A uniform asymptotic analysis of dispersive wave motion across a space-time shadow boundary

The one dimensional Klein-Gordon equation with spatially varying coefficients and with amplitude modulated high frequency signaling data is analyzed. A formal uniformly valid asymptotic expansion of the solution across a space-time shadow boundary is obtained with the help of two families of rays. These rays may also give rise to shadow regions. The asymptotic expansion involves three functions, a Fresnel function and two successive Bessel functions of integer order.     

A non-hydrostatic numerical scheme for dispersive waves generated by bottom motion

A numerical scheme based on the staggered finite volume method is presented at the aim of studying surface waves generated by a bottom motion. We address the 2D Euler equations in which the vertical domain is resolved only by one layer. The resulting non-hydrostatic scheme is used to simulate surface waves generated by bottom motion in a water tank. Here we mimic Hammack experiments numerically, in which a bed section is moved upwards or downwards, resulting in transient dispersive waves. For an impulsive downward bottom thrust, free surface responds in terms of a negative leading wave, followed with dispersive train of waves. For an upward bottom thrust, amplitude of the leading wave decays as the wave propagates, and no wave of permanent form evolves— instead, there appears a train of solitons. In this article, we show that our numerical scheme can produce the correct wave profiles, comparable with the analytical and experimental results of Hammack. Simulations using intermediate and slow bottom motions are also presented. In addition, we perform a simulation of a wave generated by submerged landslide, that compares well against previous numerical simulations. Via this simulation, we demonstrate that our scheme can incorporate a moving wet–dry boundary algorithm in the run-up simulation.     

A non-reflective spectral wave maker for SPH modeling of nonlinear wave motion

A momentum source function derived from the linear wave theory is introduced into the weakly compressible smoothed particle hydrodynamic (SPH) model for the long-time simulation of regular and irregular wave generation problems. Wave absorption is realized by adding a velocity attenuation term into the governing momentum equation. The performances of the wave maker are tested under different wave conditions. The wave maker is then applied to the study of the challenging processes, such as random wave breaking on the reef-face or the reef-crest, wave setup and spectral transfer of wave energy from the peak frequency to lower frequencies over the reef-flat. The predicted results are compared with the experimental data and a good agreement is obtained. The SPH model with a non-reflective spectral wave maker can be used as a practical tool for studying wave interaction with coastal structures.     

Finite volume methods for unidirectional dispersive wave models

We extend the framework of the finite volume method to dispersive unidirectional water wave propagation in one space dimension. In particular, we consider a KdV–BBM‐type equation. Explicit and implicit–explicit Runge–Kutta‐type methods are used for time discretizations. The fully discrete schemes are validated by direct comparisons to analytic solutions. Invariants' conservation properties are also studied. Main applications include important nonlinear phenomena such as dispersive shock wave formation, solitary waves, and their various interactions. Copyright © 2012 John Wiley & Sons, Ltd.     

A review of wave motion in anisotropic and cracked elastic-media

Recent developments in the theory and calculation of wave propagation in anisotropic media have been published in the geophysical literature and refer specifically to seismological applications. Anisotropic phenomena are comparatively common, and it is the intention of this review to present these developments to a wider audience. Few of the results are new, but the opportunity is taken to tidy up a few loose ends, and present consistent theoretical formulations for the numerical solution of a number of propagation problems. Such numerical experiments have played a large part in our increasing understanding of wave motion in anisotropic media. It now appears that the solution of most problems in anisotropic propagation can be formulated, if the corresponding solution exists for isotropic propagation, and may be solved at the cost of considerably more numerical computation.

There are two significant results from these developments: the recognition of the importance of body- and surface-wave polarizations in diagnosing and estimating anisotropy; and the recognition that many two-phase materials, particularly cracked solids, can be modelled by anisotropic elastic-constants. This last result opens up a new class of materials to wave-motion analysis, and has applications in a variety of different fields.     

Solitary waves for a nonlinear dispersive long wave equation

All the possible traveling wave solutions of Whitham-Broer-Kaup （WBK） equation are investigated in the present paper. By employing phase plane analysis, transition boundaries are derived to divide the parameter space into several regions associated with different types of phase portraits corresponding to different forms of wave solutions. All the exact expressions of bounded wave solutions are obtained as well as their existence conditions. The mechanism of bifurcation between different waves with varying Hamiltonian value has been revealed. It is pointed out that as the periods of two coexisted periodic waves tend to infinity, they may evolve to two solitary waves. Furthermore, when their trajectories pass through the common saddle point, the two solitary waves may merge into a periodic wave, and its amplitude is nearly equal to the sum of the amplitudes of the two solitary wave solutions.     

Weakly nonlinear magnetohydrodynamic wave motion

One-dimensional magnetchydrodynamic (mhd) waves are treated including nonlinearity up to the second order in the wave amplitude; particular emphasis is on the effects of the diffusivity of the electromagnetic field on plane wave propagation. Unidirectional propagation is either governed by an equation of the Varley-Rogers type in the high frequency limit or by the Burgers equation in the low frequency limit. External resonant excitation in a finite layer leads to equations well known from resonance in various other physical systems.     

Nonlinear wave motion in magnetoelasticity

Archive for Rational Mechanics and Analysis -     

Diffraction by a half-plane in uniform translational motion revisited

A problem is reconsidered of time harmonic, electromagnetic diffraction by a perfectly conducting half-plane moving in free space with a constant velocity. Similarities and differences between stationary and moving diffraction have been discussed.     

On the theory of transient wave propagation in a dispersive medium

Summary Choosing a simple parallel-plate waveguide, an exact expression is obtained for the transient response when a step-modulated carrier signal is applied. The analysis constitutes a modest extension of the early work of Sommerfeld and the more recent investigations of Rubinowicz and Knop. Numerical calculations are presented for a range of the parameters which have some relevance to propagation in the earth-ionosphere waveguide. The exact form of the transient envelope function is compared with an approximate version which is in the form of a Fresnel integral. It is shown that the approximate method gives a good qualitative estimate for the transient response characteristics. Thus, confidence is gained in applying it to other situations where an exact solution is not available.The research reported here was supported by the Advanced Research Projects Agency, Washington, D.C., under ARPA Order No. 183-62.Formerly the Central Radio Propagation Laboratory of the National Bureau of Standards.     

One dimensional,non-linear wave motion

A transformation of the quasilinear wave equation is made to study some of the geometrical properties of its solution. It is also shown that any given solution of it can always be continuously extended across a characteristic in a simple wave.     

Free vibration and wave propagation analysis of uniform and tapered rotating beams using spectrally formulated finite elements

This paper presents a formulation of an approximate spectral element for uniform and tapered rotating Euler–Bernoulli beams. The formulation takes into account the varying centrifugal force, mass and bending stiffness. The dynamic stiffness matrix is constructed using the weak form of the governing differential equation in the frequency domain, where two different interpolating functions for the transverse displacement are used for the element formulation. Both free vibration and wave propagation analysis is performed using the formulated elements. The studies show that the formulated element predicts results, that compare well with the solution available in the literature, at a fraction of the computational effort. In addition, for wave propagation analysis, the element shows superior convergence.     

Akhmediev breather signatures from dispersive propagation of a periodically phase-modulated continuous wave

We investigate in detail the qualitative similarities between the pulse localization characteristics observed using sinusoidal phase modulation during linear propagation and those seen during the evolution of Akhmediev breathers during propagation in a system governed by the nonlinear Schrödinger equation. The profiles obtained at the point of maximum focusing indeed present very close temporal and spectral features. If the respective linear and nonlinear longitudinal evolutions of those profiles are similar in the vicinity of the point of maximum focusing, they may diverge significantly for longer propagation distance. Our analysis and numerical simulations are confirmed by experiments emulating an optical fibre.     

Numerical analysis of higher-dimensional dispersive long-wave equations

In this paper, numerical analysis of the (2+1)-dimensional dispersive long-wave equation (DLWE) is studied by using the homotopy perturbation method (HPM). For this purpose, the available analytical solutions obtained by multiple traveling-wave solution will be compared to show the validity and accuracy of the presented numerical algorithm. The obtained results prove the convergence and accuracy of the HPM for the numerically analyzed (2+1)-dimensional DLWE system.     

Similarity analysis and asymptotic models

Fundamental problems of the heat transfer theory and physical hydrodynamics are associated with the most complicated processes of macrophysics: turbulence, multiphase and multi-component interactions and physico-chemical conversions of substances. In this field there are no good and sufficiently universal model equations. Moreover, the principal vista of their derivation is still obscure. The probabilistic essence of these processes in the small and almost classical determinancy on the average are the most important here. Similarity methods, mathematical and physical simulation, separation of conservative characteristics and elucidation of asymptotic processes are of the decisive character here. The German school of research workers has made a significant contribution to these studies.A compact representation of several important problems of this kind is just the subject of the present communication. Several data are the original results of the studies which have been performed by the author since 1935.The paper is preceded by a preface devoted to Professor U. Grigull on his 70th birthday.

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ähnlichkeitsanalyse und asymptotische Modelle

Zusammenfassung Die grundlegenden physikalischen Probleme der Wärmetransporttheorie und Hydrodynamik stellen sehr komplizierte Prozesse der Makrophysik dar wie: Turbulenz, die gegenseitige Einwirkung mehrphasiger und mehrkomponentiger Systeme sowie physico-chemische Umwandlung von Stoffen.Auf diesen Gebieten existieren keine guten und universellen Modellgleichungen. Darüber hinaus ist der prinzipielle Weg ihrer Herleitung noch immer unklar. Hier ist von Bedeutung, daß die Wirkung der Prozesse mit Hilfe klassischer Methoden durch mittlere Größen beschrieben werden können. ähnlichkeitsmethoden, mathematische und physikalische Simulation, Trennung von konservativen Merkmalen und die Betrachtung von asymptotischen Prozessen sind hier die entscheidenden Merkmale. Forscher der deutschen Schule haben hier einen bedeutenden Beitrag geleistet.Eine geraffte Darstellung verschiedener wichtiger Probleme dieser Art ist Thema des vorliegenden Beitrages. Einige Untersuchungen stellen Originalergebnisse dar, die vom Autor selbst seit 1935 durchgeführt wurden.

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Nomenclature U scale flow velocity (e.g., velocity outside boundary layer or average flow-rate velocity in channel) - R characteristic linear dimensions (e.g., tube radius, equivalent drop or bubble radius) - kinematic viscosity - density - surface tension coefficient - g gravitational acceleration - r latent heat of evaporation - T _s saturation temperature - C specific heat - heat fransfer coefficient. Accents and correspond to liquid and gas (vapor), respectively Dedicated Prof. Dr.-Ing. Grigull on his 70th birthday.