Airy Function Research Articles

Electro-Mechanical Buckling of Shallow Segments of Functionally Graded Porous Toroidal Shell Covered by Piezoelectric Actuators

The current work deals with the mechanical buckling behavior of saturated porous toroidal shell segments sandwiched by piezoelectric actuator layers. Material properties of the porous shells with nonuniform distributed porosities are assumed to vary as a specific function of the thickness and porosity parameters. The nonlinear equations of the sandwich toroidal shell having either positive or negative Gaussian curvature are obtained on the basis of Biot’s poroelasticity theory. The energy method as the minimum potential energy principle is applied to derive the governing equations of the sandwich shells. The buckling equations of the piezo-porous sandwich shells under various types of mechanical loading are established by employing the adjacent equilibrium criterion. The governing equations as a set of the coupled partial differential equations are solved using an analytical method including the Airy stress function. Closed-form solutions are derived for shallow segments of the sandwich toroidal shell with positive/negative Gaussian curvature under lateral pressure, axial compression, and hydrostatic pressure loadings. Novel numerical results are presented in this work to show the effects of important factors such as the piezoelectric layer thickness, applied voltage, shell geometrical ratio, and variation of the porosity coefficient on the buckling behavior of these structures.

Error estimates for Gaussian beams at a fold caustic

In this work we show an error estimate for a first order Gaussian beam at a fold caustic, approximating time-harmonic waves governed by the Helmholtz equation. For the caustic that we study the exact solution can be constructed using Airy functions and there are explicit formulae for the Gaussian beam parameters. Via precise comparisons we show that the pointwise error on the caustic is of the order O ( k − 5 / 6 ) where k is the wave number in Helmholtz.

Eccentric catastrophes & what to do with them

Analytic modeling of gravitational waves (GWs) from inspiraling eccentric binaries poses an interesting mathematical challenge. When constructing analytic waveforms in the frequency domain, one has to contend with the fact that the phase of the Fourier integral in non-monotonic, resulting in a breakdown of the standard stationary phase approximation (SPA). In this work, we study this breakdown within the context of catastrophe theory. We find that the SPA holds in the context of eccentric Keplerian orbits when the Fourier frequency satisfies , where are integer multiples of the apocenter/pericenter frequencies, respectively. For values outside of this interval, the phase undergoes a fold catastrophe, giving rise to an Airy function approximation of the Fourier integral. Using these two different approximations, we generate a matched asymptotic expansion that approximates generic Fourier integrals of Keplerian motion for bound orbits across all frequency values. This asymptotic expansion is purely analytic and closed-form. We discuss several applications of this investigation and the resulting approximation, specifically: (1) the development and improvement of effective fly-by waveforms for binary black holes, (2) the transition from burst emission in the high eccentricity limit to wave-like emission in the quasi-circular limit, which results in an analogy between eccentric GW bursts and Bose–Einstein condensates, and (3) the calculation of f-mode amplitudes in eccentric binary neutron stars and black hole-neutron star binaries in terms of complex Hansen coefficients. The techniques and approximations developed herein are generic, and will be useful for future studies of GWs from eccentric binaries within the context of post-Newtonian theory.

A four-node rectangular plate finite element using Airy functions with transverse shear

PurposeAmong different types of engineering structures, plates play a significant role. Their analysis necessitates numerical modeling with finite elements, such as triangular, quadrangular or sector plate elements, owing to the intricate geometrical shapes and applied loads. The scope of this study is the development of a new rectangular finite element for thin plate bending based on the strain approach using Airy's function. It is called a rectangular plate finite element using Airy function (RPFEUAF) and has four nodes. Each node had three degrees of freedom: one transverse displacement (w) and two normal rotations (x, y).Design/methodology/approachEquilibrium conditions are used to generate the interpolation functions for the fields of strain, displacements and stresses. The evolution of the Airy function solutions yielded the selection of these polynomial bi-harmonic functions. The variational principle and the analytical integration approach are used to evaluate the basic stiffness matrix.FindingsThe numerical findings for thin plates quickly approach the Kirchhoff solution. The results obtained are compared to the analytical solution based on Kirchhoff theory.Originality/valueThe efficiency of the strain based approach using Airy's function is confirmed, and the robustness of the presented element RPFEUAF is demonstrated. Because of this, the current element is more reliable, better suited for computations and especially intriguing for modeling this kind of structure.

Reduction of PAPR in OTFS signal using airy special function

Abstract Orthogonal time frequency space (OTFS) is a new two dimensional modulation technique that use the delay-Doppler domain. Despite the fact that OTFS can handle time varying channels and high Doppler scenarios, it has a high Peak-to-Average Power Ratio (PAPR). In this study, we developed an airy-based companding technique for the OTFS signal to address this issue. The airy function is a particular function that is commonly used in optics. We evaluate the proposed airy based companded OTFS with various companding techniques in order to assess its performance. Simulations signify that the proposed airy based companding outperforms the rooting companding, and μ − law companding in terms of both PAPR and instantaneous to average power ratio (IAPR).

A new probability density function for the surface elevation in irregular seas

To date, the predominant means for computing the probability density function (p.d.f.) for the free surface elevation of a nonlinear, irregular water wave field, free of assumptions involving narrow-bandedness and small directionality, is the approximate Gram–Charlier series solution of Longuet-Higgins (J. Fluid Mech., vol. 17, 1963, pp. 459–480, hereafter LH63). In this paper we re-visit the derivation of this p.d.f. to second order in the wave steepness, utilizing both moment and cumulant generating functions. We show that LH63's approximate solution based on the cumulant generating function, in fact, matches that derived from the moment generating function. Moreover, through a change of variables coupled with complex analysis, it is shown that the approximation employed by LH63 is unnecessary, and the second-order p.d.f. stemming from the cumulant generating function can be represented exactly in terms of the Airy function. The new second-order p.d.f. predicts increased probability of extreme positive surface elevations typical of e.g. rogue waves, relative to both second- and third-order solutions of LH63. This heavy positive tail is inherent, and is explained through comparison of the asymptotic limits of the p.d.f.s for large surface elevations. A semi-theoretical method is also proposed for remedying non-physical spurious oscillations that arise in the negative tail, based on the envelope of the Airy function with negative arguments. This modified negative tail is valid for irregular wave fields having skewness less than or equal to 0.2. The new p.d.f.s are compared against those based on data sets generated from second-order irregular wave theory as well as a fully nonlinear, spectrally accurate numerical wave model. Good accuracy is collectively demonstrated for directionally spread irregular seas in both finite and infinite water depths for a range of directional spreading.

Airy stress based nonlinear forced vibrations and internal resonances of nonlocal strain gradient nanoplates

This paper presents a novel investigation of nonlinear forced vibrations and internal resonance in nonlocal strain gradient nanoplates, taking into account in-plane motions. The study comprehensively models the nanoplate structure, employing Kirchhoff's plate theory, nonlocal elasticity theory, and strain gradient elasticity theory. The incorporation of the Airy stress function allows for the consideration of in-plane displacements. The large amplitude deformations in the nanoplate are described using the von Kármán geometric nonlinearity model, and the equations of motion are derived using the force-moment dynamic balance method. The coupled equations of motion are solved using the Galerkin scheme and a dynamic equilibrium technique. Detailed discussions are provided on the influence of nonlocal and strain gradient parameters on the nonlinear vibration response of the Airy stress nanostructure. Moreover, it is demonstrated that specific combinations of nonlocal and strain gradient parameters lead to various types of internal resonance (including two-to-one and three-to-one internal resonances), significantly affecting the nonlinear frequency responses of the nanoplate, resulting in rich nonlinear behaviours. This study contributes to the understanding of the complex dynamics of nanoplates subjected to external time-dependant loads and offers valuable insights for the design of diverse nanoplate systems, including graphene sheets, which are relevant to nanoelectromechanical systems (NEMS) applications.

Airy-Gaussian vector beam and its application in generating flexible optical chains.

In recent years, the manipulation of structured optical beam has become an attractive and promising area. The Gaussian beam is the most common beam as the output beam of the laser, and the Airy beam is recently proposed with fascinating properties and applications. In this paper, for the first time to our knowledge, the polarization is used as a tool to design a new kind of Airy-Gaussian vector beam by connecting the Gaussian and Airy functions, which opens a new avenue in designing new beams based on the existed beams. We realize the Airy-Gaussian vector beam with space-variant polarization distribution in theory and experiment, and find that the vector beam can autofocus twice during propagation. The optical chains with flexible intensity peaks are achieved with the Airy-Gaussian vector beam, which can be applied in trapping and delivering particles including biological cells and Rydberg atoms. Such optical chains can significantly improve the trapping efficiency, reduce the heat accumulation, and sweep away the impurity particles.

Airy Transform of the New Power-Exponent-Phase Vortex Beam

A new power-exponent-phase vortex beam with nonlinear phase winding has shown flexible control freedom compared with conventional vortex beams. In order to further enrich the modulation freedom and expand the ability of self-healing to meet current application requirements, we conducted a detailed study on the characteristics of the Airy transform of the new power-exponent-phase vortex beam. The influences of the Airy function, the power exponent, and the topological charge on normalized intensity and phase distributions are investigated theoretically and experimentally. More importantly, the self-healing properties of the new power-exponent-phase vortex beam with and without the Airy transform are compared. This shows that the new power-exponent-phase vortex beam with the Airy transform exhibits better self-healing ability when obstructed by obstacles. This study has potential applications in optical trapping and free-space optical communication.

Analytical estimation of lifetime of quasi-bound states in iii-v semiconductors quantum well

The lifetime of electrons in the Quasi Bound States in a quantum well formed by III-V semiconductors has been calculated numerically by several researchers in the past. In this work, we obtain an analytical expression for the lifetime by finding the poles of the scattering matrix in small-width approximation. Airy functions which are solutions of the Schrödinger equation for triangular potentials are expanded asymptotically for large arguments and in power series for small arguments. A scattering problem for the triangular potential well is solved with the help of mixed boundary conditions to derive the expression for the decay width that further gives the net tunneling current from the quantum well. Heterostructures of III-V materials are modeled with triangular wells whose experimental measurements are in close approximation with the theoretical calculations presented in this paper. The analytically calculated decay width in this work is also compared with the reported values by different numerical methods and found in close agreement with them.

Gaussian and Faddeeva broadening of the Franz-Keldysh oscillations at three-dimensional critical-points

Faddeeva broadening (FB) couples the physical characteristics of Gaussian and Lorentzian broadening into a single profile shaping the spectral response of dielectric functions. The electric field induces Franz-Keldysh oscillations (FKOs) as deviations in the dielectric function of semiconductors. We investigated the impact of Gaussian and FB on FKOs at the isotropic M0 of three-dimensional critical points. The twenty-pole Martin–Donoso–Zamudio quadrature was applied to compute the FB of FKOs in differential reflectivity based on the electro-optic function, H(z), with the complex Airy function, Ai(z). H(z) was simplified into a compact product of to reduce the truncation error and determine its asymptotics. We derived a general asymptotic expansion of order N, accounting for Stokes’ phenomenon as the arg(z) crosses The computation of H(z) is detailed for twelve digits of accuracy. The effects of higher Gaussian and FB widths on the damping of FKOs are assessed using FKO peak-to-peak attenuations. For the low-field regime, this work H(z) smoothly evolves into Aspnes’ third derivative functional form (TDFF). Between low- to intermediate-field regimes, we investigated the boundaries for the relevant use of the TDFF by assessing parameters extracted from simulated lineshapes, and the average root mean-squared error. We fitted of the Faddeeva broadened single Airy model to published experimental photoreflectance (PR) lineshapes of InGaAsP. To model degenerate light-and heavy-holes transitions, we fitted of the Faddeeva broadened double Airy model to published experimental contactless electroreflectance (CER) lineshapes of GaAs. In addition to legacy mean-squared error () and coefficient of determination (), the goodness of fit was evaluated using the unbiased Akaike information criteria. The effectiveness of Faddeeva versus Lorentzian broadening in sample PR and CER spectra analysis was assessed using Airy function FKO models. Concludingly, the FB of the differential reflectivity model outperformed the Lorentzian broadening even with the empirical energy-dependent width.

The birth of the strong components

AbstractIt is known that random directed graphs undergo a phase transition around the point . Earlier, Łuczak and Seierstad have established that as when , the asymptotic probability that the strongly connected components of a random directed graph are only cycles and single vertices decreases from 1 to 0 as goes from to . By using techniques from analytic combinatorics, we establish the exact limiting value of this probability as a function of and provide more statistical insights into the structure of a random digraph around, below and above its transition point. We obtain the limiting probability that a random digraph is acyclic and the probability that it has one strongly connected complex component with a given difference between the number of edges and vertices (called excess). Our result can be extended to the case of several complex components with given excesses as well in the whole range of sparse digraphs. Our study is based on a general symbolic method which can deal with a great variety of possible digraph families, and a version of the saddle point method which can be systematically applied to the complex contour integrals appearing from the symbolic method. While the technically easiest model is the model of random multidigraphs, in which multiple edges are allowed, and where edge multiplicities are sampled independently according to a Poisson distribution with a fixed parameter , we also show how to systematically approach the family of simple digraphs, where multiple edges are forbidden, and where 2‐cycles are either allowed or not. Our theoretical predictions are supported by numerical simulations when the number of vertices is finite, and we provide tables of numerical values for the integrals of Airy functions that appear in this study.

CONSTRUCTION OF THE SOLUTION OF THE SYSTEM OF SINGULARLY PERTURBATED DIFFERENTIAL EQUATIONS WITH A DIFFERENTIAL TURNING POINT

In this work, a system of singularly perturbed differential equations of the 4th order with a small parameter at the highest derivative and a turning point is considered. The turning point x=0 is located in the end of the segment [-l;0] under consideration. The coefficients of a given matrix are infinitely differentiable functions on a given interval. The goal is to construct a uniform solution asymptotics for a system of singularly perturbed differential equations with a differential turning point on the segment [-l;0]. In this case, the spectrum of the limit operator contains multiple and identically zero elements. The uniform asymptotic solution is constructed by the method of essential-singular functions using the Airy functions [0;l] and their derivatives. Studies have shown that the Apparatus of Airy functions and model equations is quite effective for constructing a uniform asymptotic solution of problems with turning points. It was also established that in order to select these functions along with the independent variable it is necessary to introduce a new variable t of the segment [0;l]. As a result, an extended problem will be obtained, which will make it possible to formally find the solution in the form of series with a small parameter. The constructions of the solutions are obtained by sequentially solving the systems of iterative equations, which at a certain step of the iterations make it possible to determine all the components of the sought-after vector functions with an accuracy of two scalar factors that form an arbitrary vector. The regularizing function Bi(t) is chosen in such a way that the function increases indefinitely when t approaches the infinity. The conducted studies showed that for any ratio of the signs of the matrix coefficient near which the inflection point is located, it is not possible to write down a smooth solution in the form of a single analytical expression. The solution of a degenerate vector equation in the general case has a discontinuity of the second kind at the inflection point, so it cannot be used explicitly to construct the asymptotic solution of this problem. For this purpose, a technique from the classical theory of linear differential equations was applied and solutions for a degenerate equation of the 2nd order were obtained.

Revisiting the Stress-Induced Birefringence in Polarization-Maintaining Fibers

Two conventional errors in the stress-induced birefringence (SIB) of polarization-maintaining fibers (PMFs) are revealed and corrected. The first issue refers to the fundamental expression and its calculation of the SIB, which depends on the selection of stress components. There were obvious inconsistencies in two pioneering reports, while both were widely applied in subsequent works. The second issue is the basic form of the Airy stress function (ASF) in the cross-section plane of the PMF, which dominates the complexity of the stress calculation for SIB. The ASFs in two early pioneering works were different, which led to two different expressions of SIB. Both expressions cannot be reproduced, and calculated results were just around 1/2 of the corresponding measurements. In this paper, both issues have been fixed by revising the definition of SIB and proposing a new ASF form for PMFs. The developed theory is simple and does not have any mathematical contradiction in the calculation process, compare to reported works. Results of the revised SIB based on the developed theory achieve a good agreement with the practical experimental measurements.

The long-time asymptotics of the derivative nonlinear Schr[formula omitted]dinger equation with step-like initial value

Consideration in this present paper is the long-time asymptotics of solutions to the derivative nonlinear Schrödinger (DNLS) equation with the step-like initial value q(x,0)=q0(x)=A1eiϕe2iBx,x<0,A2e−2iBx,x>0,by Deift–Zhou method. The step-like initial problem is described by a matrix Riemann–Hilbert problem. A crucial ingredient used in this paper is to introduce the g-function mechanism for solving the problem of the entries of the jump matrix growing exponentially as t→∞. It is shown that the leading order term of the long-time asymptotics solution of the DNLS equation is expressed by the Theta function Θ about the Riemann-surface of genus 3 and the subleading order term expressed by parabolic cylinder and Airy functions.

Investigating teeth root stresses of various spur gears using stress function and FEM

AbstractThe gear is the most efficient and reliable portion of the transmission system. In machine tools, gears are utilized to handle heavy loads. These gears are used indefinitely if the prerequisites are met. Gear tooth bending stress for numerous spur gears has been examined in this research. The influence of changing the pressure angles on the spur gear tooth root stress has been investigated. The purpose of this study is to look for a new way of finding the stresses generated in the roots of the gear teeth as well as the effect of changing pressure angles on these stresses. Solidworks 2021 uses parametric formulation to model gears, analyzed for deformation and maximum bending stress in ANSYS. The airy stress function is used to conduct the nominal root tooth stress. The enhancement percentages of modified cases are 5.92%, 1.70%, 10.70%, 17.22%, 10.86%, 24.32%, 20.91%, and 26.15% for the cases 2, 3, 4, 5, 6, 7, 8, and 9, respectively according to the stress function findings.

Non-homogeneous impulsive time fractional heat conduction equation

This article provides a concise exposition of the integral transforms and its application to singular integral equation and fractional partial differential equations. The author implemented an analytical technique, the transform method, for solving the boundary value problems of impulsive time fractional heat conduction equation. Integral transforms method is a powerful tool for solving singular integral equations, evaluation of certain integrals involving special functions and solution of partial fractional differential equations. The proposed method is extremely concise, attractive as a mathematical tool. The obtained result reveals that the transform method is very convenient and effective.Certain new integrals involving the Airy functions are given.

A continuum description of the buckling of a line of spheres in a transverse harmonic confining potential

A line of contacting hard spheres, placed in a transverse confining potential, buckles under compression or when tilted away from the horizontal, once a critical tilt angle is exceeded. This interesting nonlinear problem is enriched by the combined application of both compression and tilt. In a continuous formulation, the profile of transverse sphere displacement is well described by numerical solutions of a second-order differential equation (provided that buckling is not of large amplitude). Here we provide a detailed discussion of these solutions, which are approximated by analytic expressions in terms of Jacobi, Whittaker and Airy functions. The analysis in terms of Whittaker functions yields an exact result for the critical tilt for buckling without compression.

Improved airy function formalism for the study of anti-parallel double δ-magnetic-barrier nanostructure under the modulation of the applied bias

The Goos–Hänchen (GH) shift and the spin polarization are theoretically investigated under the anti-parallel double [Formula: see text]-magnetic-barrier nanostructure model by applying a bias using Airy’s function formalism and transfer-matrix technique, which is an improvement on previous calculation. The results show the magnitude and sign of GH shift and spin polarization can be easily changed by modulating the applied bias. The closer the applied bias to the critical value is, the more intensive GH shift and spin polarization will be. Furthermore, two critical values come through modulating the strength of magnetic fields, and an intensive GH shift and a spin polarization also exist near each critical value. These findings can provide a practical approach to designing and fabricating spin beam filters or splitters modulated by applied bias.

Electromagnetic field generated by tsunamigenic seabed deformation

We derive a mathematical model of an electromagnetic (EM) field generated by tsunamigenic seabed deformation over an ocean of constant depth. We solve the governing Maxwell equations for the EM field, coupled with a potential flow model of Cauchy–Poisson type for the transient fluid motion forced by seabed deformation. Our new model advances previous studies, where simplified formulae without direct forcing were assumed for the wave field. Using complex integration and large-time asymptotics, we obtain a novel analytical solution for the magnetic field propagating at large distance from the seabed deformation in two dimensions. We show that this magnetic field is made of two terms, one proportional to an Airy function, and thus propagating similarly to the surface gravity wave, and one proportional to a Scorer function, which exhibits a phase lag with respect to the surface gravity wave. Such a phase lag explains the time difference between the arrival of the EM field and the surface gravity wave generated by seabed deformation, which were observed in recent measurements and numerical results. Finally, we discuss the opportunity to detect EM fields as precursors of surface gravity waves in tsunami early warning systems. We introduce a novel non-dimensional parameter to identify the propagation regime of the magnetic field, i.e. self-induction versus diffusion-dominated. We show that tsunami early warning via EM field is possible for diffusion-dominated regimes when the water depth is less than 2 km. Our findings provide a rigorous analytical explanation of existing observations and numerical results.