Finite-amplitude Waves Research Articles

IMPACT OF COLLISIONS ON THE DYNAMICS OF WAVES OF FINITE AMPLITUDE IN A PLASMA

The problem of determining wave dynamics, considering the evolution of nonlinear waves of finite amplitude in a weakly collisional, Maxwellian plasma, is the focus of this article. Considering this medium with a certain interaction potential associated with a certain limited core of pairwise interaction, the dynamics of the distribution function was studied using the Vlasov equation. Collisions of electrons with neutral particles are described using the collision integral in the Bhatnagar - Gross- Krook form. Having constructed its particular solution and the equilibrium distribution function in the form of a series, an equation was obtained that makes it possible to determine the potential function. Considering the case of a Maxwellian plasma, an integral potential equation was obtained. Based on it, an equation was constructed that determines the perturbation of the potential relative to the spatially homogeneous one. At the same time, this perturbation appears due to the existence of some spatial-temporal stable structure. Based on this, a dispersion relation was obtained, which makes it possible to estimate the spatial scales of the coherent structure.

A VIRTUAL PISTON-TYPE WAVE MAKER BASED ON OPENFOAM

OpenFOAM is an open source CFD (Computational Fluid Dynamics) toolbox and recently attracts many researchers to develop codes based on it for their own applications. In order to numerically generate waves based on the wave-maker theory for a piston motion, numerical improvements have been done on the base of OpenFOAM by the author. In gen- eral, the present new tool can be employed to simulate wave generation as long as the piston motion is given. This paper presents the related computational procedure and simulations for generating relatively long finite-amplitude waves ac- cording to Madsen’s second-order wave-maker theory. The sensitivities of the computed incident wave profile to grid density and time step are investigated for the case of generating a wave with permanent form. The simulation accuracy is validated by comparison with the analytical solution and available experimental data.

Higher-order nonlinear and dispersive effects on dust-ion-acoustic solitary waves in magnetized dusty plasmas

We deal with the theoretical investigation of the higher-order nonlinear and dispersive effects on the basic features of dust-ion-acoustic (DIA) solitary waves in a magnetized dusty plasma. Also, we examine the impacts of the external magnetic field and adiabaticity, obliquity, etc. in changing solitary wave structures. The waves of small but finite amplitude propagate obliquely through the dusty plasma where the system comprises adiabatic positively charged ions (mobile), negatively charged trapped light ions, Maxwellian distributed electrons and stationary dust particles with arbitrary charges. The reductive perturbation method is extensively used for incorporating the higher-order terms into the modified Korteweg-de Vries (mKdV) equation, and the procedure yields an inhomogeneous mKdV-type equation. It is observed that the inclusion of higher-order nonlinear and dispersion effects increases the wave’s amplitude and width. Furthermore, in the presence of higher-order effects, the magnetic field and adiabaticity have decreasing and increasing effects respectively on the wave features. It is expected that the results will be useful to describe higher-order effects on DIA solitary waves.

Dissipative dust acoustic solitons in magnetized nonextensive warm dusty plasma

A systematic theoretical investigation has been carried out in order to explore the propagation of both linear and nonlinear dust acoustic waves (DAWs) in a collisional magnetized warm dusty plasma in the presence of nonextensive electrons and ions. A linear dispersion relation is analyzed to show that the wave frequencies of both parallel and perpendicular modes are modified by the relevant plasma parameters. The standard reductive perturbative technique leads to a Korteweg–de Vries equation with a linear damping term for the propagation dynamics of the finite amplitude waves in warm dusty plasma. It is found that the dust-neutral collision induced dissipation is responsible for the linear damping. The analysis reveals that DAW propagates in the form of a weakly dissipative compressive/rarefactive solitons and the nonlinear excitations are significantly influenced by different plasma parameters, such as dust concentration, nonextensivity, external magnetic field strength, and dust-neutral collisionality. The present results can be relevant to the DAWs in some astrophysical plasma environments, such as Saturn’s rings, interstellar dusty clouds, etc.

Bridging shock: Blackstock’s general solution for propagation of finite-amplitude sound waves

Reflections are offered on D. T. Blackstock [“Connection between the Fay and Fubini solutions for plane sound waves of finite amplitude,” J. Acoust. Soc. Am. 39, 1019–1026 (1966)]. This work treats the propagation of planar finite-amplitude sound waves. It nominally links the Fourier-series-based solution of Fubini for the pre-shock condition, and the distant, post-shock solution of Fay. However, addressing the transition region between these two required solving the entire problem, which Blackstock did with aplomb. This additionally enabled specification of undetermined factors in the Fay solution, statement of an explicit bound on the applicability of the solution, and generalization to cylindrical and spherical waves. Hallmarks of Blackstock’s work are evident: scholarly mindfulness of historical precedent, meticulous analysis that inevitably advanced understanding of nonlinear acoustic phenomena, and elegance in presentation, which was, moreover, pedagogically convincing.

Directional dependence of nonlinear piezoelectric surface acoustic waves in lithium niobate

A model for the propagation of finite-amplitude surface acoustic waves (SAWs) in a piezoelectric material [Hamilton et al., J. Acoust. Soc. Am. 100, 2567(A) (1996)] is used to investigate the dependence of SAW nonlinearity on crystal cut and propagation direction in lithium niobate (LiNbO3). The model is based on Hamiltonian formalism, in which the boundary conditions at the free surface need only be evaluated at linear order, thus simplifying computation of the nonlinear problem. In contrast with the scalar coefficient of nonlinearity that appears in evolution equations for bulk planewaves, nonlinearity of SAWs is described by a matrix for harmonic interaction due to the variation of depth dependence of the SAW field with frequency. Components of the nonlinearity matrix are generally complex-valued as a result of material anisotropy and piezoelectricity. Variation with propagation direction of the (1,1) component of the nonlinearity matrix, which governs second-harmonic generation, is presented for LiNbO3 crystals that are X-cut, Y-cut, and Z-cut. Relative contribution of each nonlinear mechanism (nonlinear elasticity, nonlinear piezoelectricity, electrostriction, and nonlinear dielectricity) to the nonlinearity matrix is examined. Distortion of an initially sinusoidal waveform during propagation is demonstrated by integrating coupled spectral evolution equations for the harmonic amplitudes.

Nonlinear wave propagation in dense vapor of Bethe–Zel'dovich–Thompson fluids subjected to temperature gradients

Flows of fluids made of complex organic molecules exhibit unconventional fluid dynamic behavior in the vapor phase if their thermodynamic state is close to that of the vapor–liquid critical point. If the molecule is sufficiently complex, this thermodynamic domain is characterized by negative values of the fundamental derivative of gasdynamics Γ, the fluid is called Bethe–Zel'dovich–Thompson (BZT) fluid, and atypical phenomena, such as rarefaction shockwaves, are theoretically admissible. The nature of the steepening of nonlinear waves in dense vapor flows of organic fluids evolving in this thermodynamic region can be significantly affected by the presence of temperature gradients in the flow. This study investigates the evolution of finite-amplitude acoustic waves in these conditions. The steepening of the wavefront is analyzed using the wavefront expansion technique, and the deformation of the wavefront is simulated numerically by solving the Westervelt equation. The results of simulations of wave propagation in dense vapors indicate that, though Γ governs the nature of steepening waves, local gradients in sound speed and density can alter the rate of steepening and can enhance or delay shock formation in the medium, a result relevant also to the envisaged experiments aimed at proving the existence of nonclassical gasdynamics phenomena in BZT vapors.

David Blackstock and the reflection and refraction of finite amplitude waves

In honor of David Blackstock, this presentation recalls his contributions regarding nonlinear acoustics in the areas of reflection and refraction. Some of this is well documented and other pieces are not. For example, in his 1960 Ph.D. dissertation David was the first to recognize Pfriem’s pressure amplification equation would apply to finite amplitude plane-wave reflection from a rigid surface. Acoustic pressure MORE than doubles for finite amplitude rigid surface reflection, and it is the variational sound speed that exactly doubles. And this result provides a basis for an understanding of oblique reflection and the formation of finite amplitude Mach stems. Further, the work of David’s Ph.D. student Frederick Cotaras on finite amplitude refraction from 1989 will be highlighted in addition to other contributions as time permits.

Illuminating finite-amplitude wave propagation: Hargrove’s use of recurrence relations

A classic problem in the theory of finite-amplitude wave propagation is that of the spectral evolution of a plane wave. An initial sinusoidal wave becomes triangular, with a vertical face at the shock distance. The wave in the pre-shock regime was represented by a Fourier series in the seminal work by Fubini, i.e., Fubini Ghiron [Alta Frequenza, 4, 530 (1935)]. Independently, Hargrove performed a similar derivation, also invoking recurrence relations for cylindrical Bessel functions, but presenting this with exceptional heuristic clarity and brevity [J. Acoust. Soc. Am. 32, 511 (1960)]. This was coincident with a third paper, by Keck and Beyer [Phys. Fluids 3, 346 (1960)], which knowingly followed Fubini. All of this work has facilitated understanding, providing cogent reminders of the power of recurrence relations when dealing with special functions.

The Peregrine Breather on the Zero-Background Limit as the Two-Soliton Degenerate Solution: An Experimental Study

Solitons are coherent structures that describe the nonlinear evolution of wave localizations in hydrodynamics, optics, plasma and Bose-Einstein condensates. While the Peregrine breather is known to amplify a single localized perturbation of a carrier wave of finite amplitude by a factor of three, there is a counterpart solution on zero background known as the degenerate two-soliton which also leads to high amplitude maxima. In this study, we report several observations of such multi-soliton with doubly-localized peaks in a water wave flume. The data collected in this experiment confirm the distinctive attainment of wave amplification by a factor of two in good agreement with the dynamics of the nonlinear Schrödinger equation solution. Advanced numerical simulations solving the problem of nonlinear free water surface boundary conditions of an ideal fluid quantify the physical limitations of the degenerate two-soliton in hydrodynamics.

Formation and evolution of roll waves in a shallow free surface flow of a power-law fluid down an inclined plane

A laminar flow of a thin layer of mud down an inclined plane under the action of gravity is considered. The instability of a film flow and the formation of finite amplitude waves are studied within the framework of both two-dimensional governing equations of a power-law fluid and its depth-averaged hyperbolic simplification. The conditions of the existence of roll waves for these models are formulated in terms of the Whitham criterion. A free surface evolution and the development of roll waves are numerically calculated. The amplitude of roll waves obtained by the 2D equations is slightly larger than that for the 1D model. Moreover, for certain flow parameters, the small perturbations of the basic solution grow for the 2D equations and decay for the depth-averaged model. A two-parameter class of exact piecewise-smooth solutions of the 1D model is obtained and a comparison with a numerical solution is made. In the region of these parameters, diagrams of the roll waves existence are constructed.

Large amplitude capillary-gravity solitary waves in a stratified fluid sandwiched between two deep homogeneous layers

The study addresses the propagation of plane capillary gravity solitary waves of permanent form in a three layer formulation. The intermediate fluid is assumed to be stratified, while the upper and lower ones are homogeneous and infinitely deep. One or both interfaces separating these layers are subject to capillarity. The research can be applied to the case of two deep fluids when one of these fluids is stratified near the interface. The latter formulation is relevant to studies of capillary gravity waves in the transitional area between sea water and liquid carbon dioxide in the deep ocean. This has become an issue of importance for the secure storage of carbon dioxide, which is an environmental/technological problem in modern days. Therefore, we address a capillary-gravity wave motion beyond the well-examined cases of a free surface or two fluid flows. It is shown that in the considered formulation, capillary-gravity solitary waves of finite amplitude obey an integro-differential equation. This equation contains both Korteweg-de Vries (KdV) and Benjamin-Ono (BO) dispersion laws and a specific nonlinearity, which depends on the properties of the stratified layer. Capillary (KdV-type) dispersion dominates if the thickness of the stratified layer is d≪d∗. When d≫d∗, the gravitational (BO-type) dispersion determines the flow. The value d∗ depends on the mode number, gravitational acceleration, and capillarity effects. Analytical solutions for the amplitude function and the streamline patterns are presented.

MATHEMATICAL MODELING OF THERMOGASDYNAMIC PROCESSES IN PISTON ENGINES

A method for calculating the non-stationary gas flow in the exhaust pipeline of a piston engine by the method of solitary waves of finite amplitude is presented. The comparison of experimental data and calculation results by the method of characteristics and obtained when considering the process of propagation of solitary waves through the exhaust pipeline of a piston engine equipped with a pulsed turbocharging system is shown.

Wave propagation of resonance multi-stripes, complexitons, and lump and its variety interaction solutions to the (2+1)-dimensional pKP equation

This study deals with the (2+1)-dimensional potential Kadomtsev-Petviashvili (pKP) equation, which is used to describe the dynamics of a wave of small but finite amplitude in two dimensions in diverse areas of physics and applied mathematics. Through symbolic computations with Maple, the resonance multi-stripe solutions in real fields, and multi-stripe complexiton solutions in complex fields are derived via the simplified linear superposition principle in conjunction with Hirota's bilinear method (HBM). Furthermore, lump, lump-stripe, and lump-triangular periodic wave solutions are derived by employing the HBM. A positive quadratic function with exponential, hyperbolic cosine and trigonometric cosine functions are considered to reach such aims. For lump solutions, it is found that the shape and amplitude of a lump wave remain unchanged during its propagation. On the other hand, lump-stripe interaction solitons present the fission and fusion interaction phenomena between the lump and stripe solitons. To illustrate the dynamical characteristics of the attained solutions, the three-dimensional (3D) and contour plots of some of the representative attained solutions are displayed with the particular choice of the free parameters. The obtained solutions and their physical features might be helpful to understand the propagation of small but finite amplitude waves in shallow water.

Finite-amplitude steady-state resonant waves in a circular basin

Abstract

Nonlinear phenomenon in laser-induced finite-amplitude acoustic waves propagating along cracks

The propagation of the longitudinal and surface waves along cracks in the plate was studied. Waves with strain up to 5×10−3 could be generated by radiation of a Q-switched laser absorbed in a layer of liquid suspended carbon particles deposited on the surface of the plate. The longitudinal and surface Rayleigh waves traveled parallel to and on the fracture surfaces, respectively, and bounced from the free surfaces of the plate. It was observed that the longitudinal echo pulses exhibited shapes, which were dependent on the magnitude of the laser excitation and, thus, on the amplitudes of the laser-generated acoustic waves. At a certain photo-excitation level, the leading tensile part of the first longitudinal echo vanished and reappeared again at higher excitation levels. This effect did not occur in the crack-free areas of the plate and was amplitude-dependent; therefore, it could be attributed to the nonlinear interaction of the longitudinal and surface Rayleigh waves with the crack.

The onset of instabilities and finite amplitude waves in a model of aluminum reduction cells with nonuniform cathode current

In aluminum reduction cells, the electric current density at the cathode is seldom uniform. This may be due to a variety of reasons. One of the reasons is that a mass of undissolved alumina may get enrobed by the cryolitic bath, and this mix freezes due to the decrease in temperature below the liquidus point. Thus, an electrically resistive solid layer (“mud”) is formed at a part of the cathode. This leads to horizontal electric currents in the liquid aluminum layer, and their magnetohydrodynamic (MHD) interaction with the vertical magnetic field induces a rotating flow of aluminum. This may cause undesirable MHD instabilities. A quasi-two-dimensional model for the laboratory facility has been presented. The results show that the dominating feature of the flow is a depression of the free surface of the liquid metal above the mud spot. This is due to strong rotation of the liquid metal around the mud spot, which causes the centrifugal force. Other effects may include superimposed conventional MHD instability, which manifests itself in a rotating interface, modulated waves, reflection of the waves from the corners of the domain, sloshing, etc. It has been shown that small mud spots do not affect the cell stability, while the large ones may cause such deep depressions that the centrifugal force completely removes the liquid metal above the spot.

Planar and non-planar nucleus-acoustic solitary waves in warm degenerate multi-nucleus plasmas

A warm degenerate plasma (containing ultra-relativistically or non-relativistically warm degenerate inertia-less electron species, non-relativistically warm degenerate inertial light nucleus species and stationary heavy nucleus species) is considered. The basic features of planar and non-planar solitary structures associated with the degenerate pressure-driven nucleus-acoustic waves propagating in such a warm degenerate plasma system are investigated. The reductive perturbation method, which is valid for small- but finite-amplitude solitary waves, is used. It is found that the effects of non-planar cylindrical and spherical geometries, non- and ultra-relativistically degenerate electron species and the temperature of degenerate electron species significantly modify the basic features (i.e. speed, amplitude and width) of the solitary potential structures associated with degenerate pressure-driven nucleus-acoustic waves. The warm degenerate plasma model under consideration is applicable not only to all cold white dwarfs, but also to many hot white dwarfs, such as DQ white dwarfs, white dwarf H1504+65, white dwarf PG 0948+534, etc.

Traveling magnetization waves in diluted magnetic material

The magnetization evolution of the paramagnetic matrix and ferromagnetic admixture ions is considered using the continuous medium model. A dynamical equation system is derived. Two possible solutions are obtained. The first is a travelling wave of finite amplitude, which is stable in the system with a ferromagnetic interaction of admixture ions. The second solution is a soliton; its amplitude and velocity are determined by the exchange energy of the admixture ions and dipole interaction between them and the bulk matrix. The dipolar coupling of the bulk lattice and admixture ions has a significant effect on the solution stability.

Development and Validation of Quasi-Eulerian Mean Three-Dimensional Equations of Motion Using the Generalized Lagrangian Mean Method

This study aims at developing a new set of equations of mean motion in the presence of surface waves, which is practically applicable from deep water to the coastal zone, estuaries, and outflow areas. The generalized Lagrangian mean (GLM) method is employed to derive a set of quasi-Eulerian mean three-dimensional equations of motion, where effects of the waves are included through source terms. The obtained equations are expressed to the second-order of wave amplitude. Whereas the classical Eulerian-mean equations of motion are only applicable below the wave trough, the new equations are valid until the mean water surface even in the presence of finite-amplitude surface waves. A two-dimensional numerical model (2DV model) is developed to validate the new set of equations of motion. The 2DV model passes the test of steady monochromatic waves propagating over a slope without dissipation (adiabatic condition). This is a primary test for equations of mean motion with a known analytical solution. In addition to this, experimental data for the interaction between random waves and a mean current in both non-breaking and breaking waves are employed to validate the 2DV model. As shown by this successful implementation and validation, the implementation of these equations in any 3D model code is straightforward and may be expected to provide consistent results from deep water to the surf zone, under both weak and strong ambient currents.