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.     

Attenuation of weak shock waves along pseudo-perforated walls

In order to attenuate weak shock waves in ducts, effects of pseudo-perforated walls were investigated. Pseudo-perforated walls are defined as wall perforations having a closed cavity behind it. Shock wave diffraction and reflection created by these perforations were visualized in a shock tube by using holographic interferometer, and also by numerical simulation. Along the pseudo-perforated wall, an incident shock wave attenuates and eventually turns into a sound wave. Due to complex interactions of the incident shock wave with the perforations, the overpressure behind it becomes non-uniform and its peak value can locally exceed that behind the undisturbed incident shock wave. However, its pressure gradient monotonically decreases with the shock wave propagation. Effects of these pseudo-perforated walls on the attenuation of weak shock waves generated in high speed train tunnels were studied in a 1/250-scaled train tunnel simulator. It is concluded that in order to achieve a practically effective suppression of the tunnel sonic boom the length of the pseudo-perforation section should be sufficiently long. Received 23 June 1997 / Accepted 16 September 1997     

Interaction of thin perforated plates and weak shock waves

The paper presents experimental results on elastic deformation of perforated thin plates made of glass-fiber laminate and subjected to weak shock waves. The characteristics of the dynamic process in the plates are determined __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 11, pp. 126–130, November 2006.     

Focusing of weak shock waves in confined axisymmetric chambers

Theoretical study of a weak shock wave focusing process on a spherical region in confined 3-D axisymmetric chambers is presented. The chambers are elliptic or parabolic in the plane cross-section containing their axis of symmetry. In the elliptic case a spherical shock wave of constant strength generated at one of the focal points will reflect off the chamber wall and converge on a spherical region around the second focus of the chamber. It is shown that the pressure distribution on the converging spherical shock wave is not homogeneous. In the parabolic case two possibilities of shock generation are considered. In the first one a plane shock wave of constant intensity is send in the inner of the chamber. This shock wave with the plane perpendicular to the symmetry axis will after the reflection off the chamber wall transform to a spherical shock with non-homogeneous pressure distribution. Alternatively, a spherical shock of constant intensity generated at the focus of the paraboloidal chamber will after the reflection transform to a plane shock with non-homogeneous pressure distribution propagating in the outer of the chamber. The above mentioned problems are solved within the frame of the geometrical acoustics approximation and the flow fields as well as the non-uniform shock strengths behind the converging wave fronts are calculated.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.     

Growth and decay of weak shock waves in magnetogasdynamics

The purpose of the present study is to investigate the problem of the propagation of weak shock waves in an inviscid, electrically conducting fluid under the influence of a magnetic field. The analysis assumes the following two cases: (1) a planar flow with a uniform transverse magnetic field and (2) cylindrically symmetric flow with a uniform axial or varying azimuthal magnetic field. A system of two coupled nonlinear transport equations, governing the strength of a shock wave and the first-order discontinuity induced behind it, are derived that admit a solution that agrees with the classical decay laws for a weak shock. An analytic expression for the determination of the shock formation distance is obtained. How the magnetic field strength, whether axial or azimuthal, influences the shock formation is also assessed.     

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.     

Mach reflection of weak shock waves from a rigid wall

On the basis of experimental observations and theoretical analysis of flow structure in the neighborhood of the triple point, it is shown that one should reject the condition for equality of the angle of deflection of flows passing through the Mach front and the two other fronts and replace it with some supplementary condition. The system of consistency equations in the indicated region is closed by an equation which is obtained under the assumption of the extremality of the deflection angle of a flow passing through the incident and reflected fronts. Calculations of the pressure drops behind the shock fronts agree with experimental data in this case.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 26–33, September–October, 1973.The authors thank S. A. Khristianovich for consideration of the work and advice.     

Short-wave asymptotics for weak shock waves and solitons in mechanics

Some recent applications of the theory of non-linear waves in smoothly inhomogeneous and weakly dissipative media are discussed in the paper. The possibilities of “linear-ray” approximation when the non-linear self-refraction effects may be neglected in comparison with the non-linear wave distortion along the rays are demonstrated for weak acoustic shocks in stratified atmospheres and ocean, the solitary waves in shallow water of variable depth and the solitons in elastic rods.     

Structure of weak shock waves in 2-D layered material systems

Shock waves in homogeneous materials in the absence of phase transitions are understood to have a one-wave structure. However, upon loading of a layered heterogeneous material system a two-wave structure is obtained––a leading shock front followed by a complex pattern that varies with time. This dual shock-wave pattern can be attributed to material architecture through which the shock wave propagates, i.e. the impedance (and geometric) mismatch present at various length scales, and nonlinearities arising from material inelasticity and failure.The objective of the present paper is to provide a better understanding of the role of material architecture in determining the structure of weak shock waves in 2-D layered material systems. Normal plate-impact experiments are conducted on 2-D layered material targets to obtain both the precursor decay and the late-time dispersion. The particle velocity at the free surface of the target plate is measured by using a multi-beam VALYN VISAR. In order to understand the effects of layer thickness and the distance of wave propagation on elastic precursor decay and late-time dispersion several different targets with various layer and target thicknesses are employed. Moreover, in order to understand the effects of material inelasticity both elastic–elastic and elastic–viscoelastic bilaminates are utilized.The results of these experiments are interpreted by using asymptotic techniques to analyze propagation of acceleration waves in 2-D layered material systems. The analysis makes use of the Laplace transform and Floquet theory for ODE’s with periodic coefficients [Asymptotic solutions for wave propagation in elastic and viscoelastic bilaminates. In: Developments in Mechanics, Proceedings of the 14th Mid-Eastern Mechanics Conference, vol. 26, no. 8, pp. 399–417]. Both wave-front and late-time solutions for step-pulse loading on layered half-space are compared with the experimental observations. The results of the study indicate that the structure of acceleration waves is strongly influenced by impedance mismatch of the layers constituting the laminates, density of interfaces, distance of wave propagation, and the material inelasticity.     

Theory of weak turbulence of waves on a fluid surface

The nonlinear interaction of waves in a fluid of finite depth is discussed. Forbidden decay processes in the gravitational portion of the spectrum are eliminated from the Hamiltonian by means of a canonical transformation. This provides an opportunity to obtain a kinetic equation which takes into account scattering of capillary waves by gravitational waves, in addition to decays in the subsystem of gravitational waves. The distribution N_k P^1/2h^1/4k^–4 is obtained for capillary waves in shallow water with constant flow of energy P with respect to the spectrum in the space of the wave numbers k. The interaction of the gravitational and capillary turbulence spectra is discussed. An induced distribution of gravitational waves is found which results from their interaction with capillary waves. It is an increasing function of the wave numbers q in the region bounded by the capillary constant k_o, N_q q^9/4 (q < k_o). The coupling of spectra in the gravitational and capillary regions and the conversion from slightly turbulent distributions to universal distributions are discussed.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 97–106, November–December, 1974.     

Structure of weak shock waves with phase transitions of the first kind

An analytic solution is found to the problem of the structure of weak shock waves with phase transitions of the first kind; it is based on the use of the Mandel ' shtam—Leontovich—D ' yakov concept of second viscosity. The kinetic and thermodynamic characteristics of the problem are investigated, and also the condition of applicability of the second viscosity concept. It is shown that the obtained solution simplifies considerably if the vapor phase is regarded as a perfect gas. The results of calculations for the water—water vapor system are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 55–62, July–August, 1980.     

Diffraction of weak shock waves by deformed bodies submerged in a liquid

Conclusion On the basis of an analysis of theoretical and experimental data obtained up to now by various investigators, we can note the following major advances in the field of the interaction of shock waves with barriers submerged in a liquid:Exact solutions have been obtained for problems in the diffraction of acoustic shock waves by rigid and stationary bodies of specified shape (plates, wedges, cones, parabolic, elliptical, and circular cylinders, spheres, paraboloids of revolution); approximate schemes have been worked out for estimating hydrodynamic loads, making it possible to investigate various stages of the interaction of shock waves with elastic shells of revolution and solid bodies; studies have been conducted in the exact formulation of the interaction of plane (spherical) nonstationary waves with elastic barriers (unbounded plates, plates in a screen, infinitely long thin-walled and thick-walled cylindrical shells, closed thin-walled and thick-walled spherical shells); an exact solution has been found for the internal problems in the case of cavities (circular and elliptical cylinders, spheres, spheroids) and elastic shells of revolution (infinitely long cylindrical and closed spherical shells); methods have been worked out for the approximate determination of the parameters of objects (elastic thin-walled infinitely long cylindrical and closed spherical shells) from reflected echo signals; estimates have been given for the influence of the structural characteristics of an object (support, concentrated masses), the nonlinear properties of interacting media, cavitation in liquid, and plastic deformations in the barrier material on the process of hydrodynamic interaction.We should also mention the main lines of further investigation and the problems which require solution: designing new experimental apparatus and measuring complexes for studying the nonstationary behavior of deformed bodies and structures in a liquid; solution of problems in diffraction by oonical and cylindrical shells of finite length, and by compound structures of complicated form in which account is taken of the structural characteristics and the internal elements; calculation of three-layer and multilayer shells acted upon by shock waves, taking account of the transverse compression of the filler; construction of more exact schemes (models) for the nonlinear and cavitation-type interaction of waves with barriers; development of numerical and combined methods for the solution of the problems in hydroelasticity.Mechanics Institute, Moscow State University. Translated from Prikladnaya Mekhanika, Vol. 16, No. 5, pp. 3–11, May, 1980.     

Experimental investigation of the interaction between weak shock waves and granular layers

 The paper describes new experimental results regarding the pressure fields in front of and inside granular layers of different materials during their collision with weak shock waves. A variety of waves result from the shock wave-granular layer interaction. The pressure behind the reflected wave from the material interface approaches the equilibrium value, P ₅, which would have been reached had the shock wave reflected from a solid end-wall. The wave succession inside the layer depends solely on two processes: the complex interaction of the compaction wave with the granular material and the gas filtration, which affects the particles by the drag forces between the two phases. Inside a material with a permeability coefficient f>0.001 mm² the transmitted wave moves with a constant velocity which is largely governed by the gas filtration. For low permeability materials ( f<0.0003 mm²) the transmitted wave trajectory strongly depends on the compaction wave propagation. In such cases the compaction wave was found to be unsteady and its acceleration was higher in material having low material densities. The maximum compressive stress values, P _c , reached at the shock tube end-wall, covered by the materials under investigation, manifested as an unsteady pressure peak twice as large as the gas pressure P ₅, measured ahead of the layer. Comparing the present data with those available in the literature showed that the amplitude of the unsteady pressure peak was higher in materials having low effective densities, γ, and small permeability coefficients f. Contrary to flexible foams where the available experimental data indicated that the compressive stress in the post peak period converges to P ₅=P _g , the results obtained in the present study indicated that during the test time the compressive stress, P _s , was well preserved in the material and for most of the sample length its value was within the range P _s >P ₅>P _g . Received: 4 March 1996 / Accepted:26 September 1996     

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.     

Asymptotic laws of damping of weak continuous and shock waves in dust-laden gas

Analytic investigations into the damping of perturbations in dust-laden gas have been restricted to self-similar flows [1, 2] and flows with a symmetry plane, it being assumed in the latter case that thermal and velocity equilibrium of the phases is established instantaneously [3–6], i.e., the relaxation time of the medium is short. In the present paper, asymptotic laws of damping are obtained for plane, cylindrical, and spherical shock and continuous waves whose amplitude and width are such that the acceleration of the particles and the change in their temperature can be ignored. It is assumed that between the phases there is heat transfer proportional to the temperature difference and frictional momentum transfer proportional to the difference between the velocities of the phases. The obtained laws of damping of plane waves are found to be entirely analogous to the laws of damping of magnetohydrodynamic waves in a medium with finite conductivity, when the presence of Joule dissipation and the additional ponderomotive force in the traveling wave or in the gas flow behind the shock wave leads to exponential damping of the wave amplitude [7–9].