Interaction between a stokes wave packet and a solitary wave

This paper is a theoretical and experimental study of the propagation of a short gravity wave packet (modulated Stokes wave) over a solitary wave. The theoretical approach used here relies on a nonlinear WKB-type perturbation method. This method yields a theory of gravity waves that can describe both short and long waves simultaneously. We obtain explicit analytical solutions describing the interaction between the soliton and the short wave packet: phase shifts, variations of wavelengths and of frequencies (Doppler effects). In the experimental part of this work the phase shift experienced by the Stokes wave is measured. The theoretical conclusions are confirmed.     

On wave families in a two-layer falling film

Using an approximate method, families of non-linear steady-traveling periodic waves in a two-layer falling film have been found for the first time. Computed waves have qualitatively similar behavior as that of those found in homogeneous films but the quantitative characteristics of the waves strongly depend on additional similarity parameters in the two-layer films. In particular, the average location of the interface affects the bifurcation scheme of the waves.     

Adjustment of spiral drift by a travelling wave perturbation

We apply a velocity-field approach to investigate the interaction between spiral waves and the travelling wave modulation of system excitability which leads to a prediction: the direction of the straight-line drift of spiral waves is linearly adjusted by the propagation direction of the travelling waves. Direct numerical computations of the Oregonator model and the formulas of drift-velocity field confirm the validity and robustness of our theoretical prediction.     

Thermodynamic restrictions and wave features of a non-linear Maxwell model

A non-linear rate-type constitutive equation, established by Rajagopal, provides a generalization of the Maxwell fluid. This note embodies such a constitutive equation within the scheme of materials with internal variables thus allowing also for solids with both dissipative and thermoelastic mechanisms. The compatibility with the second law of thermodynamics, expressed by the Clausius–Duhem inequality, is examined and the restrictions on the evolution equations are determined. Next the propagation condition of discontinuity waves is derived, for shock waves and acceleration waves, by regarding the body as a definite conductor. Infinitesimal shock waves and acceleration waves show similar effects. The effective acoustic tensor proves to be the sum of a thermoelastic tensor and a tensor arising from the rate-type equation.     

Generation mechanism of bed wave in a sand-bed channel

The previously proposed theory of bed load motion by fluid flow is developed. A plane system of equations for bed perturbations is obtained using a formula for the sediment transport rate which takes into account the effect of free-surface perturbations on sediment transport and is extended to the two-dimensional case. Dependences of the lengths and velocities of longitudinal and transverse waves with the most rapidly increasing amplitude on Froude number are determined. The effect of macroturbulent viscosity and surface waves on the generation of bed waves is determined.     

Localization of wave motions in a fluid-filled cylinder

The properties of harmonic surface waves in a fluid-filled cylinder made of a compliant material are studied. The wave motions are described by a complete system of dynamic equations of elasticity and the equation of motion of a perfect compressible fluid. An asymptotic analysis of the dispersion equation for large wave numbers and a qualitative analysis of the dispersion spectrum show that there are two surface waves in this waveguide system. The first normal wave forms a Stoneley wave on the inside surface with increase in the wave number. The second normal wave forms a Rayleigh wave on the outside surface. The phase velocities of all the other waves tend to the velocity of the shear wave in the cylinder material. The dispersion, kinematic, and energy characteristics of surface waves are analyzed. It is established how the wave localization processes differ in hard and compliant materials of the cylinder __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 4, pp. 72–86, April 2008.     

Laboratory experiments on internal wave interactions with a pycnocline

Laboratory experiments have been performed to investigate the interaction of internal waves with a pycnocline. An oscillating cylinder generated internal wave beams, which were observed using the synthetic schlieren technique. Internal waves incident on the pycnocline layer excited higher-frequency modes. In the absence of shear, a discrete spectrum of harmonic modes was generated due to nonlinear effects. These harmonic modes might play a role in the formation of internal solitary waves which have been observed in ocean pycnoclines. With shear, a continuous spectrum of excited modes was found.     

Reflection of a spherical blast wave from a planar surface

Numerous authors have carried out rather extensive studies in the last twenty to thirty years of the problem of the interaction of shock and blast waves with obstacles in their paths. Owing to the complexity of the problem, they assumed certain limiting cases for the shock wave interactions in which the parameters behind the shock wave were usually taken to be constants. The first wave diffraction studies involving variable parameters behind the front were presented in [1, 2], wherein a development of the theory of “short waves” (blast waves at a substantial distance from the center of an explosion) and their reflection from a planar surface was given. The theory of short waves assumes that the jump in pressure at the wave front and the region over which the parameters vary are small. The problem concerning reflection of a blast wave from a surface was also considered in [3, 4], wherein a solution in the region behind the reflected wave was obtained at initial times. The initial stage in the reflection of a blast wave from a planar, cylindrical, or spherical surface (the one-dimensional case) was studied in [5]. In this paper we investigate the interaction of a spherical blast wave, resulting from a point explosion, with a planar surface; we consider both regular and non-regular reflection stages. In solving this problem we use S. L. Godunov's finite-difference method. We obtain numerical solutions for various values of the shock strength at the instant of its encounter with the surface. We present the pressure fields in the flow regions, the pressure distribution over the surface at various instants of time, and the trajectories of the triple point. The parameter values at the front of the reflected wave are compared with results obtained from the theory of regular reflection of shock waves.     

Dynamic action of a tsunami wave traveling along a channel

The results of experiments, in which the propagation of a tsunami-type wave along rectangular channels with horizontal and inclined bottoms, are presented. Emphasis is placed on the mechanical action of the wave on a vertical wall. The force is shown to be appreciably dependent on the shape of the leading front of the wave. Experimental data are obtained for both smooth and breaking waves, as well as for waves in different stages of the wave-breaking process.     

The reflection of a shock wave over a cone

An investigation was made of the reflection of planar shock waves from cones. 86 cones, the half apex angle of which varied from 10° to 52° at every 0.5°, were installed in a 60 mm×150 mm diaphragmless shock tube equipped with holographic interferometry. The diaphragmless shock tube had a high degree of reproducibility with which the scatter of shock wave Mach number was within ±0.25% for shock wave Mach number ranging from 1.16 to approximately 2.0. The reflection of shock waves over cones was visualized using double exposure holographic interferometry. Whitham's geometrical shock wave dynamics was used to analyse the motion of Mach stems over cones. It is found that for relatively smaller apex angles of cones trajectory angles of resulting irregular reflections coincide with the so-called glancing incidence angles and their Mach stems appear to be continuously curved from its intersection point with the incident shock wave, which shows the chractericstic of von Neumann reflection. The domain of the existence of the von Neumann reflection was analytically obtained and was found to be broadened much more widely than that of two-dimensional reflections of shock waves over wedges.     

A spherical wave in a viscoelastic half-space

The propagation of spherical waves in an isotropie elastic medium has been studied sufficiently completely (see, e.g., [1–4]). it is proved [5, 6] that in imperfect solid media, the formation and propagation of waves similar to waves in elastic media are possible. With the use of asymptotic transform inversion methods in [7] a problem of an internal point source in a viscoelastic medium was investigated. The problem of an explosion in rocks in a half-space was considered in [8]. A numerical Laplace transform inversion, proposed by Bellman, is presented in [9] for the study of the action of an explosive pulse on the surface of a spherical cavity in a viscoelastic medium of Voigt type. In the present study we investigate the propagation of a spherical wave formed from the action of a pulsed load on the internal surface of a spherical cavity in a viscoelastic half-space. The potentials of the waves propagating in the medium are constructed in the form of series in special functions. In order to realize viscoelasticity we use a correspondence method [10]. The transform inversion is carried out by means of a representation of the potentials in integral form and subsequent use of asymptotic methods for their calculation. Thus, it becomes possible to investigate the behavior of a medium near the wave fronts. The radial stress is calculated on the surface of the cavity.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 139–146, March–April, 1976.     

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.     

Nonlinear wave effects in a fluid-saturated porous medium

Some one-dimensional nonlinear effects associated with wave propagation in weakly permeable fluid-saturated porous media are investigated. The effect of nonlinearity on the damping of monoharmonic waves is estimated and, moreover, the characteristics of the nonlinear parametric interaction of two waves excited in the medium by two monoharmonic sources of different frequencies are established.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 74–77, January–February, 1992.     

Shock wave in a gas-liquid medium

The propagation of weak shock waves and the conditions for their existence in a gas-liquid medium are studied in [1]. The article [2] is devoted to an examination of powerful shock waves in liquids containing gas bubbles. The possibility of the existence in such a medium of a shock wave having an oscillatory pressure profile at the front is demonstrated in [3] based on the general results of nonlinear wave dynamics. It is shown in [4, 5] that a shock wave in a gas-liquid mixture actually has a profile having an oscillating pressure. The drawback of [3–5] is the necessity of postulating the existence of the shock waves. This is connected with the absence of a direct calculation of the dissipative effects in the fundamental equations. The present article is devoted to the theoretical and experimental study of the structure of a shock wave in a gas-liquid medium. It is shown, within the framework of a homogeneous biphasic model, that the structure of the shock wave can be studied on the basis of the Burgers-Korteweg-de Vries equation. The results of piezoelectric measurements of the pressure profile along the shock wave front agree qualitatively with the theoretical representations of the structure of the shock wave.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 65–69, May–June, 1973.     

Stress wave propagation in a rectangular bar

The process of propagation of nonstationary waves in a rectangular bar is studied from the viewpoint of three-dimensional elasticity. The motion arises owing to the action of normal impact forces applied at the end face of a half-infinite bar all of whose four lateral surfaces are force-free. Precisely these one-type conditions complicate the solution of this problem. The already known solutions were obtained under the assumption that conditions of mixed type are partially or completely posed on the lateral sides, and precisely this fact permits separating the boundary values of distinct waves on these surfaces. In the absence of this simplifying factor, it is rather problematic to construct a solution satisfying all free lateral conditions.     

Investigations on flexural wave propagation of a periodic beam using multi-reflection method

The flexural wave propagation in a periodic beam with a propagating disturbance is studied by the use of the multi-reflection method. A propagating wave is incident upon a discontinuity and gives rise to transmitted and reflected waves. Here all of the transmitted and reflected waves of given flexural wave incident upon the beam at some specified location are found and superposed, and the method is extended to the case of incident evanescent wave. The results of incident waves at some location between discontinuities in a periodic beam are concerned. The relation between the wave-field of incident waves and the wave-field of resulting waves on any segments is expressed. As an example, the application of the results to the analysis of a finite periodic beam with a propagating disturbance is then demonstrated. The influences of the number of cells on the energy associated with propagating waves are considered.     

Self-organized wave pattern in a predator-prey model

In this paper, pattern formation of a predator-prey model with spatial effect is investigated. We obtain the conditions for Hopf bifurcation and Turing bifurcation by mathematical analysis. When the values of the parameters can ensure a stable limit cycle of the no-spatial model, our study shows that the spatially extended models have spiral waves dynamics. Moreover, the stability of the spiral wave is given by the theory of essential spectrum. Furthermore, although the environment is heterogeneous, the system still exhibit spiral waves. The obtained results confirm that diffusion can form the population in the stable motion, which well enrich the finding of spatiotemporal dynamics in the predator-prey interactions and may well explain the field observed in some areas.     

The evolution of a solitary wave in a nonhomogeneous medium

Using the example of surface waves in a heavy liquid, the article discusses the propagation of a solitary wave in a nonhomogeneous medium. Ananalysis is made of processes of the decomposition of a wave into solitary waves, as a function of differences in the depth.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 80–85, November–December, 1971.The author thanks A. V. Gaponov, A. G. Litvak, and L. A. Ostrovskii for their evaluation of the results of the work.     

Acceleration wave speeds in a hypoplastic constitutive model

The paper presents a theoretical investigation of acceleration waves in a plastic material described by an incrementally non-linear hypoplastic constitutive equation. Speeds of plane acceleration waves and their dependence on the stress state are calculated. The spectrum of possible wave speeds is found to be continuous, which is in contrast to discrete wave speed spectra in incrementally linear models. Two types of ill-posedness are revealed, known as flutter ill-posedness and stationary discontinuity. The wave speed analysis is also performed by the characteristic method, leading to different equations compared to the acceleration wave approach. It is proved that for the considered hypoplastic constitutive equation both approaches give identical wave speed spectra.     

Self-switching of a transverse plane wave propagating through a two-component elastic composite

The paper discusses the results of theoretical and numerical analysis of the interaction of nonlinear elastic plane harmonic waves in a composite material whose nonlinear properties are described by modeling it with a two-phase mixture. The interaction of two transverse vertically polarized harmonic waves is studied using the method of slowly varying amplitudes. The truncated and evolutionary equations as well as the Manley-Rowe relations are derived. The mechanism of energy pumping from a strong pumping wave with frequency ω to a weak signal wave with frequency 3ω is analyzed. The switching mechanism for hypersonic waves in a nonlinear elastic composite is similar to the switching mechanism observed in transistors __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 7, pp. 35–46, July 2007.