Propagation Of Small-amplitude Waves Research Articles

Models of fractional viscous stresses for incompressible materials

We present and review several models of fractional viscous stresses from the literature, which generalise classical viscosity theories to fractional orders by replacing total strain derivatives in time with fractional time derivatives. We also briefly introduce Prony-type approximations of these theories. Here, we investigate the issues of material frame-indifference and thermodynamic consistency for these models and find that on these bases, some are physically unacceptable. Next, we study elementary shearing and tensile motions, observing that some models are more convenient to use than others for the analysis of creep and relaxation. Finally, we compute the incremental stresses due to small-amplitude wave propagation in a deformed material, with a view to establish acoustoelastic formulas for prospective experimental calibrations.

Распространение магнитогидродинамических волн и гравитационная неустойчивость в астрофизической анизотропной и теплопроводной плазме

The hydrodynamic instability of a magnetised, self-gravitating rotating anisotropic plasma is analysed in the collisionless approximation and taking into account the heat flux vector on the basis of the modified Chew-Goldberger-Low equations. The dispersion relation is derived and simplified cases of propagation of small-amplitude perturbation waves are discussed, with modified analytical criteria for hydrodynamic instability obtained. In accordance with the general dispersion relation, three cases when the propagation of the perturbation wave passes across, along, and obliquely to the magnetic field vector are specifically considered. It is shown that anisotropic pressure and heat flux not only modify the classical Jeans instability criterion but also induce new unstable regions. It is found that the presence of uniform plasma rotation reduces the critical wave number and has a stabilising effect on the gravitational instability criterion for transverse propagation of the perturbation wave and has no effect in the case of its longitudinal propagation. The inclusion of thermal flows leads to the appearance of two additional wave modes. The results obtained are important for the construction of evolutionary magnetohydrodynamic models of collisionless astrophysical plasma.

Effect of initial conditions on a one-dimensional model of small-amplitude wave propagation in shallow water. II: Blowup for nonsmooth conditions

PurposeThe purpose of this paper is to analyze numerically the blowup in finite time of the solutions to a one-dimensional, bidirectional, nonlinear wave model equation for the propagation of small-amplitude waves in shallow water, as a function of the relaxation time, linear and nonlinear drift, power of the nonlinear advection flux, viscosity coefficient, viscous attenuation, and amplitude, smoothness and width of three types of initial conditions.Design/methodology/approachAn implicit, first-order accurate in time, finite difference method valid for semipositive relaxation times has been used to solve the equation in a truncated domain for three different initial conditions, a first-order time derivative initially equal to zero and several constant wave speeds.FindingsThe numerical experiments show a very rapid transient from the initial conditions to the formation of a leading propagating wave, whose duration depends strongly on the shape, amplitude and width of the initial data as well as on the coefficients of the bidirectional equation. The blowup times for the triangular conditions have been found to be larger than those for the Gaussian ones, and the latter are larger than those for rectangular conditions, thus indicating that the blowup time decreases as the smoothness of the initial conditions decreases. The blowup time has also been found to decrease as the relaxation time, degree of nonlinearity, linear drift coefficient and amplitude of the initial conditions are increased, and as the width of the initial condition is decreased, but it increases as the viscosity coefficient is increased. No blowup has been observed for relaxation times smaller than one-hundredth, viscosity coefficients larger than ten-thousandths, quadratic and cubic nonlinearities, and initial Gaussian, triangular and rectangular conditions of unity amplitude.Originality/valueThe blowup of a one-dimensional, bidirectional equation that is a model for the propagation of waves in shallow water, longitudinal displacement in homogeneous viscoelastic bars, nerve conduction, nonlinear acoustics and heat transfer in very small devices and/or at very high transfer rates has been determined numerically as a function of the linear and nonlinear drift coefficients, power of the nonlinear drift, viscosity coefficient, viscous attenuation, and amplitude, smoothness and width of the initial conditions for nonzero relaxation times.

Jeans Gravitational Instability of a Rotating Collisionless Magnetized Plasma with Anisotropic Pressure

The problem of self-gravitational instability of an astrophysical rotating plasma in a strong magnetic field with an anisotropic pressure tensor is studied on the basis of the Chew–Goldberger–Low (CGL) quasi-hydrodynamic equations modified by generalized polytropic laws. Using the general form of a dispersion relation obtained by the normal-mode perturbation method, a discussion is provided of the propagation of small-amplitude perturbation waves in an infinite homogeneous plasma medium for transverse, longitudinal, and oblique directions with respect to the magnetic field vector. It is shown that different polytropic indices and anisotropic pressures not only change the classical Jeans instability condition but also cause the appearance of new unstable regions. Modified Jeans instability criteria are obtained for isotropic MHD equations and anisotropic CGL equations owing to the influence of the polytropic indices on gravitational and firehose instabilities for astrophysical plasma. It is shown that in the case of a longitudinal mode of perturbation wave propagation, the Jeans instability criterion does not depend on uniform rotation. In the case of the transverse propagation regime, the presence of rotation reduces the critical wave number and exerts a stabilizing effect on the growth rate of the unstable regime.

Stratonovich–Khasminskii averaging principle for multiscale random Korteweg–de Vries-Burgers equation

In this paper, we consider the multiscale random Korteweg–de Vries-Burgers (KdVB) equation, namely, the KdVB equation perturbated by a fast time oscillating external force and a random oscillating noise, the random oscillating noise is made up of a family of strong mixing stationary processes with singular small parameters. A Stratonovich–Khasminskii type averaging principle for multiscale random KdVB equation is established, in physics, this averaging principle can describe the asymptotic behavior for the propagation of small-amplitude long waves in nonlinear dispersive and dissipative media with singular structure and the soliton propagation in the random weakly viscous media or in the random field.

Nonreciprocal vibrations of discretized finite elastic structures with spatiotemporally modulated material properties

Elastic waveguides with time- and space-dependent material properties have received great attention as a means to realize nonreciprocal propagation of small-amplitude mechanical waves in unbounded elastic media. Previous works have shown that propagating waves in a modulated medium violate reciprocity by means of asymmetric frequency and wave number conversion between two counterpropagating modes. We previously investigated nonreciprocal longitudinal and transverse vibrations in a finite elastic waveguide with time- and space-dependent material properties using a semi-analytical coupled-mode theory The model employs the mode shapes of the non-modulated beam as the orthogonal basis functions to describe the nonreciprocal vibrations of beams with spatiotemporally modulated properties. The present study extends previous work to consider vibrations of Euler beams with spatially discretized stiffness that are modulated in time and phased relative to each other to mimic spatiotemporal modulation. This is achieved by approximating the discretized stiffness and then employing the mode shapes of the beam with unmodulated discretized stiffness to predict the nonreciprocal vibrational response of the beam when subjected to spatiotemporal modulation. We perform a parametric study of the system parameters and discuss how this model can be used to define requirements of future systems that can be used for experimental demonstration.

Conserved vectors and solutions of the two-dimensional potential KP equation

Abstract This article investigates the potential Kadomtsev–Petviashvili (pKP) equation, which describes the evolution of small-amplitude nonlinear long waves with slow transverse coordinate dependence. For the first time, we employ Lie symmetry methods to calculate the Lie point symmetries of the equation, which are then utilized to derive exact solutions through symmetry reductions and with the help of Kudryashov’s method. The solutions obtained include exponential, hyperbolic, elliptic, and rational functions. Furthermore, we provide one-parameter group of transformations for the pKP equation. To gain a better understanding of the nature of each solution, we present 3D, 2D, and density plots. These obtained solutions, along with their associated physical characteristics, offer valuable insights into the propagation of small yet finite amplitude waves in shallow water.In addition, the pKP equation conserved vectors are derived by utilizing the multiplier method and the theorems by Noether and Ibragimov.

Propagation of ion-acoustic wave and its fractal representations in spin polarized electron plasma

Propagation of small-amplitude quantum ion-acoustic waves and its fractal representations is investigated in an electron-ion quantum plasma with separated spin electrons in the framework of the KdV and EKdV equations derived using reductive perturbation technique. These two equations are transformed into planar dynamical systems by using suitable traveling wave transformation. Qualitatively different phase portraits of these two systems are drawn with their respective Sagdeev’s pseudopotential curves and surface plots. Distinct orbits in the phase portraits give rise to distinct wave solutions. Periodic and superperiodic wave solutions are investigated numerically and solitary wave solutions are derived analytically. The effect of parameters such as Mach number, direction cosine, spin polarization and frequency ratio on these wave solutions are presented. For instance, it is seen that the spin density polarization ratio has an impressive effect on the amplitude of the wave solution, while the frequency ratio has no effect on its amplitude. Also, we have provided a physical explanation for our finding. Further, the dynamical features of the original system and reconstructed system using fractal interpolation function are investigated under the external forcing term. Chaotic and quasiperiodic phenomena are observed for different initial conditions. The results may help to better understand the degenerate electron gas that exists in dense astrophysical bodies.

Incremental equations and propagation of small amplitude waves for isotropic incompressible elastic bodies. The case the Hencky strain tensor is a function of the Cauchy stress tensor

For a relatively new class of constitutive equation for incompressible elastic bodies, wherein the Hencky strain tensor is a function of the Cauchy stress tensor, incremental equations are obtained, where it is assumed the presence of an initial time-independent stress that causes large deformations, to which a small time-dependent stress tensor is added. That stress tensor is assumed to cause a small (incremental) time-dependent deformation. For the case of infinite media the incremental equations are solved assuming travelling waves, and the speed of such waves is obtained for different problems considering initial homogeneous distributions of stresses and strains. Some numerical calculations of the speed of such waves are presented, considering a particular constitutive equation for rubber published recently in the literature. The speed of such small amplitude waves are compared with the predictions of the classical theory of nonlinear elasticity for similar problems.

Analysis of the propagation of small-amplitude waves in nonlinear elastic solids for problems with infinitesimal strains

A relatively new type of constitutive equation for elastic bodies, wherein the linearized strain tensor is a function of the Cauchy stress tensor, is studied for the particular case the stresses are divided into two parts. One is an initial time-independent stress to which is added an incremental (small in comparison with that initial stress) time-dependent stress. Incremental equations are derived that can be used to study the propagation of small-amplitude waves. The behaviour of such waves is studied for three boundary value problems, where the distributions of the initial stresses are inhomogeneous. The problems are: the inflation of a hollow sphere, the inflation of an infinitely long cylindrical annulus, and the circumferential shear and inflation of a cylindrical annulus. The behaviour of the small-amplitude waves is compared for different external loads, investigating how the initial stress affects such waves. A nonlinear model for a class of isotropic dry elastic rock is considered.

Lie Symmetries, Closed-Form Solutions, and Various Dynamical Profiles of Solitons for the Variable Coefficient (2+1)-Dimensional KP Equations

This investigation focuses on two novel Kadomtsev–Petviashvili (KP) equations with time-dependent variable coefficients that describe the nonlinear wave propagation of small-amplitude surface waves in narrow channels or large straits with slowly varying width and depth and non-vanishing vorticity. These two variable coefficients, Kadomtsev–Petviashvili (VCKP) equations in (2+1)-dimensions, are the main extensions of the KP equation. Applying the Lie symmetry technique, we carry out infinitesimal generators, potential vector fields, and various similarity reductions of the considered VCKP equations. These VCKP equations are converted into nonlinear ODEs via two similarity reductions. The closed-form analytic solutions are achieved, including in the shape of distinct complex wave structures of solitons, dark and bright soliton shapes, double W-shaped soliton shapes, multi-peakon shapes, curved-shaped multi-wave solitons, and novel solitary wave solitons. All the obtained solutions are verified and validated by using back substitution to the original equation through Wolfram Mathematica. We analyze the dynamical behaviors of these obtained solutions with some three-dimensional graphics via numerical simulation. The obtained variable coefficient solutions are more relevant and useful for understanding the dynamical structures of nonlinear KP equations and shallow water wave models.

The Cauchy problem for the nonlinear viscous Boussinesq equation in the Lq framework

In this paper, we study the viscous Boussinesq equation in the whole space Rn, which describes the propagation of small amplitude and long waves on the surface of water with viscous effects. Concerning the linear Cauchy problem, some qualitative properties of solutions including Lm−Lq estimates with 1⩽m⩽q⩽∞, inviscid limits and asymptotic profiles of solution with respect to the small viscosity are investigated by means of the Fourier analysis and the WKB method. For another, by applying some fractional order interpolations in the harmonic analysis, we derive the Lq well-posedness and estimates for small data solutions to the nonlinear viscous Boussinesq equation under some conditions for the parameter of nonlinearity.

Acoustoelastic analysis of soft viscoelastic solids with application to pre-stressed phononic crystals

The effective dynamic properties of specific periodic structures involving rubber-like materials can be adjusted by pre-strain, thus facilitating the design of custom acoustic filters. While nonlinear viscoelastic behaviour is one of the main features of soft solids, it has been rarely incorporated in the study of such phononic media. Here, we study the dynamic response of nonlinear viscoelastic solids within a ‘small-on-large’ acoustoelasticity framework, that is we consider the propagation of small amplitude waves superimposed on a large static deformation. Incompressible soft solids whose behaviour is described by the Fung–Simo quasi-linear viscoelasticity theory (QLV) are considered. We derive the incremental equations using stress-like memory variables governed by linear evolution equations. Thus, we show that wave dispersion follows a strain-dependent generalised Maxwell rheology. Illustrations cover the propagation of plane waves under homogeneous tensile strain in a QLV Mooney–Rivlin solid. The acoustoelasticity theory is then applied to phononic crystals involving a lattice of hollow cylinders, by making use of a dedicated perturbation approach. In particular, results highlight the influence of viscoelastic dissipation on the location of the first band gap. We show that dissipation shifts the band gap frequencies, simultaneously increasing the band gap width. These results are relevant to practical applications of soft viscoelastic solids subject to static pre-stress.

Two-phase hyperbolic model for porous media saturated with a viscous fluid and its application to wavefields simulation

A new hyperbolic two-phase model of a porous deformable medium saturated with a viscous fluid is presented and some of its features or performances are discussed. The governing equations are derived in the framework of Symmetric Hyperbolic Thermodynamically Compatible (SHTC) systems. The model accounts for such dissipative mechanisms as interfacial friction and viscous dissipation of the saturating fluid. Both dissipative mechanisms are modeled with hyperbolic equations with relaxation type source terms. The linearized version of the presented model is derived under the assumption of instantaneous relaxation of the phase pressures and its application to modeling propagation of small-amplitude waves in a porous medium saturated with a viscous fluid is discussed. We observe that accounting of the fluid viscosity may significantly affect the dispersion of fast and slow compression waves. Additionally, the shear wave attenuates rapidly due to the viscosity of the saturating fluid which makes it difficult to observe this wave in most of the test cases presented. However, some test cases are presented in which shear waves can be observed in the vicinity of interfaces between regions with different porosity.

Symmetry Analysis, Exact Solutions and Conservation Laws of a Benjamin–Bona–Mahony–Burgers Equation in 2+1-Dimensions

The Benjamin–Bona–Mahony equation describes the unidirectional propagation of small-amplitude long waves on the surface of water in a channel. In this paper, we consider a family of generalized Benjamin–Bona–Mahony–Burgers equations depending on three arbitrary constants and an arbitrary function G(u). We study this family from the standpoint of the theory of symmetry reductions of partial differential equations. Firstly, we obtain the Lie point symmetries admitted by the considered family. Moreover, taking into account the admitted point symmetries, we perform symmetry reductions. In particular, for G′(u)≠0, we construct an optimal system of one-dimensional subalgebras for each maximal Lie algebra and deduce the corresponding (1+1)-dimensional nonlinear third-order partial differential equations. Then, we apply Kudryashov’s method to look for exact solutions of the nonlinear differential equation. We also determine line soliton solutions of the family of equations in a particular case. Lastly, through the multipliers method, we have constructed low-order conservation laws admitted by the family of equations.

Numerical treatment of Benjamin-Bona-Mahony-Burgers equation with fourth-order improvised B-spline collocation method

In this work, the Benjamin-Bona-Mahony-Burgers (BBMB) equation is solved using an improvised cubic B-spline collocation technique. This equation describes the propagation of small amplitude waves in a nonlinear dispersive medium, in the modeling of unidirectional planar waves. Due to the higher smoothness and sparse nature of matrices corresponding to splines, cubic B-splines are chosen as the basis function in the collocation method. But, the optimal accuracy and order of convergence cannot be achieved using the standard B-spline collocation method. So to overcome this, improvised cubic B-splines are formed by making posteriori corrections to cubic B-spline interpolant and its higher-order derivatives. The Crank-Nicolson scheme is used to discretize the temporal domain along with the quasilinearization process to deal with the nonlinear terms. The spatial domain discretization is carried out using the improvised cubic B-spline collocation method (ICSCM). The stability analysis of the technique is performed using the von-Neumann scheme. Several test problems are solved numerically and obtained results are compared with the results available in the literature. The aim of the paper is to show that such improvised techniques which were earlier used to solve ODEs, can be applied to solve the BBMB equation also, with excellent accuracy in results.

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.

DEM analysis on the stress wave response of spherical particle assemblies under triaxial compression

It is challenging to understand the stress wave response of cohesionless particle assemblies including compression and shear wave velocities (Vp and Vs) due to complicated particle-scale interactions particularly during a loading process. This contribution adopts the discrete element method to simulate triaxial compression experiments using spherical particles. Four samples are isotropically confined at various initial packing densities and then sheared monotonically up to the critical state. Small-amplitude wave propagation simulation is performed during shearing along the axis of loading. The results reveal that Vp is affected by the major principal stress and the coordination number contributing to the vertical direction rather than horizontal direction, while Vs is more influenced by the geometric mean stress and the mean coordination number. The wave velocity ratio (Vp/Vs) is well correlated with the fabric anisotropy, having a linear relationship. This enables prediction of microscopic change in fabric from macroscopic wave velocity values. Besides, a material-specific relationship between stress levels and wave velocities is found at the critical state with a unique fabric anisotropy, independent of initial packing. Vs in the isotropic state is found to be always larger that at the critical state under an equivalent stress level.

Diffuse interface approach to modeling wavefields in a saturated porous medium

We present a model and numerical technique method for simulation a small-amplitude wave propagation in the fluid-saturated porous medium in regions with different porosity, including domains with pure fluid and pure solid. The governing equations are derived from the general hyperbolic thermodynamically compatible model of compressible fluid flow in a deformed porous medium. The resulting linear partial differential equations (PDE) form the symmetric hyperbolic system written for velocities, relative velocities, pressure and shear stress, which ensures application of a high order finite difference scheme on a staggered grid. A diffuse interface approach is applied to simulate wavefields in regions with interfaces between media with different porosity, including pure phases. The latter allows computations with the use of a single PDE system on rectangular grids without cumbersome interface tracking. Numerical experiments confirming the efficiency of the proposed approach are presented.

Small amplitude rayleigh-lamb wave propagation in a finitely deformed viscoelastic dielectric elastomer (DE) layer

In recent years, Dielectric elastomers (DEs) have received much interest for the dynamics applications as waveguide. However, predicting the wave propagation behavior in a DE medium is very challenging due to the complexity of the problem, which involves the delicate interplay between electromechanical coupling, large deformation, and particularly the intrinsic material viscoelasticity. In this paper, by integrating the state-of-art finite-deformation viscoelasticity theory into the framework of small-amplitude wave propagation superimposed on a finitely deformed medium, the Rayleigh-Lamb wave propagation in a viscoelastic DE medium is investigated. Simulation results demonstrate the effects of material viscosity, status of relaxation, external electrical load, and mechanical pre-stretch on the dispersion behavior of the wave. It is found that for both purely elastic and viscoelastic DE media, waves with certain frequencies could be filtered by actively tuning the electrical loads. Moreover, some interesting findings conclude that the material viscoelasticity may cause some significant changes in the wave dispersion behavior. Therefore, incorporating the material viscosity in modeling DE waveguide is expected to provide more accurate prediction on their performance. This work will help to better understand the fundamentals of wave propagation in DE media and trigger more innovative and optimal design for DE waveguide.