Arbitrary Initial Velocity Research Articles

Infinitely narrow soliton solutions to scalar conservation laws in Colombeau sense

We construct the infinitely narrow N-soliton solutions to scalar conservation laws in Colombeau's space of generalized functions in the association sense for arbitrary initial positions and velocities of solitons. With the additional assumptions on the initial positions and velocities of the solitons it is possible to modify the above solution such that it becomes a solution in pointwise sense.

Generation of Compton harmonics by scattering linearly polarized light of arbitrary intensity from free electrons of arbitrary initial velocity

We present an analytic classical relativistic derivation for a general expression of the harmonic power generated per unit laboratory solid angle due to Compton scattering of plane wave, linearly polarized light of arbitrary intensity from free electrons moving initially with arbitrary velocity. We show graphically the generated frequency as a function of the coordinates of the observation point for several initial electron kinetic energies and light field intensities.

Harmonic generation by scattering circularly polarized light of arbitrary intensity from free electrons of arbitrary initial velocity

We derive a general analytic expression for the harmonic power generated per unit solid angle as a result of scattering plane-wave, circularly polarized light of arbitrary intensity from free electrons moving initially with arbitrary velocity. The relativistic derivation is carried out fully in the laboratory frame.

Stability of an elastically connected two-body space station by the perturbation method

This investigation deals with the stability, in a circular orbit, of a flexible space station consisting of two inertially identical rigid end bodies connected together by an elastic structure. The earth-pointing motion and the rotation with arbitrary initial angular velocity perpendicular to the orbital plane are studied by using the multiple-scales technique. The first-order approximate analytical solution and the conditions of stability are obtained.

The Sliding Velocity Flow of Rough Collisions in Multibody Systems

Single-point rough collisions in multibody systems with perfect constraints, under the assumptions of Coulomb’s friction and infinite tangential stiffness at the collision point, require usually an integration over the normal impulse. The evolution of the sliding velocity, which is needed in the integration, is determined by an autonomous nonlinear flow. The phase-space geometry of this flow depends upon five parameters associated with the system collision configuration and the friction coefficient μ, and gives a global picture of the system behavior in collisions with the configuration considered and arbitrary initial velocities. This geometry is studied using μ, as a control parameter, and a set of threshold values of μ, associated with changes in qualitative behavior are determined.

New analytic approach to multivelocity annihilation in the kinetic theory of reactions.

A new, exact, analytic approach to multivelocity, one-species, ballistic annihilation in one dimension is proposed. For an arbitrary one-particle initial velocity distribution, the problem can be solved rigorously in terms of the two-particle conditional probability, which obeys a closed nonlinear integro-differential equation. We present a method for solving this equation for an arbitrary discrete velocity distribution. This method is applied to the three-velocity case. The outcome of numerical simulations compares well with our exact results.

Analysis of the deflection system for a magnetic-field-immersed magnicon amplifier

A linear analysis of the electron-beam deflection system in a magnicon amplifier is presented. The system consists of identical cavities, one driven and the remainder passive, separated by a drift space and immersed in an axial magnetic field. The cavities contain a rotating TM/sub 110/ mode. The length of each cavity is pi nu /sub z// omega , and that of the drift space is pi nu /sub z// omega /sub c/, where omega is the RF frequency, omega /sub c/ is the relativistic gyrofrequency in the guide field, and nu /sub z/ is the mean axial velocity of the beam electrons. The linearized electron orbits are obtained for arbitrary initial axial velocity, radial coordinate, and magnetic field. The small-signal gain and the phase shift are determined. The special case where omega /sub c// omega =2 has unique features and is discussed in detail. For the NRL magnicon design, a power gain of 10 dB per passive cavity is feasible. Results from numerical modeling of a magnicon with two passive cavities are presented. Operation of the output cavity at the fundamental and higher harmonics of the input drive frequency is briefly discussed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Spray Diffusion Flames with Arbitrary Initial Droplet Velocity Distributions

A model of a laminar Burke-Schumann type of diffusion flame is developed in which the fuel is supplied in the form of droplets having an arbitrary initial velocity distribution. The spray of droplets is described using a sectional approach. The governing equations are normalized and various nondimensional parameters associated with the spray and the motion of the droplets therein are identified. Analytic solutions are presented for different assumed droplet velocity distributions. Effects of droplet velocity, dimensionless drag and vaporization coefficients and spray angle parameter on flame height, width and type are illustrated in a discussion of a detailed parametric analysis based on numerous calculations.

Laboratory-frame and rotating-frame ion trajectories in ion cyclotron resonance mass spectrometry

The magnitude of the ion cyclotron resonance (ICR) time-domain signal detected from the oscillating differential charge induced on opposed detector electrodes depends on the number, ICR orbital and magnetron radii, and phase coherence of the trapped ions. In this paper, we present analytical solutions for the motion of magnetically confined ions of arbitrary initial position, velocity and phase during and after excitation by a spatially uniform oscillatory electric field. Resonant and non-resonant single-frequency and frequency-sweep (chirp) excitation are treated. Spatial coherence may then be incorporated by analyzing the ion ensemble into ion packets, such that ions in each packet share a common ICR orbit center. We find that resonant (single-frequency or frequency-sweep) electric field excitation not only increases the ICR orbital radius, but also induces an oscillating (for linearly polarized excitation) or rotating (for circularly polarized excitation) shift in the center of the ion cyclotron orbit of the ion packet (i.e. the magnetron radius). The amplitude of the r.f.-induced oscillating or rotating magnetron radius increases directly with ion mass-to-charge ratio and r.f. electric field amplitude. Optimal excitation for high mass ions should therefore be conducted with low magnitude long-duration excitation rather than a short high magnitude “rectangular” (or “burst” or “impulse”) waveform. None of the above excitation waveforms changes the initial spatial distribution of the ion packet: in particular, off-resonance single-frequency excitation results in oscillatory variation in ICR orbital radius, but does not induce any net spatial synchronization in the ion packet. Finally, ion trajectories during on- and off-resonance single-frequency excitation are displayed in the usual fixed laboratory coordinate frame as well as in a coordinate frame rotating about the z-axis at the ICR orbital frequency of that ion. Phase coherence and shifts in the center of the ICR orbit are much more easily visualized in the new rotating-frame representation.

On the ‘inverse phugoid problem’ as an instance of non-linear stability in pitch

SummaryThe motion of an aircraft along a straight flight path inclined to the vertical is considered from an arbitrary initial velocity which may be far from steady flight speeds. The velocity and incidence, as a function of distance or time, provide a solution to the ‘inverse phugoid problem’ of finding the control laws that keep an aircraft on a straight flight path by exactly cancelling the phugoid instability (the short period mode is omitted by neglecting the rotational inertia of the aircraft). This is a non-linear problem, namely finding the motion of an aircraft from an initial condition far removed from a stable steady state, or finding the large amplitude deviations from an unstable equilibrium. The stability and instability problems are solved analytically using the following assumptions: (i)the aircraft flies subsonically so that only form and induced drag need be considered;(ii)the thrust consists of a constant term, minus a term specifying a dependence on velocity similar to that of the total drag; and(iii)the aircraft incidence is away from stall so that lift is a linear function of incidence.The response curves show that: (i)flight at velocities below the lower steady flight speed, which is unstable, is difficult to maintain due to fast divergence towards the stall; and(ii)for an initial velocity above the lower steady flight speed, there is convergence towards the higher steady flight speed.The convergence, in the latter case (ii), may be slow along a straight flight path, although it can be hastened by suitable manoeuvres.

Energetics of Linear Geostrophic Adjustment

The potential and kinetic energy spectra following geostrophic adjustment of a barotropic fluid are determined as functions of the spectrum of arbitrary initial sea level or velocity. Remarkably, the functional dependences between initial and final spectra, and the general conclusions reached are independent of whether spatial variation in the initial field is permitted in either one or both horizontal directions. For initial sea level displacements, the potential and kinetic energy spectra obtained for the adjusted state are respectively shown to be low- and band-pass filtered versions of the initial potential energy spectrum. The maximum kinetic energy, ΔE, is obtained for initial displacements of wavelength k0−1 equal to the deformation radius, a, and is equal to 1/3 of the change in potential energy, ΔP, from initial to adjusted state. The value of ⅓ for ΔE/ΔP is also found for steplike initial displacements for which no horizontal scale is defined. However, in general ΔE/ΔP is shown to be dependent on the horizontal scale of initial displacements and bounded above by ½. The values, ΔE/ΔP≤⅓, found by van Heijst and Smeed for a variety of nonlinear multiphase fluid adjustment problem are also suggested to be products of the assumed steplike initial conditions and length scales imposed by the presence of walls. The energetics of linear adjustment for initial velocity fields are also discussed and a time scale of adjustment for narrow band initial fields is estimated to be of order (1 + k02a2)½/k02a2 inertial periods.

The brachistochrone for a material point with arbitrary initial velocity

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Hydromagnetic laminar flow of a conducting fluid in an annulus in presence of a radial magnetic field

The object of the present paper is to study the MHD effects on the laminar flow of a viscous, incompressible and conducting fluid in an annulus with arbitrary time-varying pressure gradient and arbitrary initial velocity in presence of a radial magnetic field. Using finite Hankel transform, solutions for both the unsteady and steady flows under different prescribed pressure gradients have been found out.

Solution of initial-value problems for some infinite chains of harmonic oscillators

Procedures are given for solving the equations of motion of infinite and half-infinite chains of linear spring-mass oscillators with nearest-neighbor coupling. Arbitrary initial displacements and initial velocities may be prescribed for any finite number of the masses in the chains, and external forces may be applied to any finite number of them. The solutions appear as integrals with integrands involving orthogonal polynomials generated by three-term recurrence relations whose coefficients are determined from the equations of motion. Tables of the polynomicals needed for solving the equations of motion of a number of special chains are included.

An exact solution of the spin−spin autocorrelation function for a one−dimensional system of hard rods

It is shown exactly that, in a one−dimensional system of hard rods with spins, the autocorrelation function of any function of spin F(w) decays as t−1 at long times provided that 〈F〉eq exists and that g (0) ≠ 0, where g (v) is the linear velocity distribution function. As a consequence of this, when F (w) = w, the spin diffusion coefficient defined by the Kubo relation Ds = ℱ∞0 〈w (0) w (t) 〉 dt does not exist. The results are true for arbitrary initial equilibrium velocity and spin distributions, the only restriction being that they be symmetric.

Laminar Flow of an Incompressible Fluid in a Conduit With Arbitrary Cross Section, Arbitrary Time-Varying Pressure Gradient, and Arbitrary Initial Velocity

A computer-oriented solution is given for the flow described in the title of the paper. The boundary shape is represented by specification of the coordinates of N points on the boundary; the initial velocity is represented by specification of L values of the velocity in the cross section at time zero; the arbitrary time-varying pressure gradient is implemented by use of Duhamel’s Theorem. In the solution method presented, boundary and initial conditions are satisfied in the least squares sense. The Gram determinant is used to determine eigenvalues and the Gram-Schmidt orthonormalizing procedure is used to construct a set of functions appropriate for a finite series solution. Computer programs are referenced which have been used to investigate the correctness of the solution and the accuracy obtainable with reasonable digital computational time.

VII.—Some Contributions to the Theory of Random Walk and Multiple Scattering of Particles

SynopsisA theory of the random walk with “persistence” of movement of a point in a three-dimensional cubic lattice is presented from which explicit expressions for the moments of the distribution function for the displacements of an ensemble of points after N steps for any arbitrary initial average velocity are derived. The results are applied to the problem of small angle multiple scattering of particles on their passage through a material medium, and formulae for the mean square of the lateral displacements are obtained which, in first approximation, have the form of the expressions, generally used for evaluating the experimental results but, in higher approximation, indicate a deviation from this relationship for greater thickness of matter.Another approach to the same problem of multiple scattering is further presented which is based on Kramer's stochastic differential equation for the distribution function for the position and velocities of an ensemble of particles in phase space. By this method formulae for the mean square of the scatter angles, the lateral displacements and the correlation products between these are derived. The first of these expressions shows again characteristic deviations from the usual ones for greater thickness of matter, the second coincides essentially with the expression obtained from the random walk theory.

Magnetohydrodynamic Generalized Couette Flow (With Uniform Suction; Time-Dependent Pressure Gradient; and Arbitrary Initial Velocity)

Magnetohydrodynamic Generalized Couette Flow (With Uniform Suction; Time-Dependent Pressure Gradient; and Arbitrary Initial Velocity)

Laminar Flow in an Annulus With Arbitrary Time-Varying Pressure Gradient and Arbitrary Initial Velocity

In this paper, the problem of incompressible laminar viscous flow in the annular space bounded by two coaxial infinite circular cylinders with an arbitrary time-varying pressure gradient and with an arbitrary initial distribution of velocity has been studied. The present problem generalizes the several earlier works in which the pressure gradient and the initial distribution of velocity have been taken in special forms. The analysis has been made by the use of finite Hankel transform. The case of steady flow when the pressure gradient is constant has been deduced by taking the pressure gradient to be a constant quantity and then letting the time since the start of the motion be infinite. This result has been shown in agreement with the already well-established result.

Dynamic stability of the flexural vibrations of a thin-walled beam

The stability of the free transverse vibrations of a simply supported, thin-walled elastic beam with double axes of symmetry is investigated for arbitrary initial velocity distributions. The lateral-torsional modes may be parametrically excited through the interaction of the pulsating flexural moment and the rotation of the cross section of the beam about the longitudinal axis and the vertical axis of symmetry. A criterion is presented to predict this excitation. A numerical example for a particular transverse flexural mode of vibration indicates that the critical initial velocity for lateral-torsional parametric excitation may be relatively small.