Terms Of Spherical Wave Functions Research Articles

An Alternative Plane Wave Decomposition of Electromagnetic Fields Using the Spherical Wave Expansion Technique

In this letter, we expand electromagnetic fields into plane waves by first employing a spherical wave expansion (SWE) and determining the coefficients of the plane wave expansion in terms of spherical wave functions. This allows for a more general and computationally efficient approach compared to the traditional plane wave expansion. Our technique consists of two main parts. First, SWE is used to express any electromagnetic field, which can be specified analytically or by numerical data. The unknown SWE coefficients are obtained using a numerical technique. Then, the spherical vector wave functions are converted into a series of plane waves. Results demonstrate excellent numerical accuracy and efficiency. The method can be applied to both antenna and scattering problems.

Study of axial acoustic radiation force on a sphere in a Gaussian quasi-standing field

Based on the finite series method, the Gaussian standing or quasi-standing beam is expressed in terms of spherical wave functions and a weighting parameter, which describe the beam shape and location relative to the particle. An expression is derived for the radiation force on a sphere centered on the axis of a Gaussian standing or quasi-standing wave propagating in an ideal fluid. Rigid, fluid, elastic, and viscoelastic spheres immersed in water are treated as examples. In addition, a method is proposed to compute the axial acoustic radiation force when the sphere is translated axially. Results indicate the capability of the proposed method to manipulate and separate spheres based on their mechanical and acoustical properties. The interaction of a Gaussian quasi-standing beam with a sphere can result in periodic axial force under specific operating conditions. The results presented here may provide a theoretical basis for the development of acoustical tweezers in a Gaussian standing beam, which would be useful in micro-fluidic lab-on-chip applications.

Separation of variables method for multilayered nonspherical particles

A separation of variables method based on expansions of the electromagnetic fields in terms of spherical wave functions is expanded at nonspherical (axisymmetric) particles with a rather large number of layers. Commonly used alternative approaches to systems of linear algebraic equations relative to unknown field expansion coefficients for layered particles are considered in some detail. The SVM code developed is compared with the EBCM, GMT and DDA codes designed for multilayered scatterers and some numerical results obtained for nonspherical scatterers with up to 100 layers are presented as illustrations.

Light scattering by multilayered axisymmetric particles: Solution of the problem by the separation of variables method

A new solution to the problem of light scattering by multilayered particles possessing axial symmetry is obtained. Two methods are applied for this purpose. One is the separation of variables method with expansion of fields in terms of spherical wave functions, and the other is a novel approach based on the separation of fields into axisymmetric and nonaxisymmetric parts and on the choice of specific scalar potentials for each of them. A specific feature of the new solution is that the dimension of truncated linear algebraic systems used for determining unknown expansion coefficients of fields does not increase with an increasing number of layers. Using double-and three-layer spheroidal and Chebyshev particles of different shape and size as examples, the domain of applicability of the solution presented is compared with that of the solution previously obtained by the extended boundary conditions method. Except for nearly spherical particles, the solution presented is shown to be more favorable than the previously obtained solution.

Acoustic scattering by a penetrable spheroid

The scattering of a plane acoustic wave from a penetrable prolate or oblate spheroid is considered. Two different methods are used for the evaluation. In the first, the pressure field is expressed in terms of spheroidal wave functions. In the second, a shape perturbation method, the field is expressed in terms of spherical wave functions only, while the equation of the spheroidal boundary is given in spherical coordinates. Analytical expressions are obtained for the scattered pressure field and the various scattering cross-sections when the solution is specialized to small values of the eccentricity h = d/(2a), (h ≪ 1), with d being the interfocal distance of the spheroid and 2a the length of its rotation axis. In this case, exact, closed-form expressions are obtained for the expansion coefficients g(2) and g(4) in the relation S(h) = S(0)[1 + g(2)h2 + g(4)h4 + O(h6)] expressing the scattered field and the scattering cross-sections. Numerical results are given for various values of the parameters.

An Equivalence Theorem Based on the Use of Electric Currents Radiating in Free Space

The most common formulations of the equivalence principle use both electric and magnetic currents radiating in free space or only electric (magnetic) currents radiating in presence of a perfectly magnetic (electric) conductor. In this paper, a formulation is provided for equivalent currents of only electric (magnetic) type as a function of the tangential total fields of the original problem over the equivalence surface. The formulation of these equivalent currents becomes simple for some canonical surfaces, like a plane, a cylinder and a sphere. Explicit expressions of the equivalent electric currents on a spherical surface are provided in terms of spherical wave functions.

Acoustic scattering by an impenetrable spheroid

The scattering of a plane acoustic wave from an impenetrable, soft or hard, prolate or oblate spheroid is considered. Two different methods are used for the evaluation. In the first, the pressure field is expressed in terms of spheroidal wave functions. In the second, a shape perturbation method, the field is expressed in terms of spherical wave functions only, while the equation of the spheroidal boundary is given in spherical coordinates. Analytical expressions are obtained for the scattered pressure field and the various scattering cross-sections, when the solution is specialized to small values of the eccentricity h = d/(2a) , where d is the interfocal distance of the spheroid and 2a is the length of its rotation axis. In this case, exact, closed-form expressions are obtained for the expansion coefficients g (2) and g (4) in the relation S(h) = S(0)[1 + g (2) h 2 + g (4) h 4 + O(h 6)] expressing the scattered field and the scattering cross-sections. Numerical results are given for various values of the parameters.

Extended boundary condition method in combination with field expansions in terms of spheroidal functions

The problem of light scattering by nonspherical particles, which arises in many applications, is nowadays most frequently solved by the method of extended boundary conditions in combination with the expansion of the fields in terms of spherical wave functions. However, such an approach encounters difficulties if the shape of particles is far from spherically symmetric, even in the simplest case of spheroids with the semiaxis ratio a/b > 5−10. A new approach to solving this problem is proposed, which also applies the extended boundary condition method but involves the expansion of the fields in terms of spheroidal functions. In this case, to obtain effective solutions for strongly prolate and oblate particles, the fields are divided in two parts with known properties and specific scalar potentials are used for each part. The basic relations of the approach are presented and some results of calculations of the optical properties of spheroids and spheroidal Chebyshev particles that are performed using computer codes realizing this approach are given. The convergence of the results for different cases and the domain of applicability of the method are discussed.

Capture and rendering of spatial sound over headphones

A theory for capturing an audio scene and then rendering it remotely over headphones is developed. The method relies on capture of a sound field up to a certain order in terms of spherical wave functions. Then, the captured sound field is convolved with the head‐related transfer function and rendered to provide presence in the auditory scene. The sound‐field representation is then transmitted to a remote location for immediate rendering or stored for later use. A system that implements the capture using a spherical array is developed and tested. Head‐related transfer functions are measured using the system described in [D.N. Zotkin et al., J. Acoustic. Soc. Am. (to appear)]. The sound renderer, coupled with the head tracker, reconstructs the acoustic field using individualized head‐related transfer functions to preserve the perceptual spatial structure of the audio scene. [Work partially supported by VA.]

Light scattering by multilayer axially symmetric particles

An exact solution to the problem of light scattering by multilayer axially symmetric particles is derived and some aspects of its computer-aided implementation are discussed. The main specific features of the solution are (i) separation of the incident, scattered, and internal fields into two parts and special selection of the scalar potentials for each of them; (ii) expansion of the potentials in terms of spherical wave functions; (iii) formulation of the problem in the form of surface integral equations; and (iv) solution of the reduced systems of the linear algebraic equations for the coefficients of the potential expansions. Mathematical justification of the solution is discussed, which is formulated in the recursive and nonrecursive form (for the T-matrix). The developed computer program has shown that the proposed approach makes it possible to consider axially symmetric particles with essentially different internal structures (i.e., with a spherical core, oblate spheroidal shell, or prolate spheroidal intermediate layer). The results of calculations of the optical properties of the multilayer nonspherical particles are presented and discussed.

Method of Moments Analysis of Active Thin Circular Loop Antennas With Closed-Form Currents and Validity

This paper presents an analysis of electromagnetic radiation due to thin circular loop antennas of arbitrary radii a. The electric type of dyadic Green's function expanded in terms of spherical wave functions is employed in this study. The Galerkin's method of moments is applied in the derivation of current distributions along the thin circular loop wires. With an advantage over others, the present analysis is quite general, compact and straightforward; most importantly, the current distributions are obtained analytically and expressed in a closed and simple form. Resulted from these currents, radiated fields in both near- and far-zones and both inside and outside the sphere of radius a are expressed in terms of the spherical vector wave functions. Also discussed in this paper is the validity of definitions of far-zone and electrically small size. The maximum radius of an electrically small circular loop antennas is found from a quantitative error analysis of both loop currents and the far-zone fields.

Acoustic eigenfrequencies in a spheroidal cavity with a concentric penetrable sphere

The acoustic eigenfrequencies fnsm in a spheroidal cavity containing a concentric penetrable sphere are determined analytically, for both Dirichlet and Neumann conditions in the spheroidal boundary. Two different methods are used for the evaluation. In the first, the pressure field is expressed in terms of both spherical and spheroidal wave functions, connected with one another by well-known expansion formulas. In the second, a shape perturbation method, this field is expressed in terms of spherical wave functions only, while the equation of the spheroidal boundary is given in spherical coordinates. The analytical determination of the eigenfrequencies is possible when the solution is specialized to small values of h=d/(2R2), (h≪1), with d the interfocal distance of the spheroidal boundary and 2R2 the length of its rotation axis. In this case exact, closed-form expressions are obtained for the expansion coefficients gnsm(2) and gnsm(4) in the resulting relation fnsm(h)=fns(0)[1+h2gnsm(2)+h4gnsm(4)+O(h6)]. Analogous expressions are obtained with the use of the parameter v=1−(R2/R2′)2, (|v|≪1), with 2R2′ the length of the other axis of the spheroidal boundary. Numerical results are given for various values of the parameters.

ACOUSTIC EIGENFREQUENCIES IN CONCENTRIC SPHEROIDAL–SPHERICAL CAVITIES: CALCULATION BY SHAPE PERTURBATION

The acoustic eigenfrequenciesfnsmin concentric spheroidal–spherical cavities are determined analytically, for both Dirichlet and Neumann boundary conditions, by a shape perturbation method. Two types of cavities are examined, one with spheroidal outer and spherical inner boundary and inversely for the other. The analytical determination is possible in the case of smallh=d/(2R2), (h⪡1), wheredis the interfocal distance of the spheroidal boundary and 2R2the length of its rotation axis. In this case exact, closed form expressions are obtained for the expansion coefficientsgnsm(2)andgnsm(4)in the resulting relationfnsm(h)=fns(0) [1+h2gnsm(2)+h4gnsm(4)+Oh6)]. Analogous expressions are obtained with the use of the parameterv=1−(R2/R2′)2, (∣v∣⪡1) where 2R2′ is the length of the other axis of the spheroidal boundary. There is no need for using any spheroidal wave functions and the expansion formulas connecting them with the concentric spherical ones. The pressure field is expressed in terms of spherical wave functions, while the equation of the spheroidal boundary is given in terms of the spherical co-ordinates. Numerical results are given for various values of the parameters.

Analysis of spherical radar cross-section enhancers

A method for determining the plane wave scattering by a spherical radar cross section enhancer is described. In the two-step process, the desired field representation is expressed as a superposition of vector spherical wave functions. This allows the plane wave scattering from a spherically stratified object to be determined. Love's equivalence principle is then employed to account for the metallic portions of the spherical reflector. The analysis utilizes the expansions of a plane wave, an electric current element, and a magnetic current element in terms of spherical wave functions. Measured results show fairly good agreement with predicted results.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The exact sonar cross section of an elastic spherical shell near the sea surface

The acoustic scattering by a submerged spherical elastic shell near a free surface and insonified by plane waves at arbitrary angles of incidence is analyzed in an exact fashion by the classical separation of variables method. To satisfy the boundary conditions at the free sea surface, as well as on the surfaces of the elastic shell, the mathematical problem is formulated using the method of images. The scattered wave fields are expanded in terms of spherical wave functions using the addition theorems. The final results come out in terms of Wigner 3‐j symbols or Clebsch–Gordan coefficients. Quite similar to the problem of scattering by multiple spheres, the computation of the scattered pressure field involves the solution of an ill‐conditioned complex matrix system, of size that depends on the number of terms in the modal series required for convergence. This, in turn, depends on the value of the frequency and on the proximity of the steel shell to the free surface. The matrix equation is solved by the Ga...

Scattering by a thin multicoated perfectly conducting spherical shell with a circular aperture

An analytic solution is presented for the problem of an infinitely thin perfectly conducting spherical shell with a circular aperture of arbitrary angle cut into the shell, filled with a dielectric, and coated by different thicknesses of spherical dielectric layers. The fields in all regions are expanded in terms of spherical wave functions and the boundary conditions of the continuity of the tangential fields at the dielectric–dielectric and dielectric–free-space boundaries are applied to express the expansion coefficients of the first dielectric layer in terms of the scattering coefficients. To approximate the modal expansion coefficients, the least-square error method is applied to the equations resulting from matching the fields through the aperture. Different numerical results for the simple case of a single coating layer are obtained in the form of amplitude patterns for the aperture and scattered fields versus angle as well as the backward- and forward-scattering cross sections for different loadings as functions of cavity size.

Scattering of plane harmonic SH, SV, P and rayleigh waves by non‐axisymmetric three‐dimensional canyons: A wave function expansion approach

AbstractScattering of elastic waves by three‐dimensional canyons embedded within an elastic half‐space is investigated by using a wave function expansion technique. The geometry of the canyon is assumed to be non‐axisymmetric. The canyon is subjected to incident plane Rayleigh waves and oblique incident SH, SV and P waves. The unknown scattered wavefield is expressed in terms of spherical wave functions which satisfy the equations of motion and radiation conditions at infinity, but they do not satisfy stress‐free boundary conditions at the half‐space surface. The boundary conditions are imposed locally in the least‐squares sense at several points on the surface of the canyon and the half‐space.Through a comparative study the validity and limitations of two‐dimensional approximations (antiplane strain and plane strain models) have been examined. It is shown that scattering of waves by three‐dimensional canyons may cause substantial change in the surface displacement patterns in comparison to the two‐dimensional models. These results emphasize the need for three‐dimensional modelling of realistic problems of interest in strong ground motion seismology and earthquake engineering.

Acoustic resonant frequencies in an eccentric spherical cavity

The interior boundary-value scalar (acoustic) problem in the region between two spheres of radii R1, R2 and distance d between their centers is considered for both Dirichlet and Neumann boundary conditions. Surface singular integral equations are used to formulate the problem. Their solution is obtained in terms of spherical wave functions in combination with related addition theorems. It is then specialized to the case of small values for kd=2πd/λ to yield exact, closed-form expressions for the coefficients gns in the resulting relations ωns(kd) =ωns(0) [1+gns(kd)2+⋅⋅⋅] for the resonant (natural) frequencies of the cavity. Numerical results, comparisons, and possible generalizations are also included.

Some remarks on exterior electromagnetic boundary value problems for spheres

An investigation of expansion problems in terms of spherical wave functions from different points of view is given. The possibility of having infinitely many formal series representations for a given function in terms of the associated Legendre functions <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p_{n}^{m}</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\sin \theta(dp_{n}^{m}/d\theta</tex> is also pointed out. One of the representations is derived by the use of the orthogonality relation, and the infinitely many others are derived by the use of the recurrence relation.

On the mode theory of V. L. F. ionospheric propagation

The space between the earth and the ionosphere is considered as a wave-guide with sharply bounded walls. Employing a representation in terms of spherical wave functions of complex order, the field of a vertical dipole source is calculated for very low frequencies. It is shown that the dominant mode for 16 kc is of order one and not zero as has been commonly supposed. Good agreement is obtained with the experimental results ofJ. Heritage.