Structure of fully dispersed waves in dusty gases

The structure of one-dimensional fully dispersed waves in dusty gases is investigated using the Navier-Stokes equations for the gas phase and for the particle phase. The resulting system of six ordinary nonlinear differential equations is reduced to a system of four autonomous nonlinear differential equations which is solved analytically in phase space by expanding the variables of state in power series.

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Zusammenfassung Die Struktur eindimensionaler voll dispergierter Wellen in staubhaltigen Gasen wird untersucht indem die Navier-Stokesschen Gleichungen auf die Gas-Phase und auf die Partikel-Phase angewendet werden. Das sich ergebende System von sechs gewöhnlichen nichtlineren Differentialgleichungen wird auf ein System von vier autonomen nichtlinearen Differentialgleichungen zurückgeführt. Dieses System wird durch Reihenentwicklung der Zustandsgrößen im Phasenraum analytisch gelöst.

     

Solitary waves in self-gravitating dusty plasma

The electrostatic dust acoustic wave is studied in a uniform gravitating dusty plasma, when the self-gravitation does alter the linear modes given by the dispersion relation. It is observed that instead of a single nonlinear equation we get two coupled KdV-like systems. It is observed that such a system can sustain solitary-wave-like excitations under suitable conditions on the plasma parameters. Such an analysis is very important in ascertaining the fate of a grain orbiting around a magnetized planet in space, where there is a competition between magnetic and gravitational forces. And this in turn gives rise to the question of magnetogravitational instability, which plays an important role in star formation. It is observed that the gravitational soliton's amplitude, being proportional to the Jeans frequency, is always smaller compared to the electrostatic counterpart.     

The mutual penetration of simple waves in gases

Summary The encounter of two waves in a gas is treated by considering the mutual influence on the characteristics of the simple waves as a small disturbance. The results can be represented in a very simple way by use of the particle displacement. The particularities of the case =3 are discussed.

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Zusammenfassung Die Durchdringung von zwei Wellen in der Gasdynamik wird behandelt, indem man die gegenseitige Beeinflussung der Charakteristiken der einfachen Wellen als kleine Störungen auffasst. Die Ergebnisse werden besonders einfach mit Hilfe der Teilchenverschiebung dargestellt. Besonderheiten des hypothetischen Falles =3 werden diskutiert.

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Dedicated to Professor Dr. J. Ackeret, on the occasion of his eightieth birthday     

Laser-induced shock waves in liquids

Summary The interaction of TEA-CO₂-laser pulses with liquids has been investigated using shadowgraphy and high-speed pressure transducers. It is shown that the absorption coefficient of the liquid at the laser wavelength defines the geometry of the shock wave. Peak pressures in the kbar range have been observed.

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Zusammenfassung Die Wechselwirkung von TEA-CO₂-Laserimpulsen mit Flüssigkeiten wurde untersucht mit Hilfe von Schattenphotographien und schnellen Drucksonden. Es wird gezeigt, dass der Absorptionskoeffizient der Flüssigkeit bei der Laserwellenlänge die Geometrie der Stosswelle bestimmt. Es wurden Druckmaxima im kbar-Bereich beobachtet.

     

Analytical study of the nonlinear dust-acoustic waves in an unmagnetized dusty plasma

The nonlinear dust-acoustic waves in an unmagnetized dusty plasma, including consideration of the dust charge variation, is analytically investigated by using the formally variable separation approach. The exact analytical solutions in the general case are also obtained.     

Interaction of acoustic waves with shock waves in elastic solids

Summary Transmission and reflexion of plane acoustic waves through a longitudinal shock wave in elastic isotropic solids are investigated. As a result, the amplitude of the transmitted and reflected waves and the jump of the acceleration of the shock are explicitly determined.

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Résumé On envisage la transmission et la réflexion d'une onde acoustique plane par une onde de choc longitudinale dans les solides élastiques isotropes. On détermine explicitement l'amplitude des ondes transmises et réflechies et le saut de l'acceleration du choc.

     

Higher order shock waves

Shock waves and the relevant Rankine-Hugoniot conditions may be inadequate even in simple cases. As an example, the electromagnetic field generated by a point charge, whose velocity jumps from0 to a constant value, not only jumps across a spherical surface expanding at light speed, but also includes a-distribution term. This suggests that the concept of higher order shock waves be introduced, the associated compatibility conditions being also deduced.

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Sommario Il concetto di onda d'urto può rivelarsi inadeguato per descrivere situazioni anche semplici. A titolo di esempio, si determina esplicitamente l'espressione del campo elettromagnetico generato da una carica puntiforme la cui velocità passa istantaneamente da0 a un valore costante. Il campo elettromagnetico non solo è discontinuo attraverso una superficie sferica che si espande alla velocità della luce, ma include anche un termine che coinvolge la distribuzione. Ciò suggerisce di definire le onde d'urto di ordine superiore, per le quali si determinano le corrispondenti condizioni di compatibilità.

     

Delta shock waves in chromatography equations

The previous investigations on delta shock waves were mostly focused on those with Dirac delta function in only one state variable. In this paper, we obtain another kind from the nonlinear chromatography equations, in which the Dirac delta functions develop simultaneously in both state variables. It is strictly proved to satisfy the system in the sense of distributions. The generalized Rankine-Hugoniot relation and entropy condition are clarified. The numerical results completely coinciding with the theoretical analysis are presented.     

3+1 dimensional envelop waves and its stability in magnetized dusty plasma

It is well known that there are envelope solitary waves in unmagnetized dusty plasmas which are described by a nonlinear Schrodinger equation (NLSE). A three dimension nonlinear Schrodinger equation for small but finite amplitude dust acoustic waves is first obtained for magnetized dusty plasma in this paper. It suggest that in magnetized dusty plasmas the envelope solitary waves exist. The modulational instability for three dimensional NLSE is studied as well. The regions of stability and instability are well determined in this paper.     

Solitary waves in dusty plasmas with variable dust charge and two temperature ions

Propagation of nonlinear waves in dusty plasmas with variable dust charge and two temperature ions is analyzed. The Kadomtsev–Petviashivili (KP) equation is derived by using the reductive perturbation theory. A Sagdeev potential for this system has been proposed. This potential is used to study the stability conditions and existence of solitonic solutions. Also, it is shown that a rarefactive soliton can be propagates in most of the cases. The soliton energy has been calculated and a linear dispersion relation has been obtained using the standard normal-modes analysis. The effects of variable dust charge on the amplitude, width and energy of the soliton and its effects on the angular frequency of linear wave are discussed too. It is shown that the amplitude of solitary waves of KP equation diverges at critical values of plasma parameters. Solitonic solutions of modified KP equation with finite amplitude in this situation are derived.     

Theoretic study of dust acoustic waves in a dusty plasma with dust charge variation

The nonlinear dust acoustic waves in two-dimensional dust plasma with dust charge variation is investigated by using the formally variable separation approach. New solutions for the governing equation of this system have been obtained for dust acoustic waves in a dust plasma firsthand. We derive exact mathematical expressions and numerical simulation studies for the general case of the nonlinear dust acoustic waves in two-dimensional dust plasma with dust charge variation.     

Multidimensional dust acoustic solitary waves in inhomogeneous magnetized dusty plasmas with nonthermal ions

The linear dispersion relation and a modified variable coefficients Korteweg–de Vries (MKdV) equation governing the three-dimensional dust acoustic solitary waves are obtained in inhomogeneous dusty plasmas comprised of negatively charged dust grains of equal radii, Boltzmann distributed electrons and nonthermally distributed ions. The numerical results show that the inhomogeneity, the nonthermal ions, the external magnetic field and the collision have strong influence on the frequency and the nonlinear properties of dust acoustic solitary waves and both dust acoustic solitary holes (soliton with a density dip) and positive solitons (soliton with a density hump) are excited.     

Simple waves and shock waves in a thin prestressed elastic rod

Zusammenfassung Ein halb-unendlicher dünner elastischer Stab wird einer monoton ansteigenden Störung am freien Ende unterworfen. Die unstetige Wellenbewegung für endliche Verzerrungen wird untersucht, und die Bedingungen zur Bildung von Stössen werden aufgestellt. Eine implizite Lösung für die Wellenbewegung im Gebiete einfacher Wellen wird mittels der Methode der Charakteristiken erhalten. Eine andere und explizite Lösung in diesem Gebiete wird gefunden durch Anwendung der Theorie sich fortpflanzender Diskontinuitätsflächen, als eine Taylor-Reihe um die Ankunftszeit der Wellenfront. Es wird gezeigt dass der Koeffizient des ersten Gliedes in der Entwicklung eine Lösung für eine homogene nichtlineare gewöhnliche Differentialgleichung ecster Ordnung darstellt. Die übrigen Koeffizienten befriedigen lineare, jedoch inhomogene gewöhnliche Differentialgleichungen mit veränderlichen Koeffizienten. Für den Fall einer rampenartigen Belastung und für Stossbildung an der Wellenfront wird für schwache Stösse durch Anwendung einer Methode von Friedrichs die Vereinigung von einfachen Wellen untersucht. Als Beispiel wird die Entlastung eines Stabes betrachtet, der einer anfänglichen Dehnung von 50% unterworfen ist.     

The persistence of ideal shock waves

Local-in-time piecewise smooth solutions to hyperbolic systems of conservation laws are constructed by means of Li-Yu theory. The novelty consists in the application of this approach to shock waves for which the number of outgoing modes is at least as big as the number of incoming modes (undercompressive shocks), the motivation in a possible interpretation from the zero dissipation limit point of view.     

Turbulent shock waves in a collisionfree plasma

The paper deals with the structure of collisionless shocks arising from turbulent wave-particle interactions. The conditions under which wave-particle interaction effects could become significant leading to growing waves and a shock are discussed. Using the Mott-Smith expression for the zero-order distribution functions for the ions within the shock, the dielectric constant as well as the integral representing the wave-particle interaction term in the Lenard-Balescu equation are evaluated for a collisionless plasma. An expression is given for the ion distribution function within the shock. It is shown that the component of the pressure tensor perpendicular to the direction of flow of the plasma leads to a new kind of viscosity term arising from the interaction of the particles with the growing waves and this provides a dissipative mechanism to account for the conversion of the kinetic energy of the incoming plasma into the thermal energy of the hot ionised gas behind the shock.     

On the shock dynamics of weak converging shock waves in solid materials

Ricerche di Matematica - We presented a geometrical shock dynamics model to predict the behavior of weak converging shock waves in solid materials. Taking into consideration...