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 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 unsteady weak shock waves in a vibrationally nonequilibrium gas subjected to external radiation

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 26–35, September–October, 1990.     

Plane stationary flows with ionizing shock waves in a magnetic field

The formulation of stationary, plane, and self-similar problems is considered when the flow parameters depend only on the polar angle, and the magnetic field lies in the flow plane. The case in which the magnetic field is perpendicular to the flow plane has been examined in [1]. The conditions are found under which the solution depends on an arbitrary parameter and the reasons for this nonuniqueness are explained. Self-similar solutions are constructed to describe the flow around an insulating wedge and a wall.     

Propagation of weak waves in elastic-plastic solids

Wave fronts admitting discontinuities only in the derivatives of the dependent variables are by convention called ‘weak’ waves. For the special case of discontinuous first-order derivatives, the fronts are customarily called ‘acceleration’ waves. If the governing equations are quasi-linear, then the weak waves are necessarily characteristic surfaces. Sometimes, these surfaces are also referred to as ‘singular surfaces’ of order r ? 1, where r stands for the order of the first discontinuous derivatives. This latter approach is adopted in this paper and applied to governing equations which form a set of first-order quasi-linear hyperbolic equations. When these equations are written in terms of singular surface coordinates, simplification occurs upon differencing equations written on the front and rear sides of the surface: a set of algebraic (‘connection’) equations is generated for the discontinuities in the normal derivatives of the dependent variables across the surface. When a similar operation is performed on the governing equations written for second-order derivatives, a set of first-order differential (‘transport’) equations is generated.     

The reflected pressure field in the interaction of weak shock waves with a compressible foam

This paper deals with the waves that are reflected from slabs of porous compressible foam attached to a rigid wall when impacted by a weak shock wave. The interest is in establishing possible attenuation of the pressure field after a shock or blast wave has struck the surface. Foam densities from 14 to 38 kg/m³ were tested over a range of shock wave Mach numbers less than 1.4. It is shown that the initial reflected shock wave strength is accurately predicted by the pseudo-gas model of Gelfand et al. (1983), with a pressure ratio of approximately 80% of the value for reflection off a rigid wall. Evidence is presented of gas entering the foam during the early stages of the process. A second wave emerges from the foam at a later stage and is separated from the first by a region of constant velocity and pressure. This second wave is not a shock wave but a compression front of significant thickness, which emerges from the foam earlier than predicted by the pseudo-gas model. Analysis of the origin of this wave points to much more complex flows within the foam than previously assumed, particularly in an apparent decrease in average wave front speed as the foam is compressed. It is shown that the pressure ratio across both these waves taken together is slightly higher than that for reflection off a rigid wall. In some cases this compression wave train is followed by a weak expansion wave.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1990.     

Propagation of cylindrical shock waves in a gravitating medium

We study gas motion behind the front of a cylindrical shock wave created by the motion of a piston in a gravitating medium. The problem is self-similar, but the solution cannot be obtained in closed form. A numerical calculation is made for various Mach numbers. The calculation shows that the central part of the configuration is displaced a definite distance from the axis of symmetry.Cylindrical shock waves through a compressible homogeneous medium in a gravity field have been examined by Sedov [1] and Lin [2], However, these studies contain the essential assumption that the total energy (i. e., the sum of the kinetic and thermal energies) within the region bounded by the expanding shock wave is independent of time.In the following we extend the previous studies to the case of shock waves in nonhomogenous media, which propagate in the fluctuating gravity field created by the disturbed mass itself. The shock wave is created by the motion of a piston whose velocity varies as some power of the time, i. e., v. The total energy of the configuration also depends on the time.The authors wish to thank M. P. Murgai for cooperation and assistance and C. D. Ghildyal for valuable advice, as well as L. I. Sedov and G. I. Petrov for their critical comments.     

Propagation of shock waves in free space

The problem of the exit of a shock wave from an axisymmetric channel and its propagation in a free space occupied by an ideal gas is examined. This problem has been studied earlier in [1], in which the shock wave front was considered planar, as well as in [2], in which the wave front was regarded as a surface of an ellipsoid of revolution. The solutions obtained in these studies assumed the presence of two regions in the wave-front surface: the region of the original shock wave and a region stemming from the decomposition of an infinitesimally thin annular discontinuity of the gas parameters, with the wave intensity over the front surface in each region being considered constant, i.e., the wave character of the process over the front was not considered. In this study a solution will be achieved by the method of characteristics [3–5] of the equations of motion of the shock-wave front, as obtained in [6, 7]. Flow fields are determined for the region immediately adjacent to the shock-wave front for a wide range of shock-wave Mach numbers M _a =1.6–20.0 for = 1.4. On the basis of the data obtained, by introduction of variables connected with the length of the undisturbed zone, as calculated from the channel cross-section along the x axis, together with the pressure transition at the wave front, relationships are proposed which approach self-similarity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 163–166, September–October, 1971.In conclusion, the author thanks S. S. Semenov for his valuable advice on this study.     

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 a piezoelectric layer with a weak horizontal nonhomogeneity

The process of wave propagation along a piezoelectric layer is considered. It is assumed that the properties of the medium vary slowly along the horizontal directions, and the boundaries of the layer are slightly bent. The propagation of the wave along the layer is studied by the ray method. The transport equations, which describe change of the intensity along the rays, are solved.     

Propagation of nonlinear body waves in a magnetic tube

The propagation of slow symmetrical small-amplitude body waves in a cylindrical magnetic tube is investigated on the basis of the nonlinear equation obtained in [3, 4]. The breaking of periodic disturbances of a certain type in a finite time is numerically demonstrated. It is noted that the equation in question does not have solutions in the form of solitary waves.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 81–84, January–February, 1990.The author is grateful to M. S. Ruderman for formulating the problem and to V. B. Baranov for his interest in the work.     

Theory of diffraction of weak shock waves

A numerical solution is considered to the universal nonlinear boundary-value diffraction problem which occurs in various problems of weak interaction [1, 2] in the asymptotic analysis of the flow in a region with large gradients of the parameters near the point of intersection of the incident, diffracted, and reflected waves. The analytical solutions to this type of problem usually approximately satisfy the conditions on the diffracted front, the position of which is not known beforehand, but is found along with the solution. In the present paper, the problem is solved by the numerical method of [3], which reduces the initial boundary-value problem for the system of short-wave equations with an unknown boundary to the solution of a series of boundary-value problems with a fixed boundary. The problem of the diffraction of a weak shock wave on a wedge with a finite apex angle is considered as an application of the solution. The data calculated by the asymptotic theory agree significantly better with the experimental data [5] than the theoretical data of [4].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6. pp. 176–178, November–December, 1984.     

Focusing of weak shock waves on a target in a parabolic chamber

Theoretical study of a weak shock focusing process in a confined chamber filled with liquid is presented. The chamber has a form of a thin cylinder with a parabolic cross-section, planar bottom and an arbitrary, although slowly varying, upper bounding surface. Analytical, numerical and experimental studies of weak shock wave focusing have been previously performed in the elliptic and ellipsoidal cases with a shock wave generated at one of the foci by means of an electric discharge or a microexplosion. In the present case a planar shock, perpendicular to the axis of the parabolic cross-section, sent in the inner of the chamber will converge at the focus after the reflection off the chamber wall, thus offering a different technical realization of the shock generation. The problem is solved within the frame of the geometrical acoustics approximation and a relation between the form of the upper bounding surface of the chamber and the pressure distribution behind the converging wavefront is obtained. It is shown that a desired pressure distribution may be obtained by an appropriate choice of the upper bounding surface.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1.0 and the AMS fonts, developed by the American Mathematical Society.