Zeroth Harmonics Research Articles

Recovery of a general nonlinearity in the semilinear wave equation

We study the inverse problem of recovery a nonlinearity f ( t , x , u ), which is compactly supported in x, in the semilinear wave equation u tt − Δ u + f ( t , x , u ) = 0. We probe the medium with either complex or real-valued harmonic waves of wavelength ∼ h and amplitude ∼ 1. They propagate in a regime where the nonlinearity affects the subprincipal but not the principal term, except for the zeroth harmonics. We measure the transmitted wave when it exits supp x f. We show that one can recover f ( t , x , u ) when it is an odd function of u, and we can recover α ( x ) when f ( t , x , u ) = α ( x ) u 2 m . This is done in an explicit way as h → 0.

Hamiltonian form of an Extended Nonlinear Schrödinger Equation for Modelling the Wave field in a System with Quadratic and Cubic Nonlinearities

We derive a Hamiltonian form of the fourth-order (extended) nonlinear Schrödinger equation (NLSE) in a nonlinear Klein–Gordon model with quadratic and cubic nonlinearities. This equation describes the propagation of the envelope of slowly modulated wave packets approximated by a superposition of the fundamental, second, and zeroth harmonics. Although extended NLSEs are not generally Hamiltonian PDEs, the equation derived here is a Hamiltonian PDE that preserves the Hamiltonian structure of the original nonlinear Klein–Gordon equation. This could be achieved by expressing the fundamental harmonic and its first derivative in symplectic form, with the second and zeroth harmonics calculated from the variational principle. We demonstrate that the non-Hamiltonian form of the extended NLSE under discussion can be retrieved by a simple transformation of variables.

Fourier decomposition of polymer orientation in large-amplitude oscillatory shear flow.

In our previous work, we explored the dynamics of a dilute suspension of rigid dumbbells as a model for polymeric liquids in large-amplitude oscillatory shear flow, a flow experiment that has gained a significant following in recent years. We chose rigid dumbbells since these are the simplest molecular model to give higher harmonics in the components of the stress response. We derived the expression for the dumbbell orientation distribution, and then we used this function to calculate the shear stress response, and normal stress difference responses in large-amplitude oscillatory shear flow. In this paper, we deepen our understanding of the polymer motion underlying large-amplitude oscillatory shear flow by decomposing the orientation distribution function into its first five Fourier components (the zeroth, first, second, third, and fourth harmonics). We use three-dimensional images to explore each harmonic of the polymer motion. Our analysis includes the three most important cases: (i) nonlinear steady shear flow (where the Deborah number is zero and the Weissenberg number is above unity), (ii) nonlinear viscoelasticity (where both and exceed unity), and (iii) linear viscoelasticity (where exceeds unity and where approaches zero). We learn that the polymer orientation distribution is spherical in the linear viscoelastic regime, and otherwise tilted and peanut-shaped. We find that the peanut-shaping is mainly caused by the zeroth harmonic, and the tilting, by the second. The first, third, and fourth harmonics of the orientation distribution make only slight contributions to the overall polymer motion.

Universe as a representation of affine and conformal symmetries

The evolution of the Universe is considered by means of a nonlinear realization of affine and conformal symmetries via Maurer-Cartan forms. Conformal symmetry is realized by the geometry of similarity with the Dirac scalar dilaton. We provide preliminary quantitative evidence that the zeroth harmonic of the Dirac scalar dilaton may lead to observationally viable cosmology, where the type la supernova luminosity distances-redshift relation can be explained by vacuum dilaton dark energy. The diffeo-invariance of spin connection coefficients of the affine formulation leaves only one degree of freedom of strong gravitation waves. We discuss that the dark matter effect in spiral galaxies can be described by the gravitation waves expressed through the spin connection coefficients of the affine formulation. We show that, the evolution equations of the affine gravitons with respect to the dilaton zeroth mode coincide with the equations of “squeezed oscillator”. The list of theoretical and observational arguments is given in favor of that the origin of the Universe can be described by quantum vacuum creation of these squeezed oscillators.

Oscillation spectrum of an electron gas with a small density fraction of ions

The problem is solved of the stability of a nonneutral plasma that completely fills a waveguide and consists of magnetized cold electrons and a small density fraction of ions produced by ionization of the atoms of the background gas. The ions are described by an anisotropic distribution function that takes into account the characteristic features of their production in crossed electric and magnetic fields. By solving a set of Vlasov-Poisson equations analytically, a dispersion equation is obtained that is valid over the entire range of allowable electric and magnetic field strengths. The solutions to the dispersion equation for the m = +1 main azimuthal mode are found numerically. The plasma oscillation spectrum consists of the families of Trivelpiece-Gould modes at frequencies equal to the frequencies of oblique Langmuir oscillations Doppler shifted by the electron rotation and also of the families of “modified” ion cyclotron (MIC) modes at frequencies close to the harmonics of the MIC frequency (the frequencies of radial ion oscillations in crossed fields). It is shown that, over a wide range of electric and magnetic field strengths, Trivelpiece-Gould modes have low frequencies and interact with MIC modes. Trivelpiece-Gould modes at frequencies close to the harmonics of the MIC frequency with nonnegative numbers are unstable. The lowest radial Trivelpiece-Gould mode at a frequency close to the zeroth harmonic of the MIC frequency has the fastest growth rate. MIC modes are unstable over a wide range of electric and magnetic field strengths and grow at far slower rates. For a low ion density, a simplified dispersion equation is derived perturbatively that accounts for the nonlocal ion contribution, but, at the same time, has the form of a local dispersion equation for a plasma with a transverse current and anisotropic ions. The solutions to the simplified dispersion equation are obtained analytically. The growth rates of the Trivelpiece-Gould modes and the behavior of the MIC modes agree with those obtained by numerical simulation.

Hard undulator radiation in an inverse medium

X-ray radiation from fast electrons in undulators filled with an inverse medium is studied. A formula for the spectral density of the number of photons is derived. The intensity zeroth harmonics, as well as the intensity of radiation in zero undulator field, are described by the Tamm-Frank formula for Cherenkov radiation. It is shown that the spectral density of emitted photons in a wavelength range of λ ≅ 0.4–2.0 A in such a medium can be increased by four orders of magnitude as compared to the radiation intensity in a vacuum undulator. The energy of emitted electrons in the former case must be in an interval of 5-2 GeV, while electrons with an energy of 14-6 GeV are required in vacuum undulators.

Zakharov equations with zeroth harmonic and a new type of modulation instability

It has been shown that the system of Zakharov equations for the amplitudes of the first and zeroth harmonics of the waves on the surface of an ideal liquid describes not only the known type of the modulation instability of the envelope of the main harmonic with respect to harmonic perturbations with small wave vectors κ (Benjamin-Feier modulation instability), but also the modulation instability of a combination of the main and zeroth harmonics at κ values on the order of the wave vector k 0 of the main harmonic. In contrast to the Benjamin-Feier modulation instability typical for large depths, the described modulation instability does not disappear at k 0 h < 1.363.

Efficient and consistent method for superellipse detection

Superellipses are a flexible representation for a variety of objects and the detection of such a primitive is an interesting issue in machine vision research. Least-mean-square fitting using an algebraic distance has been suggested to determine the parameters of a superellipse, but the computational cost is high and a high curvature bias problem is involved. An efficient, consistent and threshold-free scheme is derived for the estimation of superellipse parameters. The closed solutions for the centre, orientation and squareness parameters are obtained by using the zeroth harmonic of its Fourier description, the consistent symmetric axis method and the theorem of diagonal segment, respectively. Only the lengths of the major and the minor axes are repeatedly estimated by Powell's conjugate direction technique to reduce the sensitivity of noise. The proposed method is suitable for use on relatively complete, closed superellipse curves. Both convex and concave superellipses have been considered, and a compensation technique is suggested for concave superellipses. Experiments with complete and disjoint superellipses, defective superellipses and superellipses extracted from a photograph of a real object indicate the efficiency, accuracy and reliability of the proposed method, both theoretically and practically.

Inhomogeneous viscosity fluid flow in a wide-gap Couette apparatus: Shear-induced migration in suspensions

The first portion of this two-part paper investigates the short time evolution of a low Reynolds number flow characterized by a spatially inhomogeneous viscosity within the annular domain between two widely separated concentric circular cylinders undergoing relative rotation. The viscosity is regarded as a material property and as such is convected with the fluid. Any possible “diffusion” of this viscosity is supposed negligible, at least in the short times of interest in our calculation. The initial viscosity field, assumed to be only slightly inhomogeneous, is expanded into a Fourier series with respect to the polar angle, and the contributions of the zeroth and first harmonics are subsequently addressed. Approximate short-time analytic solutions for the velocity and viscosity fields are obtained. In the second part of this paper the results of the preceding analysis are employed in an attempt to gain insight into the experimentally observed shear-induced migration of particles in suspensions being sheared in a wide-gap Couette apparatus. The connection of the inhomogeneous viscosity problem studied in the first part to such shear-induced migration phenomena lies in the assumption that the local viscosity of a suspension of (non-Brownian) particles is functionally dependent only upon the local suspended particle volumetric fraction. In such circumstances, the local transport of suspended particles corresponds to a concomitant transport of the local suspension viscosity and vice-versa. Subject to the foregoing interpretation and limited by algebraic tractability to short times, a global radial migration is predicted. It increases with an increase in the annular gap size between the cylinders and depends upon the phase angle between the rotating outer and inner cylinders, but not upon their relative circumferential velocity—a conclusion consistent with experimental observations. Further, to leading order, particle migration is found to be independent of purely radial viscosity disturbances (the zeroth harmonic) and to arise entirely from coupling between circumferential disturbances in the velocity and viscosity (i.e., particle concentration) fields. The solution also indicates that the high shear-rate region, proximate to the inner wall, may either become less viscous on average (thereby predicting net radial migration away from the high shear rate region) or, conversely, more viscous (corresponding to migration toward the high shear rate region migration). The latter case arises in circumstances that involve a large positive radial gradient in the viscosity’s first harmonic.

Pulmonary arterial impedance after single lung transplantation

Single lung transplantation (SLT) is emerging as definitive therapy for end-stage pulmonary disease of varying etiology, yet a complete description of the hemodynamic properties of the transplanted lung has not been reported. In this study, Fourier analysis was used to calculate the pulmonary arterial (PA) impedance spectrum before and immediately after SLT to define precisely the pulmonary pressure-flow relationship. Median sternotomies were performed in 18 dogs (donors): an ultrasonic flow probe was placed around the PA and micromanometers were placed in the PA and left atrium (LA). Control PA pressure and flow (PAQ) and LA pressure were measured during transient occlusion of the right PA. The lungs were harvested using cold modified Euro-Collins solution for preservation. After thoracotomy and pneumonectomy, left SLT was performed in 18 recipient dogs with a mean ischemic time of 179 ± 6 min. After reperfusion for 1 hr, PA pressure and flow data were again collected. Characteristic impedance ( Z 0), a measure of resistance to pulsatile flow, was compared to input resistance ( R in), a measure of resistance to mean flow, and pulmonary vascular resistance (PVR), the conventional index. R in is defined as the zeroth harmonic of the impedance spectrum and Z 0 as the mean of impedance moduli from 2–12 Hz. All recipients survived transplantation. Both PVR and R in increased significantly after transplantation (11 ± 1 vs 19 ± 3 Wood U, P < 0.05, and 1352 ± 121 vs 1964 ± 244 dyne · sec · cm −5, P < 0.05). Moreover, Z 0 increased 144% after transplantation (377 ± 51 vs 918 ± 134 dyne · sec · cm −5, P < 0.05), indicating a proportionately greater change in resistance to pulsatile blood flow than to mean blood flow. These data suggest that characteristic impedance may be a more sensitive indicator of pathologic changes in the pulmonary vasculature secondary to transplantation. Furthermore, this study demonstrates the feasibility of impedance measurements in a lung transplant model which may be useful in evaluating preservation techniques and pharmacologic interventions.

Nonlinear effects on dissipative mhd modes

It is demonstrated that by including the viscous force in the resistive MHD equations, a simple nonlinear theory can be constructed to account for the nonlinear effects on the linearly unstable tearing modes near the stability threshold. The method for carrying out the perturbational nonlinear analysis is described in detail. The essentials of this method are exhibited by working out the theory for simple sheet pinch geometry, calculating the effects of mode coupling of the second and zeroth harmonics on the evolution of the modes. 9 references.

Abstract: Theory of lenslet interactions in a fly’s eye lens

In an electron-optical fly’s eye lens, comprised of several parallel plates with arrays of aligned holes, the field at each lenslet is distorted by the fields of the surrounding lenslets. This interaction sets a limit on how closely packed the array can be (or how large the lenslet holes can be for a given spacing) and therefore has a direct bearing on performance. In this paper a method is developed for calculating the effect of the interaction for arbitrary lens geometry. First, from the symmetry of the environment of each lenslet, Neumann boundary conditions are derived on the bounding polygon between lenslets. These boundary conditions are expanded in a Fourier-perturbation series about the circle which is the zeroth harmonic of the polygon. Laplace’s equation is then solved within the bounding circle for each harmonic to first-order in the perturbation theory, for example by numerical relaxation. Second- and higher-order solutions can be similarly obtained if necessary. From the zeroth Fourier component of the resulting axial potential, Gaussian properties of the lenslet are obtained, along with the spherical aberration coefficient. The next nonvanishing harmonic (fourth for a square array of lenslets, sixth for an hexagonal array) determines a corresponding aberration coefficient, which describes the major effect of lenslet interactions on the focused spot. Higher-order aberration coefficients are calculated from higher-harmonic potentials, if needed. Finally, as an example, the method is applied to a three-element einzel lens with a square array of lenslets. Fourfold aberrations, which are the dominant interaction effect, are found to be significant if the hole diameter in the center element is greater than about 70% of the lenslet spacing, in rough agreement with experimental results. At this diameter such aberrations amount to roughly one tenth of the spherical aberration.

High intensity Compton scattering off particles with anomalous magnetic moment

For a particle with an anomalous magnetic moment the differential cross section for the emission of a photon due to interaction with an intense external wave field of circular polarization is calculated. The scattered frequencies appear in triplets of very small separation and there is in addition a 'zeroth' harmonic of very low frequency. In the cross section the contributions from the anomalous moment add to the relativistic terms in the corresponding formula for normal particles. The transition probabilities for all three numbers of the triplets differ from each other. For neutrons only the zeroth, first and second harmonics can appear. The intensity of the zeroth harmonic turns out to be very low both for electrons and nucleons.

Second Order Conductivity and Nonlinear Plasma Excitation of an Isotropic Electron Gas

AbstractThe theory of the second order conductivity tensor of an isotropic electron gas is developed using thermodynamic Green's functions. It is shown that this quantity is exactly determined by the first order conductivity tensor and an additional term consisting of a pure three‐current correlation function. This term is found to be small in the optical and plasma regions. The second order conductivity tensor is calculated in the Hartree approximation. The poles of the second order response function relating the induced particle density to a scalar external potential are discussed and shown to yield the second and zeroth harmonics of the ordinary plasma waves as determined by the linear response of the system. For the special case of shock excitation by a Coulomb potential, semiquantitative arguments are presented in favour of a non‐negligible second order contribution to the total density response which may possibly also contribute to the energy loss of fast electrons in metals.