A Wiener–Hopf theory for a semi-infinite dielectric slab

1990 ◽  
Vol 68 (11) ◽  
pp. 1348-1351
Author(s):  
G. Mitsioulis
Keyword(s):  
Fourier Transform ◽  
Wave Equation ◽  
Fourier Series ◽  
Boundary Conditions ◽  
Series Expansion ◽  
Field Distribution ◽  
Wave Number ◽  
Fourier Series Expansion ◽  
System Of Equations ◽  
Dielectric Slab

We investigate the field distribution around a semi-infinite dielectric slab. The wave equation is transformed, through a Fourier transform, to the wave number domain. The boundary conditions are imposed and we end up with an equation of the Wiener–Hopf type. Certain functions are factorized and we find a system of equations for the unknown coefficients of the Fourier series expansion for the field at the mouth of the slab.

scholarly journals GREEN'S FUNCTION OF A FINITE CHAIN AND THE DISCRETE FOURIER TRANSFORM

2006 ◽  
Vol 20 (05) ◽  
pp. 593-605 ◽  
Author(s):  
SERGIU COJOCARU
Keyword(s):  
Fourier Transform ◽  
Fourier Series ◽  
Green’S Function ◽  
Discrete Fourier Transform ◽  
Series Expansion ◽  
Green's Function ◽  
Basis Set ◽  
Fourier Series Expansion ◽  
Open Boundary ◽  
Open Boundary Conditions

Unlike the Fourier series expansion, the discrete Fourier transform is defined on a finite basis set of harmonic functions. The first approach is widely used in condensed matter to describe the thermodynamic limit of various lattice models, while the latter did not receive sufficient development that would allow to address finite lattices. In the present paper a general expression for the Green's function of a finite one-dimensional lattice with nearest neighbor interaction is derived for the first time via discrete Fourier transform. Solution of the Heisenberg spin chain with periodic and open boundary conditions is considered as an example. Although the final expressions are completely equivalent to Bethe ansatz, the examples allow us to clarify the differences between the two approaches. On the other hand, it is explained why the well known results obtained by Fourier series expansion were incomplete and thus provides a deeper understanding of the approach.

An Exact Fourier Series Method for the Vibration Analysis of Multispan Beam Systems

2009 ◽  
Vol 4 (2) ◽  
Author(s):  
Wen L. Li ◽  
Hongan Xu
Keyword(s):  
Fourier Series ◽  
Boundary Conditions ◽  
Series Expansion ◽  
Vibration Analysis ◽  
Series Solution ◽  
Series Representation ◽  
Displacement Function ◽  
Fourier Series Expansion ◽  
Series Method ◽  
Fourier Series Method

An exact Fourier series method is developed for the vibration analysis of multispan beam systems. In this method, the displacement on each beam is expressed as a Fourier series expansion plus an auxiliary closed-form function such as polynomials. The auxiliary function is used to deal with all the possible discontinuities, at the end points, with the original displacement function and its derivatives when they are periodically extended over the entire x-axis as implied by a Fourier series representation. As a result, not only is it always possible to expand the beam displacements into Fourier series under any boundary conditions, but also the series solution will be substantially improved in terms of its accuracy and convergence. Mathematically, the current Fourier series expansion represents an exact solution to a class of beam problems in the sense that both the governing equations and the boundary/coupling conditions are simultaneously satisfied to any specified degree of accuracy. In the multispan beam system model, any two adjacent beams are generally connected together via a pair of linear and rotational springs, allowing a better modeling of many real-world joints. Each beam in the system can also be independently and elastically restrained at its ends so that all boundary conditions including the classical homogeneous boundary conditions at the end and intermediate supports can be universally dealt with by simply varying the stiffnesses of the restraining springs accordingly, which does not involve any modification of basis functions, formulations, or solution procedures. The excellent accuracy and convergence of this series solution is demonstrated through numerical examples.

An Improved Fourier–Ritz Method for Analyzing In-Plane Free Vibration of Sectorial Plates

2017 ◽  
Vol 84 (9) ◽  
Author(s):  
Siyuan Bao ◽  
Shuodao Wang ◽  
Bo Wang
Keyword(s):  
Fourier Series ◽  
Boundary Conditions ◽  
Ritz Method ◽  
Displacement Function ◽  
Accurate Solution ◽  
Previous Method ◽  
Fourier Series Expansion ◽  
Arbitrary Boundary ◽  
Fourier Cosine ◽  
Radial Variable

A modified Fourier–Ritz approach is developed in this study to analyze the free in-plane vibration of orthotropic annular sector plates with general boundary conditions. In this approach, two auxiliary sine functions are added to the standard Fourier cosine series to obtain a robust function set. The introduction of a logarithmic radial variable simplifies the expressions of total energy and the Lagrangian function. The improved Fourier expansion based on the new variable eliminates all the potential discontinuities of the original displacement function and its derivatives in the entire domain and effectively improves the convergence of the results. The radial and circumferential displacements are formulated with the modified Fourier series expansion, and the arbitrary boundary conditions are simulated by the artificial boundary spring technique. The number of terms in the truncated Fourier series and the appropriate value of the boundary spring retraining stiffness are discussed. The developed Ritz procedure is used to obtain accurate solution with adequately smooth displacement field in the entire solution domain. Numerical examples involving plates with various boundary conditions demonstrate the robustness, precision, and versatility of this method. The method developed here is found to be computationally economic compared with the previous method that does not adopt the logarithmic radial variable.

A SIMPLE MODEL FOR ORIENTATIONAL ORDERING OF A SILVER MONOLAYER ELECTRODEPOSITED ON A C(0001) SURFACE

1995 ◽  
Vol 02 (04) ◽  
pp. 489-494 ◽  
Author(s):  
E.E. MOLA ◽  
A.G. APPIGNANESSI ◽  
J.L. VICENTE ◽  
L. VAZQUEZ ◽  
R.C. SALVAREZZA ◽  
...  
Keyword(s):  
Experimental Data ◽  
Fourier Series ◽  
Simple Model ◽  
Series Expansion ◽  
Unit Cell ◽  
Fourier Series Expansion ◽  
Crystal Formation ◽  
Substrate Potential ◽  
Weber Type ◽  
Orientational Ordering

The model for the angular orientational energy (AOE) has been extended to hexagonal submonolayer domains of Ag electrodeposited at a constant overpotential on a C(0001) surface. These domains which are characterized by an epitaxy angle θ=15±5° and an Ag−Ag distance d Ag−Ag =0.330± 0.016 nm, can be considered as precursors of 3D Ag crystal formation, according to a Volmer-Weber type mechanism. Calculations are based upon a simple Hamiltonian evaluated by introducing the concept of the commensurable unit cell. A Fourier series expansion for the substrate potential was used. Results from the model predict the existence of a commensurable cell in agreement with the experimental data derived from STM imaging.

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