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

For the original article see ibid., vol.MTT-34, no.1, p.161-7 (1986). The commenters cite a previously published work which contains some solutions which are identical to those found in the above-titled work. In replying, the authors agree that the solutions are identical and express appreciation for having the previously published work called to their attention.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

All numerical calculations will fail to provide a reliable answer unless the continuous problem under consideration is well posed. Well-posedness depends in most cases only on the choice of boundary conditions. In this paper we will highlight this fact, and exemplify by discussing well-posedness of a prototype problem: the time-dependent compressible Navier–Stokes equations. We do not deal with discontinuous problems, smooth solutions with smooth and compatible data are considered. In particular, we will discuss how many boundary conditions are required, where to impose them and which form they should have in order to obtain a well posed problem. Once the boundary conditions are known, one issue remains; they can be imposed weakly or strongly. It is shown that the weak and strong boundary procedures produce similar continuous energy estimates. We conclude by relating the well-posedness results to energy-stability of a numerical approximation on summation-by-parts form. It is shown that the results obtained for weak boundary conditions in the well-posedness analysis lead directly to corresponding stability results for the discrete problem, if schemes on summation-by-parts form and weak boundary conditions are used. The analysis in this paper is general and can without difficulty be extended to any coupled system of partial differential equations posed as an initial boundary value problem coupled with a numerical method on summation-by parts form with weak boundary conditions. Our ambition in this paper is to give a general roadmap for how to construct a well posed continuous problem and a stable numerical approximation, not to give exact answers to specific problems.

A plane sound wave is assumed to be incident upon an irregular pressure release surface z=ζ (x,y). The solution for the reflected field is regarded as a superposition of plane waves having an amplitude spectrum A (k1,k2). Next, the Fourier transform G(x,y) of A (k1,k2) is introduced and subjected to the boundary condition on ζ This leads to an integral equation for G(x,y) that cannot be readily solved. However, if one causes G(x,y) to depend exponentially on a function u(x,y), then a differential equation may be derived from this integral equation, the solution of which gives an approximate form of u(x,y); the degree of approximation involved depends on the smallness of ζ. This method is applied to the problem of sound scattering from a one-dimensional sinusoidally corrugated surface and the results compared with experimental measurements of LaCasce and Tamarkin and also with the results of a theory due to Rayleigh. This comparison shows the predictions of the theory presented here to be as good as the Rayleigh theory in all cases and closer in the majority of cases.

The basis of the quasiclassical operator (QO) method is described. Application of the method for series of problems are discussed. These are processes in the superposition of plane wave and constant field, radiation in linear colliders, radiation in inhomogeneous fields, the new approach to the pair creation by a photon in a magnetic field, the theory of radiation in oriented crystals, radiation spectra taking into account the energy loss.

Long wavelength vibrational modes in the ferromagnetic semiconductor Ga_{0.91}Mn_{0.09}As are investigated using time resolved x-ray diffraction. At room temperature, we measure oscillations in the x-ray diffraction intensity corresponding to coherent vibrational modes with well-defined wavelengths. When the correlation of magnetic impurities sets in, we observe the transition of the lattice into a disordered state that does not support coherent modes at large wavelengths. Our measurements point toward a magnetically induced broadening of long wavelength vibrational modes in momentum space and their quasilocalization in the real space. More specifically, long wavelength vibrational modes cannot be assigned to a single wavelength but rather should be represented as a superposition of plane waves with different wavelengths. Our findings have strong implications for the phonon-related processes, especially carrier-phonon and phonon-phonon scattering, which govern the electrical conductivity and thermal management of semiconductor-based devices.

Ultrasonic scattering by a side-drilled hole

A new derivation for the radiation reaction on a point charge is presented. The field of the charge is written as a superposition of plane waves. The plane wave spectrum of the field consists of homogeneous plane waves which propagate away from the charge at the speed of light, and inhomogeneous plane waves which constitute the Coulomb field of the point charge. The radiation field is finite at the orbit of the point charge. The force acting on the charge due to this field is the well known Abraham-Lorentz radiation reaction.

Users data of usage and the number of devices connected to the network are increasing from years to years, simply, the internet usage is growing. Due to this phenomenon, we need a decent channel capacity to keep up with it. One of the channel characteristics that affects to the capacity is channel coefficient. By this relation, we could enlarge the channel capacity by improving its channel coefficient. In this research, we conducted an analysis towards the maximum and minimum values of a channel coefficient, simultaneously defined by upper and lower threshold. The simulation on this research is using superposition of plane wave as the basic model. Previously, this model had been used on pCell technology to enlarge the multiplexing gain. However, on this research, we conducted a different approach by using the model to achieve the maximum and minimum value of the channel coefficient. At the end, we found a relation between the upper and lower threshold formulation and the channel coefficient simulation on some signal paths, with their direction randomized by uniform distribution.

The chiral magnet Cu_{2}OSeO_{3} hosts a Skyrmion lattice that may be equivalently described as a superposition of plane waves or a lattice of particlelike topological objects. A thermal gradient may break up the Skyrmion lattice and induce rotating domains, raising the question of which of these scenarios better describes the violent dynamics at the domain boundaries. Here, we show that in an inhomogeneous temperature gradient caused by illumination in a Lorentz transmission electron microscope different parts of the Skyrmion lattice can be set into motion with different angular velocities. Tracking the time dependence, we show that the constant rearrangement of domain walls is governed by dynamic 5-7 defects arranging into lines. An analysis of the associated defect density is described by Frank's equation and agrees well with classical 2D MonteCarlo simulations. Fluctuations of boundaries show a surgelike rearrangement of Skyrmion clusters driven by defect rearrangement consistent with simulations treating Skyrmions as point particles. Our findings underline the particle character of the Skyrmion.

Summary Seismic waves obey the wave equation, and low-frequency EM waves obey the diffusion equation. Despite this, there is a deep similarity in their properties. Both wave types may be analyzed as a superposition of plane waves which are both dispersive and attenuative; the principal difference is that for seismics, these effects are weak, whereas for EM, they are strong. This means that all seismic processing algorithms which do not assume low dispersion and attenuation may be used to process properly acquired EM data; this is the majority of the seismic toolkit. In particular, EM data may be imaged, seismic-style, rather than mathematically inverted, EM style. With both data-types, the signal from the subsurface is weak, so it is best to detect it without competition from a concurrent source. This means that controlled sources of EM should be impulsive sources (“ISEM”) rather than continuous sources (“CSEM”). At low frequency, the EM phase velocity is a few km/s, comparable to seismic velocities. Hence, seismic-style acquisition, including the measurement of moveout, is feasible for EM data. For seismics, the Quality factor Qseis is a large material property, to be determined from the data; whereas QEM= ½ is a property of the equation.

In the study of E region irregularities the standard procedure is to Fourier analyze the irregularities in both time and space, that is, to describe them as a superposition of plane waves. This introduces difficulties when the amplitude of the plane waves becomes large, thereby adding nonlinear terms to the original equations and forcing all the plane waves to become coupled to one another. In the present work we stay away from Fourier analysis and use the standard fluid description of the instabilities in the limit of perturbed electric fields that are strictly perpendicular to the geomagnetic field. We obtain a nonlinear generalization of the standard results whereby the diffusion‐like operator found in linear theory is now a function of the density itself. Therefore, as the structures grow, the net electric field seen by the ambient plasma inside the structures changes in time: it rotates and its amplitude decreases. Consequently, one possible saturation mechanism for the instabilities is a reduction in the net electric field inside the structures, which brings them to threshold velocity conditions. This being stated, other nonlinear saturation mechanisms remain possible if they require smaller saturation amplitudes than the present work. Either way, our work is consistent with intermittency and implies that the largest amplitude structures in the medium should be rotated away from zero flow angle conditions by a measurable amount. Finally, we show that when compared to an irregularity‐free situation, there should be a measurable reduction in the average Hall current carried by the plasma, while the average Pederseri current should not be affected.

The aim of this study is to engage a basic understanding of the information micro-Raman spectroscopy may yield when this characterization tool is applied to carbon nanotubes. All collective vibrations that occur in crystals can be viewed as the superposition of plane waves, called phonons, that virtually propagate to infinity. The two dominant Raman features are the radial breathing mode at low frequencies, the tangential G band and the D band multi-feature at higher frequencies. Carbon nanotubes (CNT) were formed by double furnace chemical vapor deposition. This method was based on the pyrolysis of liquid aerosols containing hydrocarbons as carbon source (camphor oil), ferrocene as the catalyst source and nitrogen as the carrier gas. The samples were prepared by placing the carbon precursor on the alumina boat into the first furnace which contains the catalyst source at different alumina boat heated at 200 °C and passed through the deposition furnace. The deposition furnace was heated at 500-900°C for 1 hour depositing CNT without annealing treatment. Then, the samples were characterized using micro-Raman spectrometer obtaining the carbon G and D peaks around 1580 cm1 and 1350 cm1 respectively and the image of the CNT produced were obtained from field emission scanning electron microscope and high resolution transmission electron microscope. Keywords: micro-Raman spectroscopy, Carbon nanotubes, Camphor oil

In this paper, a straightforward approach is presented to generate Bessel beam sources in three-dimensional finite-difference time-domain (FDTD) method. Based on the angular spectrum representation (ASR), the incident Bessel beam is described as a superposition of plane waves whose wavevectors covering a conical surface. This decomposition of Bessel beam is then approximated by a finite collection of plane waves, which are injected into FDTD simulation domain using the total-field/scattered-field (TF/ST) method. The present method's correctness and accuracy are verified by comparing the reconstructed field in FDTD with the original field. Far-field scattered diagrams of a dielectric sphere and a spheroid particle illuminated by a zero-order or a higher-order Bessel beam are calculated using FDTD. The results are compared with those calculated using the generalized Lorenz-Mie theory (GLMT) and surface integral equation method (SIEM). Very good agreements have been achieved, which partially indicate the correctness of our method. Internal and near-surface field distributions for a two-layer hemisphere particle, which are illuminated by Bessel beams, are also displayed to show the potentials of this approach in solving scattering problems of complex particles. This approach can also be applied to generate other structured beam sources in FDTD, which provides an access to solve structured beam scattering by complex particles using FDTD.

Double-walled structures separated by an acoustic cavity are widely used in many applications for sound insulation purposes. Their sound transmission loss is estimated by calculating the ratio between the incident sound power on the first wall and the radiated sound power by the second. The introduction of a melamine foam associated with heavy masses brings a better sound insulation against external noise. In this context, this work deals with the numerical prediction of the sound transmission of a roller shutter box equipped with porous and viscoelastic materials. Such system can be modeled, at first approximation, as a double-walled structure. The proposed approach is based on the use of the finite element method for the modeling of the different solid and fluid domains of the problem. A mixed displacement pressure formulation of the Biot poroelasticity equations is used to model the poroelastic domain. The structure is excited by a diffuse field represented by a superposition of plane waves with random phases and directions. The acoustic power radiated by the second wall is calculated by applying the Rayleigh integral method. The results show the importance of optimizing the position of the porous and the heavy layers to maximize the sound insulation.

The fact that plane waves are solutions of the Helmholtz equation in free space allows us to write the exact solution to the diffraction problem as a superposition of plane waves. The solution of other related problems can also be expressed in similar forms. These forms are very well suited for directly importing various signal processing tools to diffraction related problems. Another signal processing-diffraction link is the application of novel sampling theorems and procedures in signal processing to diffraction for the purpose of more convenient and efficient discrete representation and the use of associated computational algorithms. Another noteworthy link between optics and signal processing is the fractional Fourier transform. Revisiting diffraction from a modern signal processing perspective is likely to yield both interesting viewpoints and improved techniques.