed new formulation. Quantitative assessments on the accuracy of the TIMIBC for scattering of acoustic waves by lined surfaces are carried out, and its effectiveness is demonstrated by a numerical example. It is found that the TIMIBC provides a good approximation to the original Ingard-Myers condition for flows of low to mid subsonic Mach numbers. As such, the proposed TIMIBC can offer a practical solution for overcoming the intrinsic instability associated with the Ingard-Myers condition. Moreover, time domain implementation of the TIMIBC is also discussed and illustrated with a numerical example using a finite difference scheme. In particular, a minimization procedure for finding the poles and coefficients of a broadband multipole expansion for the impedance function is formulated by which, unlike the commonly used vector-fitting method, passivity of the model is ensured.

Effect of dust size distribution and nonadiabatic charge variation on dispersion relation for linear waves in inhomogeneous complex plasmas

This paper explores linear waves’ instability in nonuniform complex plasmas with dust size distribution and non-adiabatic dust charge fluctuation. For this purpose, the dispersion relation is obtained with the aid of linearizing the hydrodynamic equations governing the properties of linear waves. The contribution of physical parameters such as inhomogeneity, nonadiabatic dust charge fluctuation, and dust size distribution to the features of dust acoustic waves is examined at great length. The numerical results show that the present research results are very important in the phenomenon of space astrophysical and experimental plasmas.

NON-LINEAR DISPERSION EFFECTS IN NEARSHORE WAVES: PERSPECTIVES FOR DEPTH-INVERSION APPLICATIONS

Remote-sensing technology, combined with depth-inversion algorithms, presents a promising opportunity to quantify the morphological evolution of sandy beaches during storms. Current depth-inversion algorithms such as cBathy rely on the linear wave dispersion relation to invert depth from remotely-sensed dispersion properties. In the surf zone, however, non-linear amplitude dispersion effects become important and significant deviations from the linear dispersion are expected (Martins et al., 2021). Optical imagery also suffers from other technical limitations when dealing with breaking waves (e.g., saturated pixels and phase shifts), which affects the stability and accuracy of remotely-sensed wave dispersive properties. These limitations explain the poor accuracy reached with common approaches in the surf zone (typically 100 percent error). This contribution paves the way to new depth-inversion algorithms suited to the surf zone, based on remotely-sensed free surface elevation data (e.g., by lidar scanners). We here report on the first efforts to account for non-linear amplitude dispersion effects.

Bright and dark solitons in the systems with strong light-matter coupling: Exact solutions and numerical simulations.

We theoretically study bright and dark solitons in an experimentally relevant hybrid system characterized by strong light-matter coupling. We find that the corresponding two-component model supports a variety of coexisting moving solitons including bright solitons on zero and nonzero background and dark-gray and gray-gray solitons. The solutions are found in the analytical form by reducing the two-component problem to a single stationary equationwith cubic-quintic nonlinearity. All found solutions coexist under the same set of the model parameters, but, in a properly defined linear limit, approach different branches of the polariton dispersion relation for linear waves. Bright solitons with zero background feature an oscillatory-instability threshold which can be associated with a resonance between the edges of the continuous spectrum branches. "Half-topological" dark-gray and nontopological gray-gray solitons are stable in wide parametric ranges below the modulational instability threshold, while bright solitons on the constant-amplitude pedestal are unstable.

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.

Relaxed magnetohydrodynamics with ideal Ohm's law constraint

The gap between a recently developed dynamical version of relaxed magnetohydrodynamics (RxMHD) and ideal MHD (IMHD) is bridged by approximating the zero-resistivity ‘ideal’ Ohm's law (IOL) constraint using an augmented Lagrangian method borrowed from optimization theory. The augmentation combines a pointwise vector Lagrange multiplier method and global penalty function method and can be used either for iterative enforcement of the IOL to arbitrary accuracy, or for constructing a continuous sequence of magnetofluid dynamics models running between RxMHD (no IOL) and weak IMHD (IOL almost everywhere). This is illustrated by deriving dispersion relations for linear waves on an MHD equilibrium.

Relation between Orbital Velocities, Pressure, and Surface Elevation in Nonlinear Nearshore Water Waves

AbstractThe inability of the linear wave dispersion relation to characterize the dispersive properties of nonlinear shoaling and breaking waves in the nearshore has long been recognized. Yet, it remains widely used with linear wave theory to convert between subsurface pressure, wave orbital velocities, and the free surface elevation associated with nonlinear nearshore waves. Here, we present a nonlinear fully dispersive method for reconstructing the free surface elevation from subsurface hydrodynamic measurements. This reconstruction requires knowledge of the dispersive properties of the wave field through the dominant wavenumbers magnitude κ, representative in an energy-averaged sense of a mixed sea state composed of both free and forced components. The present approach is effective starting from intermediate water depths—where nonlinear interactions between triads intensify—up to the surf zone, where most wave components are forced and travel approximately at the speed of nondispersive shallow-water waves. In laboratory conditions, where measurements of κ are available, the nonlinear fully dispersive method successfully reconstructs sea surface energy levels at high frequencies in diverse nonlinear and dispersive conditions. In the field, we investigate the potential of a reconstruction that uses a Boussinesq approximation of κ, since such measurements are generally lacking. Overall, the proposed approach offers great potential for collecting more accurate measurements under storm conditions, both in terms of sea surface energy levels at high frequencies and wave-by-wave statistics (e.g., wave extrema). Through its control on the efficiency of nonlinear energy transfers between triads, the spectral bandwidth is shown to greatly influence nonlinear effects in the transfer functions between subsurface hydrodynamics and the sea surface elevation.

Relaxation in a Fuzzy Dark Matter Halo. II. Self-consistent Kinetic Equations

Abstract Fuzzy dark matter (FDM) is composed of ultra-light bosons having a de Broglie wavelength that is comparable to the size of the stellar component of galaxies at typical galactic velocities. FDM behaves like cold dark matter on large scales. However, on the scale of the de Broglie wavelength, an FDM halo exhibits density fluctuations that lead to relaxation, a process similar to the two-body relaxation that occurs in classical gravitational N-body systems and is described by the Fokker–Planck equation. We derive the FDM analog of that kinetic equation, which describes the evolution of the velocity distribution in a spatially homogeneous FDM halo. We show that the evolution of the velocity distribution predicted by our kinetic equation matches numerical solutions of the coupled Schrödinger–Poisson equations. We also determine the dielectric function and the dispersion relation for linear waves in an FDM halo.

Dispersive characteristics of non-linear waves propagating and breaking over a mildly sloping laboratory beach

The dispersive characteristics of unidirectional irregular waves propagating and breaking over a mildly sloping beach are examined using a highly-resolved laboratory dataset. Cross-spectral analyses are used to determine the cross-shore evolution of (single-valued) dominant wavenumber κ and phase velocity c spectra, and lead to the identification of four different regimes of propagation: I - a linear regime where short waves mostly propagate as free components; II - a shoaling regime where non-linear effects at high harmonics are significant but primary components follow the linear wave dispersion relation; III - a shoaling regime near the mean breaking point location, where amplitude dispersion effects at primary components are important; IV - a surf zone regime, where all components propagate slightly faster than non-dispersive shallow water waves. Bispectral analyses performed onshore of the shoaling region show that the presence of forced energy at high harmonics, which originate from non-linear interactions between triads of frequencies, are responsible for the deviations of wavenumber and phase velocity spectral estimates from predictions by the linear dispersion relation, confirming the findings from previous field-based studies. A Boussinesq approximation of the non-linear energy exchanges between triads is then used to quantify the relative amount of forced energy at high harmonics and explain the differences in dispersion properties observed in the shoaling region between broad and narrow-band spectra. Larger relative amounts of forced energy at high frequencies, which suggest more efficient non-linear energy transfers, are found to be associated with larger deviations of dominant κ and c from predictions by the linear dispersion relation.

Characteristic velocity and dispersion relation of linear waves in extended thermodynamics of a van der Waals gas

We study the characteristic velocities of the extended thermodynamics of molecular rotational- and vibrational- relaxation processes for a van der Waals gas. The lower and upper bounds of the characteristic velocity are, respectively, estimated by the one of rarefied gases and the one evaluated on the spinodal curve. We also study the dispersion relation of linear waves, in particular, the phase velocity, attenuation factor and attenuation per wavelength in a low frequency region.

High-resolution bathymetry estimates via X-band marine radar: 2. Effects of currents at tidal inlets

In this second of two papers, the high spatial resolution cBathy depth estimate algorithm is extended to account for the existence of currents aligned with the wave directions. The extended algorithm is applied using X-band radar image time series collected at two hydrodynamically and morphologically complex environments: an ebb-tidal shoal (New River Inlet, NC) and an estuarine river mouth (Columbia River, OR/WA). Even without accounting for Doppler effects, the depth estimates are capable of discerning the location of channels and shoals over domain sizes of 4km2 at the ebb-tidal shoal and 25km2 at the estuary mouth. However, the tidal currents at these sites are shown to strongly modulate depth estimate accuracy when only the intrinsic linear wave dispersion relation (neglecting Doppler shift effects) is inverted. Low-pass filtering of the depth estimate time series, via an uncertainty-weighted tidal average, explicitly attenuates the error oscillations beyond that of the uncertainty-weighted Kalman filter. This post-hoc method of compensating for the unmodeled Doppler term achieves 0.35m RMSE and 0.02m shallow bias at the shallow ebb tidal shoal as compared to a concurrent sounding survey, but substantial bias remains at the estuary mouth due to asymmetric Doppler effects for large following or opposing currents at large relative water depths. Simultaneous estimation of depth and currents via inversion of the dispersion relation including Doppler shifts is more effective at the deep estuary mouth, where currents more strongly affect the incident wave field. In the estuary channel with locations as deep as 27m, Kalman-filtered depths achieve 0.92m RMSE and 0.38m deep bias as compared to a concurrent sounding survey (95% of estimates are within 10% of those surveyed). Independent of the Doppler compensation technique, Kalman filtered depths can be used to more tightly constrain high-resolution current velocity estimates.

Waves in porous media saturated with bubbly liquid

Wave processes in a porous medium saturated with bubbly liquid, taking into account the nonlinearity of the oscillations of the bubbles, are numerically investigated. The nonlinearity is accounted in the equation of gas state and the equation of Rayleigh - Lamb. Numerical study in the framework of nonlinear model is performed for the first time. The dispersion relations for linear waves in a porous medium containing bubbly liquid, as well as for the main acoustic mode in a cylindrical cavity in such a medium, are obtained. Frequency dependences of velocity and attenuation of harmonic waves in a cylindrical waveguide are calculated. The influence properties of the bubbly liquid saturated porous medium on the wave velocity and attenuation and the penetration depth of perturbation into the surrounding porous medium are analyzed. The passage of the step wave from fluid into a porous medium saturated with bubbly liquid is investigated. The influence of porous medium parameters and intensity of the incident wave on the wave evolution in bubbly liquid saturated porous medium is studied. The calculation results show good qualitative agreement with the experimental data of other authors.

Dispersion Relation of Linear Waves in Quantum Magnetoplasmas**supported by National Natural Science Foundation of China (No. 11447125) and the Research Training Program for Undergraduates of Shanxi University of China (Nos. 2014012167, 2015013182)

The quantum magnetohydrodynamic (QMHD) model is applied in investigating the propagation of linear waves in quantum magnetoplasmas. Using the QMHD model, the dispersion equation for quantum magnetoplasmas and the dispersion relations of linear waves are deduced. Results show that quantum effects affect the propagation of electron plasma waves and extraordinary waves (X waves). When we select the plasma parameters of the laser-based plasma compression (LBPC) schemes for calculation, the quantum correction cannot be neglected. Meanwhile, the corrections produced by the Fermi degeneracy pressure and Bohm potential are compared under different plasma parameter conditions.

GLOBAL MHD SIMULATIONS OF ACCRETION DISKS IN CATACLYSMIC VARIABLES. I. THE IMPORTANCE OF SPIRAL SHOCKS

ABSTRACT We present results from the first global 3D MHD simulations of accretion disks in cataclysmic variable (CV) systems in order to investigate the relative importance of angular momentum transport via turbulence driven by the magnetorotational instability (MRI) compared with that driven by spiral shock waves. Remarkably, we find that even with vigorous MRI turbulence, spiral shocks are an important component of the overall angular momentum budget, at least when temperatures in the disk are high (so that Mach numbers are low). In order to understand the excitation, propagation, and damping of spiral density waves in our simulations more carefully, we perform a series of 2D global hydrodynamical simulations with various equation of states, both with and without mass inflow via the Lagrangian point (L1). Compared with previous similar studies, we find the following new results. (1) The linear wave dispersion relation fits the pitch angles of spiral density waves very well. (2) We demonstrate explicitly that mass accretion is driven by the deposition of negative angular momentum carried by the waves when they dissipate in shocks. (3) Using Reynolds stress scaled by gas pressure to represent the effective angular momentum transport rate is not accurate when mass accretion is driven by non-axisymmetric shocks. (4) Using the mass accretion rate measured in our simulations to directly measure α defined in standard thin-disk theory, we find for CV disks, consistent with observed values in quiescent states of dwarf novae. In this regime, the disk may be too cool and neutral for the MRI to operate and spiral shocks are a possible accretion mechanism. However, we caution that our simulations use unrealistically low Mach numbers in this regime and, therefore, future models with more realistic thermodynamics and non-ideal MHD are warranted.

Water Depth Inversion From a Single SPOT-5 Dataset

Knowing bathymetry at intermediate depth, over large areas, and at a reasonable cost is a key issue. Spaceborne remote sensing techniques must play an essential role in retrieving such bathymetry. In this paper, a method is proposed that aims to address this issue without any in situ measurements by exploiting the characteristics of the SPOT-5 satellite dataset. The proposed method is designed to provide bathymetry from two optical SPOT-5 satellite images separated by a time lag $DT$ of 2.04 s. It relies on the estimation of several clouds of wave celerity and wavelength pairs using wavelet and cross-correlation techniques and on the linear wave dispersion relation. This method has been applied to two SPOT-5 images on a test site characterized by complex bathymetry (Saint-Pierre, La Reunion Island). A comparison of the retrieved bathymetry with in situ bathymetric measurements reveals good morphological agreement. The mean relative error is less than 30% in the 3–80-m water depth range. The methodological choices made during method development are discussed based on additional computations, and guidelines for using the proposed method on other images at other sites are provided.

Nonlinear traveling waves for the skeleton of the Madden–Julian oscillation

The Madden{Julian Oscillation (MJO) is the dominant component of intraseasonal (30{90 days) variability in the tropical atmosphere. Here, traveling wave solutions are presented for the MJO skeleton model of Majda and Stechmann. The model is a system of nonlinear partial dierential equations that describe the evolution of the tropical atmosphere on planetary (10,000{ 40,000 km) spatial scales. The nonlinear traveling waves come in four types, corresponding to the four types of linear wave solutions, one of which has the properties of the MJO. In the MJO traveling wave, the convective activity has a pulse-like shape, with a narrow region of enhanced convection and a wide region of suppressed convection. Furthermore, an amplitude-dependent dispersion relation is derived, and it shows that the nonlinear MJO has a lower frequency and slower propagation speed than the linear MJO. By taking the small-amplitude limit, an analytic formula is also derived for the dispersion relation of linear waves. To derive all of these results, a key aspect is the model's conservation of energy, which holds even in the presence of forcing. In the limit of weak forcing, it is shown that the nonlinear traveling waves have a simple sech-squared waveform.

A new statistical model of wave heights based on the concept of wave breaking critical zone

When waves propagate from deep water to shallow water, wave heights and steepness increase and then waves roll back and break. This phenomenon is called surf. Currently, the present statistical calculation model of surf was derived mainly from the wave energy conservation equation and the linear wave dispersion relation, but it cannot reflect accurately the process which is a rapid increasing in wave height near the broken point. So, the concept of a surf breaking critical zone is presented. And the nearshore is divided as deep water zone, shallow water zone, surf breaking critical zone and after breaking zone. Besides, the calculation formula for the height of the surf breaking critical zone has founded based on flume experiments, thereby a new statistical calculation model on the surf has been established. Using the new model, the calculation error of wave height maximum is reduced from 17.62% to 6.43%.

Arbitrary Amplitude Ion Acoustic Solitary Waves in An Unmagnetized Two Electron Population Ultra-Relativistic Dense Plasmas

The nonlinear wave structure of arbitrary amplitude ion acoustic solitary waves (IASWs) are studied in the Sagdeev’s pseudopotential framework for an ultra-relativistic degenerate dense plasma comprising cold and hot electrons and inertial ultra-cold ions. By employing standard normal-mode analysis the dispersion relation for linear waves is studied. The numerical results are presented to understand the features of ion acoustic solitary wave structures. It is shown that the present plasma model supports IASWs having positive potential well. Also, it is found that the small amplitude rarefactive double layer solution can exist in such a plasma system in some parametric region. It is shown that solitary structures and double layers are affected by relevant plasma parameters.