Forced Harmonic Oscillator Research Articles

Noncommutative double scalar fields in FRW cosmology as cosmical oscillators

We investigate the effects caused by noncommutativity of the phase space generated by two scalar fields that have non-minimal conformal couplings to the background curvature in the FRW cosmology. We restrict deformation of the minisuperspace to noncommutativity between the scalar fields and between their canonical conjugate momenta. Then, the investigation is carried out by means of a comparative analysis of the mathematical properties (supplemented with some diagrams) of the time evolution of variables in a classical model and the wavefunction of the universe in a quantum perspective, both in the commutative and noncommutative frames. We find that the imposition of noncommutativity causes more ability in tuning time solutions of the scalar fields and, hence, has important implications in the evolution of the Universe. We find that the noncommutative parameter in the momenta sector is the only responsible parameter for the noncommutative effects in the spatially flat universes. A distinguishing feature of the noncommutative solutions of the scalar fields is that they can be simulated with the well-known three harmonic oscillators depending on three values of the spatial curvature, namely the free, forced and damped harmonic oscillators corresponding to the flat, closed and open universes, respectively. In this respect, we call them cosmical oscillators. In particular, in closed universes, when the noncommutative parameters are small, the cosmical oscillators have an analogous effect with the familiar beating effect in the sound phenomena. Some of the special solutions in the classical model and the allowed wavefunctions in the quantum model make bounds on the values of the noncommutative parameters. The existence of a non-zero constant potential (i.e. a cosmological constant) does not change time evolutions of the scalar fields, but modifies the scale factor. An interesting feature of the well-behaved solutions of the wavefunctions is that the functional form of the radial part is the same as the commutative ones within a given replacement of constants caused by the noncommutative parameters. Furthermore, the Noether theorem has been employed to explore the effects of noncommutativity on the underlying symmetries in the commutative frame. Two of the six Noether symmetries of spatially flat universes, in general, are retained in the noncommutative case, and one out of the three in non-flat universes.

Measurement of the trap parameters of a magneto-optical trap by external periodic perturbation

We report a simple method for trap parameter measurement by realizing forced harmonic oscillation in a magneto-optical trap (MOT). Through the resonance curve of the vibrational amplitude of the harmonic oscillation on the driving frequency of the laser intensity, we measured the damping coefficient and trap frequency of trapped atoms in the MOT under various conditions and compared them with those obtained from the parametric resonance. In particular, we report a significant effect of a transverse laser on one-dimensional harmonic oscillation. This effect is expected to explain the discrepancies in other previous experiments between the experimental and theoretical results.

Improved Understanding of Vibrating-Wire Viscometer−Densimeters

The vibrating-wire technique has been widely applied in measurements of both viscosity and density. When tensioned by a suspended mass or sinker, the device may be used to measure these properties simultaneously. The sensitivity to density arises mainly from the effect of buoyancy on the tension and hence the resonance frequency of the vibrating wire. Earlier studies have used simplified working equations to relate the resonance frequency and the tension. In this work, we have employed the exact equations describing forced simple harmonic oscillations of a stiff wire under tension and also a second-order analytical solution of those equations. We show that the exact working equation describes the actual resonance frequency of tensioned tungsten wires with improved accuracy. This finding has been validated by means of experimental measurements of the resonance frequency at prescribed tensioning forces and at temperatures in the range (293 to 473) K. One consequence of this is that the true independently measured sinker volume may be used in preference to one obtained by calibration of the assembled sensor. The remaining parameters to be obtained are then the radius of the wire, Young’s modulus E for the wire, and an empirical parameter that accounts for the remaining departure from the theoretical expression. The values of wire radius obtained by calibration in a liquid of known viscosity have been compared with those from independent mechanical measurements having a relative uncertainty of 0.2 %. These comparisons have been made for both drawn wires and for a centerless ground rod with a diameter of 0.15 mm, and the differences are discussed in the light of scanning electron microscopy (SEM) studies of the circularity and uniformity of the wire. Finally, we provide experimental evidence that rough drawn wires may yield inaccurate viscosity measurements when applied to fluids having viscosities very different to the one used for calibration; conversely, a centerless-ground rod gave good results even when the viscosity to be measured was 60 times that of the calibration fluid.

Estimating site amplification factors from ambient noise

We present a methodology to obtain frequency‐dependent relative site amplification factors using ambient seismic noise. We treat a seismic network or array as a forced damped harmonic oscillator system where each station responds to a forcing function obtained from frequency‐wavenumber beams of the ambient noise field. A network or array beam is necessary to estimate the forcing function. Taken over long time periods, each station responds to the forcing function showing a frequency‐dependent resonance peak whose amplitude and spectral width depends upon the elastic and anelastic properties of the underlying medium. Our results are encouraging in that hard rock sites show little variability and have narrower resonance peaks with reduced amplitudes relative to soft rock sites in sedimentary basins. There is much more variability observed at soft rock sites and a tendency for spectral peaks to shift to higher frequencies and become broader as the site amplification increases. This could be due to due to lower densities and/or small‐strain nonlinearity at stations having high site amplification.

MULTI-MODE LINEAR COMPOSITE MOMENTUM REPRESENTATION AND ITS APPLICATION

A unitary transformation matrix, n linear-composite coordinate operators and n linear-composite momentum operators are constructed for an n-particle system, and the complete and orthonormal common eigenvectors of the multi-mode linear composite momentum operators are examined by virtue of the technique of integration within an ordered product of operators. The multi-mode linear composite momentum representation is proposed, and its application to a general two-mode forced quantum harmonic oscillator system with kinetic coupling is presented for solving some dynamic problems.

Nonequilibrium dissociation and recombination rates in nitrogen: From shock waves to discharge conditions

A discussion about the validity of state-to-state and total weighted dissociation and recombination rate coefficients under nonequilibrium conditions is presented, both for the cases of high translational temperatures, in a shock wave, and of moderate and low temperatures, in a electrical discharge. Accurate data for these processes are obtained in N 2 using the semiclassical nonperturbative analytic method known as forced harmonic oscillator (FHO) for V– V– T and V– T energy exchanges.

Increase of Boltzmann entropy in a quantum forced harmonic oscillator

Recently, a quantum-mechanical proof of the increase of Boltzmann entropy in quantum systems that are coupled to an external classical source of work has been given. Here we illustrate this result by applying it to a forced quantum harmonic oscillator. We show plots of the actual temporal evolution of work and entropy for various forcing protocols. We note that entropy and work can be partially or even fully returned to the source, although both work and entropy balances are non-negative at all times in accordance with the minimal work principle and the Clausius principle, respectively. A necessary condition for the increase of entropy is that the initial distribution is decreasing (e.g., canonical). We show evidence that for a nondecreasing distribution (e.g., microcanonical), the quantum expectation of entropy may decrease slightly. Interestingly, the classical expectation of entropy cannot decrease, irrespective of the initial distribution, in the forced harmonic oscillator.

Exact wave functions and coherent states for a forced damped harmonic oscillator

The exact wave function ψnz(t) (x, t) of Schr\:odinger equation for a forced damped harmonic oscillator is obtained using transformation of variables and factorization of a wave function, where suffix n takes values 0, 1,2, … The mean value of the position or momentum satisfies the classical equation of motion. The uncertainty of position or momentum is related to time and independent of an external force, but their product being independent of time. When n = 0, ψnz(t) (x,t) expresses both the coherent states for a forced damped harmonic oscillator and the squeezed coherent states for a standard harmonic oscillator.

Resolving a puzzle concerning fluctuation theorems for forced harmonic oscillators innon-Markovian heat baths

A harmonic oscillator that evolves under the action of both a systematictime-dependent force and a random time-correlated force can do workw. This work is a random quantity, and Mai and Dhar have recently shown,using the generalized Langevin equation (GLE) for the oscillator’s positionx, that it satisfies a fluctuation theorem. In principle, the same result could have beenderived from the Fokker–Planck equation (FPE) for the probability density function,P(x,w,t), for the oscillatorbeing at x attime t, havingdone work w. Although the FPE equivalent to the above GLE is easily constructed and solved,one finds, unexpectedly, that its predictions for the mean and variance ofw do not agree with the fluctuation theorem. We show that to resolve this contradiction,it is necessary to construct an FPE that includes the velocity of the oscillator,v, as an additionalvariable. The FPE for P(x,v,w,t) does indeed yield expressions for the mean and variance ofw that agree with the fluctuation theorem.

Measurements of turbulent magnetic diffusivity in a liquid-gallium flow

Direct measurements of the effective conductivity (magnetic diffusivity) in the turbulent flow of a liquid metal have been performed. An nonstationary turbulent flow of a gallium alloy has been excited in a closed toroidal channel with dielectric walls. The Reynolds number reaches a maximum value of Re ≈ 106, which corresponds to the magnetic Reynolds number Rm ≈ 1. The conductivity of the metal in the channel has been determined from the phase shift of forced harmonic oscillations in a series RLC circuit whose inductance is a toroidal coil wound around the channel. The maximum deviation of the effective conductivity of the turbulent medium from the ohmic conductivity of the metal is about 1%.

A concise quantum mechanical treatment of the forced damped harmonic oscillator

Abstract By selecting a right generalized coordinate X, which contains the general solutions of the classical motion equation of a forced damped harmonic oscillator, we obtain a simple Hamiltonian which does not contain time for the oscillator such that Schrödinger equation and its solutions can be directly written out in X representation. The wave functin in x representation are also given with the help of the eigenfunctions of the operator $$ \hat X $$ in x representation. The evolution of $$ \left\langle {\hat x} \right\rangle $$ is the same as in the classical mechanics, and the uncertainty in position is independent of an external influence; one part of energy mean is quantized and attenuated, and the other is equal to the classical energy.

Spatial linear stability of a hypersonic shear layer with nonequilibrium thermochemistry

We examine the spatial linear stability of a shear layer in a hypervelocity flow where high temperature effects such as chemical dissociation and vibrational excitation are present. A shock triple point is used to generate a free shear layer in a model problem which also occurs in several aerodynamic applications such as shock-boundary layer interaction. Calculations were performed using a state-resolved, three-dimensional forced harmonic oscillator thermochemical model. An extension of an existing molecular-molecular energy transfer rate model to higher collisional energies is presented and verified. Nonequilibrium model results are compared with calculations assuming equilibrium and frozen flows over a range of (frozen) convective Mach numbers from 0.341 to 1.707. A substantial difference in two- and three-dimensional perturbation growth rates is observed among the three models. Thermochemical nonequilibrium has a destabilizing effect on shear-layer perturbations for all convective Mach numbers considered. The analysis considers the evolution of the molecular vibrational quantum distribution during the instability growth by examining the perturbation eigenfunctions. Oxygen and nitrogen preserve a Boltzmann distribution of vibrational energy, while nitric oxide shows a significant deviation from equilibrium. The difference between translational and vibrational temperature eigenfunctions increases with the convective Mach number. Dissociation and vibration transfer effects on the perturbation evolution remain closely correlated at all convective Mach numbers.

Response of a ferrofluid to traveling-stripe forcing

We observe the dynamics of waves propagating on the surface of a ferrofluid under theinfluence of a spatially and temporally modulated field. In particular, we exciteplane waves by applying a traveling lamellar modulation of the magnetization. Bymeans of this external driving, both the wavelength and the propagation velocityof the waves can be controlled. The amplitude of the excited waves exhibits aresonance phenomenon similar to that of a forced harmonic oscillator. Its analysisreveals the dispersion relation of the free surface waves, from which the criticalmagnetic field for the onset of the Rosensweig instability can be extrapolated.

The propagation of terminated waves in dispersion media and the resulting time-dependent dielectric functions

In this paper we examine the physics of terminated wave propagation in a dispersive medium. A forced harmonic oscillator model is used to describe the molecule's motion. This model leads to a time-dependent dielectric function e(ν,t) and the wavefrontal velocity of a pulse identical with the light velocity c in a vacuum. The numerical calculations for the rectangular and Gaussian pulse propagation confirm this statement. Furthermore, in the limit of an ultra-short Gaussian pulse, the difference between 'front velocity' and 'group velocity' in a dispersive medium can be neglected.

Viscoelastic property measurement in thin tissue constructs using ultrasound

We present a dual-element concave ultrasound transducer system for generating and tracking of localized tissue displacements in thin tissue constructs on rigid substrates. The system is comprised of a highly focused PZT-4 5-MHz acoustic radiation force (ARF) transducer and a confocal 25-MHz polyvinylidene fluoride imaging transducer. This allows for the generation of measurable displacements in tissue samples on rigid substrates with thickness values down to 500 microm. Impulse-like and longer duration sine-modulated ARF pulses are possible with intermittent M-mode data acquisition for displacement tracking. The operations of the ARF and imaging transducers are strictly synchronized using an integrated system for arbitrary waveform generation and data capture with a shared timebase. This allows for virtually jitter-free pulse-echo data well suited for correlation-based speckle tracking. With this technique we could faithfully capture the entire dynamics of the tissue axial deformation at pulse-repetition frequency values up to 10 kHz. Spatio-temporal maps of tissue displacements in response to a variety of modulated ARF beams were produced in tissue-mimicking elastography phantoms on rigid substrates. The frequency response was measured for phantoms with different modulus and thickness values. The frequency response exhibited resonant behavior with the resonance frequency being inversely proportional to the sample thickness. This resonant behavior can be used in obtaining high-contrast imaging using magnitude and phase response to sinusoidally modulated ARF beams. Furthermore, a second order forced harmonic oscillator (FHO) model was shown to capture this resonant behavior. Based on the FHO model, we used the extended Kalman filter (EKF) for tracking the apparent modulus and viscosity of samples subjected to dc and sinusoidally modulated ARF. The results show that the stiffness (apparent modulus) term in the FHO is largely time-invariant and can be estimated robustly using the EKF. On the other hand, the damping (apparent viscosity) is time varying. These findings were confirmed by comparing the magnitude response of the FHO (with parameters obtained using the EKF) with the measured ones for different thin tissue constructs.

Extended irreversible thermodynamics and non-equilibrium temperature

We briefly review the concept of non-equilibrium temperature from the perspectives of extended irreversible thermodynamics, fluctuation theory, and statistical mechanics. The relations between different proposals are explicitly examined in two especially simple systems: an ideal gas in steady shear flow and a forced harmonic oscillator in a thermal bath. We examine with special detail temperatures related to the average molecular kinetic energy along different spatial directions, to the average configurational energy, to the derivative of the entropy with respect to internal energy, to fluctuation-dissipation relation and discuss their measurement.

On the quantum motion of a generalized time-dependent forced harmonic oscillator

In this work, we use linear invariants and the dynamical invariant method to obtain exact solutions of the Schrödinger equation for the generalized time-dependent forced harmonic oscillator in terms of solutions of a second order ordinary differential equation that describes the amplitude of the classical unforced damped oscillator. In addition, we construct Gaussian wave packet solutions and calculate the fluctuations in coordinate and momentum as well as the quantum correlations between coordinate and momentum. It is shown that the width of the Gaussian packet, fluctuations and correlations do not depend on the external force. As a particular case, we consider the forced Caldirola–Kanai oscillator.

Spectral dynamic of pulsating turbulent pipe flow

The present work investigates the effects of a forced harmonic oscillations of fixed frequency on the characteristics of a turbulent pipe flow with a mean Reynolds number based on the cross sectional bulk velocity and pipe diameter of 5900. The setup refers to an extremely thin Stokes layer so that the vorticity associated to the oscillating motion is generated in a small near wall region. When the ratio of the amplitudes of the oscillating and bulk velocities is increased from 1 to 11, the turbulence is affected by the harmonic forcing so much that the near wall coherent structures are substantially modified, although not suppressed through a reverse transition process. This results in an overall space and time averaged resistance reduction which, for the largest amplitude case, adds up to 27% of the non-pulsating flow at the same bulk Reynolds number. The present contribution is focused on the analysis of the phase locked averaged statistics of the Reynolds stress tensor components and velocity spectra. The analysis is carried out processing a set of statistically independent samples obtained from wall resolved large Eddy simulations with a spectrally accurate Navier–Stokes code.

TWO-VARIABLE HERMITE POLYNOMIALS AS TIME-EVOLUTIONAL TRANSITION AMPLITUDE FOR DRIVEN HARMONIC OSCILLATOR

For the two-variable Hermite polynomials Hm,n(β,β*) we find its new physical explanation in the dynamics of a linear forced quantum harmonic oscillator (or a dispaced oscillator), i.e. Hm,n(β,β*) can be explained as the time-evolutional transition amplitude from an initial number state |m〉 at t0 to a final state |n〉 at t of the dispaced oscillator. Two new properties of the time-evolutional operator for driven oscillator are revealed.

Ultrasonic bubbles in medicine: Influence of the shell

Ultrasound contrast agents consist of microscopically small bubbles encapsulated by an elastic shell. These microbubbles oscillate upon ultrasound insonification, and demonstrate highly nonlinear behavior, ameliorating their detectability. (Potential) medical applications involving the ultrasonic disruption of contrast agent microbubble shells include release-burst imaging, localized drug delivery, and noninvasive blood pressure measurement. To develop and enhance these techniques, predicting the cracking behavior of ultrasound-insonified encapsulated microbubbles has been of importance. In this paper, we explore microbubble behavior in an ultrasound field, with special attention to the influence of the bubble shell. A bubble in a sound field can be considered a forced damped harmonic oscillator. For encapsulated microbubbles, the presence of a shell has to be taken into account. In models, an extra damping parameter and a shell stiffness parameter have been included, assuming that Hooke’s Law holds for the bubble shell. At high acoustic amplitudes, disruptive phenomena have been observed, such as microbubble fragmentation and ultrasonic cracking. We analyzed the occurrence of ultrasound contrast agent fragmentation, by simulating the oscillating behavior of encapsulated microbubbles with various sizes in a harmonic acoustic field. Fragmentation occurs exclusively during the collapse phase and occurs if the kinetic energy of the collapsing microbubble is greater than the instantaneous bubble surface energy, provided that surface instabilities have grown big enough to allow for break-up. From our simulations it follows that the Blake critical radius is not a good approximation for a fragmentation threshold. We demonstrated how the phase angle differences between a damped radially oscillating bubble and an incident sound field depend on shell parameters.