Finite Amplitude Research Articles

DAMPING OF FINITE AMPLITUDE SOLITARY WAVES IN A FLUME

Solitary wave is a permanent wave when the dissipation is ignored. Many analytical solutions have been developed for finite amplitude solitary waves. In additional to the perturbation solutions, closed form solutions are also available (e.g., McCowan 1891, Clamond and Fructus 2003, being denoted as CF from hereon), which are more accuracy, especially for larger amplitude solitary waves which were discussed in Wang and Liu (2022). In Wang and Liu (2022) ’s laboratory experiments, solitary waves are slowly damped along the wave flume, which can be attributed to the energy dissipation inside the boundary layers on the bottom and sidewalls. Keulegan (1948) first derived an approximate solution to describe the wave damping in the wave flume for small amplitude solitary wave. The basic idea of calculating the damping effect is that the rate of energy dissipation inside the boundary layers must be the same as the rate of wave energy loss. In this work, the same principle of energy balance is adopted for calculating the wave damping for finite amplitude solitary waves in a wave flume. The closed form solutions provided in CF are employed here for analyses.

(3+1)-dimensional cylindrical dust ion-acoustic solitary waves in dusty plasma

The basic properties of (3+1)-dimensional cylindrical dust ion-acoustic (DIA) solitary waves have been studied in a multi-component plasma medium consisting of inertial ions, Maxwellian electrons, and stationary negative dust particles. The reductive perturbation method has been used to derive a cylindrical (3+1)-dimensional Kadomtsev–Petviashvili equation for small but finite amplitude of dust ion-acoustic waves. The impacts of nonplanar geometry and presence of static negative dust particles on the dust ion-acoustic solitary structures have been investigated. DIA solitary waves with both positive and negative polarity is observed to generate for this existing multi-component and complex plasma medium. The changes in the DIA wave structure in radial, azimuthal, and axial coordinates over time have been presented analytically. The results of this present investigation may contribute to a better understanding of laboratory plasma events as well as nonlinear electrostatic excitations in astrophysical environments.

Simulation of Head-on Collisions between Filamentary Molecular Clouds Threaded by a Lateral Magnetic Field and Subsequent Evolution

Filamentary molecular clouds are regarded as the place where newborn stars form. In particular, a hub region, a place where it appears as if several filaments are colliding, often indicates active star formation. To understand the star formation in filament structures, we investigate the collisions between two filaments using two-dimensional magnetohydrodynamical simulations. As a model of filaments, we assume that the filaments are in magnetohydrostatic equilibrium under a global magnetic field perpendicular to the filament axis. We set two identical filaments with an infinite length and make them collide with a zero-impact parameter (head-on). When the two filaments collide while sharing the same magnetic flux, we found two types of evolution after a merged filament is formed: runaway radial collapse and stable oscillation with a finite amplitude. The condition for the radial collapse is independent of the collision velocity and is given by the total line mass of the two filaments exceeding the magnetically critical line mass for which no magnetohydrostatic solution exists. The radial collapse proceeds in a self-similar manner, resulting in a unique distribution irrespective of the various initial line masses of the filament, as the collapse progresses. When the total line mass is less massive than the magnetically critical line mass, the merged filament oscillates, and the density distribution is well-fitted by a magnetohydrostatic equilibrium solution. The condition necessary for the radial collapse is also applicable to the collision whose direction is perpendicular to the global magnetic field.

Nonlinear evolution of viscoplastic film flows down an inclined plane.

In this article, we experimentally investigate the nonlinear behaviour of a viscoplastic film flow down an inclined plane. We focus on the nonlinear instabilities that appear as roll waves. Roll waves are generated by perturbing a permanent flow of Herschel-Bulkley fluid (Carbopol 980) at low frequencies. To determine the local thickness of the film, we used a laser sensor and a camera to globally capture the transverse shape of the waves. For a regular forcing, the results show the existence of different regimes. First, we observe primary instabilities below the cut-off frequency at the entrance of the channel. After the exponential growth of the wave in the linear regime, we recognise the nonlinear dynamics with the existence of finite amplitude waves. This finite amplitude depends on the frequency, the Reynolds number and the inclination angle. The results show that this instability is supercritical. At moderate Reynolds numbers, the finite 2-D waves become sensitive to transverse perturbations, due to a secondary instability, and become 3-D waves. The experimental results illustrate a phenomenology of viscoplastic film flows similar to Newtonian fluids, except for the capillary waves.

Laminar boundary layers and damping of finite amplitude solitary wave in a wave flume

In this paper, the laminar viscous effects on finite amplitude solitary wave propagation in a wave flume with a rectangular cross section are investigated. The laminar viscous effects are most important in the laminar boundary layers adjacent to the bottom and sidewalls of the flume. The closed-form solutions of Clamond and Fructus [“Accurate simple approximation for the solitary wave,” C. R. Mec. 331, 727 (2003)] provide the free stream velocity to derive the boundary layer flows. The rotational velocities inside the bottom and sidewall boundary layers are first obtained analytically after linearization. Numerical solutions for the nonlinear boundary layer flows are also calculated. The importance of the nonlinearity is discussed. The laminar viscous damping rate is estimated by balancing the energy dissipation inside the boundary layers and the rate of wave energy change in the entire wave. Numerical results of the damping rate show the dependence on the viscosity of the fluid, the wave amplitude, the water depth, and the width of the wave flume. New laboratory experiments on the wave damping of finite amplitude solitary waves are also performed. The theoretical results are confirmed with the laboratory data. The methodologies introduced in this paper for obtaining boundary layer solutions and laminar viscous damping rate can be applied to other transient waves with finite amplitude.

Nonlinear coherent structures of nucleus-acoustic wave excitations in multi-nuclei quantum plasmas with ultra-relativistically degenerate electrons and positrons

The propagation dynamics of the nucleus-acoustic waves (NAW) in a quantum plasma composed of nondegenerate inertial light nuclei, stationary heavy nuclei, and ultra-relativistically degenerate electrons and positrons has been theoretically investigated within the framework of the Boussinesq equation, which is valid for a bi-directional propagation of a small but finite amplitude limit. The N-soliton solution of the Boussinesq equation is derived using Hirota's method. It is found that positive potential structures exist in the sonic and supersonic regimes, whereas negative potential structures are found to be present in the subsonic regime. Pertinent plasma properties are analyzed for one-, two-, and three-soliton solutions in terms of different parameters. In addition to the typical solitary wave solutions, our findings indicate that the nonlinear NAW has breather structures. The three- and four-soliton solutions are used to construct the elastic interaction solutions of the breather–soliton and breather–breather, respectively. The findings are discussed in the context of ultra-relativistic astrophysical plasmas.

Highly nonlinear internal solitary waves with a free surface

A new mathematical model is derived for internal wave propagation in a two-layer fluid domain with a free surface in which the lower layer is of finite depth. The mathematical model consists of three coupled nonlinear equations for displacement of the interface and velocity potentials of the two layers, each of which is expressed as the convex combination of their values at two arbitrary depths. Model equations are correct up to O(μ4) terms, where μ is the ratio of harmonic mean of the depths of the two layers to double of a typical wavelength. Finite amplitude internal solitary wave profiles as well as free surface wave profiles are obtained numerically for different wave speeds, density ratios and depth ratios. Interfacial solitary wave profiles obtained here are in very good agreement with recent experimental observations of Zhao et al. (2020). The figures presented here demonstrate how the interfacial wave profile predicted by the free surface model differs from predictions of rigid-lid two-layer fluid models.

Combined effect of helical force and rotation on the double convection of a binary viscoelastic fluid in a porous medium: Linear and weakly nonlinear analysis

AbstractWe used linear stability theory based on the normal mode decomposition technique and nonlinear stability theory based on the minimum representation of double Fourier series to study the criterion of appearance of the stationary convection and the oscillatory convection; and the rate of heat and mass transfer in a binary viscoelastic fluid mixture in a rotating porous medium under the effect of helical force. We have determined the analytical expression of the Rayleigh number of the system as a function of the dimensionless parameters. Expressions for heat and mass transfer rates are determined as a function of Nusselt and Sherwood number, respectively. The transient behaviors of the Nusselt number and the Sherwood number are studied by solving the finite amplitude equations using the Runge‐Kutta method. Then, the effect of each dimensionless parameter on the system is studied pointed out interesting results. According to the analysis of the different results obtained, it appears that in a porous medium saturated by a binary mixture of viscoelastic fluid in rotation, the Taylor‐Darcy number , the delay time parameter λ2 and the mass Rayleigh‐Darcy number delay the onset of stationary and oscillatory convection. On the other hand, the helical force , the relaxation time parameter λ1, the ratio of diffusivities τ and the number of Vadasz accelerate the onset of stationary and oscillatory convection. In the case of the nonlinear stability study, at steady state, the helical force accelerates the rate of heat and mass transfer. On the other hand, the Taylor‐Darcy number , the diffusivity ratio τ and the mass Rayleigh‐Darcy number retard the rate of heat and mass transfer. In the unsteady state, the helical force , the relaxation time parameter λ1 and the mass Rayleigh‐Darcy number accelerate the rate of heat and mass transfer. This is not the case for the Taylor‐Darcy number and the delay time parameter λ2.

Interaction of Relativistic Electrons with Packets of the Electromagnetic Ion Cyclotron Waves of Finite Length and Low Amplitude

Interaction of relativistic electrons with packets of the electromagnetic ion cyclotron waves of finite length and low amplitude in the Earth’s radiation belts is analyzed. The variance of the equatorial pitch angle of electrons for wave packets located near the Earth’s geomagnetic equator is estimated analytically within the linear approximation. The analytical estimates agree with the results of numerical test-particle simulation. It is demonstrated that reduction in the packet length extends the interaction range to lower energies beyond the range of resonant energies. Such an interaction can result in precipitation of electrons with energies on the order of several hundred kiloelectronvolts into the ionosphere.

Nonlinear coupling of electromagnetic and spin-electron-acoustic waves in spin-polarized degenerate relativistic astrophysical plasma

Propagation of the finite amplitude electromagnetic wave through the partially spin-polarized degenerate plasmas leads to an instability. This instability happens due to the interaction of the electromagnetic wave with the small-frequency longitudinal spin-electron-acoustic waves. The strongest growth of the instability happens in the high-density degenerate plasmas with the Fermi momentum close to mec, where me is the mass of electron and c is the speed of light. The increase of the instability increment with the growth of the spin polarization of plasmas is found.

Highly Sensitive Detection of Microstructure Variation Using a Thickness Resonant Transducer and Pulse-Echo Third Harmonic Generation.

In nonlinear ultrasound testing, the relative nonlinear parameter is conveniently measured as a sensitive means of detecting and imaging overall variation of microstructures and damages. Compared to the quadratic nonlinear parameter (β'), the cubic nonlinear parameter (γ'), calculated as the third harmonic amplitude divided by the cube of the fundamental amplitude, has generally a higher value, providing better sensitivity in nonlinear parameter mapping. Since the third harmonic amplitude is about two orders of magnitude lower than the fundamental amplitude, efficient excitation and highly sensitive reception of third harmonic is very important. In this paper, we explore an odd harmonic thickness resonant transducer that meets the requirements for pulse-echo third harmonic generation (THG) measurements. We also address the problem of source nonlinearity that may be present in the measured amplitude of the third harmonic and propose a method to properly correct it. First, we measure γ' for a series of aluminum specimens using the through-transmission method to observe the behavior of γ' as a function of specimen thickness and input voltage, and examine the effects of various corrections such as attenuation, diffraction and source nonlinearity. Next, we apply the odd harmonic resonant transducer to pulse-echo THG measurements of precipitation heat-treated specimens. It is shown that such transducer is very effective in generation and detection of fundamental and third harmonics under finite amplitude toneburst excitation. The highly sensitive detectability of γ' are presented as a function of aging time, and the sensitivity of γ' is compared with that of β' and β'2.

Nonlinear Analysis of Cross Rolls of Electrically Conducting Fluid under an Applied Magnetic Field with Rotation

The proposed planer layer dynamo physical model has real-world applications, especially in the Earth’s liquid core. Thus, in this paper, an attempt is made to understand the finite amplitude convection when there exists a coupling between the Lorentz force and the Coriolis force. In particular, the effect of a horizontally applied magnetic field is studied on the Rayleigh–Bénard convection (RBC) that contains the electrically conducting fluid and rotates about its vertical axis. Free–free boundary conditions are assumed on the geometry. Attention is focused on the nonlinear convective flow behavior during the occurrence of cross rolls which are perpendicular to the applied magnetic field and parallel to the rotation axis. The visualization of cross rolls is achieved using the Fourier analysis of perturbations up to the O(ε8). The relationship of the Nusselt number (Nu) with respect to the Rayleigh number (R), the Ekman number (E), and the Elsasser number (Λ) is investigated. It is observed that E generates a strong damping effect on the flow velocity and on the heat transfer at high rotation rates. Using the heatline concept, it is observed that the temperature within the central regime is enhanced as the Λ increases. The results show that either E decreases or Λ increases, then the heat transfer rate increases.

Stokes wave traveling along a thin elastic plate floating at water surface

This paper addresses nonlinear deflection waves traveling along a thin elastic plate floating at the water surface. The investigations are based on both the analytical and experimental aspects. A finite amplitude wave theory for the waves traveling on the plate developed by the present authors, in which, the nonlinear wave is formulated by a perturbation with respect to the wave height/wavelength ratio, is used. The behavior of the nonlinear wave traveling on the plate is like what the Stokes wave theory predicts for the free surface. However, for the waves traveling on the plate, there is an instability frequency at which the second order wave component grows considerably. The sign of the second order wave component is inverted at the instability frequency when the incident wave frequency is changed from low to high regions. These predictions are verified in a tank test in the two-dimensional flume using a long flexural plate that was newly designed. The spatio-temporal distribution of the deflection and bending strain on the plate was measured directly using the stereo Digital Image Correlation (DIC) technique with two high-speed CCD cameras. The measured results are compared with the predictions by analytical theory. The instability frequency is found in the tank test and the nonlinear wave properties are discussed.

Complete discrimination System method for finding exact solutions, dynamical properties of combined Zakharsov-Kuznetsov-modified Zakarsov-Kuznetsov equation

The (2+1)-dimensional combined Zakharsov-Kuznetsov-modified Zakarsov-Kuznetsov (ZK-mZK) equation with constant co-efficients are solved by employing the complete discrimination system for polynomial Method (CDSPM). Use of it reduces the key Nonlinear Differential Equation (NLPDE) to an Ordinary Differential Equation (ODE) in the integral form under the traveling wave transformation, and the solutions are classified. Exact traveling wave solutions are derived using CDSPM of order four. Based on CDSPM, Liu’s approach provides a set of new solutions including Jacobi elliptic functions for the Combined-(ZK-mZK) equation. These analytic solutions exhibit different nonlinear wave structures viz., solitary wave, shock wave, and periodic wave. Among them, the finite amplitude periodic solutions are very important wave features for Combined-(ZK-mZK) equation and their existence is confirmed from the phase portrait of the said equation. Especially the finite amplitude periodic solutions are acquired from the system with the inclination of Jacobean elliptic functions. Numerical results confirm the reliability and effectiveness of the method. Finally, the phase portrait of the dynamical system ensures the entity and exactness of different topological structures of the solutions. Based on this analysis, we conclude that this method is appropriate one to classify the solutions, and can be expanded throughout the vast areas of application.

Volume-Law to Area-Law Entanglement Transition in a Nonunitary Periodic Gaussian Circuit.

We consider Gaussian quantum circuits that alternate unitary gates and postselected weak measurements, with spatial translation symmetry and time p eriodicity. We show analytically that such models can host different kinds of measurement-induced phase transitions detected by entanglement entropy, by mapping the unitary gates and weak measurements onto Möbius transformations. We demonstrate the existence of a log-law to area-law transition, as well as a volume-law to area-law transition at a finite measurement amplitude. For the latter, we compute the critical exponent ν for the Hartley, von Neumann and Rényi entropies exactly.

Steady-state waves at class II Bragg resonance

This paper examines class II Bragg resonance in the interaction between surface gravity waves and bottom ripples and derives the corresponding steady-state wave system based on the homotopy analysis method. Specifically, we study the resonance between a nonlinear oblique incident wave and a doubly periodic cosinusoidal bottom. The bottom extends infinitely in the horizontal direction, and its two bottom-wave components are collinear. We analyze the influence of the relative water depth, slope of the incident wave, slope of the bottom topography, and amplitude ratio of the two bottom-wave components on the energy distribution of the wave system. The energy distribution can be classified into two types. In type 1, the incident and resonant wave components have the same energy. In type 2, the energy of each component is different. There are two situations in both type 1 and type 2. The incident and resonant wave components account for most of the energy of the wave system, whereas the energy of other wave components cannot be ignored in the other system. These two situations were not identified in our previous study [Xu et al., “Equilibrium states with finite amplitudes at exactly and nearly class-I Bragg resonances,” J. Math. 2021, 9986114]. We then investigate the effects of different physical parameters (including the incident wave slope and bottom undulations) on the energy distribution, wave slope, and angular frequencies of the wave system. The results of this study deepen our understanding of class II Bragg resonance and suggest several directions for future research.

Locally finite two-loop QCD amplitudes from IR universality for electroweak production

We describe the implementation of infrared subtractions for two-loop QCD corrections to quark-antiquark annihilation to electroweak final states. The subtractions are given as form-factor integrands whose integrals are known. The resulting subtracted amplitudes are amenable to efficient numerical integration. Our procedure is based on the universality of infrared singularities and requires a relatively limited set of subtractions, whose number grows as the number of two-loop diagrams, rather than with the number of singular regions of integration.

Quasiparticle random-phase approximation calculations forM1transitions with the noniterative finite-amplitude method and application to neutron radiative capture cross sections

We derive the equations of quasiparticle random-phase approximation (QRPA) based on the finite amplitude method (FAM) with the Hartree-Fock+BCS (HF+BCS) single-particle states, and calculate the magnetic dipole (M1) transition for deformed gadolinium isotopes. Our QRPA calculation shows both large spin-flip transitions in the 5 to 10 MeV excitation energy and the low energy orbital transition that would correspond to the M1 scissors mode observed in nuclear experiments. Then, we calculate neutron capture reactions based on the statistical Hauser-Feshbach theory with the photoabsorption cross sections given by QRPA. We find that the capture cross section is enhanced due to the contribution from the low energy M1 transition although the calculated capture cross section still underestimates the experimental data. This issue in the calculated capture cross section could be improved by uncertainties of low energy E1 transition neglected in our QRPA calculation.

Solitary wave solution of (3+1)-dimensional cylindrical Kadomstev–Petviashvili equation in warm opposite polarity dusty plasma

A theoretical investigation is done to study the propagation of dust-acoustic (DA) solitary waves in a dusty plasma system containing dynamic positively as well as negatively charged warm dust species and nonthermally distributed ion and electron species. The standard reductive perturbation method is employed to derive the (3+1)-dimensional cylindrical Kadomstev–Petviashvili (KP) equation (also known as cylindrical Korteweg–de Vries equation), which admits solitary wave solution for small but finite amplitudes. The parametric regimes for the existence of DA solitary waves are obtained. Two modes, namely, slow and fast are observed corresponding to different phase velocities. The slow mode has negative solitary pulses, whereas the fast mode contains both positive and negative DA solitary waves. It is shown that the ratio of positively charged dust to negatively charged dust number density significantly modifies the basic features of the DA solitary waves. The noteworthy effects of other parameters (viz. the nonthermal parameter and ion-to-electron temperature ratio) on the basic properties (namely, phase speed, polarity, amplitude, and width) of the DA solitary waves are also mentioned. The fundamental features and the underlying physics of the electrostatic solitary structures that are relevant to cometary tails, upper mesosphere, Jupiter’s magnetosphere, etc., are briefly discussed.

A Two-Stage Optimization Framework for Fast Lithium-Ion Battery Impedance Measurement

Online monitoring of battery impedance is of increasing interest as it brings rich information for battery state estimation and fault diagnosis, however, the existing techniques are limited by high implementation costs and poor accuracy. This letter addresses the challenges by proposing a novel two-stage optimization framework to form a binary sequence with the optimal power proportion in specified harmonics for fast battery impedance measurements. The Optimized Binary Sequence (OBS) with finite amplitude levels can greatly simplify the implementation, and also the optimized power spectrum has superb accuracy to obtain the frequency responses. Experimental studies confirm the superiority of the proposed method in comparison with three existing techniques under various circumstances.