Long-wavelength Waves Research Articles

A hybrid adaptive low-Mach number/compressible method: Euler equations

Flows in which the primary features of interest do not rely on high-frequency acoustic effects, but in which long-wavelength acoustics play a nontrivial role, present a computational challenge. Integrating the entire domain with low-Mach-number methods would remove all acoustic wave propagation, while integrating the entire domain with the fully compressible equations can in some cases be prohibitively expensive due to the CFL time step constraint. For example, simulation of thermoacoustic instabilities might require fine resolution of the fluid/chemistry interaction but not require fine resolution of acoustic effects, yet one does not want to neglect the long-wavelength wave propagation and its interaction with the larger domain.The present paper introduces a new multi-level hybrid algorithm to address these types of phenomena. In this new approach, the fully compressible Euler equations are solved on the entire domain, potentially with local refinement, while their low-Mach-number counterparts are solved on subregions of the domain with higher spatial resolution. The finest of the compressible levels communicates inhomogeneous divergence constraints to the coarsest of the low-Mach-number levels, allowing the low-Mach-number levels to retain the long-wavelength acoustics. The performance of the hybrid method is shown for a series of test cases, including results from a simulation of the aeroacoustic propagation generated from a Kelvin–Helmholtz instability in low-Mach-number mixing layers. It is demonstrated that compared to a purely compressible approach, the hybrid method allows time-steps two orders of magnitude larger at the finest level, leading to an overall reduction of the computational time by a factor of 8.

Pulsar timing perturbations from Galactic gravitational wave bursts with memory

Pulsar timing arrays (PTAs) are used to search for long-wavelength gravitational waves (GWs) by monitoring a set of spin-stable millisecond pulsars. Most theoretical analyses assume that the relevant GW sources are much more distant from Earth than the pulsars comprising the array. Unlike ground- or solar system-based GW detectors, PTAs might well contain embedded GW sources. We derive the PTA response from sources at any distance, with a specific focus on GW bursts with memory (BWMs). We consider supernovae and compact binary mergers as potential Galactic BWM sources and evaluate the signature for an array with pulsars in globular clusters or in the Galactic center. Understanding the response of PTAs to nearby sources of BWM is a step towards investigating other more complex Galactic sources.

Characteristics of solitary waves, quasiperiodic solutions, homoclinic breather solutions and rogue waves in the generalized variable-coefficient forced Kadomtsev–Petviashvili equation

In this paper, the generalized variable-coefficient forced Kadomtsev–Petviashvili (gvcfKP) equation is investigated, which can be used to characterize the water waves of long wavelength relating to nonlinear restoring forces. Using a dependent variable transformation and combining the Bell’s polynomials, we accurately derive the bilinear expression for the gvcfKP equation. By virtue of bilinear expression, its solitary waves are computed in a very direct method. By using the Riemann theta function, we derive the quasiperiodic solutions for the equation under some limitation factors. Besides, an effective way can be used to calculate its homoclinic breather waves and rogue waves, respectively, by using an extended homoclinic test function. We hope that our results can help enrich the dynamical behavior of the nonlinear wave equations with variable-coefficient.

Time-dependent Tonks-Langmuir model is unstable.

We investigate a time-dependent extension of the Tonks-Langmuir model for a one-dimensional plasma discharge with collisionless kinetic ions and Boltzmann electrons. Ions are created uniformly throughout the volume and flow from the center of the discharge to the boundary wall due to a self-consistent, zero-order electric field. Solving this model using a particle-in-cell simulation, we observe coherent low-frequency, long-wavelength unstable ion waves which move toward the boundary with a speed below both the ion acoustic speed and the average ion velocity. The maximum amplitude of the wave potential fluctuations peaks at ≈0.09T_{e} near the wall, where T_{e} is the electron temperature in electron volts. Using linear kinetic theory, we identify this instability as slow ion-acoustic wave modes which are destabilized by the zero-order electric field.

Gravitational waves and core-collapse supernovae

A mechanism of formation of gravitational waves in the Universe is considered for a nonspherical collapse of matter. Nonspherical collapse results are presented for a uniform spheroid of dust and a finite-entropy spheroid. Numerical simulation results on core-collapse supernova explosions are presented for the neutrino and magneto-rotational models. These results are used to estimate the dimensionless amplitude of the gravitational wave with a frequency ν ~ 1300 Hz, radiated during the collapse of the rotating core of a pre-supernova with a mass of 1.2 M⊙ (calculated by the authors in 2D). This estimate agrees well with many other calculations (presented in this paper) that have been done in 2D and 3D settings and which rely on more exact and sophisticated calculations of the gravitational wave amplitude. The formation of the large-scale structure of the Universe in the Zel’dovich pancake model involves the emission of very long-wavelength gravitational waves. The average amplitude of these waves is calculated from the simulation, in the uniform spheroid approximation, of the nonspherical collapse of noncollisional dust matter, which imitates dark matter. It is noted that a gravitational wave radiated during a core-collapse supernova explosion in our Galaxy has a sufficient amplitude to be detected by existing gravitational wave telescopes.

Damping properties of plasmonic waves on graphene

By considering the friction force due to the interaction of plasmonic waves and graphene lattice, the damping properties (lifetime and propagation length) of long-wavelength plasmonic waves on a monolayer graphene are studied by means of a perturbative method. Electronic excitations on the graphene surface are modeled by an infinitesimally thin layer of massless electron gas, which is described by means of the linearized hydrodynamic theory. The analytical expressions for the frequency dependence of damping function, the propagation length and the lifetime of long-wavelength surface waves on graphene with small intrinsic damping are derived and analyzed. Also, simple expressions for the stored and dissipated energy densities of the surface waves are presented.

Anomalous Behavior of Ultra-Low-Amplitude Capillary Waves. A Glimpse of the Viscoelastic Properties of Interfacial Water?

We investigate, both theoretically and by a differential interferometric technique, the behavior of large-wavelength capillary waves (of the order of 10-4 m) selectively excited at the surface of drops and bubbles with typical eigenfrequencies of the order of 102 Hz. The resonance peaks of gas bubbles or hydrocarbon drops in water (radius less than 1 mm) highlight anomalously small dissipation in the region of ultralow (sub-nanometric) oscillation amplitudes, reaching a plateau at higher amplitudes. This is in sharp contrast to the usual oscillating systems, where an anomalous behavior holds at large amplitudes alone. Dissipation is strongly dependent on the excited vibrational modes and, in spite of remarkable numerical differences, water-vapor and water-hydrocarbon interfaces exhibit the same overall trend. A phenomenological model was developed, based on the assumption that water possesses a threshold viscoelasticity, above which it behaves like a regular viscous fluid. The well-known Deborah number was then estimated within the anomalous region and found to lie in the range of viscoelastic fluids. In agreement with previous studies of nanohydrodynamics (e.g., atomic force microscopy measurements with sub-nanometric tip motions), the present one lends support to the idea that every self-aggregating fluid exhibits yield stress behavior, including classical Newtonian fluids like water. The essential requirement is that the applied perturbation lie below a critical threshold, above which viscous behavior is recovered. Our differential interferometric technique seems particularly suitable for this type of studies, as it allows measurement of long-wavelength capillary waves with sub-nanometric resolution on the oscillation amplitudes.

A mathematical description for the scattering phenomena of plane wave from elliptical plasma antenna located in oblique static magnetic field

The problem of electromagnetic wave scattering from an elliptical plasma cylinder in the presence of an external oblique magnetic field is investigated. The electromagnetic waves are landed obliquely on the plasma column. Knowing the dielectric permittivity tensor of the plasma column, the electric potential and the electric field inside and outside the plasma column are obtained for the long-wavelength waves. The graphs of the electric field profile and pattern scattering are presented. Also, the dependence of those graphs on the incident angle, the geometrical dimensions of the plasma column and the magnetic field angle are analyzed. The physical justifications based on the theory of Fresnel’s transmission coefficients for describing the graphs, have been presented. Moreover, the effective factors on the shift of the resonance frequency are investigated. Finally, to verify the accuracy of the obtained results, some limiting cases are discussed.

Scaling laws of wave-cascading superfluid turbulence

Phenomenological model is constructed for superfluid turbulence for two distinct energy cascade scenarios, sound wave cascade and critically-balanced Kelvin wave cascade, using the method for magneto-fluid turbulence theory. Excitations along dispersion relations are used as the primary energy reservoir. The spectral indices in the inertial range are estimated as −3/2 for the long-wavelength sound wave cascade, −3 in the direction to the mean filaments for the Kelvin wave cascade, and −5/3 perpendicular to the filament direction.

Solitary wave solutions for the generalized Zakharov–Kuznetsov–Benjamin–Bona–Mahony nonlinear evolution equation

In this paper, we utilize the exp(−φ(ξ))-expansion method to find exact and solitary wave solutions of the generalized Zakharov–Kuznetsov–Benjamin–Bona–Mahony nonlinear evolution equation. The generalized Zakharov–Kuznetsov–Benjamin–Bona–Mahony nonlinear evolution equation describes the model for the propagation of long waves that mingle with nonlinear and dissipative impact. This model is used in the analysis of the surface waves of long wavelength in hydro magnetic waves in cold plasma, liquids, acoustic waves in harmonic crystals and acoustic–gravity waves in compressible fluids. By using this method, seven different kinds of traveling wave solutions are successfully obtained for this model. The considered method and transformation techniques are efficient and consistent for solving nonlinear evolution equations and obtain exact solutions that are applied to the science and engineering fields.

Wave propagation in elastic and damped structures with stabilized negative-stiffness components

Effects on wave propagation achievable by introduction of a negative-stiffness component are investigated via perhaps the simplest discrete repeating element that can remain stable in the component’s presence. When the system is elastic, appropriate tuning of the stabilized component’s negative stiffness introduces a no-pass zone theoretically extending from zero to an arbitrarily high frequency, tunable by a mass ratio adjustment. When the negative-stiffness component is tuned to the system’s stability limit and a mass ratio is sufficiently small, the system restricts propagation to waves of approximately a single arbitrary frequency, adjustable by tuning the stiffness ratio of the positive-stiffness components. The elastic system’s general solutions are closed-form and transparent. When damping is added, the general solutions are still closed-form, but so complex that they do not clearly display how the negative stiffness component affects the system’s response and how it should best be tuned to achieve desired effects. Approximate solutions having these features are obtained via four perturbation analyses: one for long wavelengths; one for small damping; and two for small mass ratios. The long-wavelengths solution shows that appropriate tuning of the negative-stiffness component can prevent propagation of long-wavelength waves. The small damping solution shows that the zero-damping low-frequency no-pass zone remains, while waves that do propagate are highly damped when a mass ratio is made small. Finally, very interesting effects are achievable at the full system’s stability limit. For small mass ratios, the wavelength range of waves prohibited from propagation can be adjusted, from all to none, by tuning the system’s damping: When one mass ratio is small, all waves with wavelengths larger than an arbitrary damping-adjusted value can be prohibited from propagation, while when the inverse of this mass ratio is small, all waves with wavelengths outside an arbitrary single adjustable value or range of values can be prohibited from propagation. All of the approximate solutions’ analytically-transparent predictions are confirmed by the exact solution. The conclusions are that a stabilized tuned negative-stiffness component greatly enhances control of wave propagation in a purely elastic system, and when adjustable damping is added, even further control is facilitated.

Convective wave breaking in the KdV equation

The KdV equation is a model equation for waves at the surface of an inviscid incompressible fluid, and it is well known that the equation describes the evolution of unidirectional waves of small amplitude and long wavelength fairly accurately if the waves fall into the Boussinesq regime. The KdV equation allows a balance of nonlinear steepening effects and dispersive spreading which leads to the formation of steady wave profiles in the form of solitary waves and cnoidal waves. While these wave profiles are solutions of the KdV equation for any amplitude, it is shown here that there for both the solitary and the cnoidal waves, there are critical amplitudes for which the horizontal component of the particle velocity matches the phase velocity of the wave. Solitary or cnoidal solutions of the KdV equation which surpass these amplitudes feature incipient wave breaking as the particle velocity exceeds the phase velocity near the crest of the wave, and the model breaks down due to violation of the kinematic surface boundary condition. The condition for breaking can be conveniently formulated as a convective breaking criterion based on the local Froude number at the wave crest. This breaking criterion can also be applied to time-dependent situations, and one case of interest is the development of an undular bore created by an influx at a lateral boundary. It is shown that this boundary forcing leads to wave breaking in the leading wave behind the bore if a certain threshold is surpassed.

Three-Dimensional Numerical Simulation of Stably Stratified Flows over a Two-Dimensional Hill

Stably stratified flows over a two-dimensional hill are investigated in a channel of finite depth using a three-dimensional direct numerical simulation (DNS). The present study follows onto our previous two-dimensional DNS studies of stably stratified flows over a hill in a channel of finite depth and provides a more realistic simulation of atmospheric flows than our previous studies. A hill with a constant cross-section in the spanwise (y) direction is placed in a 3-D computational domain. As in the previous 2-D simulations, to avoid the effect of the ground boundary layer that develops upstream of the hill, no-slip conditions are imposed only on the hill surface and the surface downstream of the hill; slip conditions are imposed on the surface upstream of the hill. The simulated 3-D flows are discussed by comparing them to the simulated 2-D flows with a focus on the effect of the stable stratification on the non-periodic separation and reattachment of the flow behind the hill. In neutral (K = 0, where K is a non-dimensional stability parameter) and weakly stable (K = 0.8) conditions, 3-D flows over a hill differ clearly from 2-D flows over a hill mainly because of the three-dimensionality of the flow, that is the development of a spanwise flow component in the 3-D flows. In highly stable conditions (K = 1, 1.3), long-wavelength lee waves develop downstream of the hill in both 2-D and 3-D flows, and the behaviors of the 2-D and 3-D flows are similar in the vicinity of the hill. In other words, the spanwise component of the 3-D flows is strongly suppressed in highly stable conditions, and the flow in the vicinity of the hill becomes approximately two-dimensional in the x and z directions.

Nonlinear coupling of interfacial instabilities with resonant wave interactions in horizontal two-fluid plane Couette–Poiseuille flows: numerical and physical observations

We investigate mechanisms governing the initial growth and nonlinear evolution of interfacial waves in horizontal two-fluid plane Couette–Poiseuille flows. Nonlinear coupling of the Kelvin–Helmholtz interfacial instability with resonant wave interactions has been shown to be capable of rapidly generating long waves through the transfer of energy from linearly unstable short waves to stable long-wave components within the context of potential flow theory. The objective of this work is to determine whether that coupled mechanism persists in laminar and turbulent viscous flows. Utilizing both theoretical and computational methods, we analyse the initial Orr–Sommerfeld instability to quantify the frequencies and growth/decay rates of each wave mode for two-fluid laminar and turbulent channel flows. The obtained dispersion relation allows for the identification of resonant and/or near-resonant triads among (unstable and damped) wave components in an interfacial wave spectrum. We perform direct numerical simulations (DNS) of the two-phase Navier–Stokes equations with a fully nonlinear interface to formally establish the validity of our theoretical predictions for viscous flows. DNS results show the existence of a nonlinear energy cascade from unstable short- to damped long-wavelength waves due to resonant subharmonic and/or triadic interactions in both laminar Couette and turbulent Poiseuille flows. Spectral analysis of the interfacial evolution confirms that the combined instability–resonance mechanism persists in the presence of viscosity despite being derived under the assumption of potential flow theory. Finally, we perform a detailed examination of experimentally measured wave power spectra from Jurman et al. (J. Fluid Mech., vol. 238, 1992, pp. 187–219) and carry out a numerical sensitivity study of the flow conditions to demonstrate and verify the existence of the coupled instability–resonance mechanism in physical systems. Our analysis accurately predicts the initial instability and the resulting nonlinear energy cascade through subharmonic and triadic interfacial wave resonances.

Solitary and periodic wave solutions of the Majda–Biello system

In this paper, we present the exact solutions of the Majda–Biello system. This system describes the nonlinear interaction of long-wavelength equatorial Rossby waves and barotropic Rossby waves with a substantial midlatitude projection, in the presence of suitable horizontally and vertically sheared zonal mean flows. The methods used to construct the exact solutions are the Kudryashov method and Jacobi elliptic function method. These two methods yield solitary wave solutions and periodic wave solutions.

Propagation of Long-Wavelength Nonlinear Slow Sausage Waves in Stratified Magnetic Flux Tubes

The propagation of nonlinear, long-wavelength, slow sausage waves in an expanding magnetic flux tube, embedded in a non-magnetic stratified environment, is discussed. The governing equation for surface waves, which is akin to the Leibovich–Roberts equation, is derived using the method of multiple scales. The solitary wave solution of the equation is obtained numerically. The results obtained are illustrative of a solitary wave whose properties are highly dependent on the degree of stratification.

Counterintuitive phenomena in the interaction of long-wavelength acoustical plane waves with a viscoelastic cylinder in a non-viscous fluid

Surprising effects occurring during the interaction of acoustical plane wavefronts with a viscoelastic cylinder in the long-wavelength (Rayleigh) limit and at normal incidence are elucidated. Those phenomena include the emergence of a negative axial radiation force acting in opposite direction of wave propagation, the abnormal production of negative absorption and extinction efficiencies as well as the anomalous enhancement of the scattering efficiency as a consequence of acoustic attenuation inside the cylinder material. Numerical results are performed for a viscoelastic phenolic polymer cylinder immersed in a non-viscous host liquid (i.e. water) with particular emphasis on the shear-wave absorption coefficient of the cylinder. The computations for the radiation force function as well as the extinction, absorption and scattering efficiencies demonstrate that the monopole partial-wave plays a sole prominent role in the generation of the counterintuitive phenomena mentioned earlier. The monopole partial-wave (n=0) mode has a contribution caused by the attenuation of shear waves inside the core material of the viscoelastic cylinder.

Clustering fossil from primordial gravitational waves in anisotropic inflation

Inflationary models can correlate small-scale density perturbations with the long-wavelength gravitational waves (GW) in the form of the Tensor-Scalar-Scalar (TSS) bispectrum. This correlation affects the mass-distribution in the Universe and leads to the off-diagonal correlations of the density field modes in the form of the quadrupole anisotropy. Interestingly, this effect survives even after the tensor mode decays when it re-enters the horizon, known as the fossil effect. As a result, the off-diagonal correlation function between different Fourier modes of the density fluctuations can be thought as a way to probe the large-scale GW and the mechanism of inflation behind the fossil effect. Models of single field slow roll inflation generically predict a very small quadrupole anisotropy in TSS while in models of multiple fields inflation this effect can be observable. Therefore this large scale quadrupole anisotropy can be thought as a spectroscopy for different inflationary models. In addition, in models of anisotropic inflation there exists quadrupole anisotropy in curvature perturbation power spectrum. Here we consider TSS in models of anisotropic inflation and show that the shape of quadrupole anisotropy is different than in single field models. In fact, in these models, quadrupole anisotropy is projected into the preferred direction and its amplitude is proportional to g* Ne where Ne is the number of e-folds and g* is the amplitude of quadrupole anisotropy in curvature perturbation power spectrum. We use this correlation function to estimate the large scale GW as well as the preferred direction and discuss the detectability of the signal in the galaxy surveys like Euclid and 21 cm surveys.

Cut-off period for slow magnetoacoustic waves in coronal plasma structures

Context. There is abundant observational evidence of longitudinal compressive waves in plasma structures of the solar corona, which are confidently interpreted in terms of slow magnetoacoustic waves. The uses of coronal slow waves in plasma diagnostics, as well as analysis of their possible contribution to coronal heating and the solar wind acceleration, require detailed theoretical modelling. Aims. We investigate the effects of obliqueness, magnetic field, and non-uniformity of the medium on the evolution of long-wavelength slow magnetoacoustic waves guided by field-aligned plasma non-uniformities, also called tube waves. Special attention is paid to the cut-off effect due to the gravity stratification of the coronal plasma. Methods. We study the behaviour of linear tube waves in a vertical untwisted straight field-aligned isothermal plasma cylinder. We apply the thin flux tube approximation, taking into account effects of stratification caused by gravity. The dispersion due to the finite radius of the flux tube is neglected. We analyse the behaviour of the cut-off period for an exponentially divergent magnetic flux tube filled in with a stratified plasma. The results obtained are compared with the known cases of the constant Alfven speed and the pure acoustic wave. Results. We derive the wave equation for tube waves and reduce it to the form of the Klein–Gordon equation with varying coefficients, which explicitly contains the cut-off frequency. The cut-off period is found to vary with height, decreasing significantly in the low-beta plasma and in the plasma with the beta of the order of unity. The depressions in the cut-off period profiles can affect the propagation of longitudinal waves along coronal plasma structures towards the higher corona and can form coronal resonators.

Wave propagation in equivalent continuums representing truss lattice materials

Stiffness scales linearly with density in stretch-dominated lattice meta-materials offering the possibility of very light yet very stiff structures. Current additive manufacturing techniques can assemble structures from lattice materials, but the design of such structures will require accurate, efficient simulation methods. Equivalent continuum models have several advantages over discrete truss models of stretch dominated lattices, including computational efficiency and ease of model construction. However, the development an equivalent model suitable for representing the dynamic response of a periodic truss in the small deformation regime is complicated by microinertial effects. This paper derives a dynamic equivalent continuum model for periodic truss structures suitable for representing long-wavelength wave propagation and verifies it against the full Bloch wave theory and detailed finite element simulations. The model must incorporate microinertial effects to accurately reproduce long wavelength characteristics of the response such as anisotropic elastic soundspeeds. The formulation presented here also improves upon previous work by preserving equilibrium at truss joints for simple lattices and by improving numerical stability by eliminating vertices in the effective yield surface.