Weakly nonlinear long waves in a prestretched Blatz-Ko cylinder: Solitary, kink and periodic waves

This paper studies nonlinear waves in a prestretched cylinder composed of a Blatz-Ko material. Starting from the three-dimensional field equations, two coupled PDEs for modeling weakly nonlinear long waves are derived by using the method of coupled series and asymptotic expansions. Comparing with some other existing models in literature, an important feature of these equations is that they are consistent with traction-free surface conditions asymptotically. Also, the material nonlinearity is kept to the third order. As these two PDEs are quite complicated, the attention is focused on traveling waves, for which a first-order system of ODEs are obtained. We use the technique of dynamical systems to carry out the analysis. It turns out that the system is three parameters (the prestretch, the propagating speed and an integration constant) dependent and there are totally seven types of phase planes which contain trajectories representing bounded traveling waves. The parametric conditions for each phase plane are established. A variety of solitary and periodic waves are found. An important finding is that kink waves can propagate in a Blatz-Ko cylinder. We also find that one type of periodic waves has an interesting feature in the profile slope. Analytical expressions for all bounded traveling waves are obtained.     

Propagation of long nonlinear waves in a ponderable fluid beneath an ice sheet

In Sec. 1 the stability of small-amplitude steady-state periodic solutions of Eq. (0.1) in the neighborhood of k=k_n are investigated. The results of the investigations are consistent with those of [1]. In Sec. 2 the stability of periodic waves not lying in the neighborhood of resonance is considered. It is shown that in the region of instability when =1 steady-state solutions of the soliton type with oscillatory structure may exist. In Sec. 3 the properties of certain exact solutions — periodic waves and solitons — are studied in relation to the nature of the singular points of the dynamical system derived from (0.1). In Sec. 4 the evolution of rapidly decreasing Cauchy data is considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 88–95, January–February, 1989.     

Propagation of long waves of finite amplitude at the interface of two viscous fluids

The propagation of long waves of finite amplitude at the interface of two viscous fluids has been studied theoretically. For plane Couette-Poiseuille flow of two superposed layers of fluids of different viscosity, an equation is derived to determine the development in time of the shape of these finite amplitude waves. The influence of the viscosity ratio, the density difference of the fluids and an imposed pressure gradient have been investigated.     

Propagation of horizontally polarized transverse waves in a solid with a periodic distribution of cracks

The propagation of time-harmonic waves in a solid containing a periodic distribution of cracks is investigated in a two-dimensional configuration. The cracks are parallel to the x-axis, and their centers are located at positions x = md, y = lh(m, l = 0, ±1, ±2,…). The wave motion is polarized in the z-direction and propagates in the y-direction (normal to the cracks). The theory of Floquet or Bloch waves, together with an appropriate Green's function and the condition of vanishing traction on the crack faces leads to a system of singular integral equations, which provides the basis for the derivation of an exact dispersion equation. Numerical results are presented for the wave number as a function of the frequency. The frequency spectrum shows a pattern of passing and stopping bands. The exact results are compared with the frequency spectrum according to a simplified theory which considers the arrays of collinear cracks in the planes y = lh (l= 0 ±1, ±2,…) as planes of homogeneous transmission and reflection. Good agreement is observed between exact and approximate results.     

Propagation of long waves into a set of parallel vertical barriers on a rotating earth

The problem of propagation of long waves into a set of parallel vertical barriers on a rotating system is considered and an exact solution is obtained showing that an appropriate Kelvin wave is created in all parts of the field.Moreover, the case of a semi-infinite rotating channel, alone, is investigated as a limiting case of the above problem.     

Propagation of weak shock waves

An asymptotic method is developed in order to treat the evolution of weak shock waves. One obtains a geometrical theory according to which weak shock waves propagate along rays and satisfy a transport law.     

Propagation of weakly nonlinear magnetoacoustic waves

Using the reductive perturbation method we show that the modified Burgers equation governs the propagation of a weakly nonlinear slow magnetoacoustic wave near the equilibrium state with zero transverse magnetic field.     

Propagation characteristics of shock waves driven by gaseous detonation waves

We experimentally investigated propagation characteristics of the shock wave driven by a gaseous detonation wave emerging from the open end of a cylindrical detonation tube. In the present study, we visualized the shock wave and exhaust flowfields using a shadowgraph optical system and we obtained peak overpressure in the tube axial direction and the continuous shape transformation of shock waves around the tube open end. We also obtained overpressure histories of the shock wave using piezo-pressure transducers within 201 m from the open end of the tube. We normalized and classified these results by four regions using non-dimensional pressure and distance which are independent of variety of mixture and tube diameter. In the vicinity of the open end of the tube, the shock wave is nearly planar and does not significantly attenuate, and the peak overpressure maintains approximately C–J pressure. Subsequently, the shock wave attenuates rapidly, transforming from quasi-spherical to spherical. Farther from the tube open end, the shock wave propagates with approximately sound characteristic so that the peak overpressure decreases proportional to 1/r. Eventually, the shock wave begins to attenuate more rapidly than ideal sound attenuation, which may be due to the viscous effect.     

Three-dimensional surface waves propagating over long internal waves

A non-uniform current, such as may be generated by long internal waves, interacts with short surface waves and causes patterns on the sea surface that are of interest. In particular, regions of steep breaking waves may be relevant to specular radar scattering.A simple approach to modelling this problem is to take a set of short, surface waves of uniform wavenumber on the sea surface, as may be caused by a gust of wind. The direction of propagation of the surface waves is firstly taken to be the same as that of the current, and surface tension and viscous effects are neglected. We have a number of methods of solution at our disposal: linear (one-dimensional) ray theory is simple to apply to the problem, a nonlinear Schrödinger equation for the modulated wave amplitude, modified to include to effect of the current, can be used and solutions can be found using a fully nonlinear irrotational flow solver. Comparisons between the ‘exact’ nonlinear calculations for two dimensions (which are too complicated/ computationally intensive to be extended to three dimensions) compare well with the two approximate methods of solution, both of which can be extended, within their limitations, to model the full three-dimensional problem; here we present three-dimensional results from the linear ray theory.By choosing such a simple (although we consider physically realistic) initial state of uniform wavenumber short waves and assuming a sinusoidal surface current, we can reduce the two-dimensional problem to dependence on three non-dimensional parameters.In three-dimensions, we consider an initial condition with a uniform wavetrain at an angle α say, to the propagating current, thus introducing a fourth parameter into the problem. Extension of the linear ray theory from one space to two space dimensions is numerically quite simple since we maintain uniformity in the direction perpendicular to the current, and the only difficulty lies with the presentation of results, due to the large number of variables now present in the problem such as initial wavenumber, angle of propagation, position in (x, y, t) space etc. In this paper we present just one solution in detail where waves are strongly refracted and form two distinct foci in space-time. There is a collimation of the short waves with the direction of the propagating current.     

Nonlinear periodic waves in a gas

A system of equations describing the one-dimensional time-dependent polytropic motion of a gas is considered. In special cases the general solutions of this system of equations are obtained and exact solutions with the initial conditions which are periodic with respect to the spatial variable are found. For an arbitrary polytropic exponent an asymptotic solution, which is uniformly suitable till the onset of the gradient catastrophe, is constructed in the form of expansions in series in a small parameter, namely, the initial wave amplitude. Asymptotic dependences of the time of onset and the location of the gradient catastrophe are obtained. The complex correspondence between the initial system of equations and the system of equations describing the motion of quasi-gas media is given. An example of using this correspondence is considered.     

Propagation of very long water waves, with vorticity, over variable depth, with applications to tsunamis

We present a theory of very long waves propagating on the surface of water. The waves evolve slowly, both on the scale ε (weak nonlinearity), and on the scale, σ, of the depth variation. In our model, dispersion does not affect the evolution of the wave even over the large distances that tsunamis may travel. We allow a distribution of vorticity, in addition to variable depth. Our solution is not valid for depth=O(ε^4/5); the equations here are expressed in terms of the single parameter ε^2/5σ and matched to the solution in deep water. For a slow depth variation of the background state (consistent with our model), we prove that a constant-vorticity solution exists, from deep water to shoreline, and that regions of isolated vorticity can also exist, for appropriate bottom profiles. We describe how the wave properties are modified by the presence of vorticity. Some graphical examples of our various solutions are presented.     

On the theory of long waves

A new method of studying plane steady wave motion of a gravity fluid is elucidated in this paper. This method succeeds in establishing the existence of a solitary wave, for example, and in giving the first complete foundation for the approximate Rayleigh theory [1], which concerns the theory of finite-amplitude long waves. Underlying the method are general boundary properties of univalent functions, used earlier by the author to construct a qualitative theory of jet fluid motions [2].     

Nonlinear dispersion of long internal waves

The propagation of long weakly nonlinear waves in an atmospheric waveguide is considered. A model system of Kadomtsev-Petviashvili equations [1], which describes the propagation of such waves, is derived. In the case of one excited wave mode the system of model equations goes over into the Kadomtsev-Petviashvili equation, in which, however, the variables x and t are interchanged. The reasons for this are clarified. In the two-dimensional case an approximate solution of the model equations is constructed, and steady nonlinear waves and their interaction in a collision are considered. The results of a numerical verification of the stability of the approximate steady solutions and of the solution to the problem of decay of the wave into quasisolitons are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 151–157, May–June, 1988.     

Propagation of shock waves in nonuniform plasma

Results of numerical simulation of the propagation of one-dimensional magnetohydrodynamic shock waves in a nonuniform plasma containing a magnetic field are discussed. Possible uses for the production of high velocities and temperatures and astrophysical applications are considered. The essential effect of the magnetic field is shown; acceleration of a shock wave is intensified in a medium with decreasing density.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 22–26, March–April, 1976.In conclusion, the authors are grateful to S. K. Godunov for a detailed discussion of the computational aspects and of the results, and to A. E. Voitenko for a discussion of experimental possibilities and of the results.     

Propagation of waves in thin elasto-viscoplastic layers

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