Elementary Wave Solutions Research Articles

The analytical solutions of the Riemann problem to 1-D non-ideal flow of dusty gas with external force

This work focuses on the analytical solution to the Riemann problem (RP) for a 1-D non-ideal flow of dusty gas with external force. Here it is presumed that external force is a continuous function of time. We explicitly obtain the elementary wave curves to 1-D non-ideal flow of dusty gas with external force and determine these wave curves in form of characteristics. Exhaustive calculations were performed for the elementary wave solutions, such as the rarefaction wave, shock wave, and contact discontinuity. We examine the influence of dust particles on density, velocity of flow, and shock speed and their implications on the solution of RP. Also, the implication of addition of external force is that all solutions are not self similar.

The Riemann problem for one-dimensional dusty gas dynamics with external forces

This article concerns the study of the Riemann Problem (RP) for a non-homogeneous system governing the one-dimensional compressible flow of dusty gas, where the external forces are assumed to be continuous function of time. The elementary wave solutions such as rarefaction wave, shock wave, and contact discontinuity are rigorously determined. The effect of dust particles on the density and flow velocity and shock speed is examined, and their consequences on the solution of the Riemann Problem are discussed. Moreover, we discussed the condition for the existence of vacuum appears in the solution of the Riemann Problem.

Solution of the Riemann problem for an ideal polytropic dusty gas in magnetogasdynamics

Abstract The main aim of this paper is, to obtain the analytical solution of the Riemann problem for a quasi-linear system of equations, which describe the one-dimensional unsteady flow of an ideal polytropic dusty gas in magnetogasdynamics without any restriction on the initial data. By using the Rankine-Hugoniot (R-H) and Lax conditions, the explicit expressions of elementary wave solutions (i. e., shock waves, simple waves and contact discontinuities) are derived. In the flow field, the velocity and density distributions for the compressive and rarefaction waves are discussed and shown graphically. It is also shown how the presence of small solid particles and magnetic field affect the velocity and density across the elementary waves. It is an interesting fact about this study that the results obtained for the Riemann problem are in closed form.

Riemann problem for non-ideal polytropic magnetogasdynamic flow

The main motive of the present paper is to derive the analytical solution of the Riemann problem for magnetogasdynamic equations governing an inviscid unsteady one-dimensional flow of non-ideal polytropic gas subjected to the transverse magnetic field with infinite electrical conductivity. By using the Lax entropy condition and R–H conditions, we derive the elementary wave solutions i.e. shock wave, simple wave and contact discontinuities without any restriction on the magnitude of initial data states and discussed about their properties. Further, the density and velocity distribution in the flow field for the cases of compressive wave and rarefaction wave is discussed. Here we also compare/contrast the nature of solution in non-ideal magnetogasdynamic flow and ideal gas flow.

Solution of Riemann problem for ideal polytropic dusty gas

The Riemann problem for a quasilinear hyperbolic system of equations governing the one dimensional unsteady flow of an ideal polytropic gas with dust particles is solved analytically without any restriction on magnitude of the initial states. The elementary wave solutions of the Riemann problem, that is shock waves, rarefaction waves and contact discontinuities are derived explicitly and their properties are discussed, for a dusty gas. The existence and uniqueness of the solution for Riemann problem in dusty gas is discussed. Also the conditions leading to the existence of shock waves or simple waves for a 1-family and 3-family curves in the solution of the Riemann problem are discussed. It is observed that the presence of dust particles in an ideal polytropic gas leads to more complex expression as compared to the corresponding ideal case; however all the parallel results remain same. Also, the effect of variation of mass fraction of dust particles with fixed volume fraction (Z) and the ratio of specific heat of the solid particles and the specific heat of the gas at constant pressure on the variation of velocity and density across the shock wave, rarefaction wave and contact discontinuities are discussed.

Solution of Riemann problem for dusty gas flow

A direct approach is used to solve the Riemann problem for a quasilinear hyperbolic system of equations governing the one dimensional unsteady planar flow of an isentropic, inviscid compressible fluid in the presence of dust particles. The elementary wave solutions of the Riemann problem, that is, shock waves, rarefaction waves and contact discontinuities are derived and their properties are discussed for a dusty gas. The generalised Riemann invariants are used to find the solution between rarefaction wave and the contact discontinuity and also inside rarefaction fan. Unlike the ordinary gasdynamic case, the solution inside the rarefaction waves in dusty gas cannot be obtained directly and explicitly; indeed, it requires an extra iteration procedure. Although the case of dusty gas is more complex than the ordinary gas dynamics case, all the parallel results for compressive waves remain identical. We also compare/contrast the nature of the solution in an ordinary gasdynamics and the dusty gas flow case.

Solution of the Riemann Problem in magnetogasdynamics

In the present paper, the Riemann Problem for a quasilinear hyperbolic system of equations, governing the one dimensional unsteady flow of an inviscid and perfectly conducting gas, subjected to transverse magnetic field, is solved analytically without any restriction on the initial states. This class of equations includes, as a special case, the Euler equation of gasdynamics. The elementary wave solutions of the Riemann problem, that is, shock waves, rarefaction waves and contact discontinuities are derived and their properties are discussed. It is noticed that although the magnetogasdynamics system is more complex than the corresponding gasdynamics system, all the parallel results remain identical. It is also assessed as to how the presence of magnetic field influences the variation of velocity and density across the shock wave, rarefaction wave and contact discontinuities.

Riemann problem for kinematical conservation laws and geometrical features of nonlinear wavefronts

A pair of kinematical conservation laws (KCL) in a ray coordinate system (ξ, t) are the basic equations governing the evolution of a moving curve in two space dimensions. We first study elementary wave solutions and then the Riemann problem for KCL when the metric g, associated with the coordinate ξ designating different rays, is an arbitrary function of the velocity of propagation m of the moving curve. We assume that m > 1 (m is appropriately normalized), for which the system of KCL becomes hyperbolic. We interpret the images of the elementary wave solutions in the (ξ, t)-plane to the (x, y)-plane as elementary shapes of the moving curve (or a nonlinear wavefront when interpreted in a physical system) and then describe their geometrical properties. Solutions of the Riemann problem with different initial data give the shapes of the nonlinear wavefront with different combinations of elementary shapes. Finally, we study all possible interactions of elementary shapes.

Wave Packets and Group Velocity in Absorbing Media: Solutions of the Telegrapher's Equation - Abstract

The propagation of high-frequency wave packets and solitary pulses in absorbing media, for which the process of wave propagation is governed by the telegrapher's equation, is investigated. The group velocity W for elementary wave solutions of telegrapher's equation is real and larger than the speed of light c for real wave numbers exceeding some critical value. The group velocity is real also for purely imaginary wave numbers and in this case W < c. The case W < c appears when the long-time asymptotics of wave packet solutions is considered, while the case W > c appears for finite times and short traveling distances. Both analytical and numerical studies of the evolution of high-frequency wave packets and aperiodic solitary pulses have been performed. This is done by analyzing and calculating the convolution integral representing a causal solution in a half-space. It has been found that the maximum of the wave packet's envelope begins to propagate from the boundary with superluminal velocity and after relatively short traveling distances its velocity becomes sublumi- nal. The maximum of a solitary pulse propagates with a subluminal velocity. Numerical calculations demonstrate strong reshaping of a solitary pulse at distances of the order of several absorption lengths. The conjunction of the local group velocity concept and superluminal propagation speeds is to be perceived as kinematical phenomenon closely related to the definitions used and does not imply a falsification of special relativity theory.

Elementary Wave Solutions of the Equations Describing the Motion of an Elastic String

The equations of planar motion of an elastic string form a $4 \times 4$ system of first order conservation laws. Two of the characteristic fields correspond to genuinely nonlinear longitudinal shocks and rarefaction waves, involving changes in the tension in the string, but not the slope. The other two fields correspond to contact discontinuities, across which the slope of the string jumps, reflecting the absence of any resistance to bending. Here, the tension T is related to the local elongation $\xi > 1$ in such a way as to ensure strict hyperbolicity: $T''(\xi ) > T{{(\xi )} / {\xi \geq 0}}$. The other main assumption is chosen to reflect properties of typical materials such as nylon and rubber. That is, $T'(\xi) $ is negative for $\xi > \xi _I $ and positive for $\xi > \xi _I $ for some $\xi _I > 1 $. The principal result of the paper is that the Riemann problem has a unique solution among combinations of centered waves, with a natural entropy condition placed on shocks. That is, any initial jump discontinuity in the tension, slope and velocity of the string can be resolved into combinations of longitudinal waves and contact discontinuities. This is illustrated for the plucked string problem, whose solution (valid until a wave first hits an end of the string) necessarily involves longitudinal waves.

AN ANALYSIS OF THREE-DIMENSIONAL UNSTEADY SUBSONIC FLOW THROUGH RECTILINEAR CASCADES

A finite pitch, finite chordlength cascade theory is constructed with elementary wave solutions of an actuatordisk analysis of a three-dimensional unsteady subsonic flow. The theory is applied to the analyses of unsteady forces due to blade vibration or sinusoidal gusts taking into account the effects of spanwise phase shifts of the disturbances. By comparing the fluctuating pressure in the flow passage between the blades with fluctuating lift curves in a complex plane, it is shown that an acoustic resonance in the space between the blades may enlarge lift fluctuation on the blades. Three-dimensionality of the disturbances decreases the lift fluctuation in case of an incompressible flow, but it has the opposite effect if the acoustic resonance occurs.

Quantized Free Unstable Particle Field

The quantized field of unstable particles without charge and spin is introduced. With this purpose, c-number solutions of the Klein-Gordon equation with complex mass m = m0 - (i/2)γ are investigated, and it is proved that a new set of non-scalar elementary wave solutions, which represent a direct idealization of states of a freely moving unstable particle, exists for the values of γ/2m0 smaller than the critical value which is evaluated to be 0.8 … With the aid of this set of elementary wave solutions, the quantized free unstable particle field is defined mathematically well. The commutator of the fields is evaluated, which turns out to be an invariant function of coordinates-difference in spite of the non-simple transformation property of the elementary waves used, and vanishes for the space-like separations. The resulting causal Green function ΔF has a good asymptotic behaviour at large distances. The set of infinitesimal generators of the inhomogeneous Lorentz group is found in terms of the creation and annihilation operators of particles, and it is shown that there exists a representation which is non-unitary, of the inhomogeneous Lorentz group. The quantized field and the generators, Pµ and Mµν have no simple transformation property like scalar, vector and tensor.