Solution of a matrix Wiener–Hopf equation connected with the plane wave diffraction by an impedance loaded parallel plate waveguide

A matrix Wiener–Hopf equation connected with a new canonical diffraction problem is solved explicitly. We consider the diffraction of a plane electromagnetic wave by an impedance loaded parallel plate waveguide formed by a two‐part impedance plane and a parallel perfectly conducting half‐plane. The representation of the solution to the boundary‐value problem in terms of Fourier integrals leads to a matrix Wiener–Hopf equation. The exact solution is obtained in terms of two infinite sets of unknown coefficients satisfying two infinite systems of linear algebraic equations. These systems are solved numerically and the influence of the parameters such as the waveguide spacing and the surface impedances of the two‐part plane on the diffraction phenomenon is shown graphically. Copyright © 2005 John Wiley & Sons, Ltd.     

Wave scattering by soft–hard three spaced waveguide

In this work we consider the sound radiation of a fundamental plane wave mode from a semi-infinite soft–hard duct. This duct is symmetrically located inside an infinite duct. This infinite waveguide consist of soft and hard plates. The whole system constitutes a three spaced waveguide. A closed form solution is obtained by using eigenfunction expansion matching method. This particular problem has been solved previously by Rawlins in closed form but without numerical work. Here the numerical results for reflection coefficient are given when the lowest mode propagates out from the semi-infinite duct. At the end we give comparison to both methods.     

On the connection of the superposition and homogeneous-solutions methods in wave propagation problems in a semi-infinite rigidly-clamped elastic layer

The process of harmonic wave propagation is investigated in a semi-infinite rigidly-clamped elastic layer. An analytic solution of the problem is obtained by the superposition method. The wave field expansion in the form of a normal mode series for a corresponding infinite waveguide is established. According to residue theory, the explicit form of the expansion coefficients is established with physical requirements of the radiation conditions taken into account.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 19, pp. 3–10, 1988.     

Nonlinear Hyperbolic Equations in Infinite Homogeneous Waveguides

ABSTRACT

In this paper we prove global and almost global existence theorems for nonlinear wave equations with quadratic nonlinearities in infinite homogeneous waveguides. We can handle both the case of Dirichlet boundary conditions and Neumann boundary conditions. In the case of Neumann boundary conditions we need to assume a natural nonlinear Neumann condition on the quasilinear terms. The results that we obtain are sharp in terms of the assumptions on the dimensions for the global existence results and in terms of the lifespan for the almost global results. For nonlinear wave equations, in the case where the infinite part of the waveguide has spatial dimension three, the hypotheses in the theorem concern whether or not the Laplacian for the compact base of the waveguide has a zero mode or not.     

A radiation condition arising from the limiting absorption principle for a closed full‐ or half‐waveguide problem

In this paper, we consider the propagation of waves in a closed full or half waveguide where the index of refraction is periodic along the axis of the waveguide. Motivated by the limiting absorption principle, proven in the Appendix by a functional analytic perturbation theorem, we formulate a radiation condition that assures uniqueness of a solution and allows the existence of propagating modes. Our approach is quite different to the known one as, eg, considered recently by Fliss and Joly and allows an extension to open wave guides. After application of the Floquet‐Bloch transform, we consider the Bloch variable α as a parameter in the resulting quasiperiodic boundary value problem and study the behaviour of the solution when α tends to an exceptional value by a singular perturbation result, which goes back to Colton and Kress.     

A simple and robust boundary treatment for the forced Korteweg–de Vries equation

In this paper, we propose a simple and robust numerical method for the forced Korteweg–de Vries (fKdV) equation which models free surface waves of an incompressible and inviscid fluid flow over a bump. The fKdV equation is defined in an infinite domain. However, to solve the equation numerically we must truncate the infinite domain to a bounded domain by introducing an artificial boundary and imposing boundary conditions there. Due to unsuitable artificial boundary conditions, most wave propagation problems have numerical difficulties (e.g., the truncated computational domain must be large enough or the numerical simulation must be terminated before the wave approaches the artificial boundary for the quality of the numerical solution). To solve this boundary problem, we develop an absorbing non-reflecting boundary treatment which uses outward wave velocity. The basic idea of the proposing algorithm is that we first calculate an outward wave velocity from the solutions at the previous and present time steps and then we obtain a solution at the next time step on the artificial boundary by moving the solution at the present time step with the velocity. And then we update solutions at the next time step inside the domain using the calculated solution on the artificial boundary. Numerical experiments with various initial conditions for the KdV and fKdV equations are presented to illustrate the accuracy and efficiency of our method.     

Asymptotic formula for an eigenvalue of the dirichlet problem in a cranked waveguide

The asymptotic behavior of an eigenvalue of the Dirichlet problem for a spectral Helmholtz equation in a two-dimensional cranked acoustic waveguide with yielding walls or in a quantum waveguide is obtained. A waveguide is thought of as a cranked strip, but the boundary value problem is posed in a straight strip of unit width with wedge-shaped notches, with appropriate conjugation conditions on the edges of the notches, which provide for a smooth wave field after the initial form of the waveguide is restored. The bend angles are assumed to be small; i.e., the wedge-shaped notches are supposed to be thin, the asymptotic behavior is built from the corresponding small geometric parameter.     

Logarithmic stability inequality in an inverse source problem for the heat equation on a waveguide

ABSTRACT

We prove logarithmic stability in the parabolic inverse problem of determining the space-varying factor in the source, by a single partial boundary measurement of the solution to the heat equation in an infinite closed waveguide, with homogeneous initial and Dirichlet data.     

Bounded solutions in a T-shaped waveguide and the spectral properties of the Dirichlet ladder

The Dirichlet problem is considered on the junction of thin quantum waveguides (of thickness h ? 1) in the shape of an infinite two-dimensional ladder. Passage to the limit as h → +0 is discussed. It is shown that the asymptotically correct transmission conditions at nodes of the corresponding one-dimensional quantum graph are Dirichlet conditions rather than the conventional Kirchhoff transmission conditions. The result is obtained by analyzing bounded solutions of a problem in the T-shaped waveguide that the boundary layer phenomenon.     

Multiplicity of eigenvalues in certain boundary value problems for the Helmholtz equation

Boundary value problems for two-layer cylindrical waveguides (namely, for a circular two-layer closed waveguide and a dielectric waveguide in an unbounded medium) are considered. It is shown that the same eigenvalues determining the wave numbers of waves in the waveguides can correspond to different solutions of the boundary value problem.     

The difference method in a diffraction problem: The technique of interior boundary condition

The diffraction problem for a plane wave on a half-plane covered by thin layer with an interface is solved by the difference method. The system of difference equations is derived from the variational principle. A boundary solution at infinity must be imposed; this is a radiation condition, which is used in the form of the limit absorption principle. The arising infinite system of difference equations is reduced to a finite part of the boundary (the interface) by using the technique of so-called interior boundary conditions in the sense of Ryaben’kii. The real conditions are found by the Fourier method with respect to one spatial variable in the form of Fourier or Laurent series in the corresponding variable, which converge either inside, outside, or on the unit circle. Above the upper boundary of the layer, all unknowns are eliminated by using the so-called grid Green function, that is, the resolving function for the half-plane satisfying the radiation condition at infinity. For the unknowns on the upper boundary of the layer, an equation in terms of a function of a complex variable of Wiener-Hopf type is obtained, which is solved by factorization. Factorization is performed numerically: the logarithm of the function is expanded in a bi-infinite series, which is replaced by a discrete Fourier series. The closing system in a neighborhood of the interface has order proportional to the number of points on the interface. Solving this system yields all of the required characteristics of the solution.     

Dynamic contact between a spherical inclusion and a matrix upon incidence of an elastic wave

Using the boundary element method, we have studied the dynamic displacements and stresses in an infinite elastic matrix with a spherical elastic inclusion, caused by the propagation of an elastic wave. The original problem has been reduced to a system of boundary integral equations for the contact displacements and tractions on the interface between the inclusion and matrix. Based on the numerical solution of these equations, we have analyzed the influence of the direction of wave propagation and frequency on the important physical parameters, depending on the elastic characteristics of composite constituents.     

A fourth-order Magnus scheme for Helmholtz equation

For wave propagation in a slowly varying waveguide, it is necessary to solve the Helmholtz equation in a domain that is much larger than the typical wavelength. Standard finite difference and finite element methods must resolve the small oscillatory behavior of the wave field and are prohibitively expensive for practical applications. A popular method is to approximate the waveguide by segments that are uniform in the propagation direction and use separation of variables in each segment. For a slowly varying waveguide, it is possible that the length of such a segment is much larger than the typical wavelength. To reduce memory requirements, it is advantageous to reformulate the boundary value problem of the Helmholtz equation as an initial value problem using a pair of operators. Such an operator-marching scheme can also be solved with the piecewise uniform approximation of the waveguide. This is related to the second-order midpoint exponential method for a system of linear ODEs. In this paper, we develop a fourth-order operator-marching scheme for the Helmholtz equation using a fourth-order Magnus method.     

Maxwell's equations in an unbounded structure

This paper is concerned with the mathematical analysis of the electromagnetic wave scattering by an unbounded dielectric medium, which is mounted on a perfectly conducting infinite plane. By introducing a transparent boundary condition on a plane surface confining the medium, the scattering problem is modeled as a boundary value problem of Maxwell's equations. Based on a variational formulation, the problem is shown to have a unique weak solution for a wide class of dielectric permittivity and magnetic permeability by using the generalized Lax–Milgram theorem. Copyright © 2016 John Wiley & Sons, Ltd.     

The Sommerfeld half-plane problem revisited,IV: Variations on a theme of Carlson and Heins

A plane wave is incident upon an infinite set of equally spaced, semi-infinite parallel and staggered plates. The boundary conditions on the plates alternate between the Dirichlet and Neumann ones. This problem is formulated as a pair of coupled Wiener-Hopf integral equations and solved by a method proposed by A. E. Heins in 1950. For the case of specular reflection, that is, a single reflected plane wave, the magnitudes of the reflection coefficient and the transmission coefficients are determined.     

Analysis of the scattering by an unbounded rough surface

This paper is concerned with the mathematical analysis of the solution for the wave propagation from the scattering by an unbounded penetrable rough surface. Throughout, the wavenumber is assumed to have a nonzero imaginary part that accounts for the energy absorption. The scattering problem is modeled as a boundary value problem governed by the Helmholtz equation with transparent boundary conditions proposed on plane surfaces confining the scattering surface. The existence and uniqueness of the weak solution for the model problem are established by using a variational approach. Furthermore, the scattering problem is investigated for the case when the scattering profile is a sufficiently small and smooth deformation of a plane surface. Under this assumption, the problem is equivalently formulated into a set of two‐point boundary value problems in the frequency domain, and the analytical solution, in the form of an infinite series, is deduced by using a boundary perturbation technique combined with the transformed field expansion approach. Copyright © 2012 John Wiley & Sons, Ltd.     

The sommerfeld half-plane problem revisited III: Parallel plate media with mixed boundary conditions

A plane wave is normally incident upon an infinite stack of equally spaced parallel plates which are semi-infinite and not staggered. The plates satisfy the so-called “Rawlins” boundary conditions. This problem is formulated as a pair of simultaneous integral equations of the Wiener-Hopf type and solved by a method proposed by A. E. Heins in 1950.     

Non-spectral Field Representations in Dielectric Fibre Guides

Open waveguiding structures, such as acoustic, piezoelectric,gyrotropic and dielectric waveguides, are used for a varietyof applications. An example of such open waveguides is the dielectricfibre guide used in the construction of optical communicationsystems. An understanding of the modal properties of this guideis therefore of fundamental interest to the understanding ofthe modal properties of similar open structures. We analysein detail the transverse electric (TE) and transverse magnetic(TM) polarized electromagnetic fields within a circularly cylindricaldielectric waveguide with arbitrary non-vanishing analytic radialpermittivity variation. It is shown that the field can be representedas an infinite sum of functions, including the finite numberof spectral (surface wave) and an infinity of non-spectral (complex)eigenfunctions. Additional functions are also included. Boththe field polarization and the behaviour of the permittivityat the waveguide boundary are found to affect the form and thevalidity of the representation.     

一类一维的辐射流体力学方程组激波的存在性

该文主要讨论一维空间中一类辐射流体力学方程组的激波. 由Rankine-Hugoniot条件及熵条件得此问题可表述为关于辐射流体力学方程组带自由边界的初边值问题. 首先通过变量代换, 将其自由边界转换为固定边界, 然后研究关于此非线性方程组的一个初边值问题解的存在唯一性. 为此先构造了此问题的一个近似解, 然后分别通过Picard迭代与Newton迭代对此非线性问题构造近似解序列. 通过一系列估计与紧性理论得到此近似解序列的收敛性, 其极限即为原辐射热力学方程组的一个激波.     

Diffraction of sound waves by a rigid cylindrical cavity of finite length with an internal impedance surface

A rigorous solution is presented for the problem of diffraction of plane harmonic sound waves by a cavity formed by a terminated rigid cylindrical waveguide of finite length whose interior surface is lined by an acoustically absorbent material. The solution is obtained by a modification of the Wiener-Hopf technique and involve an infinite series of unknowns, which are determined from an infinite system of linear algebraic equations. Numerical solution of this system is obtained for various values of the parameters of the problem and their effects on the diffraction phenomenon are shown graphically.