A note on diffraction by a right-angled wedge

The notion of impedance boundary condition is generalized and illustrated by the discussion of the electromagnetic field arising from a magnetic line dipole source located above a plane structure. The generalized impedance boundary condition contains a set of parameters whose totality offers a physical description of the configuration. By suitably selecting these parameters, the configuration may be made to correspond to a structure that supports surface waves. We give an exact solution for a plane structure that supports one, two and three surface waves. The magnitudes of the surface waves are obtained and simple formulas for the radiated far field patterns are given. We also show how the methods employed can be extended to the case of any number of surface waves. This involves an n n th order mixed boundary condition for a second order partial differential equation.

We investigate the localization properties of surface waves created at the interface between a truncated 1D photonic crystal and homogeneous medium in the presence of a chiral cap layer using the transfer matrix method. The numerical results show that the interface can support surface waves with both transverse electric and transverse magnetic polarizations. We demonstrate that the surface waves can be engineered by varying the chirality parameter of the cap layer, which plays an important role in controlling and localization of surface states. It is shown that the effect of a chirality parameter on surface waves with transverse electric polarization is more remarkable compared with surface waves with transverse magnetic polarization. In addition, the effect of thickness and material type of the cap layer and the termination layer of a photonic crystal on surface waves is investigated. The results show that these parameters have considerable effect on modification of surface states. These findings may open new opportunities in the design of cavity-based devices such as high-Q sensors and perfect absorbers.

Here, we present the study of electromagnetic waves interaction with material with single and double negative parameters having periodic surface, based on rigorous solution to corresponding boundary value problem. It is well known and proved by numerous cases that the most profound and efficient study of electromagnetic waves interaction with objects and especially the objects possessing resonant properties, can be performed by the interactive and complementary study of diffraction and spectral problem solution. Here, we use the solution to diffraction problem for periodic surface obtained by C formalism associated with regularization that is described. For the spectral problem solution the analytical continuation (within the frames of the theory of analytic functions) of the solutions to diffraction problem for the boundary between two dielectric media into the domain of complex valued frequencies is considered. The reason and benefits of such continuation consist in the fact that singularities (poles, branching points and etc.) of the diffraction problem solution, appearing after such continuation into the domain of complex frequencies, define and predetermine completely the diffraction field features, obtained in the physical (at real frequencies) domain of parameters.

The purpose of this paper, which builds on previous work (Int. J. Numer. Meth. Engng 2009; 77:1646–1669), is to improve a numerical scheme based on the partition of unity finite element method (PUFEM) for the solution of the time harmonic elastic wave equations. The approach consists to approximate the displacement field by the standard finite element shape functions, enriched locally by superimposing pressure (P) and shear (S) plane waves. The aim is to accurately model two‐dimensional elastic wave problems on relatively coarse mesh grids, capable of containing many wavelengths per nodal spacing, for wide ranges of frequencies. This allows us to relax the traditional requirement of about 10 nodal points per S wavelength. In this work, an exact integration scheme for the linear triangular finite element is developed to evaluate the oscillatory integrals arising from the use of the PUFEM. The main contribution here consists in developing an explicit closed‐form solution for two‐dimensional wave‐based integrals, when the phase variation is linear in the local coordinate element system. The evaluation of the element mass matrix is performed from appropriate edge integrals. All other element matrices, obtained by adequate splitting of the element stress tensor matrix, are simply deduced from the element mass matrix entries.The results show clearly that the proposed integration scheme evaluates accurately the entries of the global matrix with drastic reduction of the computational time. Numerical tests dealing with the scattering of S elastic plane waves by a circular rigid body show that, for the same discretization level, it is possible to improve the accuracy by using large elements associated with high numbers of approximating plane waves rather than using small elements with less plane waves. However, this increases the conditioning and the fill‐in of the global matrix. At high frequency, it is even possible to push the number of degrees of freedom per S wavelength under 2 and still achieve good accuracy. Finally, some remarks on the choice of the numbers of P and S plane waves leading to better accuracy and conditioning are discussed. Copyright © 2010 John Wiley & Sons, Ltd.

The new approach to the wavelet analysis for the solutions of the homogeneous wave equation in three spatial dimensions is presented. The approach is based on the ideas suggested by G. Kaiser but has different implementation and has some advantages versus the known approach. A new physical wavelet for this wavelet analysis is also presented, with the brief discussion of its main properties. The wavelet analysis has become widely used during the last twenty years and it has a lot of applications nowadays. However most of them are in the field of the numerical processing of the experimental data, digital images, astronomical, geophysical and medical data and other applications of that kind (see, for example, [1], [2]). The amount of the results in the application of the methods of the continuous wavelet analysis to the solutions of the differential equations is not large. In particular the continuous wavelet analysis for the solutions of the three-dimensional homogeneous wave equations with a constant wave speed was first developed by G. Kaiser in his book [3]. He suggests a method for decomposition of the solutions of the wave equation in terms of the localized solutions of the same equation based on the analytic signal transform and on the theory of the analytic functions of several variables. The wavelet for such decomposition was also suggested, and the class of such wavelets was named ’physical wavelets’. The sort of the wavelet analysis developed by Kaiser is close to the holomorphic wavelet transform (see, for example, [2]). However, this approach may be found unfamiliar by the people who deal with the wavelet analysis within the framework of the signal and image processing. The aims of this paper are as follows. First we develop the wavelet analysis for the solutions of the wave equation not involving the analytic signal transform. The ideas, which we base on, were suggested by Kaiser in [3], however their implementation here differs from his approach. The method we use is intrinsic to the common continuous wavelet transform and we hope will be more familiar to the people who work in the area of the signal and image processing. Our approach also provides some advantages in comparison to that, suggested by Kaiser. We enlarge the class of solutions which can be used as the mother wavelets for the analysis. The second aim is to find a new physical wavelet for our method, i.e., to find the solution of the wave equation which will be an admissible wavelet. The new wavelet is constructed by means of the field of point sources and of proxy wavelets using the technique suggested by G. Kaiser. This new spherically symmetric physical wavelet has good properties such as exponential localization in both the coordinate and the Fourier

Enhancement of thermal emission and control of its direction are important for applications in optoelectronics and energy conversion. A number of structures have been proposed as coherent emission sources, which exhibit a large emissivity peak within a narrow wavelength band and at well-defined directions. A commonly used structure is gratings, in which the excited surface polaritons or surface waves are coupled with propagating waves in air to produce coherent emission for p polarization. One-dimensional photonic crystals can also support surface waves but have not been applied as coherent emission sources. The present study demonstrates that coherent emission can be achieved by the use of a multilayer structure consisting of periodic layers (i.e., photonic crystal) coated with a polar material such as SiC. By excitation of surface waves at the interface between SiC and the photonic crystal, coherent emission is predicted for both p and s polarization. In addition to the excitation of surface waves, the emission from the proposed structure can be largely enhanced by the cavity resonance mode, which is very similar to that of Fabry-Perot etalon, as well as by the Brewster mode that occurs only for p polarization.

<abstract><p>Photoacoustic tomography (PAT) is a novel and rapidly developing technique in the medical imaging field that is based on generating acoustic waves inside of an object of interest by stimulating non-ionizing laser pulses. This acoustic wave was measured by using a detector on the outside of the object it was then converted into an image of the human body after several inversions. Thus, one of the mathematical problems in PAT is reconstructing the initial function from the solution of the wave equation on the outside of the object. In this study, we consider the fractional wave equation and assume that the point-like detectors are located on the sphere and hyperplane. We demonstrate a way to recover the initial function from the data, namely, the solution of the fractional wave equation, measured on the sphere and hyperplane.</p></abstract>

Based on integral representations with densities distributed along a segment of the symmetry axis, a representation of the solution of the boundary value problem of plane wave diffraction by a local penetrable body of revolution with smooth surface is constructed and justified. The resulting integral representation allows one to avoid resonances of the interior domain when analyzing the scattering frequency characteristics.

Dynamics of surface and internal long waves generated by atmospheric pressure disturbances

A boundary value problem is considered for the equations of heat and mass transfer when one of the boundary conditions contains normal derivatives of the third order, to which a certain problem of heat and mass transfer in drying processes is reduced. The solution of the boundary value problem is sought in the form of the thermal potential of a double layer. A lemma on finding the limits of the third order normal derivatives is given. Using the boundary conditions, a system of the integro-differential equations (SIDE) with various heat conduction operators is obtained. A characteristic part of SIDE is solved by the method of integrated transformations of Fourier-Laplace when performing a condition of solvability. By the method of regularization of SIDE it is reduced to the system of the integrated equations of Volterra-Fredholm. A theorem of solvability of a boundary value problem is given under the condition of solvability of the heat and mass transfer equations with normal derivatives in the boundary conditions.

New micromachining techniques now provide the technology to fabricate vacuum electron devices with dimensions suitable for operation in terahertz region. In this paper, results of theoretical and numerical simulation studies of a MW-class, large diameter terahertz wave generator based on the Cerenkov principle are presented. The device consists of a miniaturized foilless diode as well as an oversized rectangular slow wave structure (SWS) which could support surface wave and provide a strong beam-wave coupling. With the method of particle-in-cell simulation, we investigated the operating characteristics of a prototype 0.22 THz oscillator. It is found that the steady-state, single-frequency operation in the overmoded situation could be realized by the property of surface wave, and the unique voltage tunable capability proves that the device performs as a forward wave oscillator.

Eigenfunction expansions for the equations of crystal optics in a half-space

We consider boundary value problem for Laplace equation in bounded two-dimensional Lipschitz domain with transmission boundary condition given upon open curve. This conditions includes itself the jump of solution of boundary value problem and the meaning of boundary value of its normal derivative. We prove the equivalence of considered boundary value problem and obtained variational problem. As a result we prove existence and uniqueness of solution of the posed problems in appropriate functional spaces. Based on the integral representation of the solution the considered boundary value problem is reduced to the system of boundary integral equation which has unique solution.

An isotropic, elastic solid permits the propagation of one straight-crested, nondispersive surface wave, called a Rayleigh wave. If a thin film of a different material is deposited on the substrate, an infinite number of straight-crested surface waves are possible, all of which are dispersive, including the one corresponding to the Rayleigh wave. Since the solution of the three-dimensional equations of elasticity is extremely unwieldy and the films may be considered thin in the frequency range of interest, the approximate equations of low-frequency extension and flexure of thin plates are employed in the solution to account for the motion of the platings. These approximate equations enable the entire effect of the plating to be treated as a boundary condition at the surface of the substrate. The accuracy of the approximation is shown to be excellent in the frequency range of interest. It is shown that platings which load the substrate reduce the velocity of straight-crested surface waves, while platings which stiffen the substrate increase the velocity. A condition is derived which tells whether a given combination of materials will lower or raise the surface-wave velocity. The solution for straight-crested waves in isotropic solids is extended to include waves with trigonometric and hyperbolic crests. An approximation technique is introduced, which enables the use of the dispersion curves for straight-crested waves in connection with the extended solution for waves with variable crests, in order to obtain approximate solutions for surface waves guided by thin films of finite width. The results of the calculation are shown to compare favorably with experiment. The analysis shows that strips of finite width may be used to guide surface waves only when the film loads the substrate and that slots of finite width in the film must be used when the film stiffens the substrate. It is also shown that while the strip configuration has a highly dispersive surface-wave velocity, the slot configuration has a large frequency range around which the surface wave velocity is essentially nondispersive. A formula for the coupling length between two weakly coupled, adjacent strips (or slots) is obtained by means of a perturbation technique, and shows that the coupling length increases rapidly with decreasing wavelength.

We present an effective numerical method for the recovery of biharmonic functions in a disk from continuous boundary values of these functions and of their normal derivatives using wavelets that are harmonic in the disk and interpolating on its boundary on dyadic rational grids. The expansions of solutions of boundary value problems into cumbersome interpolation series in the wavelet basis are replaced by sequences of their partial sums that are compactly presentable in the subspace bases of the corresponding multiresolution analysis (MRA) of Hardy spaces $$h_{\infty}(K)$$ of functions harmonic in the disk. Effective estimates are obtained for the approximation of solutions by partial sums of any order in terms of the best approximation of the boundary functions by trigonometric polynomials of a slightly smaller order. As a result, to provide the required accuracy of the representation of the unknown biharmonic functions, one can choose in advance the scaling parameter of the corresponding MRA subspace such that the interpolation projection to this space defines a simple analytic representation of the corresponding partial sums of interpolation series in terms of appropriate compressions and shifts of the scaling functions, skipping complicated iterative procedures for the numerical construction of the coefficients of expansion of the boundary functions into series in interpolating wavelets. We write solutions using interpolating and interpolating-orthogonal wavelets based on modified Meyer wavelets; the latter are convenient to apply if the boundary values of the boundary value problem are given approximately, for example, are found experimentally. In this case, one can employ the usual, well-known procedures of discrete orthogonal wavelet transformations for the analysis and refinement (correction) of the boundary values.