Frequencies Of Small Oscillations Research Articles

Thermal enhancement of the quantum decay rate in a dissipative system

The quantum decay of a metastable system which interacts with an environment at temperatureT is considered. A general formula for the decay rate at finite temperatures is obtained by a method which is based on the framework recently described by Caldeira and Leggett. The thermal enhancement of the tunnelling rate at low temperatures is discussed for arbitrary metastable potentials, and it is found that the exponent of the rate obeys a power law in a dissipative system. The power law exponent is shown to be a characteristic feature of the dissipative mechanism. Finally, a universally valid formula for the thermal enhancement factor is given, where the form of the potential enters only through the frequency of small oscillations about the metastable minimum and the “length” of the zero temperature bounce trajectory.

On the stability of equilibria of Hamiltonian systems near the main resonances

The equilibrium point O of an autonomous Hamiltonian system of two degrees of freedom is considered for small-oscillation frequencies related as ω2=2ω1+e. If under the precise resonance (e=0) the equilibrium is unstable, the inner diameter η(e) of the domain of stability containing the point O is estimated. It is shown that for the normalized variables η(e)≦e/b where b is the corresponding resonance coefficient. The estimates η(e) for other main resonances are reported.

Collective excitations in polyacetylene in a uniform electric field

The influence of the uniform electric field on the collective mode of the one-dimensional Peierls insulator trans-(CH)x is studied. The field is shown to decrease the activation energy of the 'amplitude phonons'. In the fields E<or approximately= Delta 02/eh(cross) nu F the frequency of small oscillations tends to zero, and a damping decrement proportional to the conductivity appears.

Resonance structure in kink-antikink interactions in φ 4 theory

We present new numerical and theoretical results concerning kink-antikink collisions in the classical (nonintegrable) φ 4 field model in one-dimensional space. Earlier numerical studies of such collisions revealed that, over a small range of initial velocities, intervals of initial relative velocity for which the kink and antikink capture one another alternate with regions for which the interaction concludes with escape to infinite separation. We describe the results of a new high-precision computer simulation that significantly extends and refines these observations of escape “windows”. We also discuss a simple theoretical mechanism that appears to account for this structure in a natural way. Our picture attributes the alternation phenomenon to a nonlinear resonance between the orbital frequency of the bound kink-antikink pair and the frequency of characteristic small oscillations of the field localized at the moving kink and antikink centers. Our numerical simulation also reveals long-lived small-scale oscillatory behavior in the time dependence of kink and antikink velocity following those collisions that do not lead to capture. We account for this fine structure in terms of the interaction between kink (and antikink) motion and small amplitude “radiation” generated during and after the collision. We discuss possible implications of our results for physical systems.

The effect of a magnetic field on the adiabatic oscillation of convective stellar models with radiation pressure

Using an unrestricted second order virial tensor analysis, the characteristic frequencies of small adiabatic oscillations of a convective stellar model in the presence of a weak poloidal magnetic field, with significant radiation pressure, have been calculated. In the absence of the magnetic field, the effect of increase in radiation pressure is to decrease the frequencies of radial modes, in spite of the increase in central condensation of the system. The characteristic frequencies of the transverse shear mode, the toroidal mode, and the pulsation modes show a general increase with the increase in the field strength. The marginal stability of the configuration in the presence of the magnetic field has also been discussed.

Motion of ions in the Kingdon trap

The classical and quantum motion of ions in a Kingdon trap (Orbitron) is studied for nearly circular orbits. The frequencies of small axial and radial oscillations are derived for both the logarithmic potential and the actual potential. A numerical comparison with the asymptotic approximation and with exact energy eigenvalues shows that the small oscillation method is adequate for most purposes.

Upper and lower bounds for sloshing frequencies by intermediate problems

This article presents very good, rigorous, numerical estimates for the “sloshing” eigenvalues of two families of two-dimensional regions. For each region the eigenvalues are proportional to the square of the frequencies of small lateral oscillations of an ideal fluid under the influence of gravity in a canal or in a horizontal cylindrical tank that has the region as cross-section. Our estimates are obtained by using conformal transformations that carry rectangles onto the regions, and then by employing intermediate problems with a generalized special choice to find upper and lower bounds to eigenvalues. The transformed problems are analogous to certain nonuniform string eigenvalue problems we estimate in a similar way.

Decay rates of oscillations and effective heat transfer coefficients for bubbles pulsating radially in a liquid

Vapor (gas) bubbles executing free radial oscillations in a liquid are considered. Expressions are obtained for the frequency and decay rate of small free oscillations of the bubbles. The effective coefficients of heat transfer between the radially pulsating bubbles and the liquid are determined.

An experimental study of oscillatory pipe flow at transitional Reynolds numbers

Fully developed oil flow in a smooth circular pipe at a mean Reynolds number of about 2100 was subjected to a nominally sinusoidal flow modulation at frequencies ranging from 0·05−1·75 Hz. It was observed that flow oscillation increased the critical Reynolds number and, under certain conditions, even brought about laminarization of the flow, which would be intermittently turbulent at the mean Reynolds number under quasi-steady (infinitely small oscillation frequency) conditions. The occurrence and extent of laminarization was, however, found to depend on factors such as the intermittency of turbulent puffs in the mean quasi-steady flow, frequency of oscillation, etc. Two series of experiments were performed. In one series, the oscillatory flow was almost completely laminarized. In the other series, the oscillatory flow was fully turbulent. In both the cases, instantaneous velocities in the flow were measured using laser-Doppler anemometry (LDA). The instantaneous velocity was decomposed into time-mean, periodic and random components employing ensemble-averaging techniques. The experiments indicated that the laminarized oscillatory flow behaves very similarly to laminar oscillatory flow at either end of the Strouhal-number range studied. The oscillatory turbulent flow was found to depend on both the Strouhal number and the ratio of the oscillation frequency (f) to some characteristic frequency (ft) of turbulence in the flow. The design of the present experimental facility made it possible to study the flow at f/ft ≈ 1 (‘high’ oscillation frequency), a condition that could not be attained in most previous investigations. Another unique feature of the present experiment was that the viscous sublayer and Stokes layer were both large enough (several millimetres in thickness) to allow detailed measurements to be made in these regions. It was found, that at this high frequency of oscillation, the Reynolds stresses generally remained frozen at an average state during the entire oscillation cycle. The turbulent structure showed significant departures from equilibrium at all times during the oscillation cycle. As a result, there was a net change in the time-mean velocity profile near the wall and a net increase in the time-mean wall shear stress and power loss due to friction. The observation that unsteadiness can indeed affect the mean flow behaviour in a significant way is new and contradicts the view presently held by many researchers (based on their studies at relatively low oscillation frequencies, i.e. f/ft [Lt ] 1). The data also indicated that the direct interaction between oscillation and the turbulent structure was essentially confined to the Stokes layer. The study suggests that (again contrary to the existing belief) quasi-steady turbulence models may not be adequate to describe unsteady flows when the time scale of unsteadiness is comparable to that of dominant turbulent eddies.

Radial oscillations of nonhomogeneous, thick-walled cylindrical and spherical shells subjected to finite deformations

The infinitesimal breathing motions of long cylindrical tubes and hollow spherical shells of arbitrary wall thickness subjected to a finite deformation field caused by uniform internal and/or external pressures are investigated. A neo-Hookean material with a material constant varying continuously along the radial direction is used. The shell is first subjected to finite static deformations and is then exposed to a secondary dynamic displacement field. Based on the theory of small deformations superposed on large deformations, closed form expressions are obtained for the frequency of small oscillations about the highly prestressed state. Frequency versus initial deformation parameter curves are given for several nohomogeneity functions and for various wall thicknesses.

On the existence of solutions to optimization problems with eigenvalue constraints

The optimum tapering of Bernoulli—Euler beams, i.e. the shape for which a given total mass yields the highest possible value of the first fundamental frequency of harmonic transverse small oscillations, is determined. The question of the existence of a solution to the optimization problem is considered. It is shown that, irrespective of the relationship between the flexural rigidity and linear mass density of the cantilever beam, the necessary conditions for optimality lead to a contradiction. This result is in partial disagreement with that obtained by earlier investigators. By imposing additional constraints on the optimization variable, a numerical solution for the case of the cantilever beam is obtained, using the formulation of the maximum principle of Pontryagin.

Free oscillations of a liquid rotating in a cylindrical vessel under conditions of weightlessness

The plane problem of determination of the natural frequencies of small oscillations of a viscous liquid rotating in a partially filled cylindrical vessel under conditions of weight-lessness is examined. If the angular velocity of vessel rotation is sufficiently low, surface forces acting on the liquid-gas boundary prove to be of the same order as the centrifugal forces and significantly affect the oscillatory frequencies. Asymptotic formulas expressing the dependence of the oscillatory frequencies on the parameters of the problem are obtained by the boundary layer method, with the assumption that the ratio of viscous to centrifugal forces is low.

Small signal lumped model for IMPATT diodes†

A small signal lumped model of a general IMPATT diode is presented. By applying state space analysis techniques the model is used to predict the small signal impedance of both pn and pin IMPATT diodes as a function of frequency and d.c. bias current. A lumped element equivalent circuit for the microwave cavity in a Read diode coaxial oscillator is derived. Using a state space description of the diode-cavity configuration, the variation of small signal oscillation frequency with d.c. bias current is evaluated.

Algebraic Solution of the Kepler Problem Using the Runge-Lenz Vector

The Runge-Lenz vector is used to obtain the equation for the Kepler orbits algebraically and the frequency of small oscillations of a particle about a stable circular orbit. The invariance of the Hamiltonian under the Lie algebras generated by the components of the Runge-Lenz and angular momentum vectors is discussed.

The Spectra of Small Oscillations of Thin Disk Galactic Models

This paper gives a survey of the spectral properties of the frequencies of small oscillations, both in and perpendicular to their plane, of cold thin disk galactic models. The spectra can have discrete and continuous parts, with two quite distinct types of continua arising. One type occurs with both kinds of oscillation when the density at the outer edge of the disk becomes sufficiently rarified. It is associated with rapid fluctuations in the outer region. The other type of continuum occurs for non‐axisymmetric oscillations in the plane of non‐uniformly rotating disks. Some significant discrete oscillations of fundamental type are found that exist in addition to the various continua.

Hydrodynamics in weak force fields. Small oscillations of an ideal liquid

Several problems concerned with small oscillations of an ideal liquid, taking account of the surface-tension forces, have been considered in [1–3] (as a rule, these are cases when the equilibrium liquid surface is spherical, plane, or differs only slightly from plane). Below we formulate the problem of the natural frequencies of small oscillations of a liquid for the general case of an equilibrium liquid surface in a weak potential mass force field. It is shown that the natural frequencies and the corresponding eigenfunctions of this problem may be found by the Ritz method. We note that analogous results in a somewhat different formulation have been obtained in the recently published [3].