Propagation of nonlinear waves along a viscoelastic bar

Various types of nonlinear waves propagating along a viscoelastic bar are considered. The rheological equation of state has strong physical and geometric nonlinearities, and nonisothermal effects are included. Both weak (isentropic) and shock waves of loading and unloading are investigated. It is shown that, for certain rubber-like materials, stable shock waves of extension can exist along with the shock waves of compression at very large strains. We then consider the strike of a viscoelastic bar of finite length against a rigid obstacle. Numerical solutions to this problem illustrate the influence of stress relaxation on nonlinear wave processes. A model for sticking and bouncing off is formulated and the mass-averaged velocity of the bar at the moment when it bounces off the obstacle is calculated.     

Propagation of weak shock waves in a magnetic field

This paper gives a solution of the problem of the propagation of weak shock waves in an inhomogeneous conducting medium in the presence of a magnetic field. The width of the perturbed region is taken to be small compared with the characteristic dimensions of the problem. The magnetic Reynolds number is also assumed small, which allows one to neglect the induced magnetic field. The method of solution employed is similar to that used in [1–3],The author is grateful to B. I. Zaslavskii for useful advice and for discussing the paper.     

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 of modulated nonlinear waves in a relaxing gas-liquid mixture

The propagation of nonstationary weak shock waves in a chemically active medium is essentially dispersive and dissipative. The equations for short-wavelength waves for such media were obtained and investigated in [1–4]. It is of interest to study quasimonochromatic waves with slowly varying amplitude and phase. A general method for obtaining the equations for modulated oscillations in nonlinear dispersive media without dissipation was proposed in [5–8]. In the present paper, for a dispersive, weakly nonlinear and weakly dissipative medium we derive in the three-dimensional formulation equations for waves of short wavelength and a Schrödinger equation, which describes slow modulations of the amplitude and phase of an arbitrary wave. The coefficients of the equations are particularized for the considered gas-liquid mixture. Solutions are obtained for narrow beams in a given defocusing medium as well as linear and nonlinear solutions in the neighborhood of a diffraction beam. A solution near a caustic for quasimonochromatic waves was found in [9].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 133–143, January–February, 1980.     

Propagation of elastic waves in a rod in a longitudinal magnetic field

The propagation of harmonic waves in discussed for an ideally conducting continuous elastic cylindrical rod within an ideally conducting cylindrical rube. The annulus contains a steady homogeneous longitudinal magnetic field. The dispersion equation is derived. The case of bending vibrations is considered.     

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 combined longitudinal and torsional stress waves in a functionally graded thin-walled tube

Applied Mathematics and Mechanics - An analytical model for the propagation of combined stress waves in a functionally graded thin-walled tube subjected to combined longitudinal and torsional...     

Propagation of nonlinear waves in gas-liquid media with a gas content variable in space

A study is made of the propagation of nonlinear waves in gas-liquid systems of bubble structure with a gas content which is variable in the direction of propagation of the wave. It is shown that it is possible in a number of cases to amplify the pressure waves. Restrictions on the degree of homogeneity of the gas content are obtained such that, when they are fulfilled, this amplification is possible. The study also covers the effect of inhomogeneity in the gas content on the structure of nonlinear steady waves of the soliton and shock wave types.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 180–183, January–February, 1986.     

Propagation of thermoelastic waves in semiinfinite bodies with nonlinear heat transfer

Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 24, No. 11, pp. 98–102, November, 1988.     

Propagation of nonlinear waves in a porous medium with two-phase saturation by a liquid and a gas

The propagation of nonlinear waves through a porous medium saturated with a viscous liquid and a gas is investigated with allowance for the capillary pressure. Numerical solutions of the traveling-wave type are constructed for the generalized Korteweg-de Vries-Burgers equation for the wave amplitudes. Three types of regimes of longitudinal wave propagation, including soliton-like regimes, are found.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 86–95, July–August, 1996.     

Propagation of nonlinear waves in a gas-liquid medium. Exact and approximate analytical solutions of wave equations

A number of exact and approximate analytical solutions of the equations for one-dimensional and weakly non-one-dimensional waves propagating in a liquid with gas bubbles are presented for the case where the bubble distribution density is a continuous function of the bubble radius and spatial coordinates.     

Propagation of nonlinear Alfvén surface waves along a tangential magnetohydrodynamic discontinuity in an incompressible fluid

The paper considers the propagation of low-amplitude nonlinear waves over a surface of magnetic discontinuity. An equation for these waves is derived. The evolution of a sinusoidal perturbation is investigated. The discontinuity formation time is numerically determined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 98–105, January–February, 1985.