Shock Waves in Anisotropic Cylinders

We study small-amplitude longitudinal and torsional shock waves in circular cylinders consisting of an anisotropic medium such that the velocities of the longitudinal and torsional waves are close to each other. Previously, simple waves were considered in the same situation and conditions were found for these waves to overturn and for the corresponding shock waves to form. Here we present the study of shock waves: the shock adiabat and the evolutionary conditions. The results obtained can also be related to shock waves in unbounded media with quadratic nonlinearity.     

On the Transition from Two-Dimensional to Three-Dimensional Water Waves

There are essentially two types of three-dimensional water waves: waves that bifurcate from the state of rest (these waves are commonly called short-crested waves or forced waves), and waves that bifurcate from a two-dimensional wave of finite amplitude (these waves are sometimes called spontaneous waves). This paper deals with spontaneously generated three-dimensional waves. To understand this phenomenon better from a mathematical point of view, it is helpful to work on model equations rather than on the full equations. Such an attempt was made formally by Martin in 1982 on the nonlinear Schrödinger equation, but it is shown here that it is hard to justify his results mathematically because of the hyperbolicity of the nonlinear Schrödinger equation for gravity waves. On the other hand, in some parameter regimes, the nonlinear Schrödinger equation becomes elliptic. In that case, the appearance of spontaneous three-dimensional waves can be shown rigorously by using a dynamical systems approach. The results are extended to the Benney–Roskes–Davey–Stewartson equations when they are both elliptic. Various types of three-dimensional waves bifurcating from a two-dimensional periodic wave are obtained.     

Geometric Aspects of Spatially Periodic Interfacial Waves

Periodic waves at the interface between two inviscid fluids of differing densities are considered from a geometric point of view. A new Hamiltonian formulation is used in the analysis and restriction of the Hamiltonian structure to space-periodic functions leads to an O -invariant Hamiltonian system. Motivated by the simplest O -invariant Hamiltonian system, the spherical pendulum, we analyze the properties of traveling waves, standing waves, interactions between standing and traveling waves (mixed waves) and time-modulated spatially periodic waves. A singularity in the bifurcation of traveling waves leads to a nonlinear resonance and this is investigated numerically.     

Propagation of plane waves in microstretch fluid

The propagation of plane harmonic waves are studied in a microstretch fluid medium. It is found that five basic waves can propagate at distinct speeds in an infinite linear homogeneous isotropic microstretch fluid. Out of these five waves, one is a longitudinal micro-rotational wave, two are coupled longitudinal waves and remaining two are coupled transverse waves. The longitudinal micro-rotational wave travels independently and is not influenced by the microstretching of the medium, while the coupled longitudinal waves arise due to the presence of microstretching and coupled transverse waves arise due to the presence of micro-rotation in the medium. Speed of propagation of all the waves are found to be complex valued and dispersive at low frequency, but almost non-dispersive at high frequency. Due to complex valued speeds of propagation, all the waves are attenuating but differently. Coupled sets of longitudinal waves reduce to a longitudinal wave of micropolar fluid in the absence of microstretching. Reflection phenomena of a set of coupled longitudinal waves incident obliquely at the free surface of a microstretch fluid half-space has been investigated. Closed formulae for the reflection coefficients are presented and computed numerically for a particular medium. The real and imaginary parts of the complex speeds of all the waves and their corresponding attenuation coefficients have also been studied numerically and depicted graphically against frequency parameter.     

An Analytical Model of Periodic Waves in Shallow Water

An explicit, analytical model is presented of finite-amplitude waves in shallow water. The waves in question have two independent spatial periods, in two independent horizontal directions. Both short-crested and long-crested waves are available from the model. Every wave pattern is an exact solution of the Kadomtsev-Petviashvili equation, and is based on a Riemann theta function of genus 2. These biperiodic waves are direct generalizations of the well-known (simply periodic) cnoidal waves. Just as cnoidal waves are often used as one-dimensional models of “typical” nonlinear, periodic waves in shallow water, these biperiodic waves may be considered to represent “typical” nonlinear, periodic waves in shallow water without the assumption of one-dimensionality.     

Wave propagation in thermo-chiral elastic medium

The propagation of time harmonic waves through an infinite thermo-chiral elastic material has been investigated. The elastic field of thermo-chiral medium has been described by extending the governing equations and constitutive relations of hemitropic micropolar material to include temperature field. Seven basic waves consisting of three coupled dilatational waves and four coupled shear waves traveling with distinct speeds may exist in the medium. All the waves are found to be dispersive, however the coupled dilatational waves are attenuating and temperature dependent, while the coupled shear waves are independent of temperature field. The phase speeds and corresponding attenuation quality factors of all the coupled dilatational waves have been computed numerically for a specific model. The effect of chirality and temperature field have been shown graphically.     

Bounded traveling waves of the Burgers-Huxley equation

In order to investigate bounded traveling waves of the Burgers-Huxley equation, bifurcations of codimension 1 and 2 are discussed for its traveling wave system. By reduction to center manifolds and normal forms we give conditions for the appearance of homoclinic solutions, heteroclinic solutions and periodic solutions, which correspondingly give conditions of existence for solitary waves, kink waves and periodic waves, three basic types of bounded traveling waves. Furthermore, their evolutions are discussed to investigate the existence of other types of bounded traveling waves, such as the oscillatory traveling waves corresponding to connections between an equilibrium and a periodic orbit and the oscillatory kink waves corresponding to connections of saddle-focus.     

Exact Loop Wave Solutions and Cusp Wave Solutions of the Fujimoto-Watanabe Equation

In this paper, the theory of dynamical systems is employed to investigate loop waves and cusp waves of the Fujimoto-Watanabe equation. These waves contain solitary loop waves, periodic loop waves, peakons and periodic cusp waves. Under fixed parameter conditions, their exact explicit parametric expressions are given.     

Stability of viscous shock waves associated with a system of nonstrictly hyperbolic conservations laws

We introduce a simple model of two conservation laws which is strictly hyperbolic except for a degenerate parabolic line in the state space. Besides classical shock waves, it also exhibits overcompressive, marginal overcompressive, and marginal undercompressive shock waves. Our purpose is to study the behavior of the corresponding viscous waves, in particular the manner in which these waves are stable. There are several basic differences between classical shock waves and other types of shock waves. A perturbation of an overcompressive shock wave gives rise to a new wave. Monotone marginal overcompressive waves behave distinctly from the nonmonotone ones. Analytical techniques used in our study include characteristic-energy and weighted-energy methods, and nonlinear superposition through time-invariants. Although we carry out our analysis for a simple model, the general phenomena would hold for overcompressive waves which occur in other physical models.     

Butterfly-like waves in the nonlinear Schrödinger equation with a combined dispersion term

All possible optical waves in the nonlinear Schrödinger equation with a combined dispersion term are determined according to different parameters’ regions. First, we find this equation has some popular exotic solutions, such as peaked waves, looped and cusped waves. What is more interesting, this equation admits some very particular waves, such as double kinked waves and butterfly-like waves. These new two types of solutions have not been reported in the literature regarding the study of other nonlinear equations.     

Some Results for Surface Gravity Waves on Shear Flows

This work attempts to fill some gaps in the subject of steadysurface gravity waves on two-dimensional flows in which thevelocity varies with depth, as is the case for waves propagatingon a flowing stream. Following most previous work the theoryis basically inviscid, for the shear is assumed to be producedby external effects: the theory examines the non-viscous interactionbetween wave disturbances and the shear flow. In particular,some results are obtained for the dispersion relationship forsmall waves on a flow of arbitrary velocity distribution, andthis is generalized to include the decay from finite disturbancesinto supercritical flows. An exact operator equation is developedfor all surface gravity waves for the particular case of flowwith constant vorticity; this is solved to give first-orderequations for solitary and cnoidal waves in terms of channelflow invariants. Exact numerical solutions are obtained for small waves on sometypical shear flows, and it is shown how the theory can predictthe growth of periodic waves upon a stream by the developmentof a fully-turbulent velocity profile in flow which was originallyirrotational and supercritical. Results from all sections of this work show that shear is animportant quantity in determining the propagation behaviourof waves and disturbances. Small changes in the primary flowmay alter the nature of the surface waves considerably. Theymay in fact transform the waves from one type to another, correspondingto changes in the flow between super- and sub-critical statesdirectly caused by changes in the velocity profile.     

Do peaked solitary water waves indeed exist?

Many models of shallow water waves, such as the famous Camassa–Holm equation, admit peaked solitary waves. However, it is an open question whether or not the widely accepted peaked solitary waves can be derived from the fully nonlinear wave equations. In this paper, a unified wave model (UWM) based on the symmetry and the fully nonlinear wave equations is put forward for progressive waves with permanent form in finite water depth. Different from traditional wave models, the flows described by the UWM are not necessarily irrotational at crest, so that it is more general. The unified wave model admits not only the traditional progressive waves with smooth crest, but also a new kind of solitary waves with peaked crest that include the famous peaked solitary waves given by the Camassa–Holm equation. Besides, it is proved that Kelvin’s theorem still holds everywhere for the newly found peaked solitary waves. Thus, the UWM unifies, for the first time, both of the traditional smooth waves and the peaked solitary waves. In other words, the peaked solitary waves are consistent with the traditional smooth ones. So, in the frame of inviscid fluid, the peaked solitary waves are as acceptable and reasonable as the traditional smooth ones. It is found that the peaked solitary waves have some unusual and unique characteristics. First of all, they have a peaked crest with a discontinuous vertical velocity at crest. Especially, unlike the traditional smooth waves that are dispersive with wave height, the phase speed of the peaked solitary waves has nothing to do with wave height, but depends (for a fixed wave height) on its decay length, i.e., the actual wavelength: in fact, the peaked solitary waves are dispersive with the actual wavelength when wave height is fixed. In addition, unlike traditional smooth waves whose kinetic energy decays exponentially from free surface to bottom, the kinetic energy of the peaked solitary waves either increases or almost keeps the same. All of these unusual properties show the novelty of the peaked solitary waves, although it is still an open question whether or not they are reasonable in physics if the viscosity of fluid and surface tension are considered.     

Effect of micro-inertia in the propagation of waves in micropolar thermoelastic materials with voids

The effect of micro-inertia in the propagation of waves in micropolar thermoelastic materials with voids has been investigated. Elastic waves are reflected due to incident coupled longitudinal and coupled shear waves from a plane free boundary of micropolar thermoelastic materials with voids. The amplitude ratios corresponding to the reflected coupled longitudinal and coupled shear waves are derived by using appropriate boundary conditions. Energy partition in the free surface has been presented. The amplitude and energy ratios of the reflected waves are also computed numerically for a particular model.     

Rayleigh wave scattering due to a rigid barrier in a liquid halfspace

The paper presents a theoretical formulation for studying scattering of Rayleigh waves due to the presence of rigid barriers in oceanic waters. The Wiener-Hopf technique has been employed to solve the problem. Exact solution has been obtained in terms of Fourier integrals whose evaluation gives the reflected, transmitted and scattered waves. The scattered waves have the behaviour of cylindrical waves originating at the edge of the barrier. Numerical results for the amplitude of the scattered waves have been obtained for small depth of the barrier.     

Sur la magnétohydrodynamique anisotrope relativiste

Summary This paper is concerned with the relativistic phenomenological theory of anisotropic magnetohydrodynamics. An anisotropic fluid scheme is defined and studied. The main system of anisotropic magnetohydroldynamics is deduced. This system may describe a collisionless anisotropic plasma embedded in a strong magnetic field. The main system is shown to yield to three types of waves as in isotropic (perfect) magnetohydrodynamics: the entropic waves, the magnetosonic waves and the Alfven waves. For the rays associated respectively to the magnetosonic and Alfven waves the fundamental property concerning the propagation of infinitesimal discontinuities of variables is established. The conditions under which the velocities of propagation of magnetosonic and Alfven waves are real are derived: these conditions imply as in the classical theory the absence of fire hose and mirror instabilities in the fluid. The study of wave cones allows, on the one hand to point out some particularities of the propagation of waves in anisotropic magnetohydrodynamics, and on the other hand to clear up the hyperbolicity character of differential operators associated to various waves.

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Entrata in Redazione il 23 aprile 1975.     

Solitary waves and the structures of discontinuities in non-dissipative models with complex dispersion

Stationary solutions of reversible evolutionary equations of mechanics with higher derivatives are analysed. A two-dimensional graphical method for investigating the solutions of systems of ordinary differential equations is described, which enables one to find special types of solutions: periodic waves, solitary waves and the structures of discontinuities. At the same time, solitary waves can be obtained by taking the limit of sequences of periodic waves and the structures of discontinuities obtained by taking the limit of sequences of solitary waves. This general approach has enabled the existence of all earlier predicted structures to be verified has enabled new types of structures (three-wave structures) to be revealed and has enabled all the necessary conditions at the discontinuities to be found. All the previously known types of solitary waves are found and new types of solitary waves are revealed (generalized ordinary and 1:1 multisolitons). Methods of finding generalized solitary waves, including those with a finite amplitude of the periodic component, are determined. Examples of the solution of the following problems are given for a fourth-order system: generalized solitary waves as the limiting solutions of two-wave resonance solutions, generalized solitary waves and the structure of a discontinuity with three waves, a 1:1 soliton and the structure of a discontinuity with a single radiated wave, a solitary wave with fixed propagation velocity, and the structure of a discontinuity in the form of a kink with radiation. A generalized 1:1 soliton and the structure of a discontinuity with two radiated waves is considered in the case of sixth-order systems. The discussion is mainly based on the example of travelling waves described by the generalized Korteweg-de Vries equations. Other models with complex dispersion (a plasma and a stratified fluid) are also considered.     

Numerical simulation of oblique and multidirectional wave propagation and breaking on steep slope based on FEM model of Boussinesq equations

Creating a representative numerical simulation of the propagation and breaking of waves along slopes is an important problem in engineering design. Most studies on wave breaking have focused on the propagation of normal incident waves on gentle slopes. In practice, however, waves on steep slopes are obliquely incident or multidirectional irregular waves. In this paper, the eddy viscosity term is introduced to the momentum equation of the improved Boussinesq equations to model wave dissipation caused by breaking and friction, and a numerical model based on an unstructured finite element method (FEM) is established based on the governing equations. It is applied to simulate wave propagation on a steep slope of 1:5. Parallel physical experiments are conducted for comparative analysis that considered a large number of cases, including those featuring of normal and oblique incident regular and irregular waves, and multidirectional waves. The heights of the incident wave increase for different periods to represent different kinds of waves breaking. Based on examination, the effectiveness and accuracy of the numerical model is verified through a comprehensive comparison between the numerical and the experimental results, including in terms of variation in wave height, wave spectrum, and nonlinear parameters. Satisfactory agreement between the numerical and experimental values shows that the proposed model is effective in representing the breaking of oblique incident regular waves, irregular waves, and multidirectional incident irregular waves. However, the initial threshold of the breaking parameter η_t^(I) takes different values for oblique and multidirectional waves. This needs to be paid attention when the breaking of waves is simulated using the Boussinesq equations.     

Ground waves in atomic chains with bi-monomial double-well potential

Ground waves in atomic chains are traveling waves that corresponds to minimal non-trivial critical values of the underlying action functional. In this paper we study FPU-type chains with bi-monomial double-well potential and prove the existence of both periodic and solitary ground waves. To this end we minimize the action on the Nehari manifold and show that periodic ground waves converge to solitary ones. Finally, we compute ground waves numerically by a suitable discretization of a constrained gradient flow.     

Surface waves in electrostrictive materials under biasing fields

The theory of small incremental fields superposed on finite biasing fields in an electroelastic body is summarized first in this paper. The Stroh formalism for two-dimensional anisotropic elasticity and piezoelectricity is generalized to two-dimensional problems of an electroelastic body under biasing mechanical and electric fields. Application of this generalized Stroh formalism to study surface waves propagating in an electroelastic half-space under uniform initial fields indicates that there exist at most two surface waves even in the presence of biasing fields and these two surface waves correspond to the well-known Rayleigh and Bleustein waves in the mechanical nature. A simple procedure is established to distinguish these two types of waves. Application of the results to a model electrostrictive material is included for illustration. It is found that initial biasing fields can greatly influence propagation feature of surface waves.