Parameters of shock waves created by exploding horizontal cylindrical charges in a loam

The parameters of the shock waves created by exploding horizontal cylindrical charges in a loam have been experimentally investigated with allowance for the effect of the free surface. The effect of charge depth on the shock wave parameters is demonstrated.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 10, No. 3, pp. 161–164, May–June, 1969.     

Similitude in exploding wires

Conditions for similarity have been derived for phenomena involving an exploding wire, propagation of the resulting shock through a fluid, and inelastic deformation or fracture of a structure. This analysis considers the capacitance, voltage and material properties of the exploding wire, as well as the geometry and material properties of the fluid and structure. Model experiments involving large inelastic deformation of aluminum diaphragms agree well with the theoretical predictions. Comparison of deformations due to chemical explosives and exploding wires has been made in an effort to establish an ‘equivalence’ between chemical and electrical explosions. The purpose of this is to provide a basis for using small-scale experiments with exploding wires to predict the performance of larger systems using chemical explosives. The exploding wire appears to provide a precise loading for small-scale structural-model experiments and explosion-forming experiments.     

Simulation of converging cylindrical GPa-range shock waves generated by wire array underwater electrical explosions

Results of one-dimensional numerical simulations of the parameters of the converging strong shock wave generated by electrical underwater explosions of a cylindrical wire array with different array radii and different deposited energies are presented. It was shown that for each wire array radius there exists an optimal duration of the energy deposition into the exploding array, which allows one to maximize the shock wave pressure and temperature in the vicinity of the implosion axis. The simulation results agree well with the 130-GPa pressure in the vicinity of the implosion axis that was recently obtained, which strongly indicates the azimuthal symmetry of the converging shock wave at these extreme conditions. Also, simulations showed that using a pulsed power generator with a stored energy of ~200 kJ, the pressure and temperature at the shock wave front reaches ~220 GPa and 1.7 eV at 0.1 mm from the axis of implosion in the case of a 2.5 mm radius wire array explosion. It was found that, in spite of the complicated equation of state of water, the maximum pressure at the shock wave front at radius r can be estimated as P ≈ (P*(r*/r) ^α , where P* is the known value of pressure at the shock wave front at radius r* ≥ r and α is a parameter that equals 0.62±0.02. A rough estimate of the implosion parameters of the hydrogen target after the interaction with the converging strong shock wave is presented as well.     

Thermally generated stress waves in a dispersive elastic rod

The propagation of thermally generated stress waves in a dispersive elastic rod was investigated both experimentally and analytically. In the experimental investigation, the end of a circular colored-glass rod was heated very rapidly by the deposition of luminous energy from a Q-switched ruby laser. The light from the laser was directed parallel to the axis of the rod and deposited on the polished end of the rod. The depth of deposition was of the same order as the radius of the rod. The length of the energy pulse from the laser was 20 nsec. This results in heating at such a rate that it can be considered as instantaneous when compared to the mechanical response of the material used. The resulting stress wave was measured using a thin quartz crystal in a Hopkinson pressure-bar arrangement. Radial inertia precluded the use of the simple wave equation; Love's modified wave equation was used to describe the motion. The thermoelastic problem was reduced to a homogeneous partial differential equation with appropriate initial and boundary conditions which is solved by the separation of variables technique. The experimental results are in good agreement with Love's theory. The amplitude of the stress waves was found to be directly proportional to the total energy deposited. The very short stress pulses generated by Q-switched laser deposition on the end of the thin rod gave rise to the higher modes of longitudinal wave propagation. The existence of wave propagation in a thin rod at near dilatational velocities was experimentally confirmed. It is concluded that the experimental techniques developed can be used to model stress-wave generation due to electromagnetic-energy depositions. Also, laser deposition provides an efficient means for generating the higher modes of longitudinal wave propagation in thin rods. Paper was presented at 1968 SESA Spring Meeting held in Albany, N. Y., on May 7–10. This work was supported by the U. S. Atomic Energy Commission at University of California, Lawrence Radiation Laboratory, Livermore, Calif.     

Slot diffraction of waves generated by an underwater explosion on the axis of a circular cylindrical shell

The problem of diffraction of a cylindrical shock wave generated by an electrical explosion in water on a longitudinal slot in a cylindrical rigid shell is considered. The numerical method of discontinuity breakdown is modified in relation to the polar coordinates used, which simplifies significantly the technical aspect of the solution. In the calculations pressure fields with diffraction maxima and minima are obtained and the evolution of the minima into ruptures is traced.Donetsk. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 3–8, November–December, 1996.     

Computation of waves generated by submarine landslides

Waves generated by submarine landslides are treated as three-dimensional flows of a perfect incompressible fluid. For the solution of the Cauchy-Poisson problem a time-discretization is applied which leads at each time step to a non-homogeneous free surface condition; the solution is then divided into two parts. The first part, subject to the true free surface condition, is computed in a simplified domain with constant depth. The second part involves a homogeneous free surface condition, a corrected bottom condition and the true bathymetry. In the case of constant depth, unconditional stability of the time discretization is derived. In the case of variable depth, mass and energy conservation is derived. Numerical results are presented. Comparison is made with other methods for the generation of axisymmetric waves. The transient propagation along a rectilinear coast is studied, including a comparison between two different bathymetries; trapping of energy is observed.     

Peripheral waves generated in a cylindrical shell with hemispherical endcaps by a plane acoustic wave at axial incidence

The recently proposed approach [B. Dubus, A. Lavie, N.D. Veksler, J. Acoust. Soc. Am. 102 (1997) 3523-3529] of resonance order determination is applied to a steady-state problem of acoustic wave scattering by a cylindrical shell with hemispherical endcaps. In the intermediate frequency range, the main contribution in the form function is brought by two peripheral waves: zero-order symmetric Lamb-type wave S₀ and water-borne bending-type A wave. The resonance frequencies of these waves are found and the dispersion curves of the phase velocities computed. The influence of the relative length of the cylindrical part of the shell on the characteristics of peripheral waves is discussed.     

Focusing shock waves generated by an accelerating projectile

Unsteady flow fields past a projectile either in parabolic motion or in accelerated motion are investigated numerically, with special attention to shock wave focusing phenomena induced by the motions. The two-dimensional Euler equations are solved by a moving-cell, MUSCL TVD finite-volume method with Steger-Warming flux-vector splitting. The pattern of shock wave focusing induced by the projectile is found to be similar to that of shock waves reflected from a concave wall observed in a laboratory experiment using a shock tube. Effects of projectile shapes and trajectories on focusing are also examined.     

Propagation of magnetoelastic stress waves from a cylindrical cavity in a conducting medium

We study the influence of a constant axial magnetic field on the propagation of magnetoelastic compression waves from a cavity containing a magnetoacoustic medium with a jump of the surface force given at the wall. The problem is examined in [1] in the case in which there is a vacuum in the cavity and an ideal conductor outside, without any study of the effect of a magnetic field.Here we examine the problem for both weak and ideal conductivities. The equations are linearized and Laplace transformed. Approximate asymptotic solutions are constructed which are valid in the vicinity of the wave fronts. The solutions are studied analytically and numerically.In conclusion the author wishes to thank Ya. S. Uflyand for discussions of certain questions.     

Diffraction of time-harmonic elastic waves by a cylindrical obstacle

The diffraction of two-dimensional elastic waves by a cylindrical obstacle is investigated. In two-dimensional wave motions, the geometrical configuration as well as the field quantities involved are assumed to be independent of one of the Cartesian coordinates. As such, the two-dimensional theory can be considered as a special case of the three-dimensional one. Following this line of reasoning, the integral-equation formulation of two-dimensional elastodynamic diffraction problems is derived. For a number of configurations, the resulting integral equations are solved numerically. Also, numerical results pertaining to normalized power scattering characteristics and extinction cross-sections are presented.The research reported in this paper has been supported by the Netherlands organization for the advancement of pure research (Z.W.O.).     

Surface characters of internal waves generated by Rankine ovoid

A linear theory on the internal waves generated in the stratified fluid with a pycnocline is presented in this paper. The internal wave fields such as the velocity fields in the stratified fluid and velocity gradient fields at the free surface are also investigated by means of the theoretical and numerical method. From the numerical results, it is shown that the internal wave generated by horizontally moving Rankine ovoid is a sort of trapped wave which propagates in a wave guide, and its waveform is a kind of Mach front-type internal wave in the pycnocline. Influence of the internal wave on the flow fields at the free surface is represented by the velocity gradient fields resulted from the internal waves generated by motion of the Rankine ovoid. At the same time, it is also shown that under the hypothesis of inviscid fluid, the synchronism between the surface velocity gradient fields at the free surface and the internal wave fields in the fluid is retained. This theory opens a possibility to study further the modulated spectrum of the Bragg waves at the free surface.The project supported by the National Natural Science Foundation of China (40576010). The English text was polished by Keren Wang.     

Large-scale edge waves generated by a moving atmospheric pressure

Long waves generated by a moving atmospheric pressure distribution, associated with a storm, in coastal region are investigated numerically. For simplicity the moving atmospheric pressure is assumed to be moving only in the alongshore direction and the beach slope is assumed to be a constant in the on-offshore direction. By solving the linear shallow water equations we obtain numerical solutions for a wide range of physical parameters, including storm size (2a), storm speed (U), and beach slope (α). Based on the numerical results, it is determined that edge wave packets are generated if the storm speed is equal to or greater than the critical velocity, U_cr, which is defined as the phase speed of the fundamental edge wave mode whose wavelength is scaled by the width of the storm size. The length and the location of the positively moving edge wave packet is roughly Ut/2 ≤ y ≤ Ut, where y is in the alongshore direction and t is the time. Once the edge wave packet is generated, the wavelength is the same as that of the fundamental edge wave mode corresponding to the storm speed and is independent of the storm size, which can, however, affect the wave amplitude. When the storm speed is less than the critical velocity, the primary surface signature is a depression directly correlated to the atmospheric pressure distribution.     

Characteristics of aerodynamic sound sources generated by coiled wires in a uniform air-flow

This study deals experimentally with aerodynamic sounds generated by coiled wires in a uniform air-flow. The coiled wire is a model of the hair dryer's heater. In the experiment, the effects of the coil diameter D, wire diameter d and coil spacing s of the coiled wire on the aerodynamic sound have been clarified. The results of frequency analyses of the aerodynamic sounds show that an Aeolian sound is generated by the coiled wire, when s/d is larger than 1. Also the peak frequencies of Aeolian sounds generated by the coiled wires are higher than the ones generated by a straight cylinder having the same diameter d. To clarify the characteristics of the aerodynamic sound sources, the directivity of the aerodynamic sound generated by the coiled wire has been examined, and the coherent function between the velocity fluctuation around the coiled wire and the aerodynamic sound has been calculated. Moreover, the band overall value of coherent output power between the sound and the velocity fluctuations has been calculated. This method has clarified the sound source region of the Aeolian sound generated by the coiled wire. These results show that the Aeolian sound is generated by the arc part of the coiled wire, which is located in the upstream side of the air-flow.     

Blast waves from cylindrical charges

Comparisons of explosives are often carried out using TNT equivalency which is based on data for spherical charges, despite the fact that many explosive charges are not spherical in shape, but cylindrical. Previous work has shown that it is possible to predict the over pressure and impulse from the curved surface of cylindrical charges using simple empirical formulae for the case when the length-to-diameter (L/D) ratio is greater or equal to 2/1. In this paper, by examining data for all length-to-diameter ratios, it is shown that it is possible to predict the peak over pressure, P, for any length-to-diameter ratio from the curved side of a bare cylindrical charge of explosive using the equation $P=K_PM(L/D)^{1/3}/R^3$ , where M is the mass of explosive, R the distance from the charge and $ K_P$ is an explosive-dependent constant. Further out where the cylindrical blast wave ‘heals’ into a spherical one, the more complex equation $P=C_1(Z^{\prime \prime })^{-3}+C_2(Z^{\prime \prime })^{-2}+C_3(Z^{\prime \prime })^{-1}$ gives a better fit to experimental data, where $ Z^{\prime \prime } = M^{1/3}(L/D)^{1/9}/D$ and $C_1,\, C_2 $ and $ C_3$ are explosive-dependent constants. The impulse is found to be independent of the L/D ratio.     

Water waves generated by disturbance at an inertial surface

Laplace transform technique is used to solve an initial value problem describing waves generated by a disturbance created at the surface of water covered by an inertial surface composed of a thin but uniform distribution of floating particles. Green's integral theorem produces the transformed potential function from which the form of the inertial surface is obtained as an infinite integral after taking Laplace inversion. The method of stationary phase is then employed to evaluate this integral approximately for large time and distance.     

Scattering of pulsed Rayleigh surface waves by a cylindrical cavity

A pulsed Rayleigh surface wave of prescribed shape is incident on a cylindrical cavity which is parallel to both the plane free surface and the plane wave front. Multiple reflections at the cylindrical and plane free surface are considered and the resulting displacements and stress components are calculated in the surrounding of the cavity by approximately summing infinite double sums. Use is made of the stationary loading case simulated by a periodic train of wave pulses and its time Fourier series representation and of expansions of all incident and reflected waves in terms of cylindrical wave functions. For reflection, the free surface of the half-space is approximated by a fictitious convex (or concave) cylindrical surface of “large” radius. The wave pattern due to a single pulse loading is constructed from the stationary solution by enforcing homogeneous initial conditions in the half-space ahead of the single loading pulse and by prescribing a wide spacing in the periodically set-forth train of pulses. The numerical results for stresses and dynamic stress magnification factors are especially useful for the interpretation of recent measurements in dynamic photoelasticity.     

On the scattering of spherical waves by a cylindrical object

Summary The connection between plane wave and spherical wave scattering from an infinitely cylindrical object is investigated. If the distance of the source and the observer from the axis of the cylinder are denoted by ϱ₀ and ϱ, respectively, the ratio of the scattered field to that for plane wave excitation [ϱ0/(ϱ0 + ϱ)]^1/2.     

Landslide generated impulse waves.

Landslide generated impulse waves were investigated in a two-dimensional physical laboratory model based on the generalized Froude similarity. Digital particle image velocimetry (PIV) was applied to the landslide impact and wave generation. Areas of interest up to 0.8 m by 0.8 m were investigated. The challenges posed to the measurement system in an extremely unsteady three-phase flow consisting of granular matter, air, and water were considered. The complex flow phenomena in the first stage of impulse wave initiation are: high-speed granular slide impact, slide deformation and penetration into the fluid, flow separation, hydrodynamic impact crater formation, and wave generation. During this first stage the three phases are separated along sharp interfaces changing significantly within time and space. Digital masking techniques are applied to distinguish between phases thereafter allowing phase separated image processing. PIV provided instantaneous velocity vector fields in a large area of interest and gave insight into the kinematics of the wave generation process. Differential estimates such as vorticity, divergence, elongational, and shear strain were extracted from the velocity vector fields. The fundamental assumption of irrotational flow in the Laplace equation was confirmed experimentally for these non-linear waves. Applicability of PIV at large scale as well as to flows with large velocity gradients is highlighted.List of symbols a wave amplitude (L) - c wave celerity (LT^–1) - d_diff diffraction limited minimum particle image diameter (L) - d_e diffracted particle image diameter (L) - d_g granulate grain diameter (L) - d_p seeding particle diameter (L) - d recorded particle image diameter (L) - f focal length (L) - f_# f number (-) - F slide Froude number (-) - g gravitational acceleration (LT^–2) - h still-water depth (L) - H wave height (L) - l_s slide length (L) - L wavelength (L) - M magnification (-) - m_s slide mass (M) - n refractive index (-) - n_por slide porosity (-) - N_iw number of seeding particles in an interrogation window (-) - N_pair number of detected particle image pairs in window (-) - p interrogation window size p×p pixels; 1 pixel=9 m (L) - P probability (-) - P_il probability of in-plane loss of particle (-) - P_ol probability of out-of-plane loss of particle (-) - s slide thickness (L) - S relative slide thickness (-) - t time after impact (T) - T wave period (T) - v velocity (LT^–1) - v_p particle velocity (LT^–1) - v_px streamwise horizontal component of particle velocity (LT^–1) - v_py crosswise horizontal component of particle velocity (LT^–1) - v_pz vertical component of particle velocity (LT^–1) - v_s slide centroid velocity at impact (LT^–1) - V dimensionless slide volume (-) - V_iw interrogation volume (L³) - V_s slide volume (L³) - x streamwise coordinate (L) - x_ip area of view x dimension in image plane (L) - z vertical coordinate (L) - slide impact angle (°) - bed friction angle (°) - y depth of field (L) - t laser pulse separation (T) - x mean particle image x displacement in interrogation window (L) - _x random displacement x error (L) - _v random velocity v error (LT^–1) - _tot total random velocity v error (LT^–1) - _bias velocity v error due to biased correlation analysis (LT^–1) - _optics velocity v error due to optical imaging errors (LT^–1) - _track velocity v error due to particle flow tracking error (LT^–1) - _xx streamwise horizontal elongational strain component (1/T) - _xz shear strain component (1/T) - _zx shear strain component (1/T) - _zz vertical elongational strain component (1/T) - water surface displacement (L) - wavelength (L) - dynamic viscosity (ML^–1T^–1) - density (ML^–3) - _g granulate density (ML^–3) - _p particle density (ML^–3) - _s mean slide density (ML^–3) - _w water density (ML^–3) - granulate internal friction angle (°) - _y vorticity vector component (out-of-plane) (1/T)