Bessel Function Expansion Research Articles

Globality of the DPW construction for Smyth potentials in the case of SU1,1

We construct harmonic maps into SU1,1/U1 starting from Smyth potentials ξ, by the DPW method. In this method, harmonic maps are obtained from the Iwasawa factorization of a solution L of L−1dL=ξ. However, the Iwasawa factorization in the case of a noncompact group is not always global. We show that L can be expressed in terms of Bessel functions and from the asymptotic expansion of Bessel functions we solve a Riemann-Hilbert problem to give a global Iwasawa factorization. In this way we give a more direct proof of the globality of our solution than in the work of Dorfmeister-Guest-Rossman [5], while avoiding the general isomonodromy theory used by Guest-Its-Lin [11], [12].

Higher Harmonic Generation Arising from the Bessel Function Expansions of the Carrier-Envelope-Phase Driven by the Femtosecond Optical Kerr Effect

Higher Harmonic Generation Arising from the Bessel Function Expansions of the Carrier-Envelope-Phase Driven by the Femtosecond Optical Kerr Effect

Anti‐plane stress analysis for V‐notch in a piezomagnetic half space under SH wave

AbstractIn this paper, the anti‐plane stress analysis of a V‐notch with complex boundary conditions in a piezomagnetic half space is studied. Firstly, SH wave is considered as an external load acting on piezomagnetic half space, on the basis of repeated image superposition, the analytical expression of scattering wave is conducted, which satisfies the boundary conditions on the boundary of the half space. Then, the analytical expression of standing wave is established, which satisfies the stress free and magnetic insulation conditions on the boundaries of V‐notch by the fractional Bessel function expansion method and Graf addition theorem. Finally, Green's function method is applied, the half space is divided into two parts along the vertical interface, a pair of in‐plane magnetic field and out‐plane forces are applied on the vertical interface, and the first kind of Fredholm integral equations are set up and solved by applying orthogonal function expansion technique and effective truncation. Results clarified the influence on the dynamic stress concentration factor and magnetic field intensity concentration factor under proper conditions. Besides, the analytical solutions are compared with the finite element solutions to verify the accuracy of the conclusions in this article.

Analysis on scattering characteristics of SH guided wave due to V-notch in a piezoelectric bi-material strip

In this paper, the problem of a V-notch with complex boundary conditions in a piezoelectric bi-material strip is studied. First, SH guided wave is considered as an external load acting on the piezoelectric bi-material strip; on the basis of repeated image superposition, the analytical expression of scattering wave is conducted, which satisfies the stress-free and electric insulation conditions on the upper and lower horizontal boundaries of the strip. Then, the analytical expression of standing wave is established, which satisfies the stress-free and electric insulation conditions on the boundaries of V-notch by the fractional Bessel function expansion method and Graf addition theorem. Finally, Green’s function method is applied, the bi-material strip is divided into two parts along the vertical interface, a pair of in-plane electric field and out-plane forces is applied on the vertical interface, and the first kind of Fredholm integral equations is set up and solved by applying the orthogonal function expansion technique and effective truncation. Results clarified the influence on the dynamic stress concentration factor and electric field intensity concentration factor under proper conditions. Besides, the analytical solutions are compared with the finite element solutions to verify the accuracy of the conclusions in this article.

The Memory Function of an Impurity in Mass and Hooke Constant in a Diatomic Chain

The memory function of an impurity with different mass and Hooke constant in a classical diatomic chain is studied by means of the recurrence relations method. The Laplace transform of the memory function has two pairs of resonant poles and three branch cuts. The poles contribute a cosine and the cuts contribute acoustic and optic branches, which are expressed as a convolution of a difference of two sines and an expansion of even-order Bessel functions. The expansion coefficients are integrals of Jacobian elliptic function sn(u) along the real axis in a complex u+iv plane for the acoustic branch, and integrals of nd(v) along a contour parallel to the imaginary axis of the plane for the optical branch, respectively. Different special cases are discussed in detail. It shows that a perfect monatomic or diatomic chain and such a chain with a mass impurity share the same memory function except for a constant factor, and that the pole contribution exists only if the impurity has both different mass and Hooke constant.

Dynamic analysis for V-notch in a piezomagnetic bi-material strip under SH guided wave

In this paper, the problem of a V-notch with complex boundary conditions in a piezomagnetic bi-material strip is studied. Firstly, SH guided wave is considered as an external load acting on piezomagnetic bi-material strip, on the basis of repeated image superposition, the analytical expression of scattering wave is conducted, which satisfies the stress free and magnetic insulation conditions on the upper and lower horizontal boundaries of the strip. Then, the analytical expression of standing wave is established, which satisfies the stress free and magnetic insulation conditions on the boundaries of V-notch by the fractional Bessel function expansion method and Graf addition theorem. Finally, Green’s function method is applied, the bi-material strip is divided into two parts along the vertical interface, a pair of in-plane magnetic field and out-plane forces are applied on the vertical interface, and the first kind of Fredholm integral equations are set up and solved by applying orthogonal function expansion technique and effective truncation. Results clarified the influence on the dynamic stress concentration factor and magnetic field intensity concentration factor under proper conditions. Besides, the analytical solutions are compared with the finite element solutions to verify the accuracy of the conclusions in this article.

Pre-asymptotic dispersion of active particles through a vertical pipe: the origin of hydrodynamic focusing

When motile algal cells are exposed to gyrotactic torques, their swimming directions are guided to form radial accumulation, well known as hydrodynamic focusing. The origin of hydrodynamic focusing from the effects of active swimming, ambient flow and particle anisotropy is elucidated in the present study on the pre-asymptotic dispersion of active particles through a vertical pipe. With an extension of the Galerkin method to pipe flows, time-dependent solutions directly from the Smoluchowski equation in the position and orientation space are derived by series expansions of spherical harmonics and Bessel functions. Ballistic and diffusive scaling laws are examined with the predominance of self-propelled swimming, and computation is validated against an explicit benchmark solution and Lagrangian particle simulation. In the limit of extreme shear, the competitive roles of shear dispersion and Brownian rotation are reflected concretely in the pre-asymptotic phase of hydrodynamic focusing. For flows with various shear strengths, a concentration peak in near-wall regions with a smooth transition to hydrodynamic focusing is illustrated with richer phenomena in upwelling and downwelling flows. A newly observed regime through a vertical pipe, named transient effective trapping, is revealed as a transitional mode towards hydrodynamic focusing. The pre-asymptotic approach to hydrodynamic focusing is elaborated intensively through extensive solutions of concentration moments and macroscopic transport coefficients characterised by swimming and flow Péclet numbers. The unique findings for the origin of hydrodynamic focusing could provide insight into related micro-algae reactor technology and contribute to flow control and biomass transfer in confined environments.

The Madelung constant in N dimensions

We introduce two convergent series expansions (direct and recursive) in terms of Bessel functions and the number of representations of an integer as a sum of squares for N -dimensional Madelung constants, M N ( s ) , where s is the exponent of the Madelung series (usually chosen as s = 1 / 2 ). The convergence of the Bessel function expansion is discussed in detail. Values for M N ( s ) for s = 1 2 , 3 2 , 3 and 6 for dimension up to N = 20 are presented. This work extends Zucker’s original analysis on N -dimensional Madelung constants for even dimensions up to N = 8 .

Expansion of some well-known functions and classical polynomials in series of the generalized Laguerre polynomials.1

The generalized Laguerre polynomials form a complete set (orthogonal and normalized) in the space ** with respect to a certain weighting function because they are the Eigen functions of a second-order differential operator. Here we shall show how to expand some well-known classical polynomials such as the Legendre and the Hermite polynomials in series of generalized Laguerre polynomials. Since the generalized Laguerre orthogonal polynomials are set of polynomials that are mutually orthogonal to each other with respect to a measure of weighting function that is just the integrand of the gamma function, thus it grants us the guarantee of the ability to expand the first kind of Bessel functions and the gamma function in terms of the generalized Laguerre polynomials. The series expansion in terms of the generalized Laguerre polynomials can be achieved by following various approaches. For instance, the series expansion of the first kind Bessel functions is gained by the generalized hypergeometric function approach, whereas the series expansion of gamma function was obtained directly by the usual way that is by calling the orthogonality and orthonormality properties of the generalized Laguerre polynomials. Another powerful technique to gain the series expansion of the classical polynomials is the generating function approach as it has been followed here to obtain a series expansion of the Legendre and the Hermite polynomials in terms of the generalized Laguerre polynomials. The series expansion in series of the generalized Laguerre polynomials has a variety of applications in mathematics, physics, and engineering

The linear and nonlinear inverse Compton scattering between microwaves and electrons in a resonant cavity

The new scheme of the energy measurement of the extremely high energy electron beam with the inverse Compton scattering between electrons and microwave photons requires the precise calculation of the interaction cross section of electrons and microwave photons in a resonant cavity. In the local space of the cavity, the electromagnetic field is expressed by Bessel functions. Although Bessel functions can form a complete set of orthogonal basis, it is difficult to quantify them directly as fundamental wave functions. Fortunately, with the Fourier expansion of Bessel functions, the local electromagnetic field can be considered as the superposition of a series of plane waves. Therefore, with corresponding corrections of the cross section formula of the classical Compton scattering, the cross section of the linear or nonlinear microwave Compton scattering in the local space can be described accurately. As an important application of our results in astrophysics, corresponding ground verification devices can be designed to perform experimental verifications on the prediction of the Sunyaev–Zeldovich (SZ) effect of the cosmic microwave background radiation. Our results could also provide a new way to generate wave sources with strong practical value, such as the terahertz waves, the ultra-violet (EUV) waves, or the mid-infrared beams.

Favorable Propagation for Massive MIMO With Circular and Cylindrical Antenna Arrays

Massive MIMO systems with uniform circular and cylindrical antenna arrays are studied. Favorable propagation property is rigorously shown to hold asymptotically in LOS environment for a fixed antenna spacing and under some mild conditions. The analysis is based on a novel method using a Bessel function expansion, a new bounding technique and a simplified representation of inter-user interference for array geometries obeying Kronecker structure.

Properties of Sakaguchi Kind Functions Associated with Bessel Function

The aim of the paper is to obtain the First Hankel Determinant and the Second Hankel determinant. We shall make use of few lemmas which are based on Caratheodory's class of analytic functions. We establish a new Sakaguchi class <img src=image/13424969_01.gif> of univalent function, further we estimate the sharp bound for initial coefficients <img src=image/13424969_02.gif> and <img src=image/13424969_03.gif> using the Bessel function expansion. We have discussed about the coefficient <img src=image/13424969_04.gif> as well for the Second Hankel Determinant. The results are obtained for Sakaguchi kind. Our results travel along exploring the stages of Hankel Determinants. Various types of technologies like wire, optical or other electromagnetic systems are used for the transmission of data in one device to another. Filters play an important role in the process that can remove disorted signals. By using different parameter values for the function belongs to Sakaguchi kind of functions the Low pass filter and High pass filter can be designed and that can be done by the coefficient estimates.

Scaling, Complexity, and Design Aspects in Computational Fluid Dynamics

With the availability of more and more efficient and sophisticated Computational Fluid Dynamics (CFD) tools, engineering designs are also becoming more and more software driven. Yet, the insights in temporal and spatial scaling issues are still with us and very often imbedded in complexity and many design aspects. In this paper, with a revisit to a so-called leakage issue in sucker rod pumps prevalent in petroleum industries, the author would like to demonstrate the need to use perturbation approaches to circumvent the multi-scale challenges in CFD with extreme spatial aspect ratios and temporal scales. In this study, the gap size between the outer surface of the plunger and the inner surface of the barrel is measured with a mill (one thousandth of an inch) whereas the plunger axial length is measured with inches or even feet. The temporal scales, namely relaxation times, are estimated with both expansions in Bessel functions for the annulus flow region and expansions in Fourier series when such a narrow circular flow region is approximated with a rectangular one. These engineering insights derived from the perturbation approaches have been confirmed with the use of full-fledged CFD analyses with sophisticated computational tools as well as experimental measurements. With these confirmations, new perturbation studies on the sucker rod leakage issue with eccentricities have been presented. The volume flow rate or rather leakage due to the pressure difference is calculated as a quadratic function with respect to the eccentricity, which matches with the early prediction and publication with comprehensive CFD studies. In short, a healthy combination of ever more powerful modeling tools along with the physics, mathematics, and engineering insights with dimensionless numbers and classical perturbation approaches may provide a balanced and more flexible and efficient strategy in complex engineering designs with the consideration of parametric and phase spaces.

Boundary integral formulation of the standard eigenvalue problem for the 2-D Helmholtz equation

In this paper, a boundary integral formulation is presented for obtaining the standard eigenvalue problem for the two-dimensional (2-D) Helmholtz equation. The formulation is derived by using the series expansions of zero-order Bessel functions for the fundamental solution to the Helmholtz equation. The proposed approach leads to a series of new fundamental functions which are independent of the wave number k of the Helmholtz equation. The coefficient matrix of the resulting homogeneous system of boundary element equations is of the form of a polynomial matrix in k which allows a much faster search for the eigenvalues by scanning k over an interval of interest or the standard eigenvalue problem to be formulated for directly solving for the eigenvalues without resort to iterative methods. The proposed technique was used to solve some known problems with available analytical solutions: 2-D domains with circular and rectangular geometries under Dirichlet and/or Neumann boundary conditions. The outcomes demonstrate that the proposed approach is computationally efficient and highly accurate.

Pade Approximant ELF VED Fields in the Earth‐Ionosphere Cavity

AbstractElectromagnetic (EM) fields in the extremely low‐frequency (ELF) domain that can be excited by the vertical electric dipole (VED) and propagated in the Earth‐ionosphere spherical cavity have received tremendous attention owing to their exploitation in many fields such as communication, earthquake prediction, and geophysical investigations. Some researchers use the planar model and curvature correction terms to simulate EM fields in a spherical cavity, while others directly solve the differential functions and estimate the excited EM fields represented by spherical Bessel and associated Legendre functions. Regarding the divergence problem of a series when simulating EM fields, several numerical methods are available such as Watson transformation method, W. K. B. J method, and speeding algorithm. In this study, Pade approximant is utilized to estimate EM fields propagating in the Earth‐ionosphere spherical cavity, where new asymptotic expansions of spherical Bessel functions are introduced for estimating their products for high orders. These EM fields are then interpreted as a sum of the dominant incident fields and the increase of multiple scattering fields. The convergence of series expressions of EM fields estimated by Pade approximant is theoretically guaranteed, with its computational complexity lower than that of the speeding algorithm. The EM fields are further verified by numerical tests. The effects of the height and conductivity of the ionosphere, the conductivity of air, the VED height, and the frequencies of the resonance phenomenon are also highlighted. The proposed method is helpful to capture the true nature of the field distribution within a spherical cavity.

Strichartz estimates in Wiener amalgam spaces and applications to nonlinear wave equations

In this paper we obtain some new Strichartz estimates for the wave propagator eit−Δ in the context of Wiener amalgam spaces. While it is well understood for the Schrödinger case, nothing is known about the wave propagator. This is because there is no such thing as an explicit formula for the integral kernel of the propagator unlike the Schrödinger case. To overcome this lack, we instead approach the kernel by rephrasing it as an oscillatory integral involving Bessel functions and then by carefully making use of cancellation in such integrals based on the asymptotic expansion of Bessel functions. Our approach can be applied to the Schrödinger case as well. We also obtain some corresponding retarded estimates to give applications to nonlinear wave equations where Wiener amalgam spaces as solution spaces can lead to a finer analysis of the local and global behavior of the solution.

Pulsed Harmonic Doppler Radar and Passive RF Tags for Tracking the Dynamic Motion of Held Objects

A new method for detecting microwave harmonic Doppler signatures of held objects in cluttered environments is presented. Based on a passive harmonic tag combined with a pulsed microwave radar, the motion of tagged objects held by a person can be detected in high-clutter environments. In this letter, we present a new derivation of the dynamic harmonic signal response of a harmonic tag in motion. Based on a truncated Bessel function expansion, the model predicts the time-frequency signatures of sinusoidally moving objects with high accuracy. We present a pulsed harmonic radar transmitting at 2.51 GHz and receiving at 5.02 GHz, which is capable of measuring the harmonic micro-Doppler response of the tag with better signal-to-noise since the clutter response only exists at 2.51 GHz. The pulse waveform provides detection of the harmonic tag at larger distances than continuous-wave systems by increasing the peak power of the signal incident on the tag, while the average power is fixed with a low level. Measurements of a held tag at a distance of 1 m in a highly cluttered environment are presented, showing a strong match to the simulated results.

High-speed Mueller matrix ellipsometer with microsecond temporal resolution.

A high-speed Mueller matrix ellipsometer (MME) based on photoelastic modulator (PEM) polarization modulation and division-of-amplitude polarization demodulation has been developed, with which a temporal resolution of 11 µs has been achieved for a Mueller matrix measurement. To ensure the accuracy and stability, a novel approach combining a fast Fourier transform algorithm and Bessel function expansion is proposed for the in-situ calibration of PEM. With the proposed calibration method, the peak retardance and static retardance of the PEM can be calibrated with high accuracy and sensitivity over an ultra large retardance variation range. Both static and dynamic measurement experiments have been carried out to show the high accuracy and stability of the developed MME, which can be expected to pave the way for in-situ and real-time monitoring for rapid reaction processes.

A monatomic chain with an impurity in mass and Hooke constant

In this paper, a diatomic chain composed of classical harmonic oscillators and an impurity with different mass and Hooke constant is studied by means of the recurrence relations method. The Laplace transform of the momentum autocorrelation function of the impurity has three pairs of resonant poles and three separate branch cuts; however, one pair of poles was found nonphysical, so only two pairs of poles contribute to the momentum autocorrelation function. The three branch cuts give the acoustic and optical branches that are given by a convolution of a sum of sines and an expansion of even-order Bessel functions. The expansion coefficients are integrals of a Jacobian elliptic function along the real axis in a complex plane for the acoustic branch and those along a contour parallel to the imaginary axis for the optical branch, respectively. The ergodicity of momentum of the impurity is also discussed.

Methods to ascertain the resistance of stranded conductors in the frequency range of 40 Hz–150 kHz

Although a few years have passed since the first electromagnetic compatibility problem occurred in the supraharmonic frequency range (2–150 kHz), a lot of questions are still to be answered before the completion of appropriate electromagnetic compatibility (EMC) standards. An exciting question is how supraharmonic disturbances spread and aggregate on the network. To get a better picture of this, the frequency dependent characteristics of the network elements must be examined. This paper deals with the resistance of three stranded conductors with different cross section areas typically used in the low voltage (LV) and medium voltage (MV) distribution lines. From the methodology point of view, the problem was approached by laboratory measurements, numerical simulations with Filament technique, which was realized in MATLAB. In addition the results are compared with a few other analytical calculation methods (e.g. series expansion of Bessel functions, skin factor method, Hankel method, K. Simonyi’s approximation) found in the literature. These were performed, assessed and compared. Besides that, a new method combining two analytical methods is proposed to approximate the internal impedance of conductors. It must be noted that our intention was to cover not only the supraharmonics area, but also the power harmonics range; therefore, investigations start from 40 Hz instead of 2 kHz.