Sommerfeld Integrals Research Articles

Dynamic response mechanism of the hole subjected to the 3D point source disturbance in an isotropic formation

Accurate estimation of the effects of dynamic disturbances on stress concentration is crucial for the stability of rock engineering and the corresponding analytical approaches are needed. This study presents an analytical approach to calculate the relative stress distribution with a point source inside and outside a hole using the linear theory of elasticity. The Helmholtz potentials and Sommerfeld integral are employed to describe the displacement and stress components, and then formulate the equilibrium equations to solve the equivalent stress distribution around the hole. Numerical examples demonstrate the impact of model parameters on the equivalent stress, such as frequency, hole radius, source location, etc. It is found that sometimes high frequencies can make the equivalent stress greater far from the source than that close to the source. Additionally, when the ratio of the distance between the source and the hole axis to the hole radius exceeds ten, the equivalent stress distribution around the hole remains nearly constant. This approach can be used for the design and assessment of underground engineering structures' stability under dynamic disturbances.

Improvement of optical wave propagation simulations: the scaled angular spectrum method for far-field and focal analysis

In the field of optics, accurately simulating wave propagation is essential for both theoretical insights and practical applications. This study introduces the scaled angular spectrum method (Sc-ASM) for simulating far-field and focal patterns, overcoming the limitations inherent in the standard angular spectrum method (ASM) by allowing variable sampling intervals between source and observation planes. Comparative analysis with the Rayleigh–Sommerfeld integral demonstrates Sc-ASM's superior accuracy in estimating the far-field patterns of beams with complex phase structures. Our results underscore Sc-ASM’s potential to set a new benchmark in optical simulations, significantly advancing optical system design and the study of wave propagation.

Evaluation of the Inductive Coupling between Coplanar Concentric Coils in the Presence of the Ground

An analytical approach is presented that allows deriving an exact series-form representation for the flux linkage between two physically large concentric circular coils located on a lossy soil. The expression comes from a three-step analytical procedure. First, the integral expression for the flux linkage is converted into a double integral consisting of a finite and a semi-infinite integral. Next, the semi-infinite integral is recognized to be a well-known tabulated Sommerfeld integral, which may be analytically evaluated straightforwardly. Finally, applying Lommel’s expansion allows rewriting the remaining finite integral as a sum of elementary integrals amenable to analytical evaluation. As a result, the flux linkage between the two coils is given as a sum of spherical Hankel functions of the wavenumber in the air and in the ground, multiplied by a coefficient depending on the geometrical dimensions of the coils. The accuracy and robustness of the proposed formulation is tested by comparing its outcomes with those generated by numerical integration of the complete integral representation for the flux linkage and with the results provided by previous analytical approaches to the same problem. It is found that the use of the derived expression for the inductance makes it possible to obtain significant time savings as compared to numerical quadrature schemes.

Estimation of the Born data in inverse scattering of layered media

The first term in the Born series, as we call the Born data, is linear with the scatterers. Here we present a scheme to map the total field data to the Born data in layered media using only the single-input single-output (SISO) setup. This nonlinear mapping is based on the reduced order model (ROM) approach, which constructs ROMs of the original wave operator. Normally, the construction of ROMs requires multi-input multi-output data. By introducing fictitious sensors, we estimate the Born data with SISO data in layered media. We give a simple way of using the time-domain Green’s function to estimate the received data for other fictitious sensors without calculating the complicated Sommerfeld integral. The resulting Born data contains only the single-scattering component, which can be helpful for many imaging applications. A numerical example is given incorporating the direct imaging back-propagation method. It validates the linearity of the Born data by providing an artifact-free image without the optimization process.

Calculation of Sommerfeld Integrals in Dipole Radiation Problems

This article proposes asymptotic methods for calculating Sommerfeld integrals, which enable us to calculate the integral using the expansion of a function into an infinite power series at the saddle point, where the role of a rapidly oscillating function under the integral can be fulfilled either by an exponential or by its product by the Hankel function. The proposed types of Sommerfeld integrals are generalized on the basis of integral representations of the Hertz radiator fields in the form of the inverse Hankel transform with the subsequent replacement of the Bessel function by the Hankel function. It is shown that the numerical values of the saddle point are complex. During integration, reference or so-called standard integrals, which contain the main features of the integrand function, were used. As a demonstration of the accuracy of the technique, a previously known asymptotic formula for the Hankel functions was obtained in the form of an infinite series. The proposed method for calculating Sommerfeld integrals can be useful in solving the half-space Sommerfeld problem. The authors present an example in the form of an infinite series for the magnetic field of reflected waves, obtained directly through the Sommerfeld integral (SI).

The electromagnetic field of a current conductor in the presence of lossy half-space

In this paper, the electromagnetic field problem of a line current conductor in the presence of homogenous and ISO-tropic lossy half-space is analyzed. The main difficulty in this analysis is to take correctly into account the influence of ground finite conduct since the use of integral transformations in Maxwell's equations leads to Sommerfeld integral, hard to evaluate even numerically. Charge Simulation Method, based on the equivalence theorem of different electromagnetic systems, is proposed here for the numerical approach. Two independent, equivalent systems created of appropriately chosen, shaped, and arranged fictitious sources are suggested for the determination of EM field components , in the air and in the lossy half-space. Point-matching method based on the boundary conditions between media is applied for the determination of the fictitious source intensities. The proposed procedure has the advantage that it takes into account the influence of the finite ground conductivity, without calculation of Sommerfeld integral and it ensures excellent accuracy with a small number of fictitious sources. Also, the features of the proposed method are simple impel-mentation on a standard PC, fast and accurate simulation procedure, and easy generalization for some other cases of arbitrarily oriented wires of finite lengths above a lossy half space.

A Burton–Miller-type boundary element method based on a hybrid integral representation and its application to cavity scattering

This study builds on a recent paper by Lai et al. (2018) in which a novel boundary integral formulation is presented for scalar wave scattering analysis in two-dimensional layered and half-spaces. The seminal paper proposes a hybrid integral representation that combines the Sommerfeld integral and layer potential to efficiently deal with the boundaries of infinite length. In this work, we modify the integral formulation to eliminate the fictitious eigenvalues by employing the integral equation of the Burton–Miller type. We also discuss reasonable parameter settings for the hybrid integral equation to ensure efficient and accurate numerical solutions. Furthermore, we extend the modified formulation for the scattering from a cavity in a half-space whose boundary is locally perturbed. To address the cavity scattering, we introduce a virtual boundary enclosing the cavity and couple the integral equation on it with the hybrid equation. The effectiveness of the proposed method is demonstrated through numerical examples.

Calculation of the intensity at the sharp focus of a cylindrical vector beam by three methods

We conduct a comparative analysis of diffraction fields upon sharply focusing vortex and non-vortex incident beams, calculated using three non-paraxial methods. The methods employed are a finite difference time domain (FDTD) method, a Rayleigh–Sommerfeld integral, and a Richards–Wolf transformation. The Richards–Wolf transformation is used with two apodization functions of the exit pupil, for a spherical lens and a thin diffractive lens. The numerical simulation shows that the Rayleigh–Sommerfeld integral and the Richards–Wolf transformation can produce almost the same result and save time significantly. Meanwhile, the root-mean-square deviation of the results of both methods from the FDTD method can be as low as 2%. If an ultra-short focal length is used, the Richards–Wolf transformation is found to be more accurate, whereas with increasing distance from the initial plane and outside the focal plane, the Rayleigh-Sommerfeld integral is more accurate.

Near-Field Gain Expression for Tapered Circular Aperture Antennas

Near-field electromagnetic wave, which is radiated by tapered circular aperture antennas, has found more engineering applications in many diverse fields. Through the simulation of a reflector antenna whose efficiency is up to 51.4%, it can be found that the aperture distribution cannot be regarded as uniform distribution but as tapered distribution. Thus, expressions for near fields of uniform circular apertures cannot be applied to tapered circular aperture antennas in practical engineering. To alleviate such problem, near-field calculation formulas are proposed based on the Sommerfeld integrals with steepest descents technique, which ensures the accurate and efficient analysis of the near field of the tapered circular aperture antenna. And near-field characteristics of tapered circular aperture antennas are displayed. The maximum difference of normalized near-field amplitudes between tapered and uniform distribution is 0.35V/m. Three-dimensional near-field gain of circular aperture antennas is developed from the near-field radiation pattern. The proposed expression extends the range of calculation from the axial to more directions compared with the previous expression. Also, this expression is valid in both the near field and the far field. Using full-wave simulations, results obtained from the proposed expression are compared with the simulated results. And the deviation in near-field gain is less than 0.3dB, which indicates the convenience and effectiveness of our expression. The proposed near-field gain expression can be applicable to the near-field measurement and design of millimeter or terahertz radar systems that work in the near-field region.

Electromagnetic Field Variation of ELF Near-Region Excited by HED in a Homogeneous Half-Space Model

Great attention has been paid to the propagation of electromagnetic (EM) waves across the sea surface due to its important applications. Most of the previous research, however, focuses on the half-space model illustrating the deep sea environment. In this paper, EM field distribution in the extremely low frequency (ELF) near-region under horizontal electric dipole (HED) excitation in homogeneous half-space seawater is analyzed based on the general expression of the Sommerfeld integral using the quasistatic approximation method. The focus is on deriving complete and effective solutions in air and seawater regions under the cylindrical coordinates for the EM near-field, which is generated by an HED in a shallow sea. The resulting formulas can be given by a few summands in closed form as the well-known Fourier–Bessel integrals. The analytical approximate expression of ELF Sommerfeld EM field integral excited by the HED in the homogeneous half-space seawater is deduced under the condition that the propagation distance ρ satisfies kρ << 1. To this end, the EM field distribution in the range close to the HED antenna in seawater is simulated, the results have shown that the minimum attenuation value of the vertical electric component Ez is about 15 dB, and that of the radical magnetic components Hφ is about 30 dB, and these values are found to be of greatest potential for the near-field region propagation among the electric and magnetic components. Finally, the correctness of the proposed method is verified by comparison with Pan’s approximation method and Margetis’s exact expression approximation method, which demonstrated the correctness of the proposed method.

Constituents of electromagnetic 2-D layered media Green's functions for all material types and radiation conditions

Layered media Green's functions (LMGFs) are obtained for line sources with complex longitudinal wavenumbers in planarly layered media by numerically evaluating Sommerfeld integrals. The results obtained in this article can be applied to the numerical analysis of structures in layered media composed of any combination of right-handed materials (RHMs) and left-handed materials (LHMs). In the derivation, due to the complex-valued longitudinal wavenumber values, two different boundary conditions are used to include plane waves with: (i) decaying and (ii) outgoing Poynting vectors in the LMGFs. As a result, two completely different LMGFs are obtained by using two different radiation conditions and corresponding Sommerfeld integration paths (SIP) that are rigorously tailored on the complex integrand plane. In this work, all the necessary numerical tools for calculating the 2-D LMGFs are provided to be implemented in a computer program from scratch.

Complete Nonuniform Asymptotic Expansion of Sommerfeld Integral for Dielectric Half-Space

Generally, the Sommerfeld integral (SI) for a dielectric half-space is approximated based on the steepest descent method. A higher order approximation of the integral requires higher order derivatives of the integrand in the steepest descent path in the complex domain, whose explicit formulation may be cumbersome. Recently, a new finite-range integral representation of the SI has been proposed, whose integrand is written in terms of the SI for an impedance half-plane. Based on the known complete nonuniform asymptotic expansion of the SI for an impedance half-plane, the complete nonuniform asymptotic expansion can be analytically formulated for the dielectric half-space. Moreover, the completeness is mathematically proven such that the expansion satisfies the recurrence relation for the Wilcox expansion. The accuracy of the proposed expansion is numerically examined for several propagation scenarios. Furthermore, a numerical procedure is proposed to calculate the higher order terms of the expansion by partially using the impedance approximation for a large relative permittivity of the half-space.

A Low-Frequency Approximation PEEC Model for Thin-Wire Grounding Systems in Half-Space

This paper presents a PEEC model of thin-wire structures for lightning grounding analysis based on low-frequency approximation of dyadic Green's functions. To avoid the time-consuming numerical Sommerfeld integrals, the primary and quasi-dynamic terms are extracted from the spectral-domain Green's function and used as the low-frequency approximation closed-form of the full-wave solution, which is appropriate in the near-field region. The corresponding equivalent circuit is constructed with this closed-form approximated Green's function and the formulas for the partial inductance and coefficient of potential is given. Comparison in terms of the Green's function and circuit parameters between the proposed method and the conventional modified-image method is made. The validity of the proposed method is confirmed with experimental results gathered from published literature and MOM-calculated results from other researchers. A systematic parametric analysis is conducted to evaluate the accuracy and applicability of the different approximate models, which clearly shows the superiority of the proposed method over the traditional modified-image method.

A Full-Wave PEEC Model for Thin-Wire Structure in the Air and Homogeneous Lossy Ground

This article presents an efficient full-wave partial element equivalent circuit (PEEC) formulation of thin-wire structure in the full space of air-homogeneous lossy ground for lightning transient analysis. First, the Green's functions for the case of an air-uniform ground are derived, which involve Sommerfeld-type integrals. Next, using the singularity subtraction technique, a new form of Green's functions is obtained, which has the advantage of being less singular and smooth compared with the original form. In the full-wave method, the efficient and accurate computation of partial elements is crucial for the practical application of the PEEC formulation. A three-dimensional cubic interpolation scheme is employed to accelerate the computation of Sommerfeld integrals. To improve the computational efficiency and accuracy of the interpolation, the continuity and symmetry of the Green's functions are investigated in a comprehensive way, which is important for a successful interpolation. Numerical and experimental examples from the literature are presented to validate the accuracy of the proposed model.

The Sommerfeld Integral in Problems of Simulating the Diffraction of Acoustic Waves Using a Triangular Lattice

Analytical solutions are obtained for two problems for a discrete analogue of the Helmholtz equation on a triangular lattice: (1) the problem of radiation from a point source on a plane; (2) the problem of diffraction on a half-line with Dirichlet boundary conditions. It is shown that in these problems, the complete field can be represented as an integral of an algebraic function over a family of contours located on some complex manifold. The solution to the first problem is found as an integral of some differential form over this manifold, and the asymptotics of the far field for this solution is obtained. The second problem is solved using an analogue of the Sommerfeld integral. It is checked that the obtained solution coincides with the solution of this problem by the Wiener–Hopf method.

Explicit solution of a Dirichlet problem in nonconvex angle

In the present work, we give an explicit solution of the Dirichlet boundary-value problem for the Helmholtz equation in a nonconvex angle with periodic boundary data. We present uniqueness and existence theorems in an appropriate functional class and we give an explicit formula for the solution in the form of the Sommerfeld integral. The method of complex characteristics [14] is used.

An Adaptive Rational Fitting Technique of Sommerfeld Integrals for the Efficient MoM Analysis of Planar Structures

The integral equation method is one of the most successful computational models for microwave devices or integrated circuits in planar layered media. However, the efficient and accurate evaluation of the associated Green’s function consisting of Sommerfeld integrals (SIs) is still a remaining challenge. To mitigate this difficulty, this work proposes a spatial domain rational function fitting technique (RFFT) for SIs so that the approximation accuracy is controllable. In conjunction with an adaptive sampling strategy, the proposed RFFT minimizes the orders of rational functions, and the resultant SI evaluation efficiency is optimized. In addition, we investigate the semi-analytical singularity treatment for the rational expression of SIs in method of moment (MoM) implementation. Extensive simulation of representative planar devices validates the correctness of the proposed method and demonstrates its superior performance over conventional SI approximation methods.

Schelkunoff Formulation of the Sommerfeld Integral for the Hertzian Dipole Located in the Vicinity of Cylindrical Structures

AbstractA numerical integration of the Sommerfeld integral is performed using the Schelkunoff formulation for cylindrical media. The Schelkunoff kernel for cylindrical media involves higher order modified Bessel functions with azimuthal summation over higher order modes. As such, the convergence characteristics of the cylindrical integral kernel are strongly dependent on complex linear combinations of higher order Bessel/Hankel/modified Bessel functions, compared to the case of the planar media where only a single Bessel/modified Bessel function of zeroth order is present. Two cylindrical configurations are analyzed using the new formulation, viz. a conducting cylinder and a dielectric‐coated conducting cylinder. The branch‐point singularity in the first configuration is removed using the angular transformation for the Sommerfeld/Schelkunoff formulations. A path deformation technique is used for the second configuration to address the problem of poles and branch‐point singularities on the real axis of integration. The in‐depth analysis of the cylindrical kernels and the integrals with variation in the location of the observation point clearly bring out the relative merits of both formulations for the cylindrical configurations, with the TE/TM coupling for the coated cylinder considered.

Compact expression of dyadic Green's function for homogeneous dielectric half‐space and its efficient double integral representation

The dyadic Green's function can be represented via spatial derivatives of two Sommerfeld integrals when an infinitesimal electric dipole and an observation point are located above an infinite planar dielectric interface. The two dielectric spaces are assumed to be homogeneous. Recently, new representations have been proposed for two Sommerfeld integrals and their complete uniform asymptotic expansions. The new formulation for the dyadic Green's function contains fourth-order spatial derivatives, so its explicit expression is lengthy and not compact. Thus, a compact form of the dyadic Green's function was formulated by converting the fourth-order spatial derivatives into second-order ones. Based on the new representation, a numerically efficient double integral expression of Green's function was obtained and numerically verified. In addition, five components among nine total electric field components were analytically evaluated in terms of the incomplete Hankel function when the source and observation point both are on the interface.

Closed-Form Evaluation of Mixed Potential Shielded Layered Media Green’s Functions With Spectral Differential Equation Approximation Method

A novel, robust, and computationally efficient approach is proposed for the evaluation of the Michalski–Zheng’s mixed-potential Green’s functions of general shielded layered media. A high-order variant of the spectral differential equation approximation method (SDEAM) is used to cast the spectra of Green’s functions’ components into pole-residue forms. The latter allows for closed-form evaluation of the Sommerfeld integrals (SIs) defining spatial components of the mixed-potential Green’s functions required by Method-of-Moment (MoM) discretization of different types of integral equations. The method produces Green’s functions for all elevations of interest at a fixed cost, making it substantially faster than popular alternative approaches. Numerical results validate the developed high-order SDEAM scheme and demonstrate its extended range of validity compared to the discrete complex image method (DCIM).