Infinite Set Of Linear Equations Research Articles

Impact of Radon transform on plane wave diffraction by an impedance step geometry

This work presents a Radon transform analysis of plane wave diffraction by an impedance step, such that the planes and the step have different impedances. Application of Radon transform on diffraction analysis is of vital importance in many real-time applications i.e., medical imaging, geosciences, astronomy, optics, stress analysis etc. Radon transform offers the advantage of introducing freedom in the usage of observer angle, during reconstruction of signal using plane wave diffraction. Radon transform along with Wiener–Hopf technique and asymptotic method of integration are used for computing the far field of the diffracted plane wave. The solution of plane wave diffraction by an impedance step consists of an infinite set of linear equations, which is obtained with the help of a software. Sinograms showing the effect of varying step height and impedance on the scattered field are plotted, compared and discussed. The mathematical analysis and graphical discussion show that Radon transform enables the observer angle to be varied between 0 to 2π. The results are comparable to previous works on plane wave diffraction by an impedance step, when the observer angle is fixed.

ELECTROMAGNETIC WAVE DIFFRACTION PROBLEM BY PERIODICAL GRATING IN PLANE WAVEGUIDE

The problem of an electromagnetic wave diffraction by a periodical grating made of thin conducting screens between two conducting planes is reduced to an infinite set of linear algebraic equations. Three-dimensional case was considered when a field depends on all three spatial coordinates. According to the theorem of unique diffraction problem solution if an original wave is Floquet wave, then the desired solution of the Maxwell equations set can also be only the Floquet wave. We used the expansion of the desired functions along the orthogonal system of functions depending on a transverse coordinate, and the representation in the form of a quasiperiodic function along a longitudinal coordinate. The case when a field does not depend on a transverse coordinate is considered separately. From the boundary conditions and the field conjugation conditions in a grating plane, a pair-wise summatory equation is obtained with respect to the coefficients of the electromagnetic field expansion over an eigenwave of a waveguide. This equation is reduced to an infinite set of linear algebraic equations by the method of integral-summatory identity. An infinite set of linear equations is represented in vector-matrix form: the desired values are two-dimensional vectors, and their coefficients are the square matrices of the second order. Its approximate solution can be obtained by truncation.

Reliable computation of scattering from metallic binary gratings using Fourier-based modal methods

Convergence problems in modal methods for TM polarized fields are often attributed to the inaccuracies in computing the modes of a grating. We report that even in the absence of these inaccuracies convergence problems persist. These arise because of the truncation of the infinite set of linear equations resulting from matching the fields at the grating-substrate and grating-superstrate interfaces with a square matrix. We show that dramatic improvement in convergence can be achieved if the infinite set of linear equations is truncated with a rectangular matrix and by seeking a solution with minimum least squared error.

Piezo Control of Free Vibrations of Damped Beams with Time Delay in the Sensor Feedback

The effect of the time delay between the sensor output and actuation is studied on the displacement and velocity feedback control of a cantilever beam. The system consists of a piezoelectric sensor and actuator pair bonded to an Euler-Bernoulli beam to form an actively controlled laminated structure. The effects of Kelvin-Voigt damping are also included in the problem. The piezoelectric actuators and sensors are taken as patches of partial length which leads to a differential equation formulation with discontinuities. The numerical solution is obtained by replacing the original formulation with an integral equation which is solved by expanding the state function in terms of the eigenfunctions of the freely vibrating beam. This approach leads to an infinite set of linear equations which can be solved with a high degree of accuracy by computing the characteristic equation using a small number of terms. Numerical results are given to analyze the control effectiveness in terms of changes in the natural frequencies for various gains, damping coefficients and time delays.

Time dependence of groundwater pumping from a well near a river

The time-dependent flow of groundwater through an aquifer, generated from rest by pumping from a well close to a river, is calculated without making Dupuit's approximation. The governing equations reduce to the diffusion equation for the pressure head with mixed boundary conditions at the surface of the aquifer and at the base of the river. Using transform techniques, the problem is reduced to an infinite set of linear equations. The steady-state solution provides a guide to the numerical solution of the time-dependent problem. Results are presented for the flux from the river into the aquifer as well as for the flux from the aquifer into the river and out again. Explicit expressions for these fluxes are obtained in the case of rivers that are narrow compared with the aquifer depth. The steady-state flux is then much smaller than the transient flux. Results are significantly different from those obtained using the Dupuit approximation.

Bending of an elastic slab having a spherical cavity

This paper presents the three-dimensional analysis of the stress distribution arising in an isotropic infinite slab with a symmetrically located spherical cavity under a bending moment. To satisfy both boundary conditions on the surface of slab and the surface of the cavity, harmonic functions in rectangular and spherical coordinates are used and double Fourier transform is applied. The problem is reduced to the solution of an infinite set of linear equations which is suitable for numerical solution. Finally some numerical examples are given for various values of the radius of the cavity.

A new method to predict the evolution of the power spectral density for a finite-amplitude sound wave.

A method to predict the effect of nonlinearity on the power spectral density of a plane wave traveling in a thermoviscous fluid is presented. As opposed to time-domain methods, the method presented here is based directly on the power spectral density of the signal, not the signal itself. The Burgers equation is employed for the mathematical description of the combined effects of nonlinearity and dissipation. The Burgers equation is transformed into an infinite set of linear equations that describe the evolution of the joint moments of the signal. A method for solving this system of equations is presented. Only a finite number of equations is appropriately selected and solved by numerical means. For the method to be applied all appropriate joint moments must be known at the source. If the source condition has Gaussian characteristics (it is a Gaussian noise signal or a Gaussian stationary and ergodic stochastic process), then all the joint moments can be computed from the power spectral density of the signal at the source. Numerical results from the presented method are shown to be in good agreement with known analytical solutions in the preshock region for two benchmark cases: (i) sinusoidal source signal and (ii) a Gaussian stochastic process as the source condition.

Solving matrix models using holomorphy

We investigate the relationship between supersymmetric gauge theories with moduli spaces and matrix models. Particular attention is given to situations where the moduli space gets quantum corrected. These corrections are controlled by holomorphy. It is argued that these quantum deformations give rise to non-trivial relations for generalized resolvents that must hold in the associated matrix model. These relations allow to solve a sector of the associated matrix model in a similar way to a one-matrix model, by studying a curve that encodes the generalized resolvents. At the level of loop equations for the matrix model, the situations with a moduli space can sometimes be considered as a degeneration of an infinite set of linear equations, and the quantum moduli space encodes the consistency conditions for these equations to have a solution.

In-phase and out-of-phase break-up of two immersed liquid threads under influence of surface tension

The dynamical behaviour of two infinitely long adjacent parallel liquid threads immersed in a fluid is considered under influence of small initial perturbations. Assuming all fluids to behave Newtonian, we used the creeping flow approximation, which resulted in Stokes equations. Applying cylindrical coordinates and separation of variables, and writing the dependence on the azimuthal direction in the form of a Fourier expansion, we obtained general representations of the equations for both the threads and the surrounding fluid. Substitution of these expressions into the boundary conditions leads to an infinite set of linear equations for the unknown coefficients. Its solutions for the lowest two orders of the Fourier expansion, the so-called zero- and first-order solutions, are presented. Much attention is paid to the (in)stability of the configuration, in terms of the so-called growth rate of the disturbance amplitudes. The growth rate of these amplitudes determines the behaviour of the break-up process of the threads. It turns out that this breaking up occurs either in-phase or out-of-phase. This depends on the viscosity ratio of the fluids and on the distance between the threads. These findings agree with experimental observations. The results of the present work also show that the zero-order solution yields the qualitatively correct insight in the break-up process. The extension to a one order higher expansion only leads to relatively small quantitative corrections.

Piston flow in a two-dimensional channel

A solution using biorthogonal eigenfunctions is presented for viscous flow caused by a piston in a two-dimensional channel. The resulting infinite set of linear equations is solved using Spence’s optimal weighting function method [IMA J. Appl. Math. 30, 107 (1983)]. The solution is compared to that with a shear-free piston surface; in the latter configuration the fluid more rapidly approaches the Poiseuille flow profile established away from the face of the piston.

Numerical modeling of the optical response of supported metallic particles

The present work reports a general method for the calculation of t he polarizability of a truncated sphere on a substrate. A multipole ex pansion is used, where the multipoles are not necessarily localized in the center of the sphere but can freely move on the revolution axis. From the weak formulation of the boundary conditions, an infinite set of linear equations for the multipole coefficients is derived. To obta in this set, the interaction between the island and the substrate is t aken into account by the technique of image multipoles. For numerical implementation, this set is truncated at an arbitrary mu ltipole order. The accuracy of the method is jugded through the stabil ity of the truncated sphere polarizability and the the fulfilment of t he boundary conditions which are demonstrated to be satisfied in large regions of the parameter space. This method brings an improvement wit h respect to the Bedeaux's case \cite{Wind87a,Wind87b} where the multi poles are located in the center of the sphere.

Tunnelling splitting and relaxation in the multimode Jahn-Teller system

The complex problem of the multimode Jahn-Teller effect for an impurity ion at a cubic site is studied in terms of tunnelling of oscillator states localized in trigonal minima on the lowest sheet of the adiabatic potential energy surface. The low-symmetry trigonal wells are interpreted in terms of the Jahn-Teller polaron. Its similarity to and difference from the traditional polaron in ionic crystals is discussed. The harmonic approximation at the bottom of the wells is used in the infinite-coupling limit. Phonon Green functions are introduced in order to solve the infinite set of linear equations for the corresponding Fourier transforms. From the solutions, an expression is derived for the tunnelling splitting energy gap. It is also shown that the results obtained coincide with the exact calculation reported in the literature in the particular case of no phonon dispersion (when all the frequencies coincide). The effect of relaxation broadening of the tunnelling splitting energy levels is also discussed together with an analysis of the relation between coherent and incoherent tunnelling. It is shown that the tunnelling splitting is not smoothed by the relaxation broadening and therefore is observable if the Jahn-Teller stabilization energy is bigger than , where is the Debye frequency of the vibrations of the crystal lattice. Relaxation effects for a Jahn-Teller impurity ion are compared with the so-called ohmic-dissipative effect for tunnelling in a two-level system.

Reflection coefficients for scattering from a pressure-release, sinusoidal surface

This paper deals with the scattering of an incident plane wave from a pressure-release, sinusoidal rough surface. As was shown by Uretsky [Ann. Phys. 33, 400–427 (1965)] this problem is equivalent to solving an infinite set of linear equations (involving the matrix V0m,n) for the quantities ψm, the Fourier components (apart from a phase factor) of the normal derivative of the pressure on the surface. Holford showed that the ψm satisfied an infinite set of linear equations of the ‘‘second kind’’ (involving the matrix V1m,n) and deduced the existence and uniqueness of solutions. This paper shows that the matrix elements V0m,n and V1m,n can be expressed as a single sum over products of integer index Bessel functions, subject only to the slope restriction Kd<0.6627. These series representations of the matrices V0m,n and V1m,n are used to prove the result that Rayleigh’s equations are exact up to the slope limit Kd<0.6627. It is also shown that the reflection coefficients Rn satisfy an infinite set of linear equations of the second kind, valid provided the maximum slope Kd<0.6627. The matrix elements involved in the equations for the reflection coefficients Rn are proportional to a single integer Bessel function. Consequently, these equations may be easily solved by the method of reduction.

Polling systems with station breakdowns

None of the works in the literature on Polling Systems deals with the important phenomenon of failing nodes. This paper addresses this issue. We assume that there are two modes of station breakdowns: node-dependent and general. Each node is subject to a random failure process and a failure of a node during a visit of the server causes the latter to move to the next station. We carry performance analyses of both the Gated and the Exhaustive service regimes and derive probability generating functions, Laplace transforms and means of variables such as queue sizes at polling and at arbitrary instants, durations of busy periods, number of jobs left behind at server's station-departure moments, and cycle times. The analysis of the Exhaustive case is achieved via an explicit solution of an infinite set of linear equations, where the unknowns are the state-dependent joint transforms of two key characteristics: the number of arrivals to a station during a busy period starting with a given number of jobs, and the number of jobs successfully served during that period. Stability conditions are indicated for each regime.

Polling systems with station breakdowns

Abstract None of the works in the literature on Polling Systems deals with the important phenomenon of failing nodes. This paper addresses this issue. We assume that there are two modes of station breakdowns: node-dependent and general. Each node is subject to a random failure process and a failure of a node during a visit of the server causes the latter to move to the next station. We carry performance analyses of both the Gated and the Exhaustive service regimes and derive probability generating functions, Laplace transforms and means of variables such as queue sizes at polling and at arbitrary instants, durations of busy periods, number of jobs left behind at server's station-departure moments, and cycle times. The analysis of the Exhaustive case is achieved via an explicit solution of an infinite set of linear equations, where the unknowns are the state-dependent joint transforms of two key characteristics: the number of arrivals to a station during a busy period starting with a given number of jobs, and the number of jobs successfully served during that period. Stability conditions are indicated for each regime.

Scattering of TM waves from a cylindrical scatterer buried inside a two-layer lossy ground with sinusoidal surface

Scattering of TM waves from a cylindrical scatterer embedded inside a two-layer lossy ground under the assumption of a sinusoidal air-ground interface is investigated. The problem is formulated via an integral equation approach combined with the extended boundary conditions method. The integral equation over the cross section of the scatterer is transformed into an infinite set of linear equations for the expansion coefficients of the unknown magnetic field inside the scatterer. This set is truncated and solved numerically. The numerical results obtained for the far-zone scattered field show that it is is mainly affected by the characteristics and the position of the scatterer with respect to the dips and lifts of the rough surface as well as by the conductivity of the surrounding ground layers. In certain cases the scattering pattern provides useful information about the position and the type of possible buried scatterers.

Modenanalyse des Stegrundhohlleiters

A modal analysis of a ridged circular waveguide is presented. The method of Kuhn [1] is used to find suitable expressions for the vectorpotentials, wherefore the ridged waveguide is intersected into a lot of regions. With the mode-matching method an infinite set of linear equations is derived. This set of equations is truncated and the behaviour of the eigenvalues is shown.

The viscous flow of charged particles through a charged cylindrical tube

An analysis is given for the electro-kinetic transport properties in a system consisting of a line of identical spheres placed equidistantly with their centres on the axis of a cylindrical tube containing a viscous fluid. Both the spheres and the wall of the tube are charged and a two-species symmetrical electrolyte with valence Z is present in the system. As a result of the charges on the surface of the spheres and on the surface of the tube electrical double layers will develop. When an electrical field is applied to the system an electrokinetic motion is induced. We will use the thin double layer theory (Dukhin & Derjaguin 1974; O'Brien 1983), valid for sufficiently high electrolyte concentration and where the polarization of the electrical double layer is included. Using a multipole expansion an infinite set of linear equations for the multipoles will be derived from which the electro-kinetic transport coefficients may be determined. These coefficients depend on the system parameters, such as the radius of the tube R, the radius of the sphere a, the separation between the spheres d, the Debije radius κ-1, the zeta-potentials of the spheres ζp and of the wall of the tube ζw and the valency Z of the electrolyte. From these coefficients a relation is found between the pressure drop Δp per unit length and the drag force D on the spheres on one side and with the velocity U of the spheres, the total discharge Q and the applied electrical field E0 on the other side. For some values for the system parameters we have numerically solved the infinite set of linear equations by truncation and calculated the transport coefficients. We have also calculated the streamlines for some situations. The plots of these streamlines show that depending on the conditions on the system vortices may appear.

Optical response of a thin film with arbitrary deterministic roughness of the interfaces.

We present a theoretical study of the optical response of a thin film with arbitrary, deterministic roughness of the interfaces (in one dimension). The three layers of the film are characterized each by a spatially nondispersive dielectric constant and magnetic permeability. The incident light may have TE(s) or TM(p) polarization. Using the Rayleigh hypothesis, we derive an integral matrix equation which relates the reflected fields, the transmitted fields, and the fields inside the thin film to the incident wave. This equation is applied to the special case of periodic corrugation, leading to a solution in terms of an infinite set of linear equations for the amplitudes of the diffracted partial waves in the three media. For aperiodic roughness the numerical solution is still given by a similar set of equations. For periodic (aperiodic) films the solution involves Fourier coefficients (Fourier integrals) of functions related to the roughness profiles. We also derive the secular equations for the polariton eigenmodes of periodic and aperiodic films.

Stress analysis of a stretched slab having a spherical cavity under unidirectional tension

This paper presents the asymmetric analysis of the stress distribution arising in an isotropic infinite slab with a symmetrically located spherical cavity under a unidirectional uniform tension. To satisfy both boundary conditions on the surface of the slab and the surface of the cavity, harmonic functions in rectangular and spherical coordinates are used and the double Fourier transform is applied. The problem is reduced to the solution of infinite sets of linear equations which are suitable for numerical solution by an iterative technique. Finally some numerical examples are given for various values of the radius of the cavity.