Boundary Value Problem Of Diffraction Research Articles

Solution of inverse non-stationary boundary value problems of diffraction of plane pressure wave on convex surfaces based on analytical solution

Research in the field of unsteady interaction of shock waves propagating in continuous media with various deformable barriers are of considerable scientific interest, since so far there are only a few scientific works dealing with solving problems of this class only for the simplest special cases. In this work, on the basis of analytical solution, we study the inverse non-stationary boundary-value problem of diffraction of plain pressure wave on convex surface in form of parabolic cylinder immersed in liquid and exposed to plane acoustic pressure wave. The purpose of the work is to construct approximate models for the interaction of an acoustic wave in an ideal fluid with an undeformable obstacle, which may allow obtaining fundamental solutions in a closed form, formulating initial-boundary value problems of the motion of elastic shells taking into account the influence of external environment in form of integral relationships based on the constructed fundamental solutions, and developing methods for their solutions. The inverse boundary problem for determining the pressure jump (amplitude pressure) was also solved. In the inverse problem, the amplitude pressure is determined from the measured pressure in reflected and incident waves on the surface of the body using the least squares method. The experimental technique described in this work can be used to study diffraction by complex obstacles. Such measurements can be beneficial, for example, for monitoring the results of numerical simulations.

Axially Symmetric Compact Range Reflectors: Application of the Analytic Regularization Method

An axially symmetric compact range reflector with a blended rolled edge was analyzed and optimized in a rigorous formulation of the diffraction problem. The corresponding boundary-value diffraction problem is solved with the analytic regularization method, which reduces the problem to an operator equation of the second kind, thus guaranteeing a numerically stable and effective solution. The distribution of the surface density and the fields at the aperture and in the near-field zone were obtained and analyzed for different types of the reflector-edge curvature. In addition, a “blending function” was used that esures an infinitely smooth contour across the junction between the paraboloid part of the reflector and its rolled edge. The procedure for determining the optimal edge is carried out in the rigorous formulation of the diffraction problem by minimizing the deviation from a plane wave.

Interaction of terahertz electromagnetic waves with periodic gratings of graphene micro- and nanoribbons

An original mathematical model of the interaction of terahertz (THz) electromagnetic waves with periodic gratings of graphene micro- and nanoribbons is based on the solution to the boundary-value problem of diffraction for the Maxwell equations with electrodynamic boundary conditions and material equations. The electrodynamic calculations of the transmission coefficients of the TEM wave versus frequency are performed for the 2D grating of graphene micro- and nanoribbons at several chemical potentials, grating periods, and geometrical sizes of ribbons. The results of the calculations show that the transmission spectrum exhibits a minimum in the THz range if the electric field of the wave is perpendicular to the graphene ribbons. The minimum is due to the plasmon resonance of the fundamental mode in graphene, and the absorption peaks at higher frequencies in the upper part of the THz range are related to the highorder plasmon modes.

Mixed boundary value problems of diffraction by a half‐plane with an obstacle perpendicular to the boundary

AbstractThe paper is devoted to the analysis of wave diffraction problems modeled by classes of mixed boundary conditions and the Helmholtz equation, within a half‐plane with a crack. Potential theory together with Fredholm theory, and explicit operator relations, are conveniently implemented to perform the analysis of the problems. In particular, an interplay between Wiener–Hopf plus/minus Hankel operators and Wiener–Hopf operators assumes a relevant preponderance in the final results. As main conclusions, this study reveals conditions for the well‐posedness of the corresponding boundary value problems in certain Sobolev spaces and equivalent reduction to systems of Wiener–Hopf equations. Copyright © 2013 John Wiley & Sons, Ltd.

A subhierarchic method for the solution of a pseudodifferential equation in the problem of diffraction in layers coupled through an aperture

The boundary-value problem of diffraction for the system of Maxwell equations in layers coupled through an aperture is investigated. The layers are formed by three infinitely thin perfectly conducting planes. The method of the Green’s functions is applied to reduce the boundary-value problem to a pseudodifferential equation on the aperture. The numerical results obtained with the use of a subhierarchic method are presented.

The method of autonomous blocks with magnetic nanoinclusions and floquet channels applied for simulation of magnetic nanostructures with allowance for the exchange and boundary conditions

A decomposition method is developed for simulation of microwave- and infrared-band devices containing magnetic nanomaterials and magnetophotonic crystals. The method is based on autonomous blocks that are rectangular parallelepipeds with magnetic nanoinclusions and virtual Floquet channels (referred below as magnetic Floquet autonomous blocks (MFABs)) on the faces. A computational algorithm of solution of a nonlinear diffraction boundary value problem by means of the Galerkin method is developed for constructing MFAB descriptors. The coefficient of TEM wave transmission through a lattice of ferrite nanospheres is calculated. The threshold values of the pumping-wave amplitude such that exchange spin waves are parametrically excited in ferrite-nanosphere lattices are determined from bifurcation points of the nonlinear Maxwell operator. This excitation is observed when distances between magnetic nanoparticles are reduced to the exchange-interaction length.

SURFACE RESONANCES OF METAL STRIPE GRATING ON THE PLANE BOUNDARY OF METAMATERIAL

The paper is devoted to the study of the interaction of the electromagnetic waves with the structure composed of perfectly conducting strip grating, situated on the plane boundary of metamaterial with effective permittivity, depending on the frequency of the wave of excitation. The rigorous solution to the relevant diffraction boundary value problem is developed. The extensive numerical experiments, performed with a help of corresponding algorithm constructed, allowed to establish several regularities in the complicated process of interaction of electromagnetic waves with grating on dispersive metamaterial. The efficient association of analytical and numerical study has provided the understanding of the nature of resonant phenomena appearing in this process.

Hydrodynamic properties of two vertical truncated cylinders in waves

The radiation and diffraction boundary value problem arising from the interaction of linear water waves with two vertical truncated cylinders is investigated by use of the method of separation of variables and the method of matched eigenfunction expansion. Analytical expressions for the radiated and diffracted potentials are obtained as infinite series of orthogonal functions. The unknown coefficients in the obtained expressions are determined by use of the matched eigenfunction expansion method. To verify the obtained expressions, the Green's second identity and the symmetry of the matrices for the added masses and damping coefficients are used. The results show that the analytical expressions presented in this paper are correct. By use of the present analytical solution, the hydrodynamic coefficients and wave forces for some specific cases are calculated and the hydrodynamic effects of the cylinders' radii on the hydrodynamic properties of the cylinders are investigated which will supply some useful information for the design of this kind of system.

On the radiation and diffraction of water waves by a rectangular structure with a sidewall

A radiation and diffraction boundary value problem is investigated. It arises from the interaction of linear water waves with a freely floating rectangular structure in a semi-infinite fluid domain of finite water depth with the leeward boundary being a vertical wall. Analytical expressions for the radiated potentials and the diffracted potential are obtained by use of the method of separation of variables and the eigenfunction expansion method. The added masses and damping coefficients for the structure heaving, swaying and rolling in calm water are obtained by use of the corresponding radiated potentials and the wave excitation forces are calculated by use of the diffracted potential. To verify the correctness of the method, a boundary element method is used. A comparison of the analytical results with those obtained by the boundary element method is made and good agreement is achieved, which shows that the analytical expressions for the radiated and diffracted potentials are correct. By use of the present analytical solution, the added mass, damping coefficients, wave excitation force, together with the hydrodynamic effects of the draft, width of the structure and the clearance between the structure and the sidewall are also investigated.

The Regularization of Boundary Value Diffraction Problem of an Arbitrary Scalar Wave on a Spherical Segment. Dirichlet Problem

The Regularization of Boundary Value Diffraction Problem of an Arbitrary Scalar Wave on a Spherical Segment. Dirichlet Problem

The vector-valued Wiener-Hopf equation for a mixed boundary-value problem of diffraction at a wedge intersected by a half-plane

For a mixed boundary-value problem in a nonregular region we obtain the vector-valued Wiener-Hopf equation, which is then reduced to infinite systems of linear algebraic equations using the factorization method and Liouville's theorem. It then becomes possible to solve the equation with prescribed precision for arbitrary values of the parameters of the problem. In the stationary case the solution is obtained in closed form.

Diffraction of elastic P waves by four equal coaxial circular rigid strips

The two-dimensional boundary value problem of diffraction of obliquely incident low-frequency elastic P waves by four equal coaxial circular rigid strips embedded in an infinite isotropic and homogeneous elastic medium is solved. Approximate expressions are derived for the far-field amplitudes and scattering cross section when the wavelengths are large compared to the radius of the circular strips.

Theory of diffraction of weak shock waves

A numerical solution is considered to the universal nonlinear boundary-value diffraction problem which occurs in various problems of weak interaction [1, 2] in the asymptotic analysis of the flow in a region with large gradients of the parameters near the point of intersection of the incident, diffracted, and reflected waves. The analytical solutions to this type of problem usually approximately satisfy the conditions on the diffracted front, the position of which is not known beforehand, but is found along with the solution. In the present paper, the problem is solved by the numerical method of [3], which reduces the initial boundary-value problem for the system of short-wave equations with an unknown boundary to the solution of a series of boundary-value problems with a fixed boundary. The problem of the diffraction of a weak shock wave on a wedge with a finite apex angle is considered as an application of the solution. The data calculated by the asymptotic theory agree significantly better with the experimental data [5] than the theoretical data of [4].

The scattering of regular surface waves by a fixed, half-immersed, circular cylinder

A train of regular surface waves is incident upon a fixed, half-immersed, circular cylinder; the waves are partially reflected and partially transmitted, and also induce hydrodynamic forces on the cylinder. In order to give a theoretical study of this problem, we make the familiar assumptions of classical hydrodynamics and then solve the linear, two-dimensional, diffraction boundary-value problem, using Ursell's multipole method. Accurate numerical results are presented (in the form of tables) for four important (complex) quantities; these are the reflection and transmission coefficients, and two dimensionless coefficients which describe the horizontal and vertical forces on the cylinder. We have also made an experimental study, in which we measured the forces on the cylinder, and the reflection coefficient. These measurements are compared with the linear theory, and also with other experimental data; discrepancies are noted and an attempt to analyse them is made. We have also measured the mean horizontal forces on the cylinder; these results are compared with the predictions of a simple formula obtained by Longuet-Higgins.

The stability of an economic difference scheme for solving the boundary value problem of diffraction

The stability of an economic difference scheme for solving the boundary value problem of diffraction

Theory of diffraction efficiency and anomalies of shallow metal gratings of finite conductivity

The boundary-value problem of diffraction from shallow metal gratings with finite conductivity is solved numerically by using an integral equation method. The electromagnetic field in the material is described by means of a series expansion of evanescent waves. As suggested by Hessel and Oliner, grating anomalies can be divided into two types: a Rayleigh type and a resonance type. The Rayleigh-type anomaly characterized by its appearance just at the Rayleigh wavelength λR occurs for both polarizations, but its appearance has some differences, depending on polarization. In the case of E∥ polarization, the grazing mode is suppressed, while in case of H∥ polarization, it is not. The resonance-type anomaly can be observed only for H∥ polarization as a strong peak or a dip at the longer-wavelength side of λR. It can be explained as the result of the resonance excitation of one surface mode which corresponds to a surface plasmon, and its marked features are the extra absorption of light and the relative phase shift of the diffracted field, which are not observed in Rayleigh-type anomalies.

Diffraction of plane electromagnetic wave from arrays of conducting rectangular cylinders

The boundary value problem of diffraction of parallel and perpendicular polarised waves from gratings of conducting rectangular cylinders is solved by a rigorous mode-expansion formulation and numerical solution. Comparison of computed results with the limited theoretical and experimental results available in the literature demonstrates the accuracy of the numerical method. Further numerical results inducate the usefulness of these gratings for a variety of applications, and provide additional information on the physical phenomena which cause the unexpected transmission and reflection properties. Such results are also of considerable value in serving as standards to which various approximate solutions may be compared for accuracy and interpretation.

A constructive method, based on factorization, for solving boundary value problems of diffraction at bodies of finite size

A METHOD is outlined for reducing Wiener-Hopf integral equations to an infmite system of linear algebraic equations, which can be solved by a reduction method in which the approximations converge well. The method is applied to the solution of some boundary value problems of diffraction at obstacles of finite size.

Diffraction of a Parallel- and Perpendicular-Polarized Wave from an Echelette Grating*

The boundary-value problem of diffraction of a parallel- or perpendicular-polarized wave from an echelette grating is numerically solved. Parallel and perpendicular Wood’s anomalies are investigated. A very strong anomaly for negative incident angle into a back-scattering order is found.

Panel discussion on boundary value problems of diffraction and scattering theory (I)

Panel discussion on boundary value problems of diffraction and scattering theory (I)