Eigenfunction Method Research Articles

Аппроксимация решения внутренней задачи электродинамики методом собственных функций

The problem of constructing approximation models of radiating and reradiating structures based on the eigenfunction method is considered. The ambiguity of the solution of this problem is noted. Techniques are proposed that can be used in the development of approximation models. An expression is presented for estimating the error introduced by the approximation model. As an example, one of the possible options for constructing an approximation model for a tubular vibrator is given. The results of numerical simulation are presented. Keywords: eigenfunction method, method of moments, singular integral equations, approximation model, tubular selector.

Влияние неоднородной теплозащиты на характеристики излучения антенны космического аппарата

Introduction: Knowledge of the electrical characteristics of the onboard antenna on the trajectory of the descent of the spacecraft makes it possible to assess the presence or absence of radio communication. Under conditions of aerodynamic heating, the thermal protection of the onboard antenna heats up unevenly in thickness and becomes electrically inhomogeneous. Purpose of the work: Determination of the radio technical characteristics of the onboard antennas of the returning spacecraft with the effect of high-temperature heating on the thermal protection of the antenna based on calculations using the developed mathematical model of the antenna with thermal protection. Methods: Expressions for the structure of the radiation field of a rectangular waveguide under the above conditions were derived by the WKB method. Of the well-known analytical methods of solution, it is possible to use the method of integral transformations and the method of eigenfunctions. Both of these methods were used in this work. Results: The obtained theoretical results are new, they allow predicting the radio technical characteristics of the onboard antenna, taking into account uneven heating across the thickness of the thermal protection at low temperature gradients. A mathematical model of the spacecraft's onboard antenna on the descent trajectory has been developed, taking into account the inhomogeneity of thermal protection under conditions of aerodynamic heating. Practical significance: The development of a mathematical model of the main radio technical characteristics of onboard antennas, taking into account the impact of high-temperature aerodynamic heating, as well as the results of numerical calculations, can be applied in the development of recommendations for choosing thermal protection and recommendations for reducing the effect of temperature changes in the electrical parameters of thermal protection on the characteristics of onboard antennas and reducing time loss of radio communication or loss recovery.

Mathematical Model of the On-Board Antenna of Reentry Spacecraft Taking into Account Surface Waves

Introduction. The onboard antennas of the returned hypersonic aircraft are weakly directional, which is achieved by radiation from the open end of the waveguide. When passing through dense layers of the atmosphere, they are exposed to aerodynamic heating, for protection from which a heat-resistant radio-transparent thermal protection is used. The case of uniform thermal protection of the antenna can be interpreted as the absence of heating or heating uniform in the thickness of the thermal protection.Aim. The problem of obtaining an analytical description of the radiation characteristics of a circular waveguide closed by a flat homogeneous dielectric plate is solved. Since in such a formulation it is necessary to consider the resonant domain, a strict solution of Maxwell's equations is required.Materials and methods. Of the known analytical methods of solution, it is possible to use the method of integral transformations and the method of eigen functions. Both of these methods are used in the work. In this case, the assumption is used that the electrical parameters of the dielectric plate (thermal protection) and the geometric dimensions do not depend on time.Results. The relations describing the directional pattern of a circular waveguide with dielectric thermal protection and taking into account the electrical parameters of thermal protection and its thickness are obtained. Expressions are also obtained for the fields of lateral, surface and outflow waves, from which it is possible to calculate the power output by these fields. Relations for the separation of singular points of integrand expressions into poles corresponding to surface, outflow and lateral waves are obtained. Surface, outflow and lateral waves can have a certain effect on the radiation pattern. To determine this conclusion, analytical relations are obtained for determining the poles of integrand expressions that fully describe surface, outflow and side waves. All the analytical results obtained correspond to Numerical calculations were carried out on some of the obtained ratios.Conclusion. The results showed that the power of the side waves is zero. It also follows from the calculations carried out that there is no radiation field of surface and outgoing waves, i.e. there is no contribution of them to the radiation pattern.

К исследованию влияния дефекта покрытия на поверхностный волновой процесс

The paper presents an analytical and numerical factorization method for studying the interaction of coating plates in the contact area on a rectilinear fault. Using the method of eigenfunctions and the block element method, we obtained new analytical representations of the solution for the problem, which makes it possible to estimate the behavior of the surface of the medium. We have calculated the wave field on the surface of a homogeneous elastic layer with a pinched bottom face and a composite coating with varying physical and mechanical characteristics of the plates and substrate. We have also illustrated the results of the numerical implementation for the developed algorithm and given a brief discussion about the influence of the fault boundary conditions, as well as the physical and mechanical properties of the inhomogeneous coating and substrate on the wave field formed under the influence of a vertical localized surface source.

Long wave absorption by a dual purpose Helmholtz resonance OWC breakwater

In this paper, a novel hybrid breakwater-WEC system consisting of a Helmholtz-resonance oscillating water column (OWC) caisson array and a perforated wall is proposed to attenuate long waves. The principle of Helmholtz-resonance OWC is analogous to the acoustic Helmholtz resonator for noise reduction. A semi-analytical model was developed based on potential flow theory using the matching eigenfunction method to investigate wave interaction with the OWC-breakwater system. The velocity singularity at the tip of the wall is resolved using the Galerkin method to improve the model convergence. The semi-analytical model of the present wave-air-structure interaction problem is verified using Haskind relation and energy conservation rule. Furthermore, we validate the model using numerical and experimental data in terms of wave reflection and wave power absorption. Hydrodynamic characteristics of the proposed hybrid breakwater-WEC system and the influence of key geometrical and wave parameters were investigated. Theoretical results indicate that long waves can be effectively absorbed by the OWC caisson through Helmholtz resonance for both normal and oblique waves. The resonance frequency of OWC varies with the chamber wall draft and chamber width in the along-shore direction. It was found that installing a perforated wall in front of the OWC array could broaden the frequency bandwidth of significant wave attenuation. The along-shore length of each OWC chamber ( 2l ) and the incident wave angle ( θ ) are identified as the key parameters that affect the hydrodynamic performance of the system at high frequency range due to the along-shore sloshing of water column, but have a relatively minor effect on long waves attenuation. The present OWC-breakwater system demonstrates advantage and superior performance over conventional rubble mound breakwaters and breakwaters of multiple perforated walls in attenuating long waves. The findings of this paper provide guidance for a novel design of breakwaters for absorbing long waves. • Novel Helmholtz-resonance OWC breakwater with a perforated wall is proposed to absorb longer waves．. • Semi-analytical model of wave interaction with OWC array with perforated wall is developed and validated．. • Present hybrid OWC-breakwater system is more effective in absorbing long waves than conventional breakwaters. • Significant effect of incident wave angle on wave reflection is observed at higher frequency range but not at low frequency range.

Hydrodynamic performance of dual-chamber Oscillating Water Column array under oblique waves

A multiple Oscillating Water Column (OWC) device may provide better wave absorption over a wider frequency bandwidth than a single-chamber OWC due to multiple resonances. The scattering and radiation of three-dimensional oblique waves by an array of periodic dual-chamber OWCs are considered here along a coastal cliff. A semi-analytical model was developed based on potential flow theory and matching eigenfunction method to investigate the oblique wave interaction with a dual-chamber OWC array system. The velocity singularity at the tip of a chamber wall is resolved by introducing the Galerkin technique to accelerate the convergence. The semi-analytical solution is verified by the Haskind relation and energy conservation law. Hydrodynamics of the proposed system and the influence of wave and geometric parameters were investigated. Theoretical results indicate that a dual-chamber OWC array has a broader capture bandwidth than a single-chamber OWC array for both normal and oblique waves. The presence of the along-shore and cross-shore sloshing resonance is theoretically confirmed in each subchamber of OWC unit, which decreases the hydrodynamic efficiency and increases the wave reflection drastically. Although the wave loading on the chamber wall decreases with increasing incident wave angle θ, the wave loading on chamber/partition wall may increase sharply due to sloshing resonance at critical frequency kc. To our knowledge, this is the first attempt to investigate the hydrodynamics of dual-chamber OWC array under oblique waves. The present theoretical results indicate the potential risks of structural damage and total wave reflection due to sloshing resonance, which should be an important design consideration.

SWITCH-FILTER ON A RECTANGULAR WAVEGUIDE PARTIALLY FILLED BY DIELECTRIC

At article investigates a broadband switch for microwave technology, antenna-feeder paths of which are implemented on a rectangular waveguide partially filled by dielectric. Modern means of communication of the microwave range can operate for transmission through two independent antenna channels, each of which includes its own microwave transmitter. It is also provided the operation of one microwave transmitter for two antennas. The transmission of high-power signals requires the implementation devices based on rectangular waveguide partially filled by dielectric for such antenna-feeder paths. The active element is an open nonlinear structure included in a dielectric plate which located in a rectangular waveguide. The electrodynamic problem is solved by the method of eigenfunctions. In this article, the transfer coefficient T11 and plotted the graphs of the dependence of the electrical length of the waveguide segment with the open nonlinear structure on the value of the reactive conductivity of the inductive loop at fixed reactive conductivity of the open nonlinear structure is determine. The results of this article can be used in the development of broadband switches for mobile digital combined troposcatter-radio relay stations with space-diversity transmission, antenna-feeder paths of which are implemented on the rectangular waveguide partially filled by dielectric. The article developed a filter on waveguide tees partially filled by dielectric. A design based on an H-tee on a rectangular waveguide partially filled by dielectric is proposed. The equivalent circuit of the H-tee is based on the equivalent circuit in the longitudinal slot in the narrow wall of the waveguide. Such a gap interrupts the lines of the transverse surface current of the main quasi-H10 wave of a rectangular waveguide partially filled by dielectric. Numerical results are obtained for transformation ratios.

The Eigenfunction Method to Calculate the Klein-Gordon Propagator in an Inhomogeneous Magnetic Field

In this paper, we use the Eigenfunction method (Ritus'method) to calculate the Klein-Gordon propagator under an external magnetic field with an exponential damping factor b, in a four-dimensional Euclidean space. We write the propagator of scalar field in terms of plane waves and Laguerre polynomials. Also the eigenvalues associated to differential operator show the quantization in the xy-plane through the Landau levels. We apply the calculated propagator to the mass parameter of an interacting boson system under an exponentially decaying magnetic field.

Eigenfunction method for splitting of energy levels in quantum mechanics

In this work, we illustrate the use of eigenfunction method (EFM) in quantum mechanics by taking a specific example namely the splitting of energy levels under perturbation. Initially, the symmetry group of the Hamiltonian is taken to be [Formula: see text] which under the action of a perturbing potential changes to a lower symmetry namely [Formula: see text]. We have constructed an irreducible basis using the EFM for both the symmetries and show the simplifications that ensure the calculation of the expectation value of the perturbing Hamiltonian.

An analytical solution for contaminant extraction from multilayered soil using PVD-enhanced system

Prefabricated vertical drains (PVDs) have been employed to enhance the in situ remediation of contaminated fine-grained soils. Subsurface heterogeneity can interfere with the distribution and extraction of contaminants during the remediation process. In the present study, an analytical solution for contaminant extraction from multilayered soil using a PVD-enhanced system is developed based on an equivalent planar two-dimensional model. The analytical solution is derived using a procedure that combines the Laplace transform, eigenfunction method and numerical Laplace inversion. The validity and accuracy of the solution are verified by comparison with an existing analytical solution and the results obtained using a numerical model. The effects of several key parameters on the performance of a PVD remediation system in a triple-layered contaminated soil are evaluated. The results indicate that the remediation efficiency for clay layers decreases with increasing hydraulic conductivity or thickness of the sand layer. The PVD remediation system may be unfeasible for contaminated sites with high subsurface heterogeneity caused by permeability contrast. The remediation process for clay layers can be accelerated by increasing the clay vertical dispersivity.

Influences of wave resonance on hydrodynamic efficiency and loading of an OWC array under oblique waves

In this paper, a semi-analytical solution was derived from the linear potential flow theory using the matching eigenfunction method to investigate hydrodynamic characteristics of a rectangular oscillating water column (OWC) array under oblique waves. The theoretical results indicate that the sloshing resonance in the alongshore direction varies with the incident wave angle significantly. The presence of the along-shore sloshing resonance decreases the hydrodynamic efficiency and increases the wave reflection. The critical wavenumber kc at which the along-shore sloshing resonance occurs is identified as a key controlling factor of the performance of the OWC array. It is dependent on the chamber length in the along-shore direction. The peak efficiency occurs at the lower frequency for the larger incident wave angle. To our knowledge, this study provides the first theoretical evidence that the peak wave loading of the OWC array is triggered by the along-shore resonance under oblique wave action. Our theory confirms that the optimum coastal protection can be achieved by the peak efficiency of the OWC array in the frequency region of the piston resonance of the OWC.

Hydroelastic analysis of a coupled porous structure in finite water depth

Scattering of oblique waves by a system consisting of a thin floating porous-elastic plate and a rectangular submerged porous structure is investigated. The plate is modeled by combining the thin plate theory and a complex porous-effect parameter, and the Sollitt and Cross model is used for flow through the submerged porous structure. A matched eigenfunction method is employed and roots of dispersion relations associated with the system and its components are investigated using contour plots. Scattering coefficients are evaluated and analyzed for various wave and structure parameters and the influence of each component on wave forces acting on the other is examined. The study reveals that plate porosity, along with variation in height and width of the system have a significant impact on scattering and mitigation of wave forces. A phenomenon of reversal in pattern of reflection coefficients with respect to incident angle for varying heights of the submerged structure is observed. A comparative analysis highlights the key features of scattering and energy dissipation by plate, rubblemound and coupled structure. The results of this study can be used for designing a composite breakwater or to analyze the performance of a floating breakwater in the presence of inhomogeneous bottom topography.

An Efficiency-Enhanced 0.5-THz BWO Using Double-Slot Embedded Electron Beams to Drive a Grating Loaded Rectangular Waveguide

The backward wave oscillator (BWO) is one of the best choices for generating high-power radiation waves in the terahertz (THz) region. In order to improve the efficiency of beam–wave interaction and output power, a novel THz BWO scheme is proposed in this article, which uses double-slot embedded electron beams (DSEEB) to drive a grating loaded rectangular waveguide (GLRW) slow wave structure (SWS). The dispersion relation and the coupling impedance of the proposed SWS are derived by using an eigenfunction method (EFM) and solved with a numerical method. The results show that the upper limit frequency and the coupling impedances are remarkably improved, which are greater than those of traditional grating SWS. Based on this scheme, a BWO operating at the frequency of 0.5 THz is designed. The particle-in-cell simulations show that the output power reaches 8.86 W with an efficiency of 0.9%, which is 2.5 times the electronic efficiency of the conventional BWO. Thus, the proposed scheme provides a promising aim to develop the high-power THz source, which may be used in THz biomedical and material studies, and so on.

Inverse magnetorotational catalysis and the phase diagram of a rotating hot and magnetized quark matter

We study the properties of a hot and magnetized quark matter in a rotating cylinder in the presence of a constant magnetic field. To do this, we solve the corresponding Dirac equation using the Ritus eigenfunction method. This leads to the energy dispersion relation, Ritus eigenfunctions, and the quantization relation for magnetized fermions. To avoid causality-violating effects, we impose a certain global boundary condition, and study its effect, in particular, on the energy eigenmodes and the quantization relations of fermions. Using the fermion propagator arising from this method, we then solve the gap equation at zero and nonzero temperatures. At zero temperature, the dynamical mass $\bar{m}$ does not depend on the angular frequency, as expected. We thus study its dependence on the distance $r$ relative to the axis of rotation and the magnetic field $B$, and explore the corresponding finite size effect for various couplings $G$. We then consider the finite temperature case. The dependence of $\bar{m}$ on the temperature $T$, magnetic field $B$, angular frequency $\Omega$, and distance $r$ for various $G$ is studied. We show that $\bar{m}$ decreases, in general, with $B$ and $\Omega$. This is the ''inverse magneto-rotational catalysis (IMRC)'' or the ''rotational magnetic inhibition'', previously discussed in the literature. To explore the evidence of this effect in the phase diagrams of our model, we examine the phase portraits of the critical temperature $T_c$ as well as the critical angular frequency $\Omega_c$ with respect to $G, B,\Omega$, and $r$ as well as $G, B, T$, and $r$, respectively. We show that $T_{c}$ and $\Omega_c$ decrease, in particular, with $B$. This is interpreted as clear evidence for IMRC.

Inclusion of a pitching mid-wall for a dual-chamber oscillating water column wave energy converter device

The concept of a dual-chamber oscillating water column (OWC) device consisting of a pitching mid-wall (restrained by an angle spring stiffness) is proposed and the corresponding theoretical model is established under the framework of potential flow theory. By employing the matched eigenfunction method along the common interfaces in terms of the velocity and pressure continuity, the present model can be numerically solved. A numerical strategy of successive approximation is utilized to determine the optimal turbine parameters for the optimal energy extraction. Some potential influential factors, including the linear density, installed location and draft of the mid-wall, chamber breadth and angle spring stiffness, are explored to analyze their impacts on the power extraction performance. It is found that the mid-wall in the dual-chamber structure with a relatively larger linear density and smaller draft is more beneficial for energy extraction. Under the condition that the dual-chamber structure is designed with a relatively wider size, an asymmetry configuration with a narrower front but a wider rear chamber is more recommended. In addition, a relatively smaller angle spring stiffness is advisable due to the occurrence of more satisfactory peak efficiency.

Wave power extraction for an oscillating water column device consisting of a surging front and back lip-wall: An analytical study

Traditional oscillating water column (OWC) devices are mostly fixed with almost no components allowed to move. To complement the research of OWC consisting of moving components, the capability in extracting wave energy for a device with a surging front and back lip-wall restricted by different spring stiffness values is investigated. A theoretical model is developed based on potential flow theory and solved by the matched eigenfunction method, and the pros and cons between the proposed model and the traditional fixed model are compared. The influential factors, including the wall drafts and chamber breaths, are explored to study their effects on the interested hydrodynamic parameters. To make full use of the inclusion of surging motion of lip-walls, a progressive optimized process for the combination of spring stiffness values is proposed to obtain a superior efficiency curve. The results show that a relatively larger lip-wall draft and wider chamber for the dual-surging-motion model is more beneficial for wave energy absorption compared with the stationary model. In addition, an appropriate combination of the employed two spring stiffness values can expand the high-efficiency frequency bandwidth, with the peak efficiency improved to a more preferable extent in contrast to the maximum value 0.5 for a traditional fixed device. Moreover, through tuning the adopted spring stiffness, the performance enhancement can be attained by triggering multiple peak values within the efficiency curves for the mdoel allowed dual surging motions.

On the significance of three-dimensional imaging for interpreting vocal fold dynamics

Physical mechanisms of regular and irregular vocal fold vibration were first studied in a finite element model of vocal fold vibration using the method of empirical eigenfunctions. Later the same method was used to successfully study physical mechanisms of vocal fold vibration in laboratory hemilarynx experiments (in which the medial surface of vocal fold was imaged). In subsequent clinical studies, however, the method exhibited significantly less interpretive power, presumably because of the limitations imposed by imaging from a 2D superior view. Our hypothesis is that the interpretive power of the method of empirical eigenfunctions will be significantly enhanced in clinical settings if 3D imaging is used instead of 2D imaging. To test this hypothesis, the method of empirical eigenfunctions was employed on the same finite element model implemented previously, as dynamical information was systematically removed from the analysis (interior tissue dynamics, medial surface dynamics, vertical dynamics of the superior surface, etc.). The natural modes of vibration were re-computed at each stage, and the deformations from the original natural modes were observed. The results suggest that in clinical settings, the method of empirical eigenfunctions retains significant interpretive power regarding physical mechanisms of irregular vibration when 3D imaging is used.

Extraordinary optical characteristics of one-dimensional double anti-PT-symmetric ring optical waveguide networks

In this paper, a unique one-dimensional double anti-PT-symmetric ring optical waveguide network (1D DAPTSROWN) is designed whose refractive index satisfies n(x)=−n∗(−x). The extraordinary optical characteristics of the network are studied by means of the generalized Floquet–Bloch theorem, three-material network equation and generalized eigenfunction method. Due to the physical difference between the anti-PT-symmetric and PT-symmetric optical waveguide networks, the previous method of the PT-symmetric optical waveguide network is not applicable to the anti-PT-symmetric optical waveguide network designed in this paper. Thus, based on the previous method and the network equation, we propose a connection function to identify the extremum mode points in the anti-PT-symmetric optical waveguide network. At the extremum mode point, the optical waveguide network can produce extraordinary ultra-strong transmission, ultra-strong reflection, and ultra-strong photonic localizations. The corresponding calculated values in this paper arrives at 1028, which is much larger than 1012 (the best result reported previously). This shows that the anti-PT-symmetric network may have potential in applications in designing optical filters, optical amplifiers and so on.

On Describing the Radiation and Diffraction of Electromagnetic Waves by the Eigenfunction Method

On Describing the Radiation and Diffraction of Electromagnetic Waves by the Eigenfunction Method

Semi-analytical study on the wave power extraction of a bottom-seated oscillating water column device with a pitching front lip-wall

A bottom-seated oscillating water column (OWC) device with a front lip-wall which is allowed to oscillate in a pitching mode is explored under the framework of potential flow theory. By utilizing the matched eigenfunction method and employing the continuity conditions along the adjacent common interfaces, the theoretical model can be transformed into a set of linear equations. Some potential influential factors, including the linear density (ρs) and draft (d) of the front lip-wall, the chamber breadth (b) and the imposed angle spring stiffness (K), are examined to study their effects on the wave power extraction performance. The results show that the inclusion of a pitching front lip-wall can effectively expand the frequency bandwidth of high-efficiency, especially under the geometry condition of a relatively larger front lip-wall draft and larger chamber breath. In addition, the examination of different spring stiffness K demonstrates that it is more beneficial for the energy extraction when K is designed as a relatively lower value or even set to 0.