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Three dimensional dielectric systems are simulated to obtain the relaxational polarization responses. The dielectric system consists of randomly distributed permanent dipoles that fluctuate due to thermal activation in double well potentials between two plane-parallel electrodes. The barrier height between the wells is assumed to be proportional to the dipole length. The electrodes' effect is considered with the method of images. The relaxational phenomena of two dielectric systems, one with fixed and the other with distributed dipole lengths having single and distributed relaxation times, respectively, are calculated with a Monte-Carlo simulation method. The local field at every dipole and consequently the transition probabilities are computed to determine the transient polarization response iteratively. Both dielectric systems are simulated over a wide range of temperatures for interacting and non-interacting cases. The mean relaxation times exhibit a Vogel-Fulcher law for interacting systems and an Arrhenius law for non-interacting systems.

Mechanics of defects in solids across a wide span of length scales is commonly formulated using the dislocations theory. This paper revisits the classical problem of interaction between an elastic edge dislocation and a circular cavity. A heuristic, yet, mechanistic approach is taken to obtain the stress solution to this problem. The approach uses complex variable theory of elasticity, along with method of images. For this purpose, a definition and formulation of elastic dipole singularities similar to dipole charges in electrostatics is developed. It is shown that an image dislocation with Burger’s vector of the same strength as the real dislocation but in opposite direction, as well as a set of four singularities including a dislocation dipole, a moment-dilatation dipole, and two centers of dilatation would establish a circular, traction-free boundary in an infinite elastic medium. Adding a Volterra dislocation to the finite-length edge dislocation from this study would recover the related problem of interaction between an infinite-length edge dislocation and circular cavity. The interesting analogy between the considered elastic problem and the electrostatic problem of interaction between a line electric charge and a cylindrical conductor is discussed.

For a point charge between two grounded conductor planes forming a 60° angle, the potential and electric field generated by point charge for Yukawa’s potential (e-μr/r) and Coulomb’s potential (1/r) are modeled and simulated. The expression for the potential that generalizes the cases discussed in Lopez-Marino, M. and Trujillo Caballero, J. (2017) Point Charges and Conducting Planes for Yukawa’s Potential and Coulomb’s Potential. Journal of Electromagnetic Analysis and Applications, 9, 135-146. https://doi.org/10.4236/jemaa.2017.910012 is presented. Graphs for the potential and electric field for both cases are showed using Maple, that of Coulomb and that of Yukawa for different values of μ . The purpose of this work is to offer students a practical guide for problem analysis of electrostatics using Maple’s capabilities as a computational tool.

We calculate exact two-dimensional analytic expressions for the electric and magnetic fields and their potentials generated by a linear beam of relativistic charged particles between infinite ideally conducting planes and ferromagnetic poles. We use the method of summing an infinite sequence of fields of linear image charges and image currents to obtain solutions and also calculate the density distribution of the electric charge on the conductor surface induced by the charged particle beam. We obtain new formulas for the linear approximation of the fields near the beam and discuss the mathematical features of the exact solutions and the applicability constraints for the linear approximations.

We derive an analytic solution to the problem of a screw dislocation interacting with a parabolic cavity and a semi-infinite sharp crack using conformal mapping techniques and the method of images. Closed-form expressions for the image force acting on the screw dislocation, the mode III stress intensity factor at the crack tip and the generalized mode III stress intensity factor for the parabolic cavity are obtained. The correctness of the solution is validated by comparison with existing solutions in the literature.

In this article, a new analytical method is proposed to estimate eddy currents inside linear conductive materials. The novel closed-form formulation takes into account the effects of both conductor reaction field that dominates at high frequencies as well as the spatially non-homogeneous nature of the magnetic field penetrating the material. Although the model is suitable for most kinds of permanent magnet machines, it is particularly useful for Surface-mounted Permanent Magnet (SPM) machines where both of these phenomena are prominent. The numerical implementation of the proposed model, which consists of a combination of the method of images and 4-D Fourier transform, is presented. The model is then validated against 2-D and 3-D finite element models (FEMs) for a simple magnetic circuit, showing good agreement. Finally, the eddy current magnet loss in an SPM synchronous machine is evaluated and results are discussed.

Exact 2D analytic expressions for E and B fields and their potentials created by a linear beam of relativistic charged particles between infinite perfectly conductive plates and ferromagnetic poles are derived. The solutions are obtained by summing an infinite sequence of fields from linear charge-images and current-images in complex space. Knowledge of the normal component of the E field on the conductor surface makes it possible to calculate the induced electric charge surface density. In addition, we derive within an improved linear approximation new analytical expressions for fields near the beam in the case of an arbitrary beam offset from the median plane. The mathematical features of exact solutions and limitations for the applicability of linear approximations are specified.The primary goals of the future high-luminosity p-p and heavy-ion LHC programme are the search for yet unobserved effects of physics beyond the SM, searches for rare or low-sensitivity processes in the Higgs sector, and probing in more detail the mechanism of EW symmetry breaking. This programme relies on the stable operation of the accelerator. However, as the beam luminosity increases, a number of destabilizing phenomena occur, in particular field emission, enhancing the electron cloud effect. For the case of a proton beam, we apply the exact 2D solution for estimating the intensity of electron field emission activated by the electric field of the beam in collimators of the future high-luminosity LHC. Calculation shows that the field emission intensity is very sensitive to a collimator surface roughness. In addition, with a relatively small and accidental beam displacement from the median path, about 20% of the collimator half-gap, the emission intensity increases by a factor of 1.E+7. This will partially neutralize the beam space charge, violating acceleration dynamics and enhancing instability effects.

We perform the analysis of the focusing nonlinear Schr\"odinger equation on the half-line with time-dependent boundary conditions along the lines of the nonlinear method of images with the help of B\"acklund transformations. The difficulty arising from having such time-dependent boundary conditions at $x=0$ is overcome by changing the viewpoint of the method and fixing the B\"acklund transformation at infinity as well as relating its value at $x=0$ to a time-dependent reflection matrix. The interplay between the various aspects of integrable boundary conditions is reviewed in detail to brush a picture of the area. We find two possible classes of solutions. One is very similar to the case of Robin boundary conditions whereby solitons are reflected at the boundary, as a result of an effective interaction with their images on the other half-line. The new regime of solutions supports the existence of one soliton that is not reflected at the boundary but can be either absorbed or emitted by it. We demonstrate that this is a unique feature of time-dependent integrable boundary conditions., Comment: 38 pages. 5 figures. Authors' version of the published article. Accessiblefrom https://onlinelibrary.wiley.com/doi/10.1111/sapm.12456

Wave power absorption with Wave Energy Converters (WECs) arrays in front of a vertical wall is predicted with an in-house transient wave-multibody numerical tool of ITU-WAVE which uses time marching scheme to solve a boundary integral equation for the analyses of hydrodynamic radiation and exciting forces. The perfect reflection of incident waves from a vertical wall is considered with method of images. Mean interaction factor, which can have constructive or destructive effect and determines the performances of WECs, is approximated with different array configurations. The vertical wall effect plays significant role over hydrodynamic parameters as the radiation and exciting forces show quite different behaviour in the case of WECs with and without vertical wall in an array system. The numerical results show that the performance and wave power absorption with WECs arrays in front of vertical wall are much greater compared to WECs arrays without vertical wall effect. This is mainly due to standing and nearly trapped waves between a vertical wall and WECs arrays in addition to strong interactions between WECs. The satisfactory agreements are obtained when the present ITU-WAVE numerical results for different hydrodynamic parameters in an array system are compared with other published analytical and numerical results.

We develop a fast method for computing the electrostatic energy and forces for a collection of charges in doubly-periodic slabs with jumps in the dielectric permittivity at the slab boundaries. Our method achieves spectral accuracy by using Ewald splitting to replace the original Poisson equation for nearly-singular sources with a smooth far-field Poisson equation, combined with a localized near-field correction. Unlike existing spectral Ewald methods, which make use of the Fourier transform in the aperiodic direction, we recast the problem as a two-point boundary value problem in the aperiodic direction for each transverse Fourier mode, for which exact analytic boundary conditions are available. We solve each of these boundary value problems using a fast, well-conditioned Chebyshev method. In the presence of dielectric jumps, combining Ewald splitting with the classical method of images results in smoothed charge distributions which overlap the dielectric boundaries themselves. We show how to preserve spectral accuracy in this case through the use of a harmonic correction which involves solving a simple Laplace equation with smooth boundary data. We implement our method on Graphical Processing Units, and combine our doubly-periodic Poisson solver with Brownian Dynamics to study the equilibrium structure of double layers in binary electrolytes confined by dielectric boundaries. Consistent with prior studies, we find strong charge depletion near the interfaces due to repulsive interactions with image charges, which points to the need for incorporating polarization effects in understanding confined electrolytes, both theoretically and computationally., Comment: 60 pages, 7 figures

In the present study, the problems of diffraction and radiation of water waves by a cylindrical heaving wave energy converter (WEC) placed in front of a reflecting V-shaped vertical breakwater are formulated. The idea conceived is based on the possible exploitation of amplified scattered and reflected wave potentials originating from the presence of V-shaped breakwater, towards increasing the WEC’s wave power absorption due to the wave reflections. An analytical solution based on the method of images is developed in the context of linear water wave theory, taking into account the hydrodynamic interaction phenomena between the converter and the vertical wall. Numerical results are presented and discussed concerning the hydrodynamic forces on the absorber and its wave power efficiency for various examined parameters, namely, the breakwaters’ forming angle, the distance between the converter and the vertical walls and the wave heading angle. The results show that the amount of the harvested wave power by the WEC in front of the walls is amplified compared to the wave power absorbed by the same WEC in the open sea.

The polarization characteristics of diffraction radiation appearing during the uniform motion of a nonrelativistic charged particle near an ideally conducting sphere and also a hemispherical bulge in a conducting plane are calculated. Our previously developed approach based on the method of images, which is known in electrostatics, is used. It is shown that, in the low-frequency range, radiation at a hemisphere is always linearly polarized, while radiation at a sphere is generally characterized by elliptical polarization.

The radiation resulting from the uniform motion of a charged particle near (or through) a perfectly conducting sphere as well as near a hemispheric bulge on a metal plane has been considered previously using the method of images known from electrostatics. Here we discuss three topics: polarization of the diffraction radiation on the sphere as well as on the hemispherical bulge, transition radiation on the hemispherical bulge, and the coherent diffraction radiation from a bunch of charged particles.

The present study investigates the combined wave field that is induced by the continuous interaction of plane waves with an array of truncated circular cylinders in front of a rigid wall. The long-term goal of the study is the investigation of possible increase in the efficiency of cylindrical Wave Energy Converters (WECs) by putting in the vicinity of the array a barrier to propagation, a wall, that could assist the reflection of the incoming waves. The main task is to develop a generic solution method that is free of conceptual simplifications employed, e.g. by the method of images and the assumption of “pure” wave reflection. To cope with the set task, the proposed method relies on the semi-analytical formulation of the velocity potentials, while the solution is sought by combined expressions that involve polar and elliptical harmonics. The wall is represented as an elliptical cylinder with zero semi-minor axis. This assumption has eventually a beneficial effect to the underlying formulation given that it simplifies significantly the expansions of the involved diffraction potentials.

This article presents unsteady wall pressure measurements from a low-speed, low-solidity rotor operating in a circular duct. A circular microphone array was placed within the duct downstream of the rotor to record the unsteady surface pressure. These data were decomposed into spatial modes and then further decomposed into hydrodynamic and acoustic pressure fluctuations. The acoustic pressure was then further separated into the acoustic source function and an acoustic transfer function. The acoustic source function from the rotor compared well with a previous formulation for turbulence interaction noise. The transfer function showed a complicated relationship with frequency, with significant local maxima and minima. These features of the duct acoustics were studied using a simplified Green's function type approach, using the method of images to represent the duct end effects. The simple analytical method was able to capture many of the detailed features of the measurements with qualitative agreement.

Far-field patterns of wire antennas on a PEC sphere are calculated from quasi-static near fields using spherical harmonic expansion. Three cases of the wire antennas—F-antenna, dipole, and resonant loop on the PEC sphere—are considered. With the assumed sinusoidal current distribution on the antenna, the method of images is employed to obtain currents and charges on the antenna's image against the sphere. Only from the charge density and current distributions of the wire antenna and its image, ignoring the sphere, the quasi-static near fields are calculated using Coulomb's and Biot–Savart's laws. The fields in a source-free region are expressed in spherical harmonic expansion with the coefficients obtained from the radial components of the calculated quasi-static electric and magnetic near fields. The far fields thus obtained from the expansion match well with that of the simulated patterns. Deviations between simulation and theory are discussed and explained.

The hydroelastic vibration analysis of clamped rectangular plates both verticallyand horizontally submerged in fluid is studied by isogeometric FE/BE approachin this paper. By adopting the linear hydroelasticity theory, the computationalprocedure of the fluid-structure interaction problem is divided into two parts.In the first part, dynamic analysis of the structure in vacuo conditions and inthe absence of external forces is carried out by NURBS-based isogeometric -finite element method (IGAFEM). Then, in the second part, it is assumed that fluid is inviscid, incompressible and irrotational, and fluid forces directly correspond to inertial effects of the structure. Thus, displacements of the elasticstructure due to modal vibrations are in same phase with the acceleration of thefluid particles, and fluid forces on the freely vibrating structure are obtained byNURBS-based isogeometric boundary element method (IGABEM). In order toverify the accuracy of the present methodology, a convergence study is carriedout with the available numerical and experimental data providing the generalized added mass coefficients, and the effects of the thickness and aspect ratio are also investigated. It is found that the numerical results are in good agreement with the previous results, thus validating the proposed approach.

The presence of a nearby boundary is likely to be important in the life cycle and evolution of motile flagellate bacteria. This has led many authors to employ numerical simulations to model near-surface bacterial motion and compute hydrodynamic boundary effects. A common choice has been the method of images for regularized Stokeslets (MIRS), however, the method requires discretization sizes and regularization parameters that are not specified by any theory. To determine appropriate regularization parameters for given discretization choices in MIRS, we conducted dynamically similar macroscopic experiments and fit the simulations to the data. In the experiments, we measured the torque on cylinders and helices of different wavelengths as they rotated in a viscous fluid at various distances to a boundary. We found that differences between experiments and optimized simulations were less than 5% when using surface discretizations for cylinders and centerline discretizations for helices. Having determined optimal regularization parameters, we used MIRS to simulate an idealized free-swimming bacterium constructed of a cylindrical cell body and a helical flagellum moving near a boundary. We assessed the swimming performance of many bacterial morphologies by computing swimming speed, motor rotation rate, Purcell’s propulsive efficiency, energy cost per swimming distance, and a new metabolic energy cost defined to be the energy cost per body mass per swimming distance. All five measures predicted that the optimal flagellar wavelength is eight times the helical radius independently of body size and surface proximity. Although the measures disagreed on the optimal body size, they all predicted that body size is an important factor in the energy cost of bacterial motility near and far from a surface.

We perform a scaling analysis of the terms composing the Ffowcs-Williams and Hawkings (FWH) equation, which rules the propagation of noise generated by a rigid body in motion. Our analysis extends the seminal work of Lighthill (Proc. R. Soc. Lond. A, vol. 211, 1952, pp. 564–587) and the dimensional analysis of classical sources (monopole, dipole and quadrupole) considering all the FWH integral terms. Scaling properties are analysed in light of perfect/imperfect similarity when laboratory-scale data are used for full-scale predictions. As a test case we consider a hydrodynamic example, namely a laboratory-scale ship propeller. The data, obtained numerically in a previous study, were post-processed according to the scaling analysis presented herein. We properly scale the speed of sound to obtain perfect similarity and quantify the error with respect to the imperfect scaling. Imperfect similarity introduces errors in the acoustic response related both to the linear terms and to the nonlinear terms, the latter of great importance when the wake is characterized by robust and organized vorticity. Successively, we analyse the effect of a free surface, often present in hydrodynamic applications. We apply the method of images to the FWH equation. The free surface may generate a frequency-dependent constructive/destructive interference. The analysis of an archetypal acoustic field (monopole) provides robust explanation of these interference effects. Finally, we find that imperfect similarity and the absence of a free surface may introduce errors when model-scale data are used to obtain the full-scale acoustic pressure. The error is small for microphones placed in the near field and becomes relevant in the far field because of the nonlinear terms.

Purpose This paper aims to propose a new 3D electromagnetic model to compute translational motion eddy current in the conducting plate of a novel linear permanent magnet (PM) induction heater. The movement of the plate in a DC magnetic field created by a PM inductor generates induced currents that are at the origin of a heating power by Joule effect. These topologies have strong magnetic end effects. The analytical model developed in this work takes into account the finite length extremity effects of the conducting plate and the reaction field because of induced currents. Design/methodology/approach The developed model is based on the combination of the sub-domain’s method and the image’s theory. First, the magnetic field expressions because of the PMs are obtained by solving the three-dimensional Maxwell equations by the method of separation of variables, using a magnetic scalar potential formulation and a magnetic field strength formulation. Then, the motional eddy currents are computed using the Ampere law, and the finite length extremity effects of the conducting plate are taken into account using the image’s method. To analyze the accuracy of the proposed model, the obtained results are compared to those obtained from 3D finite element model (FEM) and from experimental tests performed on a prototype. Findings The results show that the developed analytical model is very accurate, even for geometries where the edge effects are very strong. It allows directly taking into account the finite length extremity effects (the transverse edge effects) of the conducting plate and the reaction field because of induced currents without the need of any correction factor. The proposed model also presents an important reduction in computation time compared to 3D finite element simulation, allowing fast analysis of linear PM induction heater. Practical implications The proposed electromagnetic analytical model can be used as a quick and accurate design tool for translational motion PM induction heater devices. Originality/value A new 3D analytical electromagnetic model, to find the induced power in the conducting plate of a novel translational motion induction heater has been developed. The studied heating device has a finite length and a finite width, which create edge effects that are not easily considered in calculation. The novelty of the presented method is the accurate 3D analytical model, which allows finding the real power heating and real distribution of the induced currents in the conducting plate without the need to use correction factor. The proposed model also takes into account the reaction field because of induced currents. In addition, the developed model improves an important reduction in the computation time compared with 3D FEM simulation.

We consider the one-dimensional diffusion equation in a finite region, with reflecting boundary conditions and with absorbing boundary conditions. Solutions using the method of separation of variables are presented. The survival probability in a finite region in the case of absorbing boundaries is determined. An alternative approach to the solution of the diffusion equation, based on the method of images, is also described. The remarkable relationship between the representations for the PDF p(x; t) obtained by the two methods is clarified using Poisson’s summation formula.

Objective This study extends the scope of a previous analysis on the time-averaged acoustic radiation force on a rigid (sound impenetrable) cylinder near a flat boundary [F.G. Mitri, J. Phys. Commun. 2 (2018) 045019] to the case of a viscous compressible fluid (sound penetrable) particle, and determine the time-averaged acoustic radiation torque as well. Motivation and novelty Previous analytical formalisms did not consider the case of a sound penetrable cylindrical particle insonified at an arbitrary angle of incidence (in the polar plane) near a reflecting boundary. This work fills this gap, and provides exact expressions and computations for the acoustic radiation force and torque components. Method The partial-wave series expansion method, in conjunction with the method of images and the translational addition theorem of cylindrical wave functions are used to derive the analytical expressions for the longitudinal and transverse acoustic radiation force components. Moreover, the emergence of a radiation torque that causes the particle to rotate around its center of mass is computed using an exact partial-wave series expression. Results, key conclusion and some perspectives Attractive (pulling), repulsive (pushing) and neutral (zero) forces arise depending on the particle-boundary distance, the cylinder size parameter as well as the angle of incidence (in the polar plane) of the insonifying waves. Emphasis is also given on the emergence of an acoustic radiation torque (that vanishes for a rigid or non-viscous circular cylinder). Computations for the axial radiation torque efficiency anticipate the generation of positive radiation torque, its reversal, in addition to a zero efficiency, leading, respectively, to counter-clockwise, clockwise or lack of particle rotation as the angle of the incident waves deviates from normal incidence with respect to the boundary surface. The extension to the case of an elliptical/oval cylinder near a boundary is mentioned, and replies to some misleading and obtuse comments on the paper [F.G. Mitri, Phys. Fluids 28 (2016) 077104] are provided.

We show that, in strongly chaotic dynamical systems, the average particle velocity can be calculated analytically by consideration of Brownian dynamics in phase space, the method of images and use of the classical diffusion equation. The method is demonstrated on the simplified Fermi-Ulam accelerator model, which has a mixed phase space with chaotic seas, invariant tori and Kolmogorov-Arnold-Moser (KAM) islands. The calculated average velocities agree well with numerical simulations and with an earlier empirical theory. The procedure can readily be extended to other systems including time-dependent billiards., Submitted to Chaos

CA Jandieri Functional post-wall waveguides and waveguide based compact passive circuits - namely bandpass filters - that are formed by introducing additional metallic or dielectric posts inside the post-wall waveguide are analyzed. Scattering parameters (S-parameters) of the waveguide-based bandpass filters are calculated using the method of images combined with the lattice sums technique. The developed formulation is numerically very fast, hence it can be considered as one of the best-suited approaches for the analysis of multiple scattering in post-wall waveguides and for numerical structural optimization. A breeder genetic algorithm is employed as a global search heuristic in order to track down the optimal geometrical parameters for the introduced posts yielding a bandpass filter characteristic in the desired frequency bands. Comparison with other references validates the correctness and utility of our formalism. Different configurations for the specifications of post-wall waveguide-based bandpass filters are discussed.

The purpose of this study is to derive exact partial wave series expansions for the longitudinal and transverse radiation force components, for a circular cylinder in the proximity of a rigid corner space, and illuminated by incident plane waves with arbitrary orientation in the polar plane. Based on the multipole expansion method in cylindrical coordinates, the method of images as well as the translational addition theorem, an effective incident field (resulting from the primary waves as well as the multiple scattered fields from the image sources) is determined first, and used subsequently with the scattered field to derive the mathematical expressions for the radiation force components, stemming from the integration of the radiation stress in a non-viscous fluid. Numerical computations illustrate the analysis for rigid and soft cylinders with particular emphasis on the distances from the particle edges to the neighbouring walls, the size of the cylinder and the angle of incidence. Depending on the choice of these parameters, the radiation force components can vanish, rendering complete “invisibility”; i.e., the cylinder becomes unresponsive to the transfer of linear momentum carried by the incident effective field. Moreover, the radiation force components alternate between positive and negative values, suggesting a force of repulsion or attraction. The results find potential applications in acoustofluidics design and optimization as they shed light on the anechoic radiation force effect on a particle nearby a rigid corner. Other applications in noise and vibration control could also benefit from the results of the present investigation along with further related topics.

A semi-empirical analytical model is developed that predicts the noise produced by a rotor ingesting a boundary layer in proximity to a hard-wall. The rotor boundary layer ingestion noise source is an important source to include when a rotor is installed close to an aircraft fuselage. This is the case for a tail mounted counter rotating open rotor for example. This paper presents three extensions to Amiet's simplified rotor noise model to predict this noise source. The first extension is the method of images, which is used to model the acoustic reflections of the hard-wall. The second extension is an anisotropic velocity spectrum, which is used to model the boundary layer turbulence. The third extension is a numerical switch to account for the partial loading of the rotor as only a part of it is immersed in the boundary layer. The homogeneous anisotropic turbulence model used is a simplification of the actual turbulence the rotor encounters in the boundary layer. In reality, the turbulence in the boundary layer is not homogeneous in the wall-normal direction. Therefore, while the integral length scale in the streamwise direction can be determined from experimental or numerical data, the integral length scale in the wall-normal direction must be chosen empirically. The rotor noise model is validated by comparing its predictions to experimental data at three different advance ratios. The proposed rotor noise model is then used to investigate the effect of the hard-wall on the noise spectrum. The hard-wall affects the downstream directivity more significantly than the upstream directivity for all the rotor operating conditions. Changing the advance ratio of the rotor does not significantly alter the effect that the hard-wall has on the noise spectrum.

The study of the electromagnetic scattering by a particle located near a perfectly conducting corner is a fundamental topic in radar imaging, communications, remote sensing and other applications. In earlier works, the analytical modeling considered the linear field and its scattering properties. Nonetheless, energy efficiencies (or cross-sections) are meaningful fundamental physical observables connected with quadratic quantities of the linear field, which reveal additional properties of the scatterer not obtained from the linear scattered field. The aim of this work is therefore directed toward developing an improved analytical formalism for the energy efficiency factors (or cross-sections) of a cylindrical conducting particle, illuminated by electromagnetic plane waves, and located near a perfectly conducting corner-space. Incident plane progressive waves with an arbitrary angle of incidence are considered in a homogeneous, non-dissipative and non-magnetic fluid. The multiple scattering effects occurring between the electrically-conducting cylindrical particle and corner are described rigorously using the modal expansion method in cylindrical coordinates, the classical method of images and the translational addition theorem. Exact closed-form series expansions are derived and numerical examples illustrate the analysis with particular emphases on the distances from the corner-space, particle size and polarization of the incident field. The results are useful to predict the energetic scattering for various applications, and can serve to validate the results of purely numerical methods in experimental design. Some analogies with the acoustical cross-section theory are also noted.

Herein, we investigate the dynamic anti-plane problem of an elliptic-arc canyon in the corner of an elastic quarter space. The method of wavefunction expansions and the method of images are applied to derive an exact analytical solution for plane-wave incidence. Values of field quantities are evaluated via closed-form expressions involving Mathieu functions. Results are given as benchmarks for numerical studies.

This article addresses efficient implementation of the method of images for acoustic multiple scattering models (MSM) with perfectly reflecting flat boundaries. Time-harmonic problems are first solved in the polar coordinate system for circular scatterers; then the models are extended to the cylindrical coordinate system with (semi-)infinitely long circular cylinders. The MSM in this article is based on the method of separation of variables and Graf’s addition theorem. Derivations are provided for ‘image conditions’ which relate the unknown coefficients of outgoing waves from image scatterers with those of real counterparts. The method of images is applied to wedge-shaped domains with apex angles of π/n rad for a positive integer n. Image conditions make the MSM numerically more efficient: the system of linear equations for unknown coefficients is formulated 2n times faster; its memory requirements are reduced by 4n to the power of 2 times for direct solvers. The proposed model is applied to a benchmark wedge in ocean environment with n = 64. Good agreement is observed between the MSM with image conditions and the boundary element method. Furthermore, half- and quarter-space measurements in an anechoic chamber are in accordance with the correct use of image conditions. Incorrect image conditions reported elsewhere for polar coordinates are also discussed.

The photophoretic asymmetry factors (PAFs) for an absorptive dielectric circular cylinder, located near a perfectly conducting and totally reflecting corner space are derived and computed. The method used in this analysis relies on the modal expansion method in cylindrical coordinates, the classical method of images, and the translational addition theorem of cylindrical wave functions. Initially, the components of the internal electric field vector are obtained stemming from an analysis of the scattering. Subsequently, the solution is used to integrate to the normalized intensity function over the cylinder’s volume to obtain the longitudinal (L) and transverse (T) PAFs. Both TM- and TE-polarized plane progressive waves with arbitrary incidence (in the polar plane) are considered. Attention is given to varying the dimensionless size parameter of the cylinder, the angle of incidence of the incoming waves, and the dimensionless distance parameters from the corner space. Numerical examples illustrate the analysis and demonstrate the net effect of the totally reflecting corner space on the L- and T-PAFs, where negative, positive, and neutral values have been predicted. The results are relevant in applications related to the emergence of the photophoretic force and torque on an absorptive particle located near surfaces and topics in electromagnetic/optical scattering, particle manipulation and assembly, optically bound matter, light–matter interactions, and photopheresis.

Contact analyses of heterogeneous materials may find crucial applications in many tribological studies. However, the simulations of such analyses may be difficult to be implemented since the nonlinear behaviors caused by the material heterogeneity and contact may be more complicated. Based on Eshelby’s equivalent inclusion method (EIM), an arbitrarily shaped inhomogeneity interacting with a contact load may be evaluated by an equivalent inclusion model with a contact load, where the influences of the boundary surface are taken into account through the method of images. In order to improve the computational efficiency, a two-dimensional fast Fourier transform based algorithms (2D-FFT) with zero padding and wrap-around order are employed to solve the contact problem. To benchmark the present solutions, an elastic half-space containing inhomogeneities under Hertzian-type contact load is considered, and parametric studies are further performed to demonstrate the effects of elastic modulus and depth on the stress field.

Efficient implementation of the method of images is addressed for 3D multiple scattering models (MSM) for spheres with perfectly-reflecting flat surfaces. The Helmholtz equation is solved by the spherical-coordinate separation-of-variables approach and addition theorems for spherical wavefunctions. The unknown coefficients of outgoing waves from image objects are related with those of real counterparts through ‘image conditions’ which are derived in this article. The method of images is applied to concave part of wedge-shaped acoustic domains with apex angles of π/n rad for a positive integer n, which includes half-space and corners. Image conditions make the MSM numerically more efficient: memory space is reduced by 4n2 times; matrix is populated 2n- times faster for infinite or 2D wedges. Savings are 16n2 times in memory and 4n times in speed for semi-infinite or 3D wedges. Image conditions are valid regardless of the type of scatterers as long as they are spheres and submerged in acoustic domains; they are also suitable for the modified Helmholtz equation and radiation problems. However, specific formulae of image conditions depend on definitions of the spherical harmonics. Image conditions for rigid flat surfaces are verified by measurements of 13 balls in an anechoic chamber for configurations of half-space, 2D & 3D corners and 3D wedge with n=3. Image conditions for pressure-release flat interfaces are validated by the boundary element method (BEM) for the pulsation mode of an underwater air sphere in half-space and for scattering by a sphere in an ocean environment with the wedge angle of 1.2° by n=150. Agreement is very good between the MSM and measurements and is impeccable between the MSM and BEM.

We studied the force and potential for the vortex around moat corner as an example of the concave edge of the superconducting film by the numerical calculation method. Vortex behavior is one of the significant factors that determine the characteristics of superconducting devices. In some simple cases, such as a straight line with a constant width, the potential for the vortex can be calculated analytically using the method of images. On the other hand, it is difficult to apply such method to arbitrary-shaped films, for example, constricted nanobridges, and moats in single-flux-quantum circuits. Control of the vortex motion is very important for such devices. We have been studying the vortex behavior by numerical calculation using the time-dependent Ginzburg–Landau equation. Vortex around the straight edge moved to the moat and accelerated with a trajectory vertical to the edge. On the other hand, vortex around the corner (concave edge) moved to the moat avoiding the corner. We calculated the force and potential distribution for the vortex inside the superconducting film. The force acting to the vortex was found to be smaller around the corner compared with that around the moat side. Such behavior was understood by the difference of the vortex distortion.

A fast time-domain method is developed in this paper for the real-time prediction of the six degree of freedom motions of a vessel traveling in an irregular seaway in infinitely deep water. The fully coupled unsteady ship motion problem is solved by time-stepping the linearized boundary conditions on both the free surface and body surface. A velocity-based boundary integral method is then used to solve the Laplace equation at every time step for the fluid kinematics, while a scalar integral equation is solved for the total fluid pressure. The boundary integral equations are applied to both the physical fluid domain outside the body and a fictitious fluid region inside the body, enabling use of the fast Fourier transform method to evaluate the free surface integrals. The computational efficiency of the scheme is further improved through use of the method of images to eliminate source singularities on the free surface while retaining vortex/dipole singularities that decay more rapidly in space. The resulting numerical algorithm runs 2–3 times faster than real time on a standard desktop computer. Numerical predictions are compared to prior published results for the transient motions of a hemisphere and laboratory measurements of the motions of a free running vessel in oblique waves with good agreement.

Within a first-order time-dependent perturbation approach, we compute the spontaneous decay rate of a two-level system placed in the vicinity of a perfectly reflecting spherical surface. We consider a model system on which the emitter is represented by a two-level monopole coupled to a Hermitian massless scalar field. Using the method of images, we determine the appropriate Green’s function evaluated in the world line of the atom. The change in the spontaneous decay rate results from the interaction of the atom with its image. We provide a detailed analysis of the dependence of the decay rate on the sphere’s radius, the atom’s location, and the emitted radiation frequency. Both exterior and interior problems are discussed.

The applied potential, or mise‐a‐la‐masse, method is used in mineral exploration and environmental applications to constrain the shape and extent of conductive anomalies. However, few simple calculations exist to help gain understanding and intuition regarding the pattern of measured electrical potential at the ground surface. While it makes intuitive sense that the conductor must come close to the ground surface in order for the lateral extent of the potential anomaly to be affected by the dimensions of the conductor rather than simply by the depth, no simple calculation exists to quantify this effect. In this contribution, a simple method of images solution for the case of a sphere of constant electrical potential in a conducting half‐space is presented. The solution consists of an infinite series where the first term is the same as the method of images solution for a point current source in an infinite half‐space. The higher order terms result from the interaction of the constant potential sphere with the no‐flux boundary condition representing the ground surface and cause the change in the shape of the potential anomaly that is of interest in the applied potential method. The calculation is relevant to applied potentials when the conductive anomaly is limited in all three space dimensions and is highly conductive. Using the derived formula, it is shown that, while the electrical potential at the ground surface caused by the sphere is affected even when the sphere is quite deep, the ratio of the potential to the current, a quantity that is more relevant to the applied potential method, is not affected until the centre of the sphere is within two radii of the ground surface. An expression for the contact resistance of the sphere as a function of depth is also given, and the contact resistance is shown to increase by roughly 45% as the sphere is moved from great depth to the ground surface.

Room impulse responses can be simulated using various methods that differ in their computational complexity and accuracy. While boundary element methods (BEM) potentially can provide accurate simulations in complex geometries including effects such as diffraction and absorption, their use is limited due to relatively large values of parameter kD (k is the wavenumber and D is the room diameter). The size of the mesh also grows as the square of this parameter. We show that the fast multipole (FMM) accelerated BEM realized on multicore workstations can handle problems for rooms of D ∼ 10 m for the frequency range up to ~5 kHz (kD ∼ 1000). For such problems, needing meshes with several million elements, stable and efficient methods for high frequency FMM are needed. We discuss these issues and demonstrate results of such computations both in frequency and time domains. In simple cases numerical solutions are compared with analytical solutions obtained by the method of images. For more complex cases including rooms of different shapes with baffles and openings two methods, the direct and indirect BEM, are studied and their performance discussed. [This work is supported by VisiSonics Corporation.]

A new class of traversable wormholes was recently constructed which relies only on local bulk dynamics rather than an explicit coupling between distinct boundaries. Here we begin with a four-dimensional Weyl fermion field of any mass $m$ propagating on a classical background defined by a ${\mathbb Z}_2$ quotient of (rotating) BTZ $\times \, S^1$. This setup allows one to compute the fermion stress-energy tensor exactly. For appropriate boundary conditions around a non-contractible curve, perturbative back-reaction at any $m$ renders the associated wormhole traversable and suggests it can become eternally traversable at the limit where the background becomes extremal. A key technical step is the proper formulation of the method of images for fermions in curved spacetime. We find the stress-energy of spinor fields to have important kinematic differences from that of scalar fields, typically causing the sign of the integrated null stress-energy (and thus in many cases the sign of the time delay/advance) to vary around the throat of the wormhole. Similar effects may arise for higher-spin fields., Comment: 30 pages, 5 figures

To allow wearable magnetoencephalography (MEG) recordings to be made on unconstrained subjects the spatially inhomogeneous remnant magnetic field inside the magnetically shielded room (MSR) must be nulled. Previously, a large bi-planar coil system which produces uniform fields and field gradients was used for this purpose. Its construction presented a significant challenge, six distinct coils were wound on two 1.6 × 1.6 m2 planes. Here, we exploit shared coil symmetries to produce coils simultaneously optimised to generate homogenous fields and gradients. We show nulling performance comparable to that of a six-coil system is achieved with this three-coil system, decreasing the strongest field component Bx by a factor of 53, and the strongest gradient dBx/dz by a factor of 7. To allow the coils to be used in environments with temporally-varying magnetic interference a dynamic nulling system was developed with a shielding factor of 40 dB at 0.01 Hz. Reducing the number of coils required and incorporating dynamic nulling should allow for greater take-up of this technology. Interactions of the coils with the high-permeability walls of the MSR were investigated using a method of images approach. Simulations show a degrading of field uniformity which was broadly consistent with measured values. These effects should be incorporated into future designs.

We consider a chiral fermion at non-zero temperature on a circle (i.e., on a torus in the Euclidean formalism) and compute the modular Hamiltonian corresponding to a subregion of the circle. We do this by a very simple procedure based on the method of images, which is presumably generalizable to other situations. Our result is non-local even for a single interval, and even for Neveu-Schwarz boundary conditions. To the best of our knowledge, there are no previous examples of a modular Hamiltonian with this behavior., Comment: 6 pages, 3 figures

Diffusion processes with boundaries are models of transport phenomena with wide applicability across many fields. These processes are described by their probability density functions (PDFs), which often obey Fokker-Planck equations (FPEs). While obtaining analytical solutions is often possible in the absence of boundaries, obtaining closed-form solutions to the FPE is more challenging once absorbing boundaries are present. As a result, analyses of these processes have largely relied on approximations or direct simulations. In this paper, we studied two-dimensional, time-homogeneous, spatially-correlated diffusion with linear, axis-aligned, absorbing boundaries. Our main result is the explicit construction of a full family of closed-form solutions for their PDFs using the method of images (MoI). We found that such solutions can be built if and only if the correlation coefficient $\rho$ between the two diffusing processes takes one of a numerable set of values. Using a geometric argument, we derived the complete set of $\rho$'s where such solutions can be found. Solvable $\rho$'s are given by $\rho = - \cos \left( \frac{\pi}{k} \right)$, where $k \in \mathbb{Z}^+ \cup \{ +\infty\}$. Solutions were validated in simulations. Qualitative behaviors of the process appear to vary smoothly over $\rho$, allowing extrapolation from our solutions to cases with unsolvable $\rho$'s., Comment: 11 pages

We present a new class of electrostatics problems that are exactly solvable by adding finitely many image charges. Given a charge at some location inside a cavity bounded by up to four conducting grounded segments of spheres: if the spheres have a symmetry derived via a stereographic projection from a 4D finite reflection group, then this is a solvable generalization of the familiar problem of a charge inside a spherical cavity. There are 19 three-parametric families of finite groups formed by inversions relative to at most four spheres, each member of each family giving a solvable problem. We solve a sample problem which derives from the reflection group $\mathbf{D}_{4}$ and requires 191 image charges., 26 pages, 6 figures Submitted to Foundations of Physics

An analytical method is proposed for computing the low-Reynolds-number hydrodynamic mobility function of a small colloidal particle asymmetrically moving inside a large spherical elastic cavity, the membrane of which is endowed with resistance toward shear and bending. In conjunction with the results obtained in the first part [Daddi-Moussa-Ider, L\"{o}wen, and Gekle, Eur. Phys. J. E 41, 104 (2018)], in which the axisymmetric motion normal to the surface of an elastic cavity is investigated, the general motion for an arbitrary force direction can be addressed. The elastohydrodynamic problem is formulated and solved using the classic method of images through expressing the hydrodynamic flow fields as a multipole expansion involving higher-order derivatives of the free-space Green's function. In the quasi-steady limit, we demonstrate that the particle self-mobility function of a particle moving tangent to the surface of the cavity is larger than that predicted inside a rigid stationary cavity of equal size. This difference is justified by the fact that a stationary rigid cavity introduces additional hindrance to the translational motion of the encapsulated particle, resulting in a reduction of its hydrodynamic mobility. Furthermore, the motion of the cavity is investigated, revealing that the translational pair (composite) mobility, which linearly couples the velocity of the elastic cavity to the force exerted on the solid particle, is solely determined by membrane shear properties. Our analytical predictions are favorably compared with fully-resolved computer simulations based on a completed-double-layer boundary integral method., Comment: 14 pages, 4 figures. This is a pre-print of an article published in The European Physical Journal E. The final authenticated version is available online at: https://doi.org/10.1140/epje/i2019-11853-4

Although driven Brownian particles are ubiquitous in stochastic dynamics and often serve as paradigmatic model systems for many aspects of stochastic thermodynamics, fully analytically solvable models are few and far between. In this paper, we introduce an iterative calculation scheme, similar to the method of images in electrostatics, that enables one to obtain the propagator if the potential consists of a finite number of steps. For the special case of a single potential step, this method converges after one iteration, thus providing an expression for the propagator in closed form. In all other cases, the iteration results in an approximation that holds for times smaller than some characteristic timescale that depends on the number of iterations performed. This method can also be applied to a related class of systems like Brownian ratchets, which do not formally contain step potentials in their definition, but impose the same kind of boundary conditions that are caused by potential steps., 19 pages, 6 figures

This study investigates the effect of an orthogonal-shaped reflecting breakwater on the hydrodynamic characteristics of a vertical cylindrical body. The reflecting walls are placed behind the body, which can be conceived as a floater for wave energy absorption. Linear potential theory is assumed, and the associated diffraction and motion radiation problems are solved in the frequency domain. Axisymmetric eigenfunction expansions of the velocity potential are introduced into properly defined ring-shaped fluid regions surrounding the floater. The hydrodynamic interaction phenomena between the body and the adjacent breakwaters are exactly taken into account by using the method of images. Results are presented and discussed concerning the exciting wave forces on the floater and its hydrodynamic coefficients, concluding that the hydrodynamics of a vertical cylindrical body in front of an orthogonally shaped breakwater differ from those in unbounded waters.

This paper is concerned with bubble dynamics at a corner formed by two flat rigid boundaries associated with applications in ultrasonic cleaning and cavitation damage. This phenomenon is modeled using the potential flow theory and the boundary integral method. The Green’s function is obtained to satisfy the impenetrable conditions at the rigid boundaries using the method of images with the corner angle α = π/k, where k is a natural number. To evaluate the numerical model, experiments were carried out with a spark-generated bubble in water and recorded using a high-speed camera. The predicted bubble shapes are in excellent agreement with those from the experiments. A jet forms toward the end of the collapse, pointing to the corner when initiated at the bisector of the two walls but pointing to the near wall and inclined to the corner when initiated near one of the two walls. The Kelvin impulse theory predicts the jet direction well. As compared to a bubble near a flat wall, the oscillation period and the jet width increase but the jet velocity decreases. The bubble migrates away from the near wall and the corner during its expansion and moves back toward them during its collapse, but at a much larger speed and amplitude. A velocity stagnation point forms at the start of the collapse, and a high-pressure zone is generated at the base of the jet during the late stages of the collapse, which drives the jet and the bubble toward the near wall and the corner.

The typical approach to studying decoherence begins by examining a combined system and interacting environment (termed bath), and then deriving a master equation to examine the behavior of the density matrix after tracing over the bath, typically by invoking the Markovian approximation. This approach is quite successful for a large variety of systems however is frequently a non-intuitive picture that can mask the fundamental behavior of the system. We examine here a simple model that captures the essence of decoherence and study it using the wavefunction picture, invoking the analytic concepts of the correspondence principle, the method of images, the phase jitter concept, and the scattering matrix formalism. We complement this with a numerical study using the split operator FFT method, the forced harmonic oscillator method, and the thawed Gaussian approximation.