Axial Wavenumber Research Articles

Standing Sausage Perturbations in Solar Coronal Slabs with Continuous Transverse Density Profiles: Cutoff Wavenumbers, Evanescent Eigenmodes, and Oscillatory Continuum

The lack of observed sausage perturbations in solar active region loops is customarily attributed to the relevance of cutoff axial wavenumbers and the consequent absence of trapped modes (called “evanescent eigenmodes” here). However, some recent eigenvalue problem studies suggest that cutoff wavenumbers may disappear for those equilibria where the external density varies sufficiently slowly, thereby casting doubt on the rarity of candidate sausage perturbations. We examine the responses of straight, transversely structured coronal slabs to small-amplitude sausage-type perturbations that excite axial fundamentals, by solving the pertinent initial value problem with eigensolutions for a closed domain. The density variation in the slab exterior is dictated by some steepness parameter μ, and cutoff wavenumbers are theoretically expected to be present (absent) when μ ≥ 2 (μ < 2). However, our numerical results show no qualitative difference in the system evolution when μ varies, despite the differences in the modal behavior. Only oscillatory eigenmodes are permitted when μ ≥ 2. Our discrete eigenspectrum becomes increasingly closely spaced when the domain broadens, and an oscillatory continuum results for a truly open system. Oscillatory eigenmodes remain allowed and dominate the system evolution when μ < 2. We show that the irrelevance of cutoff wavenumbers does not mean that all fast waves are evanescent. Rather, it means that an increasing number of evanescent eigenmodes emerge when the domain size increases. We conclude that sausage perturbations remain difficult to detect, even for the waveguide formulated here.

Acoustic resonance mechanism for axisymmetric screech modes of underexpanded jets impinging on an inclined plate

In this paper, the acoustic resonance mechanism for different axisymmetric screech modes of the underexpanded jets that impinge on an inclined plate is investigated experimentally. The ideally expanded Mach number of jets ( $M_j$ ) ranges from 1.05 to 1.56. The nozzle-to-plate distance at the jet axis and the impingement angle are respectively set as 5.0 $D$ and $30^{\circ }$ , where $D$ is the nozzle exit diameter. The acoustic results show that the $M_j$ range for the A2 screech mode of impinging jets is broader than that of underexpanded free jets, and a new axisymmetric screech mode A3 appears. With the increase of $M_j$ , the effect of the impinging plate on the shock cell structures of jets becomes obvious gradually, and the second suboptimal peaks are evident in the axial wavenumber spectra of mean shock structures. The coherent flow structures at screech frequencies are extracted from time-resolved schlieren images via the spectral proper orthogonal decomposition (SPOD). The axial wavenumber spectra of the selected SPOD modes suggest that the A1, A2 and A3 screech modes are respectively closed by the guided jet modes that are energized by the interactions between the Kelvin–Helmholtz wavepacket and the first three shock wavenumber peaks. The upstream- and downstream-propagating waves that constitute the screech feedback loop are analysed by applying wavenumber filters to the wavenumber spectra of SPOD modes. The frequencies of these three screech modes can be predicted by the phase constraints between the nozzle exit and the rear edge of the third shock cell. For the A3 mode, the inclined plate invades the third shock cell with the increase of $M_j$ , and the phase constraint cannot be satisfied at the lower side of the jets, which leads the A3 mode to fade away. The present results suggest that external boundaries can modulate the frequency and mode of jet screech by changing the axial spacings of shock cells.

Performance evaluation for the domain decomposition method in nonlinear vibration of the composite hard-coating cylindrical shell.

The applications of the modified domain decomposition method in nonlinear vibration analysis of the composite hard-coating cylindrical shells are still at a relatively superficial level, owing to the fact that its performance under different decomposition parameters has not been thoroughly investigated for achieving sufficient precision. A parametric domain decomposition method is developed to facilitate self-performance evaluation in nonlinear vibration analysis of the shell. Correspondingly, in order to avoid a mass of redundant computation of the segment stiffness and material damping matrices during iterations, a specialized preprocessing scheme is designed by pre-establishing the parametric analytical expressions and matrix databases. The resonant response is sensitive to the circumferential segment number, but weakly affected by the axial segment number. The optimum circumferential segment number in the present study is suggested to be Nθ = 70, which can achieve good calculation accuracy and efficiency. Highly consistency is shown for the distributions of axial equivalent strain under different axial segment numbers. Smaller circumferential segment numbers would result in larger equivalent strain and bad solution accuracy. The sufficient solution accuracy of nonlinear vibration of the composite hard-coating cylindrical shell can't be achieved by increasing the axial segment number with constant segment width, but only by enough circumferential segment number, which is fundamentally determined by its equivalent strain distributions and gradients, and is with close relation to the axial and circumferential wave numbers of the shell.

Ultrasonic sensing directivity of π-phase-shifted fiber Bragg grating hydrophone

&lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}${\text{π }}$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20222154_M1.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20222154_M1.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;-phase-shifted fiber Bragg grating with a short effective sensing length becomes one of research hotspots in ultrasonic sensing, because light undergoes strong localization centered at its phase shift position. To investigate the directional sensing characteristics of &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}${\text{π }}$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20222154_M2.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20222154_M2.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;-phase-shifted fiber Bragg grating as hydrophone, the theory of sound propagation in layered media is used to calculate the strain of fiber core, then the transfer matrix method based on the coupled-mode theory in optics is used to calculate the shift of central wavelength in optical reflection spectrum. Results of strain and wavelength shift under obliquely incident ultrasonic from 1－10 MHz are divided into A area, B area, and C area, and analyzed by numerical calculation and simulation calculation. Axial strain and elasto-optical strain change the grating period and effective refractive index by the mechanical effect and elasto-optical effect, respectively, thereby resulting in wavelength shift. In A area (frequency below 5 MHz, incident angle below &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$15^\circ $\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20222154_M3.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20222154_M3.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;), the axial strain nearly equals zero, thus elasto-optical effect plays a predominant role in wavelength shift. The maximal response occurs at vertical incidence, and then obviously declines with angle increasing. The maximum is essentially unchanged with grating length. In B area and C area (angle above &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$15^\circ $\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20222154_M4.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20222154_M4.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;), both mechanical effect and elasto-optical effect contribute to wavelength shift. In B area (frequency below 5 MHz), the amplitude of strain is the largest in three areas. A peak of wavelength shift appears at the same angle of the peak of strain, where exists the interference of the guided wave in fiber with the direct ultrasonic wave form water. The peak amplitude of wavelength shift decreases with grating length increasing. In C area (frequency below 5 MHz), the amplitude of strain is larger than in A area, but the wavelength shift is smaller, which is correlated to its higher axial wave number. Comparing the results in three areas, it is clear that the wavelength shift is larger at lower frequency and at vertical incidence. Experiments on 3 MHz and 5 MHz are then performed with a π-phase-shifted fiber Bragg grating. The experimental result accords well with the theoretical result. The research is important in practically using the &lt;inline-formula&gt;&lt;tex-math id="M5"&gt;\begin{document}${\text{π }}$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20222154_M5.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20222154_M5.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;-phase-shifted fiber Bragg grating in ultrasonic sensing.

Long-wavelength electromagnetic waves of surface type in circular metal waveguides partially filled by plasma in presence of an axial static magnetic field

The method of successive approximations is applied to solve the Maxwell equations in cylindrical plasma waveguide geometry for electromagnetic waves with arbitrary azimuthal wave index and small axial wavenumber. The theory of surface flute waves is used as zeroth approximation. This study generalizes previous investigations whose results are utilized for verification of newly obtained conclusions. The influences of several plasma waveguide parameters such as magnitude and sign of the azimuthal wave index, the width of the dielectric layer between a plasma and a waveguide wall and the magnitude of its dielectric constant, the radii of the plasma column and the metal wall, and the external axial static magnetic field on the wave dispersion properties are analyzed.

Canceling and Inverting Normal and Anomalous Group‐Velocity Dispersion Using Space‐Time Wave Packets

AbstractAngular dispersion can counterbalance normal group‐velocity dispersion (GVD) that increases the wave‐vector length in a dispersive medium. By tilting the wave vector, angular dispersion reduces the axial wave number in this case to match the pre‐GVD value. By the same token, however, angular dispersion fails to counterbalance anomalous GVD, which in contrast reduces the wave‐vector length. Consequently, GVD‐cancellation via angular dispersion has not been demonstrated to date in the anomalous dispersion regime. Here, structed femtosecond pulsed beams, known as ‘space‐time’ wave packets, are designed to realize dispersion‐cancellation symmetrically in either the normal‐ or anomalous‐GVD regimes by virtue of non‐differentiable angular dispersion inculcated into the pulsed field. Furthermore, GVD‐inversion is also verified by reversing the GVD sign experienced by the field with respect to that dictated by the chromatic dispersion of the medium itself.

Dynamic Analysis for High-Speed Locomotive in Distinct Depths

The study uses coupled finite or boundary element models expressed in terms of the axial wavenumber. It is found to be important to include the track in the model as this determines how the load is distributed at the soil’s surface, which significantly affects the insertion loss of the barrier. Calculations are presented for a range of typical layered grounds in which the depth of the upper soil layer is varied. Variations in the width and depth of the trench or barrier are also considered. The results show that in all ground conditions considered, the notional rectangular open trench performs best.

Sonic assessment of physical ageing of plastic pipes

This article explores the potential of vibro-acoustics to detect physical ageing of plastic pipes. For this purpose, two different topics are combined: the ability of vibro-acoustics to estimate the storage modulus of a plastic pipe, and the sensitivity of the estimated storage modulus to changes due to ageing. Concerning the first topic, a vibro-acoustic method was applied to two water-filled HDPE pipes, one surrounded by air and another by sand. The excitation was achieved via an impact hammer and the propagating signal was recorded with the aid of hydrophones. Signal analysis led to the estimation of the axial wavenumber of the propagating axisymmetric fluid-borne wave. This value was used in the dispersion equation for the propagating mode to evaluate the storage modulus of the pipe material for a given experimental setup. Results revealed that the vibro-acoustic method gives consistent and reliable estimations of the storage modulus. Concerning the second topic, samples from two PVC pipes with an age difference of 41 years were subjected to dynamic mechanical analysis to study the behaviour of the storage modulus as a function of frequency. Results showed that it is feasible to distinguish discrepancies in the magnitude of the storage modulus due to ageing, provided that the measurement uncertainty is small. The uncertainty analysis highlighted the parameters that need to be more accurately known in order to lower the overall uncertainty of the estimated storage modulus when the proposed vibro-acoustic method is used. Irrespectively of the medium surrounding the pipe (air or soil), the distance between the points of the recording signals should be sufficiently long to measure the signal phase accurately. It was found that the accurate knowledge of the pipe’s geometry, i.e. the wall thickness and internal radius, was more or equally important for controlling the overall uncertainty than that of the parameters of surrounding soil.

Thermal strain energy induced wave propagation for imperfect FGM sandwich cylindrical shells

An investigation is developed to analyze the heat-induced wave propagation characteristics of imperfect functionally graded material (FGM) sandwich cylindrical shells composed of a metal core layer and two functionally graded materials (FGMs) surface layers in a thermal environment by using the first-order shear deformation (FSD) shell theory. The temperature-dependent material properties are assumed to vary continuously along the thickness direction. Two types of porosity in FGMs layers of sandwich cylindrical shells are taken into account. To describe the porosity effects, the new porosity function composed of the porosity distribution function, core-to-thickness ratio, and porosity volume fraction is proposed. A novel thermal strain energy approach is established for cylindrical shell structures in a high-temperature environment by introducing Green’s nonlinear strains. The Hamilton’s principle is applied to derive the wave motion equations that govern the wave propagation behaviors. The analytical solutions of dispersion relations are determined by solving a generalized eigenvalue problem. In addition, a parametric investigation is conducted to study the effects of axial and circumferential wave number, porosity type and volume fraction, core-to-thickness ratio, power-law exponent, temperature changes, and radius-to-thickness ratio on the wave propagation characteristics of imperfect FGM sandwich cylindrical shells in high-temperature environments.

Quasi-electrostatic instability in non-gyrotropic tri-ion electron plasma

The plasma modes are significantly influenced by the simple or/and molecular anions. The reciprocity of beam and tri-ion electron (TIE) plasmas is modeled by the kinetic theory. The set of Vlasov–Poisson equation is solved and decomposed by the Laguerre–Gaussian function under paraxial approximation. The beam plasma expedites unstable twisted modes (UTMs) in the TIE plasma. The UTM or instability is calibrated via axial and azimuthal wave numbers, wind speed, and its growth essence. The qualitative and quantitative anatomy of the instability threshold is also predicted. Our model is eminently applicable to the space plasmas and has been applied to the Ar+–F−–SF6− and Xe+–F−–SF6− electron plasmas, specifically.

An Axial Magnetic Field Effect on Frequency Analysis of Rotating Sandwich Cylindrical Shells with FG Graphene/AL Face Sheets and Honeycomb Core

This article concerns with free vibration analysis of spinning sandwich cylindrical shells with functionally graded (FG) graphene/aluminum (Al) face sheets and honeycomb core exposed to an axial magnetic field. Lorentz magnetic force is derived by using Maxwell’s relations. The face layers are made of multi-nanocomposite sheets. Each sheet is composed of an Al matrix reinforced with graphene platelets (GPLs) that are uniformly distributed through the sheet thickness. The effective material properties of the face layers of the spinning sandwich cylindrical shells are derived employing the modified Halpin–Tsai model. The honeycomb core layer is made of hexagonal aluminum cells. According to the first-order shear deformation theory and Hamilton’s principle, five governing equations are obtained involving Lorentz force. Frequencies of the present model are analytically derived from the equations of motion. The present outcomes are examined by introducing some comparison examples. The effects of the geometric parameters, magnetic field parameter, GPLs weight fraction, core-to-face thickness ratio, circumferential wave number, axial wave number and spinning speed on the vibration of spinning sandwich honeycomb cylindrical shells are numerically discussed.

Vortex and cavity dynamics for the tip-leakage cavitation over a hydrofoil

Large eddy simulations were used to investigate unsteady tip-leakage cavitating flow over a National Advisory Committee for Aeronautics 0009 hydrofoil. The proper orthogonal decomposition (POD) was combined with fast Fourier transforms to help better understand the deformation dynamics of the vortex and cavity. The results show three typical vortices, i.e., the tip-leakage vortex (TLV), tip-separation vortex (TSV), and shedding–trailing vortex (STV) with their own helical core lines in the averaged cavitating flows. Upstream of x/C = 1.8, the mean TLV core carries the breathing mode waves through the vortex dilatation effect, while the TSV core generates the dilatation, shrinkage, and bending distortions near the cavity interface by vortex stretching and deflection effects. Further downstream, the TLV starts to intertwine with the TSV, and the STV gradually diffuses. The TLV wandering encourages large cavity deformations dominated by the breathing and bending modes. However, in the lower-order POD modes, the contribution of bending modes to the cavity fluctuation energy is greatly weakened due to the variation of the relative cavity radius, r*, which indicates that the breathing mode and the double helix mode play major roles in the cavity interfacial waves. The results also show that the cavity interfacial oscillations are highly correlated with the surrounding velocity fluctuations. Moreover, a modified theoretical dispersion equation can well reproduce the quantitative relation between the vibration frequencies and the axial wavenumbers of the TLV cavity, especially for the wave dynamics of the breathing modes and the double helix modes.

Propagation-invariant space-time supermodes in a multimode waveguide

When an optical pulse is spatially localized in a highly multimoded waveguide, its energy is typically distributed among a multiplicity of modes, thus giving rise to a speckled transverse spatial profile that undergoes erratic changes with propagation. It has been suggested theoretically that pulsed multimode fields in which each wavelength is locked to an individual mode at a prescribed axial wavenumber will propagate invariantly along the waveguide at a tunable group velocity. In this conception, an initially localized field remains localized along the waveguide. Here, we provide proof-of-principle experimental confirmation for the existence of this class of pulsed guided fields, which we denote “space-time supermodes,” and verify their propagation invariance in a planar waveguide. By superposing up to 21 modes, each assigned to a prescribed wavelength, we construct space-time (ST) supermodes in a 170-µm-thick planar glass waveguide with group indices extending from ≈ 1 to ≈ 2 . The initial transverse width of the field is 6 µm, and the waveguide length is 9.1 mm, which is ≈ 257 × the associated Rayleigh range. A variety of axially invariant transverse spatial profiles are produced by judicious selection of the modes contributing to the ST supermode, including single-peak and multi-peak fields, dark fields (containing a spatial dip), and even flat uniform intensity profiles.

Perturbation Solution for Anisotropic Circular Waveguides Loaded With Eccentric Rods

A perturbation method is developed for the analysis of anisotropic circular waveguides partially filled with eccentric cylinders. The method allows to compute axial wavenumbers and field patterns for hybrid modes supported in waveguides with small eccentricities via a perturbation series. The proposed method is validated against analytical results and numerical results based on the finite-element method (FEM) and the finite-integration technique (FIT) in different configurations. The results show that the presented approach can solve, for example, the longitudinal wavenumbers of anisotropic waveguides with an eccentricity of 23% with a relative error less than 1.78% at a much lower computational cost when compared to numerical techniques.

Optimal control of growth of instabilities in Taylor–Couette flow

The present work aims to achieve optimal control of instabilities in a standard Taylor–Couette flow. The motivation of the present study is to reduce the disturbance growth and delay the transition process to turbulence. We numerically employ control using a stability modifier, namely, wall transpiration. In the non-modal stability framework, we form a state-space model employing control actuation by means of periodic suction/blowing of fluid from the walls. The study is conducted for two cases of flow rotations: (i) counter-rotating cylinders and (ii) the stationary outer cylinder with inner cylinder rotating. The parametric study was performed with varying radii ratios, Reynolds numbers (Re), axial (α), and azimuthal (n) wavenumbers. The time evolution of governing equation is written in terms of perturbation velocities in radial (r) and azimuthal (θ) directions. The optimal feedback control is obtained using a linear quadratic regulator controller and feed backed to the system to reduce the maximum optimal growth of the instabilities in the flow. The perturbation kinetic energy is taken as the measure of the amplification of disturbances and used as the cost function to be minimized. We use Chebyshev spectral collocation method to discretize the equations and variational method to calculate the optimal growth. We studied four different parametric cases of radii ratios (η=r1r2= 0.1, 0.25, 0.5, 0.75), with angular velocity (Ω2Ω1) ratio fixed as μ=−1 and μ = 0. We choose the subcritical wavenumbers that led to a maximum transient energy growth corresponding to a Reynolds number ≈ 0.65 times the critical Reynolds number for the case of counter-rotation. For the case of the stationary outer cylinder, we showed the effect of the control in the modal analysis framework. The presented control technique resulted in a maximum of 72% reduction in the growth rate and the typical growth of perturbation energy.

Numerical modal analysis of bulk-reactive lined duct with shear flow

In this paper, the eigenmodes of bulk-reactive lined duct with shear flow are calculated and analyzed by a wave and finite element (WFE) method. Using this method, the eigenmodes can be calculated based on commercially available finite element (FE) matrices, which is of considerable practical interest when dealing with complicated cases. The proposed method is validated by comparing results with Chebyshev collocation method. It is shown that the axial wavenumbers obey the convergence rate of Se4 for all kinds of modes, where Se is the mesh size. The calculated eigenmodes can be divided into air-borne modes, lining-borne modes and surface modes, where air-borne modes mainly propagate in air domain; lining-borne modes propagate in both air and lining domains; surface modes propagate near the interface between air and lining domains. It is found that, for air-borne and lining-borne modes, boundary layer thickness has small effect on lower order modes, but has relatively large effect on mode shapes of higher order modes. For the surface modes, boundary layer thickness can greatly change their mode properties. In the calculated example, with perforated panel considered, two coupled modes emerge. With perforation ratio evolving from large to small, one coupled mode evolves from a lower-order air-borne mode to a higher-order air-borne mode, and the other evolves from a lining-borne mode to an air-borne mode. Generally, the perforation ratio has small effect on air-borne and lining-borne modes; while it has large effect on surface modes and the coupled mode, especially in the range of [3%, 13%].

Standing Sausage Perturbations in Solar Coronal Loops with Diffuse Boundaries: An Initial Value Problem Perspective

Abstract Working in pressureless magnetohydrodynamics, we examine the consequences of some peculiar dispersive properties of linear fast sausage modes (FSMs) in one-dimensional cylindrical equilibria with a continuous radial density profile (ρ 0(r)). As recognized recently on solid mathematical grounds, cutoff axial wavenumbers may be absent for FSMs when ρ 0(r) varies sufficiently slowly outside the nominal cylinder. Trapped modes may therefore exist for arbitrary axial wavenumbers and density contrasts, their axial phase speeds in the long-wavelength regime differing little from the external Alfvén speed. If these trapped modes indeed show up in the solutions to the associated initial value problem (IVP), then FSMs have a much better chance to be observed than expected with classical theory and can be invoked to account for a considerably broader range of periodicities than practiced. However, with axial fundamentals in active region loops as an example, we show that this long-wavelength expectation is not seen in our finite-difference solutions to the IVP, the reason for which is then explored by superposing the necessary eigenmodes to construct solutions to the same IVP. At least for the parameters we examine, the eigenfunctions of trapped modes are characterized by a spatial extent well exceeding the observationally reasonable range of the spatial extent of initial perturbations, meaning a negligible fraction of energy that a trapped mode can receive. We conclude that the absence of cutoff wavenumbers for FSMs in the examined equilibrium does not guarantee a distinct temporal behavior.

The effects of rotation on the frequencies and critical speed of CNTs/fiber/polymer/metal laminates cylindrical shell

This work focuses on the frequencies of rotating carbon nanotubes (CNTs)/fiber/polymer/metal laminates (CNTFPMLs) thin circular cylindrical shell. The cylindrical shell is studied on the basis of Love’s first approximation shell theory with simply supported boundary condition. In this manuscript, Eshelby-Mori-Tanaka is used to define the modulus of carbon nanotubes reinforced composites (CNTRCs) cylindrical shell. In addition, fibers can be reinforced using the obtained matrix by means of extended rule of mixture. The influence of various parameters for example characteristics of fiber phase material, such as circumferential and axial wave numbers, lay-ups, metal volume fraction, composite volume fraction and CNTs mass fraction on the frequencies of rotating CNTFPMLs cylindrical shell have been studied. The outputs illustrated by growing rotational speed, the frequencies of CNTFPMLs cylindrical shell change differently for different fiber volume fractions. Also, the backward and forward frequencies of functionally graded ( F G ) - X and F G - O distributions are more and less than the uniformly distributed ( U D ) for rotational speed equals to 1, respectively, while this process is reversed for rotational speed equals to 5. In 10 and 20 rotational speeds, while the frequencies of backward and forward modes for F G - X distribution are more and less than the U D , respectively, but this procedure is reversed in F G - O distribution.

Hopf index and the helicity of elliptically polarized twisted light

Here, we describe a systematic derivation of the general form of the optical helicity density of ellipticaly polarized paraxial Laguerre–Gaussian modes L G ℓ , p , σ . The treatment incorporates the contributions of the longitudinal field components for both the paraxial electric E and magnetic B fields, which satisfy Maxwell’s self-consistency condition in the sense that E is derivable from B and vice versa. Contributions to the helicity density to leading order in ( k 2 w 0 2 ) − 1 (where k is the axial wavenumber and w 0 the beam waist) include terms proportional to optical spin σ and topological charge ℓ , as well as a spin-orbit σ | ℓ | term. However, evaluations of the space integrals leading to the total helicity confirm that the space integral of the ℓ -dependent term in the density (which is due entirely to the longitudinal fields) vanishes identically for all ℓ and p , so that, in general, only σ determines the Hopf index, with the optical vortex L G ℓ p character featuring only in the action constant.

Linear stability of a rotating liquid column revisited

We revisit the somewhat classical problem of the linear stability of a rigidly rotating liquid column in this article. Although the literature pertaining to this problem dates back to 1959, the relation between inviscid and viscous stability criteria has not yet been clarified. While the viscous criterion for stability, given by$We &lt; n^2 + k^2 -1$, is both necessary and sufficient, this relation has only been shown to be sufficient in the inviscid case. Here,$We = \rho \varOmega ^2 a^3 / \gamma$is the Weber number and measures the relative magnitudes of the centrifugal and surface tension forces, with$\varOmega$being the angular velocity of the rigidly rotating column,$a$the column radius,$\rho$the density of the fluid and$\gamma$the surface tension coefficient;$k$and$n$denote the axial and azimuthal wavenumbers of the imposed perturbation. We show that the subtle difference between the inviscid and viscous criteria arises from the surprisingly complicated picture of inviscid stability in the$We$–$k$plane. For all$n &gt; 1$, the viscously unstable region, corresponding to$We &gt; n^2 + k^2-1$, contains an infinite hierarchy of inviscidly stable islands ending in cusps, with a dominant leading island. Only the dominant island, now infinite in extent along the$We$axis, persists for$n=1$. This picture may be understood, based on the underlying eigenspectrum, as arising from the cascade of coalescences between a retrograde mode, that is the continuation of the cograde surface-tension-driven mode across the zero Doppler frequency point, and successive retrograde Coriolis modes constituting an infinite hierarchy.