Solitary waves in a peridynamic elastic solid

The propagation of large amplitude nonlinear waves in a peridynamic solid is analyzed. With an elastic material model that hardens in compression, sufficiently large wave pulses propagate as solitary waves whose velocity can far exceed the linear wave speed. In spite of their large velocity and amplitude, these waves leave the material they pass through with no net change in velocity and stress. They are nondissipative and nondispersive, and they travel unchanged over large distances. An approximate solution for solitary waves is derived that reproduces the main features of these waves observed in computational simulations. It is demonstrated by numerical studies that the waves interact only weakly with each other when they collide. Wavetrains composed of many non-interacting solitary waves are found to form and propagate under certain boundary and initial conditions.     

Non-linear waves in a viscous fluid contained in an elastic tube with variable cross-section

In the present work, treating the large arteries as a thin-walled, long and circularly cylindrical, prestressed elastic tube with variable cross-section and using the reductive perturbation method, we have studied the amplitude modulation of non-linear waves in such a fluid-filled elastic tube. By considering the blood as an incompressible viscous fluid, the evolution equation is obtained as the dissipative non-linear Schrödinger equation with variable coefficients. It is shown that this type of equations admit a solitary wave solution with a variable wave speed. It is observed that, the wave speed increases with distance for narrowing tubes while it decreases for expanding tubes.     

Solitary waves on the surface of a viscoelastic fluid running down an inclined plane

In this paper, we study the existence and the role of solitary waves in the finite amplitude instability of a layer of a second-order fluid flowing down an inclined plane. The layer becomes unstable for disturbances of large wavelength for a critical value of Reynolds number which decreases with increase in the viscoelastic parameter M. The long-term evolution of a disturbance with an initial cosinusoidal profile as a result of this instability reveals the existence of a train of solitary waves propagating on the free surface. A novel result of this study is that the number of solitary waves decreases with in crease in M. When surface tension is large, we use dynamical system theory to describe solitary waves in a moving frame by homoclinic trajectories of an associated ordinary differential equation.     

Acceleration waves in constrained elastic rods

The problem of the propagation and decay of acceleration waves in nonlinear hyperelastic rods, subject to a general class of constraints, is treated herein. The growth-decay equation, clearly showing the effect of the constraints, is derived for all waves which can propagate in the rod. For a certain class of constraints, general enough to include most of the practical applications, the wave decay equation is found to have the same form as for a rod without constraints, provided the rod is initially at rest.     

Helical traveling waves in elastic rods

We pose the problem on free harmonic bending vibrations of an infinite rotating tubular elastic rod with an internal fluid flow, prestressed by a torque and a longitudinal force. We show that these vibrations can only be realized as traveling circular helical waves. It is shown that, for any wavelength, there exist four waves, two having the form of a left-handed helix and the other two having the form of a right-handed helix. Each of these waves propagates in the positive and negative directions of the longitudinal rod axis at different velocities. These phenomena can manifest themselves in deep-hole drill columns.     

Numerical study of formation of solitary strain waves in a nonlinear elastic layered composite material

Propagation of nonlinear strain waves through a layered composite material is considered. The governing macroscopic wave equation for the long-wave case was obtained earlier by the higher-order asymptotic homogenization method (Andrianov et al., 2013). Non-stationary dynamic processes are investigated by a pseudo-spectral numerical procedure. The time integration is performed by the Runge–Kutta method; the approximation with respect to the spatial co-ordinate is provided by the Fourier series expansion. The convergence of the Fourier series is substantially improved and the Gibbs–Wilbraham phenomenon is reduced with the help of Padé approximants. As result, we explore how fast and under what conditions the solitary strain waves can be generated from an initial excitation. The numerical and analytical solutions (when the latter can be obtained) are in good agreement.     

Decay of acceleration waves in elastic rods

The problem of the propagation and decay of acceleration waves in nonlinear hyperelastic rods is treated herein. The general growth-decay law governing wave strength is obtained for all waves which can propagate in the rod. An expression for the induced higher order wave is also obtained. The forms taken by the law of growth or decay in a number of special cases are given.     

Geometrical nonlinear waves in finite deformation elastic rods

IntroductionSomenewphenomenaofnonlinearwavesinthesolidmediumsuchasshockwave ,solitarywaveetc.arepaidmoreattentiontoincreasinglybyresearchersbecausetheytakeonalotofimportantproperties.ItistheoreticallyanalyzedinRefs.[1 -6]thattheformationmechanismsofshockwaveandsolitarywaveintheelasticthinrodsaswellastheirpropagationproperties.TheexistenceofsolitarywaveintheelasticmediumsuchasarodandaplatehasbeenverifiedinRef.[7]byexperiments.Shockwaveandsolitarywavearesteadilypropagatingtraveling_wavesgenerat…     

Frequency related localisation of harmonic elastic waves in stratified welds

The paper presents a computational model for elastic waves in a structured weld adjacent to the free surface of an elastic solid. The main emphasis is on the interaction of waves with the micro-structure of the weld. Effects of localisation and channeling of waves are addressed. A model of a grain structure within the weld is also considered.     

Non-linear waves in heterogeneous elastic rods via homogenization

We consider the propagation of a planar loop on a heterogeneous elastic rod with a periodic microstructure consisting of two alternating homogeneous regions with different material properties. The analysis is carried out using a second-order homogenization theory based on a multiple scale asymptotic expansion.     

Nonlinear elastic shear waves

Summary The dynamics of one-dimensional two-component shear motion in elastic-isotropic homogeneous media is studied assuming isentropic finite displacements. Wave breaking of initially continuous waves on the infinite interval is discussed for weakly nonlinear waves.The problem of a resonating finite-thickness shear layer in primary resonance for single-component motion exhibits jump discontinuities of particle velocity, shear strain and stress in a finite frequency band near primary resonance.Under certain conditions two-component motion can be reduced to a quasi-single-component motion.     

A unified intrinsic functional expansion theory for solitary waves

A new theory is developed here for evaluating solitary waves on water, with results of high accuracy uniformly valid for waves of all heights, from the highest wave with a corner crest of 120°down to very low ones of diminishing height. Solutions are sought for the Euler model by employing a unified expansion of the logarithmic hodograph in terms of a set of intrinsic component functions analytically determined to represent all the intrinsic properties of the wave entity from the wave crest to its outskirts. The unknown coefficients in the expansion are determined by minimization of the mean-square error of the solution, with the minimization optimized so as to take as few terms as needed to attain results as high in accuracy as attainable. In this regard, Stokes‘s formula, F^2μπ= tanμπ, relating the wave speed (the Froude number F) and the logarithmic decremen t# of its wave field in the outskirt, is generalized to establish a new criterion requiring (for minimizing solution error) the functional expansion to contain a finite power series in M terms of Stokes‘s basic term (singular in #), such that 2Mμ is just somewhat beyond unity, i.e. 2Mμ≌ 1. This fundamental criterion is fully validated by solutions for waves of various amplitude-to-water depth ratio α= a/h, especially about α≌0.01, at which M = 10by the criterion.In this pursuit, the class of dwarf solitary waves, defined for waves with ≌≤ 0.01, is discovered as a group of problems more challenging than even the highest wave. For the highest wave, a new solution is determined here to give the maximum height αhst = 0.8331990, and speed Fhst = 1.290890, accurate to the last significant figure, which seems to be a new record.     

Efficient computation of steady solitary gravity waves

An efficient numerical method to compute solitary wave solutions to the free surface Euler equations is reported. It is based on the conformal mapping technique combined with an efficient Fourier pseudo-spectral method. The resulting nonlinear equation is solved via the Petviashvili iterative scheme. The computational results are compared to some existing approaches, such as Tanaka’s method and Fenton’s high-order asymptotic expansion. Several important integral quantities are computed for a large range of amplitudes. The integral representation of the velocity and acceleration fields in the bulk of the fluid is also provided.     

A novel method for investigation of acoustic and elastic wave phenomena using numerical experiments

The emergence of new types of composite materials, the depletion of existing hydrocarbon deposits, and the increase in the speed of trains require the development of new research methods based on wave scattering. Therefore, it is necessary to determine the laws of wave scattering in inhomogeneous media. We propose a method that combines the advantages of a numerical simulation with an analytical study of the boundary value problem of elastic and acoustic wave equations. In this letter we present the results of the study using the proposed method: the formation of a response from a shear wave in an acoustic medium and the formation of shear waves when a vertically incident longitudinal wave is scattered by a vertical gas-filled fracture. We have obtained a number of analytical expressions characterising the scattering of these wave types.     

Degenerate weakly non-linear elastic plane waves

Weakly non-linear plane waves are considered in hyperelastic crystals. Evolution equations are derived at a quadratically non-linear level for the amplitudes of quasi-longitudinal and quasi-transverse waves propagating in arbitrary anisotropic media. The form of the equations obtained depends upon the direction of propagation relative to the crystal axes. A single equation is found for all propagation directions for quasi-longitudinal waves, but a pair of coupled equations occurs for quasi-transverse waves propagating along directions of degeneracy, or acoustic axes. The coupled equations involve four material parameters but they simplify if the wave propagates along an axis of material symmetry. Thus, only two parameters arise for propagation along an axis of twofold symmetry, and one for a threefold axis. The transverse wave equations decouple if the axis is fourfold or higher. In the absence of a symmetry axis it is possible that the evolution equations of the quasi-transverse waves decouple if the third-order elastic moduli satisfy a certain identity. The theoretical results are illustrated with explicit examples.     

Explicit conditions for the existence of exceptional body waves and subsonic surface waves in anisotropic elastic solids

It is known that a subsonic surface (Rayleigh) wave exists in an anisotropic elastic half-space x₂  0 if the first transonic state is not of Type 1. If the first transonic state is of Type 1 but the limiting wave is not exceptional, a subsonic surface wave exists. If the first transonic state is of Type 1 and the limiting wave is exceptional, a subsonic surface wave exists when . It is shown that an exceptional body wave is necessarily an exceptional transonic wave, and could be an exceptional limiting wave. Only two wave speeds are possible for an exceptional body wave. We present explicit conditions in terms of the reduced elastic compliances for the existence of an exceptional body wave. If an exceptional body wave exists, conditions are given for identifying whether the transonic state is of Type 1. Hence, through the existence of an exceptional body wave we provide explicit conditions for the existence of a subsonic surface wave with the exception when needs to be computed.     

Water waves in an elastic vessel

Linear and nonlinear analyses of water waves in an elastic vessel are carried out to study the dramatic phenomena of Dragon Wash as well as related controllable experiments. It is proposed that the capillary edge waves are generated by parametric resonance, which is shown to be a possible mechanism for both rectangular an circular vessels. For circular vessel, the normal geometric resonance is also operating, thus greatly enhance the dramatic effect. The mechanism of nonlinear mode-mode interaction is proposed for the generation of axisymmetric low-frequency gravity waves by the high- frequency external excitation. A simple model system is studied numerically to demonstrate explicitly this interaction mechanism.     

Analytic solutions of simple waves

B.Riemann furnished the general solution of simple waves in1860.But it is difficult to find out the exact forms of the arbitrary function contained in the general solution which must satisfy boundary or initial conditions.For this reason it is inconvenient to probe into the characteristics of concrete problems.In this paper the analytic solutions of simple waves are afforded according to the geometric theory of quasi-linear partial differential equation,and they are determined with boundary or initial conditions.By using these solutions the specific properties of certain flows are discussed and novel results are obtained.     

The stability of finite-amplitude interfacial solitary waves

The linear stability of finite-amplitude interfacial gravity solitary waves propagating in a two-layer fluid is investigated analytically focusing on the occurrence of an exchange of stability. We make an asymptotic analysis for small growth rates of infinitesimal disturbances, and explicitly obtain their growth rates near an exchange of stability. The result indicates that an exchange of stability occurs at every stationary value of the total energy of the solitary waves. It also gives us information whether the number of growing modes increases or decreases after experiencing the exchange of stability. We apply these analytical results to specific interfacial solitary waves, and find various features on their stability that are not seen in the case of surface solitary waves.