Journal articles: 'Shock waves'

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MITROVIĆ, DARKO, and MARKO NEDELJKOV. "DELTA SHOCK WAVES AS A LIMIT OF SHOCK WAVES." Journal of Hyperbolic Differential Equations 04, no. 04 (December 2007): 629–53. http://dx.doi.org/10.1142/s021989160700129x.

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We discus the existence of delta shock waves obtained as a limit of two shock waves. For that purpose we perturb a prototype of weakly hyperbolic 2 × 2 system (sometimes called the "generalized pressureless gas dynamics model") by an additional term (called the "generalized vanishing pressure"). The obtained perturbed system is strictly hyperbolic and its Riemann problem is solvable. Since it is genuinely nonlinear, its solution consists of shocks and rarefaction waves combination. As perturbation parameter vanishes, the solution converges in the space of distribution. Specially, a solution consisting of two shocks converge to a delta function. Also, we give a formal definition of approximate solution and prove a kind of entropy argument. The paper finishes by a discussion about delta shock interactions for the original system.

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Zhang, Congyao, Eugene Churazov, and Irina Zhuravleva. "Pairs of giant shock waves (N-waves) in merging galaxy clusters." Monthly Notices of the Royal Astronomical Society 501, no. 1 (November 30, 2020): 1038–45. http://dx.doi.org/10.1093/mnras/staa3718.

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ABSTRACT When a subcluster merges with a larger galaxy cluster, a bow shock is driven ahead of the subcluster. At a later merger stage, this bow shock separates from the subcluster, becoming a ‘runaway’ shock that propagates down the steep density gradient through the cluster outskirts and approximately maintains its strength and the Mach number. Such shocks are plausible candidates for producing radio relics in the periphery of clusters. We argue that, during the same merger stage, a secondary shock is formed much closer to the main cluster centre. A close analogue of this structure is known in the usual hydrodynamics as N-waves, where the trailing part of the ‘N’ is the result of the non-linear evolution of a shock. In merging clusters, spherical geometry and stratification could further promote its development. Both the primary and the secondary shocks are the natural outcome of a single merger event and often both components of the pair should be present. However, in the radio band, the leading shock could be more prominent, while the trailing shock might conversely be more easily seen in X-rays. The latter argument implies that for some of the (trailing) shocks found in X-ray data, it might be difficult to identify their ‘partner’ leading shocks or the merging subclusters, which are farther away from the cluster centre. We argue that the Coma cluster and A2744 could be two examples in a post-merger state with such well-separated shock pairs.

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Vimercati, Davide, Giulio Gori, and Alberto Guardone. "Non-ideal oblique shock waves." Journal of Fluid Mechanics 847 (May 21, 2018): 266–85. http://dx.doi.org/10.1017/jfm.2018.328.

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From the analysis of the isentropic limit of weak compression shock waves, oblique shock waves in which the post-shock Mach number is larger than the pre-shock Mach number, named non-ideal oblique shocks, are admissible in substances characterized by moderate molecular complexity and in the close proximity to the liquid–vapour saturation curve. Non-ideal oblique shocks of finite amplitude are systematically analysed, clarifying the roles of the pre-shock thermodynamic state and Mach number. The necessary conditions for the occurrence of non-ideal oblique shocks of finite amplitude are singled out. In the parameter space of pre-shock thermodynamic states and Mach number, a new domain is defined which embeds the pre-shock states for which the Mach number increase can possibly take place. The present findings are confirmed by state-of-the-art thermodynamic models applied to selected commercially available fluids, including siloxanes and hydrocarbons currently used as working fluids in renewable energy systems.

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Леонович, Анатолий, Anatoliy Leonovich, Цюган Цзун, Qiugang Zong, Даниил Козлов, Daniil Kozlov, Юнфу Ван, and Yongfu Wang. "Alfvén waves in the magnetosphere generated by shock wave / plasmapause interaction." Solar-Terrestrial Physics 5, no. 2 (June 28, 2019): 9–14. http://dx.doi.org/10.12737/stp-52201902.

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We study Alfvén waves generated in the magnetosphere during the passage of an interplanetary shock wave. After shock wave passage, the oscillations with typical Alfvén wave dispersion have been detected in spacecraft observations inside the magnetosphere. The most frequently observed oscillations are those with toroidal polarization; their spatial structure is described well by the field line resonance (FLR) theory. The oscillations with poloidal polarization are observed after shock wave passage as well. They cannot be generated by FLR and cannot result from instability of high-energy particle fluxes because no such fluxes were detected at that time. We discuss an alternative hypothesis suggesting that resonant Alfvén waves are excited by a secondary source: a highly localized pulse of fast magnetosonic waves, which is generated in the shock wave/plasmapause contact region. The spectrum of such a source contains oscillation harmonics capable of exciting both the toroidal and poloidal resonant Alfvén waves.

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Doorly, D. J., and M. L. G. Oldfield. "Simulation of the Effects of Shock Wave Passing on a Turbine Rotor Blade." Journal of Engineering for Gas Turbines and Power 107, no. 4 (October 1, 1985): 998–1006. http://dx.doi.org/10.1115/1.3239847.

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The unsteady effects of shock waves and wakes shed by the nozzle guide vane row on the flow over a downstream turbine rotor have been simulated in a transient cascade tunnel. At conditions representative of engine flow, both wakes and shock waves are shown to cause transient turbulent patches to develop in an otherwise laminar (suction-surface) boundary layer. The simulation technique employed, coupled with very high-frequency heat transfer and pressure measurements, and flow visualization, allowed the transition initiated by isolated wakes and shock waves to be studied in detail. On the profile tested, the comparatively weak shock waves considered do not produce significant effects by direct shock-boundary layer interaction. Instead, the shock initiates a leading edge separation, which subsequently collapses, leaving a turbulent patch that is convected downstream. Effects of combined wake- and shock wave-passing at high frequency are also reported.

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Draine, B. T. "MagnetoHydrodynamic shock waves in molecular clouds." Symposium - International Astronomical Union 147 (1991): 185–96. http://dx.doi.org/10.1017/s007418090023951x.

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The fluid dynamics of MHD shock waves in magnetized molecular gas is reviewed. The different types of shock solutions, and the circumstances under which the different types occur, are delineated. Current theoretical work on C∗- and J-type shocks, and on the stability of C-type shocks, is briefly described. Observations of the line emission from MHD shocks in different regions appear to be in conflict with theoretical expectations for single, plane-parallel shocks. Replacement of plane-parallel shocks by bow shocks may help reconcile theory and observation, but it is also possible that the observed shocks may not be “steady”, or that theoretical models have omitted some important physics.

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Draine, B. T. "MagnetoHydrodynamic shock waves in molecular clouds." Symposium - International Astronomical Union 147 (1991): 185–96. http://dx.doi.org/10.1017/s0074180900198894.

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The fluid dynamics of MHD shock waves in magnetized molecular gas is reviewed. The different types of shock solutions, and the circumstances under which the different types occur, are delineated. Current theoretical work on C∗- and J-type shocks, and on the stability of C-type shocks, is briefly described. Observations of the line emission from MHD shocks in different regions appear to be in conflict with theoretical expectations for single, plane-parallel shocks. Replacement of plane-parallel shocks by bow shocks may help reconcile theory and observation, but it is also possible that the observed shocks may not be “steady”, or that theoretical models have omitted some important physics.

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8

Vieu, T., S. Gabici, and V. Tatischeff. "Particle acceleration at colliding shock waves." Monthly Notices of the Royal Astronomical Society 494, no. 3 (April 24, 2020): 3166–76. http://dx.doi.org/10.1093/mnras/staa799.

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ABSTRACT We model the diffusive shock acceleration of particles in a system of two colliding shock waves and present a method to solve the time-dependent problem analytically in the test-particle approximation and high energy limit. In particular, we show that in this limit the problem can be analysed with the help of a self-similar solution. While a number of recent works predict hard (E−1) spectra for the accelerated particles in the stationary limit, or the appearance of spectral breaks, we found instead that the spectrum of accelerated particles in a time-dependent collision follows quite closely the canonical E−2 prediction of diffusive shock acceleration at a single shock, except at the highest energy, where a hardening appears, originating a bumpy feature just before the exponential cut-off. We also investigated the effect of the reacceleration of pre-existing cosmic rays by a system of two shocks, and found that under certain conditions spectral features can appear in the cut-off region. Finally, the mathematical methods presented here are very general and could be easily applied to a variety of astrophysical situations, including for instance standing shocks in accretion flows, diverging shocks, backward collisions of a slow shock by a faster shock, and wind–wind or shock–wind collisions.

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9

Lee, Kha Loon. "Shock Waves." CFA Institute Magazine 20, no. 3 (May 2009): 16–19. http://dx.doi.org/10.2469/cfm.v20.n3.8.

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10

Butler, Michael. "Shock Waves." Cinema Journal 44, no. 4 (2005): 79–85. http://dx.doi.org/10.1353/cj.2005.0025.

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11

Horiuchi, Noriaki. "Shock waves." Nature Photonics 8, no. 7 (June 27, 2014): 499. http://dx.doi.org/10.1038/nphoton.2014.159.

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Kawamura, Yosuke, and Masafumi Nakagawa. "OS21-2 Experimental Study on the Oblique Shock Waves and Expansion Waves in the Supersonic Carbon Dioxide Two-phase Flow(Multiphase Shock Wave,OS21 Shock wave and high-speed gasdynamics,FLUID AND THERMODYNAMICS)." Abstracts of ATEM : International Conference on Advanced Technology in Experimental Mechanics : Asian Conference on Experimental Mechanics 2015.14 (2015): 260. http://dx.doi.org/10.1299/jsmeatem.2015.14.260.

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Singh, Manpreet, Federico Fraschetti, and Joe Giacalone. "Electrostatic Plasma Wave Excitations at the Interplanetary Shocks." Astrophysical Journal 943, no. 1 (January 1, 2023): 16. http://dx.doi.org/10.3847/1538-4357/aca7c6.

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Abstract Over the last few decades, different types of plasma waves (e.g., the ion acoustic waves (IAWs), electrostatic solitary waves, upper/lower hybrid waves, and Langmuir waves) have been observed in the upstream, downstream, and ramp regions of the collisionless interplanetary (IP) shocks. These waves may appear as short-duration (only a few milliseconds at 1 au) electric field signatures in the in-situ measurements, with typical frequencies of ∼1–10 kHz. A number of IAW features at the IP shocks seem to be unexplained by kinetic models and require a new modeling effort. Thus, this paper is dedicated to bridging this gap in understanding. In this paper, we model the linear IAWs inside the shock ramp by devising a novel linearization method for the two-fluid magnetohydrodynamic equations with spatially dependent shock parameters. It is found that, for parallel propagating waves, the linear dispersion relation leads to a finite growth rate, which is dependent on the shock density compression ratio, as Wind data suggest. Further analysis reveals that the wave frequency grows towards the downstream region within the shock ramp, and the wave growth rate is independent of the electron-to-ion temperature ratio, as Magnetospheric Multiscale (MMS) in-situ measurements suggest, and is uniform within the shock ramp. Thus, this study helps in understanding the characteristics of the IAWs at the collisionless IP shocks.

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HOSSEINI, Hamid. "1C05 Shock waves in regenerative medicine." Proceedings of the Bioengineering Conference Annual Meeting of BED/JSME 2013.25 (2013): 87–88. http://dx.doi.org/10.1299/jsmebio.2013.25.87.

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15

Xu, Y. F., S. C. Hu, Y. Cai, and S. N. Luo. "Origins of plastic shock waves in single-crystal Cu." Journal of Applied Physics 131, no. 11 (March 21, 2022): 115901. http://dx.doi.org/10.1063/5.0080757.

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We investigate shock wave propagation in single-crystal Cu with large-scale molecular dynamics simulations. Plastic shock waves propagate via dislocation nucleation or growth. With decreasing particle velocity, a remarkable drop in plastic shock wave velocity relative to the linear shock velocity–particle velocity relation is observed in the elastic–plastic two-wave regime for different loading directions. This reduction can be attributed to the changes in the mechanisms of plastic shock wave generation/propagation, from the dislocation nucleation-dominant mode, to the alternating nucleation and growth mode, and to the growth-dominant mode. For weak shocks, the plastic shock advances at the speed of the growth of existing dislocations (below the maximum elastic shock wave speed), considerably slower than the dislocation nucleation front for strong shocks (above the maximum elastic shock wave speed).

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Lehmann, Andrew, Benjamin Godard, Guillaume Pineau des Forêts, and Edith Falgarone. "Lyα generation in intermediate velocity shock waves." Proceedings of the International Astronomical Union 15, S352 (June 2019): 73–74. http://dx.doi.org/10.1017/s1743921320000654.

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AbstractWe update the Paris-Durham shock model, a state-of-the-art magnetohydrodynamic (MHD) shock code developed with a focus on molecular chemistry, in order to account for the self-generated UV field produced in shocks at velocities in the range 25-50 km/s. In these shocks there is significant excitation of atomic Hydrogen, with a large flux of Lyα photons escaping ahead of the shock to heat, ionize and drive molecular chemistry in a large slab of preshock gas.

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SCHWENDEMAN, DONALD W. "On converging shock waves of spherical and polyhedral form." Journal of Fluid Mechanics 454 (March 10, 2002): 365–86. http://dx.doi.org/10.1017/s0022112001007170.

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The behaviour of converging spherical shock waves is considered using Whitham's theory of geometrical shock dynamics. An analysis of converging shocks whose initial shape takes the form of regular polyhedra is presented. The analysis of this problem is motivated by the earlier work on converging cylindrical shocks discussed in Schwendeman & Whitham (1987). In that paper, exact solutions were reported for converging polygonal shocks in which the initial shape re-forms repeatedly as the shock contracts. For the polyhedral case, the analysis is performed both analytically and numerically for an equivalent problem involving shock propagation in a converging channel with triangular cross-section. It is found that a repeating sequence of shock surfaces composed of nearly planar pieces develops, although the initial planar surface does not re-form, and that the increase in strength of the shock at each iterate in the sequence follows the same behaviour as for a converging spherical shock independent of the convergence angle of the channel. In this sense, the shocks are stable and the result is analogous to that found in the two-dimensional case. A numerical study of converging spherical shocks subject to smooth initial perturbations in strength shows a strong tendency to form surfaces composed of nearly planar pieces suggesting that the stability result is fairly general.

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Lin, Xiao-Biao, and Stephen Schecter. "Traveling waves and shock waves." Discrete & Continuous Dynamical Systems - A 10, no. 4 (2004): i—ii. http://dx.doi.org/10.3934/dcds.2004.10.4i.

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Bulat, Pavel, Konstantin Volkov, and Igor Volobuev. "Interaction between Shock Waves Travelling in the Same Direction." Fluids 6, no. 9 (September 3, 2021): 315. http://dx.doi.org/10.3390/fluids6090315.

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In this paper, we study the intersection (interaction) between several steady shocks traveling in the same direction. The interaction between overtaking shocks may be regular or irregular. In the case of regular reflection, the intersection of overtaking shocks leads to the formation of a resulting shock, contact discontinuity, and some reflected discontinuities. The type of discontinuity depends on the parameters of incoming shocks. At the irregular reflection, a Mach shock forms between incoming overtaking shocks. Reflected discontinuities come from the points of intersection of the Mach stem with the incoming shocks. We also consider the possible types of shockwave configurations that form both at regular and irregular interactions of several overtaking shocks. The regions of existence of overtaking shock waves with different types of reflected shock and the intensity of reflected shocks are defined. The results obtained in the study can potentially be useful for designing supersonic intakes and advanced jet engines.

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Lowe, R. E., and D. Burgess. "The properties and causes of rippling in quasi-perpendicular collisionless shock fronts." Annales Geophysicae 21, no. 3 (March 31, 2003): 671–79. http://dx.doi.org/10.5194/angeo-21-671-2003.

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Abstract. The overall structure of quasi-perpendicular, high Mach number collisionless shocks is controlled to a large extent by ion reflection at the shock ramp. Departure from a strictly one-dimensional structure is indicated by simulation results showing that the surface of such shocks is rippled, with variations in the density and all field components. We present a detailed analysis of these shock ripples, using results from a two-dimensional hybrid (particle ions, electron fluid) simulation. The process that generates the ripples is poorly understood, because the large gradients at the shock ramp make it difficult to identify instabilities. Our analysis reveals new features of the shock ripples, which suggest the presence of a surface wave mode dominating the shock normal magnetic field component of the ripples, as well as whistler waves excited by reflected ions.Key words. Space plasma physics (numerical simulation studies; shock waves; waves and instabilities)

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Aghajani, A., and M. Hesaaraki. "On the structure of ionizing shock waves in magnetofluiddynamics." International Journal of Mathematics and Mathematical Sciences 29, no. 7 (2002): 395–415. http://dx.doi.org/10.1155/s0161171202010827.

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Ionizing shock waves in magnetofluiddynamics occur when the coefficient of electrical conductivity is very small ahead of the shock and very large behind it. For planner motion of plasma, the structure of such shock waves are stated in terms of a system of four-dimensional equations. In this paper, we show that for the above electrical conductivity as well as for limiting cases, that is, when this coefficient is zero ahead of the shock and/or is infinity behind it, ionizing fast, slow, switch-on and switch-off shocks admit structure. This means that physically these shocks occur.

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Cattell, C., S. Elliott, E. L. M. Hanson, and A. Skogsberg. "Juno Observations of Langmuir Waves in Association with Interplanetary Shocks from ~1.3 to ~5.4 au." Astrophysical Journal Letters 979, no. 2 (January 28, 2025): L38. https://doi.org/10.3847/2041-8213/ad99aa.

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Abstract Langmuir waves are often observed upstream of planetary bow shocks and interplanetary (IP) shocks. Waveform capture (WFC) measurements, obtained during Juno’s cruise phase, reveal the occurrence of Langmuir waves, as well as other wave modes, in association with IP shocks and IP coronal mass ejections (ICMEs) at distances from ∼1.3 to ∼5.4 au. This is a unique database of ICME/IP shock events with WFCs in the equatorial plane in these radial distances. We identified 12 events when WFCs were obtained within 36 hr upstream of the IP shock/ICME event (10 within less than 10 hr). Langmuir waves were observed at 9, a significantly higher occurrence than has previously been reported using spectral observations from other spacecraft. The larger rate is likely due to the bursty nature of Langmuir waves in combination with the higher time resolution of WFC data compared to spectral data. Observed peak amplitudes of ∼16 mV m−1 (peak to peak) are comparable to those observed upstream of outer planetary bow shocks. The highest peak amplitudes occurred in the strongest shocks (largest magnetic compression) and when waveforms were obtained closest to the shock. Langmuir waves were observed as much as 8 hr upstream. The WFCs provide electron density on rapid time scales, including an increase from ∼0.05 to 0.25 cm−3 in ∼0.01 s across an IP shock ramp. Although electron measurements were not obtained, these observations provide evidence for the energization of electrons at the shock or ICME front and, in some events, for propagation over large distances while maintaining the features in the distributions necessary to destabilize Langmuir waves.

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ZAMFIRESCU, CALIN, ALBERTO GUARDONE, and PIERO COLONNA. "Admissibility region for rarefaction shock waves in dense gases." Journal of Fluid Mechanics 599 (March 6, 2008): 363–81. http://dx.doi.org/10.1017/s0022112008000207.

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In the vapour phase and close to the liquid–vapour saturation curve, fluids made of complex molecules are expected to exhibit a thermodynamic region in which the fundamental derivative of gasdynamic Γ is negative. In this region, non-classical gasdynamic phenomena such as rarefaction shock waves are physically admissible, namely they obey the second law of thermodynamics and fulfil the speed-orienting condition for mechanical stability. Previous studies have demonstrated that the thermodynamic states for which rarefaction shock waves are admissible are however not limited to the Γ<0 region. In this paper, the conditions for admissibility of rarefaction shocks are investigated. This results in the definition of a new thermodynamic region – the rarefaction shocks region – which embeds the Γ<0 region. The rarefaction shocks region is bounded by the saturation curve and by the locus of the states connecting double-sonic rarefaction shocks, i.e. shock waves in which both the pre-shock and post-shock states are sonic. Only one double-sonic shock is shown to be admissible along a given isentrope, therefore the double-sonic states can be connected by a single curve in the volume–pressure plane. This curve is named the double sonic locus. The influence of molecular complexity on the shape and size of the rarefaction shocks region is also illustrated by using the van der Waals model; these results are confirmed by very accurate multi-parameter thermodynamic models applied to siloxane fluids and are therefore of practical importance in experiments aimed at proving the existence of rarefaction shock waves in the single-phase vapour region as well as in future industrial applications operating in the non-classical regime.

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Lehmann, A., B. Godard, G. Pineau des Forêts, A. Vidal-García, and E. Falgarone. "Self-generated ultraviolet radiation in molecular shock waves." Astronomy & Astrophysics 658 (February 2022): A165. http://dx.doi.org/10.1051/0004-6361/202141487.

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Context. The energetics and physical conditions of the interstellar medium and feedback processes remain challenging to probe. Aims. Shocks, modelled over a broad range of parameters, are used to construct a new tool to deduce the mechanical energy and physical conditions from observed atomic or molecular emission lines. Methods. We compute magnetised, molecular shock models with velocities Vs = 5–80 km s−1, pre-shock proton densities nH = 102–106 cm−3, weak or moderate magnetic field strengths, and in the absence or presence of an external UV radiation field. These parameters represent the broadest published range of physical conditions for molecular shocks. As a key shock tracer, we focus on the production of CH+ and post-process the radiative transfer of its rovibrational lines. We develop a simple emission model of an ensemble of shocks for connecting any observed emission lines to the mechanical energy and physical conditions of the system. Results. For this range of parameters, we find the full diversity (C-, C*-, CJ-, and J-type) of magnetohydrodynamic shocks. H2 and H are dominant coolants, with up to 30% of the shock kinetic flux escaping in Lyα photons. The reformation of molecules in the cooling tail means H2 is even a good tracer of dissociative shocks and shocks that were initially fully atomic. The known shock tracer CH+ can also be a significant coolant, reprocessing up to 1% of the kinetic flux. Its production and excitation is intimately linked to the presence of H2 and C+. For each shock model we provide integrated intensities of rovibrational lines of H2, CO, and CH+, and atomic H lines, and atomic fine-structure and metastable lines. We demonstrate how to use these shock models to deduce the mechanical energy and physical conditions of extragalactic environments. As a template example, we interpret the CH+(1−0) emission from the Eyelash starburst galaxy. A mechanical energy injection rate of at least 1011 L⊙ into molecular shocks is required to reproduce the observed line. We find that shocks with velocities as low as 5 km s−1 irradiated by a strong UV field are compatible with the available energy budget. The low-velocity, externally irradiated shocks are at least an order magnitude more efficient than the most efficient shocks with no external irradiation in terms of the total mechanical energy required. We predict differences of more than two orders of magnitude in the intensities of the pure rotational lines of CO, Lyα, and the metastable lines of O, S+, and N between representative models of low-velocity (Vs ~ 10 km s−1) externally irradiated shocks and higher-velocity shocks (Vs ≥ 50 km s−1) with no external irradiation. Conclusions. Shock modelling over an extensive range of physical conditions allows for the interpretation of challenging observations of broad line emission from distant galaxies. Our new method opens up a promising avenue to quantitatively probe the physical conditions and mechanical energy of galaxy-scale gas flows.

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Trakhinin, Yuri. "On weak stability of shock waves in 2D compressible elastodynamics." Journal of Hyperbolic Differential Equations 19, no. 01 (March 2022): 157–73. http://dx.doi.org/10.1142/s0219891622500035.

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By using an equivalent form of the uniform Lopatinski condition for 1-shocks, we prove that the stability condition found by the energy method in [A. Morando, Y. Trakhinin and P. Trebeschi, Structural stability of shock waves in 2D compressible elastodynamics, Math. Ann. 378 (2020) 1471–1504] for the rectilinear shock waves in two-dimensional flows of compressible isentropic inviscid elastic materials is not only sufficient but also necessary for uniform stability (implying structural nonlinear stability of corresponding curved shock waves). The key point of our spectral analysis is a delicate study of the transition between uniform and weak stability. Moreover, we prove that the rectilinear shock waves are never violently unstable, i.e. they are always either uniformly or weakly stable.

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Igra, O., and G. Ben-Dor. "Dusty Shock Waves." Applied Mechanics Reviews 41, no. 11 (November 1, 1988): 379–437. http://dx.doi.org/10.1115/1.3151872.

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The flow field developed behind shock waves in a pure gaseous medium is well known and documented in all gasdynamics textbooks. This is not the case when the gaseous medium is seeded with small solid particles. The present review treats various cases of shock waves propagation into a gas-dust suspension (dusty shock waves). It starts (chapter 1) with basic definitions of two-phase (gas-dust) suspensions and presents a general form of the conservation equations which govern dusty shock wave flows. In chapter two, the simple case of a steady flow of a suspension consisting of an inert dust and a perfect gas through a normal shock wave is studied. The effect of the dust presence, and of changes in its physical parameters, on the post-shock wave flow are discussed. Obviously, these discussions are limited to relatively weak shock waves (perfect gas). For stronger normal shock waves, the assumption of a perfect gas no longer holds. Therefore, in chapter three, real gas effects (ionization or dissociation) are taken into account when calculating the post-shock flow field. In chapter four, the dust chemistry is included and its effects on the post-shock flow is studied. In order to emphasize the role played by the dust chemistry, a comparison between a reactive and a similar inert suspension is presented. The case of an oblique shock wave in a dusty gas is discussed in chapter five. In all cases treated in chapters two to five the flow is steady; however, in many engineering applications this is not the case. In reality, even for the simplest case of a one-dimensional flow (normal shock wave propagation into quiescent suspension—the dusty shock tube) the shock wave attenuates and the flow field behind it is not steady. This case is treated in chapter six. The cases treated in chapters two to six deal with planar shock waves. However, all explosion generated shock waves in the atmosphere are spherical. Due to the engineering importance of this case, the post-shock flow for spherical shock waves in a dusty gas is studied, in detail, in chapter seven. It is shown in the present review that the dust presence has significant effects on the post-shock flow field. In all cases studied, a relaxation zone is developed behind the shock wave front. Throughout this zone momentum and energy exchange between the two phases of the suspension takes place. Through these interactions a new state of equilibrium is reached. The extent of the relaxation zone depends upon the dust loading ratio, the dust particle diameter, its specific heat capacity, and the dust spatial density. Due to the complexity of conducting experimental investigations with dusty shock waves, the number of published experimental results is very limited. As a result most of the present review contains numerical studies. However, in the few cases where experimental data are available, (e.g. dusty shock tube flow; see chapter six) a comparison between the numerical and experimental results is given.

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Zank, G. P., and J. F. Mckenzie. "The interaction of long-wavelength compressive waves with a cosmic ray shock." Journal of Plasma Physics 37, no. 3 (June 1987): 363–72. http://dx.doi.org/10.1017/s0022377800012241.

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This paper investigates the stability of a cosmic ray shock to long-wavelength perturbations. The problem is formulated in terms of finding the transmission coefficient for compressive waves across a cosmic ray shock by solving the generalized, two-fluid Rankine-Hugoniot relations. For strong shocks, the transmission coefficient confirms that compressive waves can undergo considerable amplification on passage through such shocks. The resonances of the transmission coefficient provides us with the dispersion equation governing the stability of the shock to long-wavelength ripple-like distortions. By using the principle of the argument method, it is established that cosmic ray shocks are stable.

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Kaur, Rupinder, and Nareshpal Singh Saini. "Ion Acoustic Shocks in a Weakly Relativistic Ion-Beam Degenerate Magnetoplasma." Galaxies 9, no. 3 (September 6, 2021): 64. http://dx.doi.org/10.3390/galaxies9030064.

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A theoretical investigation is carried out to study the propagation properties of ion acoustic shocks in a plasma comprising of positive inertial ions, weakly relativistic ion beam and trapped electrons in the presence of a quantizing magnetic field. By using the reductive perturbation technique, the Korteweg–de Vries-Burgers (KdVB) equation and oscillatory shocks solution are derived. The characteristics of such kinds of shock waves are examined and discussed in detail under suitable conditions for different physical parameters. The strength of the magnetic field, ion beam concentration and ion-beam streaming velocity have a great influence on the amplitude and width of the shock waves and oscillatory shocks. The results may be useful to study the characteristics of ion acoustic shock waves in dense astrophysical regions such as neutron stars.

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Fine, Harold J. "Seismic Shock Waves." Contemporary Psychology: A Journal of Reviews 34, no. 5 (May 1989): 497–98. http://dx.doi.org/10.1037/028056.

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Sagdeev, Roald Z., and Charles F. Kennel. "Collisionless Shock Waves." Scientific American 264, no. 4 (April 1991): 106–13. http://dx.doi.org/10.1038/scientificamerican0491-106.

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31

Stewart, J. M. "Gravitational shock waves." General Relativity and Gravitation 38, no. 6 (May 18, 2006): 1017–27. http://dx.doi.org/10.1007/s10714-006-0284-3.

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32

Bykov, A. M., K. Dolag, and F. Durret. "Cosmological Shock Waves." Space Science Reviews 134, no. 1-4 (February 2008): 119–40. http://dx.doi.org/10.1007/s11214-008-9312-9.

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33

Hoefer, Mark, and Mark Ablowitz. "Dispersive shock waves." Scholarpedia 4, no. 11 (2009): 5562. http://dx.doi.org/10.4249/scholarpedia.5562.

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34

Forrester, C. "Washington's shock waves." Communications Engineer 2, no. 2 (April 1, 2004): 8. http://dx.doi.org/10.1049/ce:20040212.

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35

MCKEE, C. F., and B. T. DRAINE. "Interstellar Shock Waves." Science 252, no. 5004 (April 19, 1991): 397–403. http://dx.doi.org/10.1126/science.252.5004.397.

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36

Nof, Doron. "Geostrophic Shock Waves." Journal of Physical Oceanography 16, no. 5 (May 1986): 886–901. http://dx.doi.org/10.1175/1520-0485(1986)016<0886:gsw>2.0.co;2.

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Dewar, William K. "Planetary Shock Waves." Journal of Physical Oceanography 17, no. 4 (April 1987): 470–82. http://dx.doi.org/10.1175/1520-0485(1987)017<0470:psw>2.0.co;2.

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38

Anile, A. M. "Relativistic shock waves." Il Nuovo Cimento A 87, no. 2 (May 1985): 139–50. http://dx.doi.org/10.1007/bf02902340.

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39

Delius, M., N. Weiss, S. Gambihler, A. Goetz, and W. Brendel. "Tumor therapy with shock waves requires modified lithotripter shock waves." Naturwissenschaften 76, no. 12 (December 1989): 573–74. http://dx.doi.org/10.1007/bf00462866.

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40

Apazidis, N., and M. B. Lesser. "On generation and convergence of polygonal-shaped shock waves." Journal of Fluid Mechanics 309 (February 25, 1996): 301–19. http://dx.doi.org/10.1017/s0022112096001644.

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A process of generation and convergence of shock waves of arbitrary form and strength in a confined chamber is investigated theoretically. The chamber is a cylinder with a specifically chosen form of boundary. Numerical calculations of reflection and convergence of cylindrical shock waves in such a chamber filled with fluid are performed. The numerical scheme is similar to the numerical procedure introduced by Henshaw et al. (1986) and is based on a modified form of Whitham's theory of geometrical shock dynamics (1957, 1959). The technique used in Whitham (1968) for treating a shock advancing into a uniform flow is modified to account for non-uniform conditions ahead of the advancing wave front. A new result, that shocks of arbitrary polygonal shapes may be generated by reflection of cylindrical shocks off a suitably chosen reflecting boundary, is shown. A study is performed showing the evolution of the shock front's shape and Mach number distribution. Comparisons are made with a theory which does not account for the non-uniform conditions in front of the shock. The calculations provide details of both the reflection process and the shock focusing.

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41

Gloag, J. M., and A. Balogh. "Shock parameter calculations at weak interplanetary shock waves." Annales Geophysicae 23, no. 2 (February 28, 2005): 545–52. http://dx.doi.org/10.5194/angeo-23-545-2005.

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Abstract. A large set of interplanetary shock waves observed using the Ulysses spacecraft is analysed in order to determine their local parameters. For the first time a detailed analysis is extended to the thermodynamic properties of a large number of events. The intention is to relate the shock parameters to the requirements set by MHD shock theory. A uniform approach is adopted in the selection of up and downstream regions for this analysis and applied to all the shock waves. Initially, the general case of a 3 component adiabatic plasma is considered. However, the calculation of magnetosonic and Alfvénic Mach numbers and the ratio of downstream to upstream entropy produce some unexpected results. In some cases there is no clear increase in entropy across the shock and also the magnetosonic Mach number can be less than 1. It is found that a more discerning use of data along with an empirical value for the polytropic index can raise the distribution of downstream to upstream entropy ratios to a more acceptable level. However, it is also realised that many of these shocks are at the very weakest end of the spectrum and associated phenomena may also contribute to the explanation of these results.

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Mond, M., and I. M. Rutkevich. "Spontaneous acoustic emission from strong ionizing shocks." Journal of Fluid Mechanics 275 (September 25, 1994): 121–46. http://dx.doi.org/10.1017/s0022112094002302.

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Dyakov (1954) and Kontorovich (1957) formulated the conditions for corrugation instability of shock waves as well as for spontaneous emission of sound and entropy-vortex waves from them. For the first time since then, it is shown here that physical circumstances do exist under which shocks in gases spontaneously emit sound waves. Such circumstances are provided by strong ionizing shocks. In order to see that, the coefficient of reflection of an acoustic wave from a shock is derived as a function of the wave's frequency and the ionization degree. Spontaneous emission of sound occurs when the reflection coefficient becomes infinitely large. It is shown that the relevant frequency range for the occurrence of spontaneous emission is that for which the electrons are not in local thermodynamic equilibrium with the heavy particles. The special properties of acoustic perturbations behind the ionizing shock are considered for this frequency range and the sound velocity in a partially ionized gas is derived. In addition, the condition for spontaneous emission of sound is modified in order to take into account the difference between the electrons and heavy-particle perturbed temperatures. It is shown, by numerical calculations, that the criterion for spontaneous emission is satisfied behind ionizing shocks in argon. In particular, for an initial pressure of 5 Torr, the threshold for the occurrence of the spontaneous emission is found to be M1 = 15. This critical value of the shock Mach number, as well as other calculated physical features, agree very well with those obtained experimentally by Glass & Liu (1978) who observed the occurrence of instability behind shocks in argon.

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Kumar Ray R, Amit. "Imploding Shock Waves on Explosive Driven Cylinder." International Journal of Science and Research (IJSR) 12, no. 8 (August 5, 2023): 779–81. http://dx.doi.org/10.21275/mr23806213411.

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MYONG, R. S., and P. L. ROE. "Shock waves and rarefaction waves in magnetohydrodynamics. Part 1. A model system." Journal of Plasma Physics 58, no. 3 (October 1997): 485–519. http://dx.doi.org/10.1017/s002237789700593x.

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The present study consists of two parts. Here in Part 1, a model set of conservation laws exactly preserving the MHD hyperbolic singularities is investigated to develop the general theory of the nonlinear evolution of MHD shock waves. Great emphasis is placed on shock admissibility conditions. By developing the viscosity admissibility condition, it is shown that the intermediate shocks are necessary to ensure that the planar Riemann problem is well-posed. In contrast, it turns out that the evolutionary condition is inappropriate for determining physically relevant MHD shocks. In the general non-planar case, by studying canonical cases, we show that the solution of the Riemann problem is not necessarily unique – in particular, that it depends not only on reference states but also on the associated internal structure. Finally, the stability of intermediate shocks is discussed, and a theory of their nonlinear evolution is proposed. In Part 2, the theory of nonlinear waves developed for the model is applied to the MHD problem. It is shown that the topology of the MHD Hugoniot and wave curves is identical to that of the model problem.

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45

Li, Dening. "Shock reflection and oblique shock waves." Journal of Mathematical Physics 48, no. 12 (December 2007): 123102. http://dx.doi.org/10.1063/1.2821982.

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Vuorinen, Laura, Rami Vainio, Heli Hietala, and Terry Z. Liu. "Monte Carlo Simulations of Electron Acceleration at Bow Waves Driven by Fast Jets in the Earth’s Magnetosheath." Astrophysical Journal 934, no. 2 (August 1, 2022): 165. http://dx.doi.org/10.3847/1538-4357/ac7f42.

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Abstract The shocked solar wind flows around the Earth’s magnetosphere in the magnetosheath downstream of the Earth’s bow shock. Within this region, faster flows of plasma, called magnetosheath jets, are frequently observed. These jets have been shown to sometimes exhibit supermagnetosonic speeds relative to the magnetosheath flow and to develop bow waves or shocks of their own. Such jet-driven bow waves have been observed to accelerate ions and electrons. We model electron acceleration by magnetosheath jet-driven bow waves using test-particle Monte Carlo simulations. Our simulations suggest that the energy increase of electrons with energies of a few hundred eV to 10 keV can be explained by a collapsing magnetic trap forming between the bow wave and the magnetopause with shock drift acceleration at the moving bow wave. Our simulations allow us to estimate the efficiency of acceleration as a function of different jet and magnetosheath parameters. Electron acceleration by jet-driven bow waves can increase the total acceleration in the parent shock environment, most likely also at shocks other than the Earth’s bow shock.

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Spiegelman, Marc. "Flow in deformable porous media. Part 2 Numerical analysis – the relationship between shock waves and solitary waves." Journal of Fluid Mechanics 247 (February 1993): 39–63. http://dx.doi.org/10.1017/s0022112093000370.

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Using numerical schemes, this paper demonstrates how viscous resistance to volume changes modifies the simplest shock wave solutions presented in Part 1. For an initial condition chosen to form a step-function shock, viscous resistance causes the shock to disperse into a rank-ordered wavetrain of solitary waves. Large obstructions in flux produce large-amplitude, slow-moving wavetrains while smaller shocks shed small-amplitude waves. While the viscous resistance term is initially important over a narrow boundary layer, information about obstructions in the flux can propagate over many compaction lengths through the formation of non-zero wavelength porosity waves. For large-amplitude shocks, information can actually propagate backwards relative to the matrix. The physics of dispersion is discussed and a physical argument is presented to parameterize the amplitude of the wavetrain as a function of the amplitude of the predicted shock. This quantitative relationship between the prediction of shocks and the development of solitary waves also holds when mass transfer between solid and liquid is included. Melting causes solitary waves to decrease in amplitude but the process is reversible and freezing can cause small perturbations in the fluid flux to amplify into large-amplitude waves. These model problems show that the equations governing volume changes of the matrix are inherently time dependent. Perturbations to steady-state solutions propagate as nonlinear waves and these problems demonstrate several initial conditions that do not relax to steady state. If these equations describe processes such as magma migration in the Earth, then these processes should be inherently episodic in space and time.

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Morris, Paul J., Artem Bohdan, Martin S. Weidl, and Martin Pohl. "Preacceleration in the Electron Foreshock. I. Electron Acoustic Waves." Astrophysical Journal 931, no. 2 (June 1, 2022): 129. http://dx.doi.org/10.3847/1538-4357/ac69c7.

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Abstract To undergo diffusive shock acceleration, electrons need to be preaccelerated to increase their energies by several orders of magnitude, else their gyroradii will be smaller than the finite width of the shock. In oblique shocks, where the upstream magnetic field orientation is neither parallel nor perpendicular to the shock normal, electrons can escape to the shock upstream, modifying the shock foot to a region called the electron foreshock. To determine the preacceleration in this region, we undertake particle-in-cell simulations of oblique shocks while varying the obliquity and in-plane angles. We show that while the proportion of reflected electrons is negligible for θ Bn = 74.°3, it increases to R ∼ 5% for θ Bn = 30°, and that, via the electron acoustic instability, these electrons power electrostatic waves upstream with energy density proportional to R 0.6 and a wavelength ≈2λ se, where λ se is the electron skin length. While the initial reflection mechanism is typically a combination of shock-surfing acceleration and magnetic mirroring, we show that once the electrostatic waves have been generated upstream, they themselves can increase the momenta of upstream electrons parallel to the magnetic field. In ≲1% of cases, upstream electrons are prematurely turned away from the shock and never injected downstream. In contrast, a similar fraction is rescattered back toward the shock after reflection, reinteracts with the shock with energies much greater than thermal, and crosses into the downstream.

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HAIM, L., M. GEDALIN, A. SPITKOVSKY, V. KRASNOSELSKIKH, and M. BALIKHIN. "Nonlinear waves and shocks in relativistic two-fluid hydrodynamics." Journal of Plasma Physics 78, no. 3 (February 10, 2012): 295–302. http://dx.doi.org/10.1017/s002237781200013x.

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AbstractRelativistic shocks are present in a number of objects where violent processes are accompanied by relativistic outflows of plasma. The magnetization parameter σ = B2/4πnmc2 of the ambient medium varies in wide range. Shocks with low σ are expected to substantially enhance the magnetic fields in the shock front. In non-relativistic shocks the magnetic compression is limited by nonlinear effects related to the deceleration of flow. Two-fluid analysis of perpendicular relativistic shocks shows that the nonlinearities are suppressed for σ≪1 and the magnetic field reaches nearly equipartition values when the magnetic energy density is of the order of the ion energy density, Beq2 ~ 4πnmic2γ. A large cross-shock potential eφ/mic2γ0 ~ B2/Beq2 develops across the electron–ion shock front. This potential is responsible for electron energization.

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Matsukiyo, Shuichi, and Yosuke Matsumoto. "Injection Process of Pickup Ion Acceleration at an Oblique Heliospheric Termination Shock." Astrophysical Journal Letters 970, no. 2 (July 29, 2024): L37. http://dx.doi.org/10.3847/2041-8213/ad5d73.

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Abstract The injection process of pickup ion acceleration at a heliospheric termination shock is investigated. Using two-dimensional fully kinetic particle-in-cell simulation, accelerated pickup ions are self-consistently reproduced by tracking long time evolution of shocks with an unprecedentedly large system size in the shock normal direction. Reflected pickup ions drive upstream large-amplitude waves through resonant instabilities. Convection of the large-amplitude waves causes shock surface reformation and alters the downstream electromagnetic structure. A part of pickup ions are accelerated to tens of upstream flow energy in the timescale of ∼100 times inverse ion gyrofrequency. The initial acceleration occurs through the shock surfing acceleration (SSA) mechanism followed by the shock drift acceleration mechanism. Large electrostatic potential accompanied by the upstream waves enables the SSA to occur.

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Journal articles on the topic 'Shock waves'

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