Use Of Addition Theorem Research Articles

Analytical solution for acoustic radiation force and torque on a spheroid near a rigid or free planar boundary.

An analytical solution is developed for the acoustic radiation force and torque caused by an arbitrary sound field that is incident on a compressible spheroid of any size near a planar boundary that is either rigid or pressure release. The analysis is an extension of a recent solution for a compressible sphere near a planar boundary [Simon and Hamilton, J. Acoust. Soc. Am. 153, 627-642 (2023)]. Approximations that account for a boundary formed by a two-fluid interface may be incorporated as in the previous analysis for a sphere. The present solution is based on expansions of the total acoustic pressure field in spheroidal wave functions and the use of addition theorems. Verification of the solution is accomplished by comparison with a finite element model. Examples are presented for incident fields that are either plane or spherical waves. Effects resulting from the presence of the boundary are studied by comparing the full theory with a simplified model in which multiple scattering is neglected. Numerical implementation of the proposed solution is also discussed.

Rayleigh Scattering from a Sphere Located Near a Planar Rigid Boundary

Rayleigh scattering from a spherical object located near a planar rigid boundary at distances smaller than the wavelength is calculated. Low frequency analysis reduces a scattering problem to a sequence of potential problems. An analytical solution based on expansion in spherical solid harmonics and the use of addition theorem is presented. Analytical perturbation approach is validated by comparison with numerical calculations. The velocity of the center of the particle and the scattering amplitude are determined. In the lowest order in wavenumber, the scattering amplitude is expressed in terms of the monopole and dipole components. In contrast to the behavior of a bubble, under the same conditions, dipole oscillations of the particle in the direction normal to the boundary are not excited and the monopole component of the scattering amplitude does not depend on the location of the particle relative to the boundary.

A Novel Design of a SIW-Fed Antenna Array Using an Accelerated Full-Wave Methodology

Arrays fed by substrate-integrated waveguides (SIWs) are a current topic of interest. However, most designers have to rely on general-purpose commercial software, usually inefficient for multiple full-wave simulations in a design process. This work proposes a new fast full-wave strategy, with very low iteration times, based on the simultaneous use of addition theorems for spherical and cylindrical modes. After a domain decomposition of the array into fixed or modifiable (to be optimized) sections, each section is analyzed using a finite-element/modal analysis to obtain its general scattering matrix. Fixed sections are only once coupled at the beginning, while each optimization iteration is calculated with full-wave precision by adding the interactions with the modifiable sections. A complete design process of a novel 16-slot progressive-wave SIW-fed array is performed to validate the strategy usefulness. 112800 iterations, with 47 optimization variables, are calculated in under 4 h (speed-up factor of over 2000 compared to general-purpose commercial software). The designed device is manufactured and experimentally characterized, measuring a 1.05 GHz <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\vert S_{11} \vert $ </tex-math></inline-formula> bandwidth under −10 dB and a maximum gain of 15.6 dBi at <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\theta =-11^\circ $ </tex-math></inline-formula> in the array plane.

Acoustic radiation force and torque on a spheroid near an interface

Acoustic radiation force and torque on a spheroid of arbitrary size near a rigid or pressure release interface are calculated using expansions of the pressure field in terms of both spherical and spheroidal wave functions. The spheroidal expansion coefficients are obtained by satisfying the boundary conditions at the interface and at the surface of the spheroid. Conditions at the interface are satisfied using a virtual (image) spheroid, while satisfying conditions at the spheroid surface require the use of addition theorems for spheroidal wave functions. The radiation force and torque are expressed as a summation of terms involving products of the coefficients in spherical wave expansions of the incident and scattered fields [Ilinskii et al., J. Acoust. Soc. Am. 144, 568–576 (2018)]. Far-field asymptotes are used to relate the spheroidal and spherical harmonic expansion coefficients analytically. Results from the present analytical model are compared with those from a finite-element model. The effect of the interface on the radiation force and torque relative to that on a spheroid in a free field is discussed. [B.E.S. is supported by the Applied Research Laboratories Chester M. McKinney Graduate Fellowship in Acoustics.]

Simultaneous Use of Addition Theorems for Cylindrical and Spherical Waves for the Fast Full-Wave Analysis of SIW-Based Antenna Arrays

This paper introduces a fast full-wave methodology for the analysis of antenna arrays placed on the top plate of substrate integrated waveguides (SIWs). The methodology is based on the generalized scattering matrix (GSM) of each array element in terms of spherical and cylindrical modes and plane-port feeding modes, and the simultaneous use of addition theorems for spherical and cylindrical waves. Addition theorems allow the analytical translation of spherical and cylindrical modes, so that the GSMs can be simultaneously connected in the SIW layer and in the radiation region, to obtain the full-wave response of the array. The proposed methodology is validated by means of full-wave simulations with commercial and in-house softwares.

FEATURES OF ELECTROMECHANICAL ACOUSTIC ENERGY CONVERSION BY CYLINDRICAL PIEZOCERAMIC TRANSDUCERS WITH INTERNAL SCREENS IN COMPOSITION OF FLAT SYSTEMS

The problem of sound emission is considered by a system formed from cylindrical piezoceramic radiators with internal acoustically soft screens. Longitudinal axis of emitters lie in one plane. This system is characterized by the interaction of electric, mechanical and acoustic fields in the process of conversion electrical energy to acoustical energy and acoustic fields in the process of forming them in the environments. The purpose of the work is to determine the peculiarities of the electromechanical acoustic transformation of energy by cylindrical piezoceramic radiators with internal screens in the composition of flat systems, taking into account all types of interaction.The research was carried out by the method of bound fields in multiply connected domains with the use of addition theorems for the cylindrical wave functions. The physical fields arising from the emission of sound by such a system are determined by the joint solution of the system of differential equations: the wave equation; equations of motion of thin piezoceramic shells with circular polarization in displacements; the equations of forced electrostatics for piezoceramics at given boundary conditions, the conditions of conjugation of fields at the boundaries of the division of domains and electric conditions.The solution of the problem is reduced to the solution of an infinite system of linear algebraic equations with respect to unknown coefficients of field expansions.An analysis of the results of numerical calculations, performed on the basis of the obtained analytical relations, called to establish a number of features in the electromechanical acoustic transformation of energy by emitters in the composition of flat systems. They include: the role of acoustic interaction in the process of energy conversion; determination of the mechanism of quantitative assessment of the influence of interaction on these processes; the dependence of the degree of violation of the radial symmetry of the acoustic loading of the emitters on the amount of acoustic interaction; the appearance of multimodality of the mechanical field of emitters in the structure of the plane system and the dependence of the redistribution of energy between all modes on the degree of disturbance of the radial symmetry of the acoustic loading of the emitters.

Electromagnetic Scattering by 2-D PEC Cylinder of General Cross Section Under &lt;inline-formula&gt; &lt;tex-math notation="LaTeX"&gt;$\text {TM}^{z}$ &lt;/tex-math&gt; &lt;/inline-formula&gt; Illumination

A semianalytical method is presented for the evaluation of scattered fields from arbitrarily shaped 2-D Perfect Electric Conductor (PEC) scatterers under $\text {TM}^{z}$ wave incidence. Surface equivalence principle for scattered fields is used in conjunction with modal expansion of fields in terms of cylindrical eigenfunctions. It is assumed that no field variation exists in axial ( $z$ ) direction. Implementation of boundary conditions on scatterer surface leads to a system of linearly independent equations to solve for equivalent surface currents. Scattered fields are calculated by equivalent surface currents leading to evaluation of scattering widths (Radar Cross-Section of 2-D objects) for different observation angles. $\text {TM}^{z}$ polarized incident wave does not excite any $\text {TE}^{z}$ component. The novelty of proposed method is use of addition theorem of Bessel functions for representation of scattered fields in terms of equivalent surface currents, giving an integral equation with an analytic integrand and a well-behaved impedance matrix. Orthogonality of cylindrical modes results in excitation matrix to be a discrete function of mode numbers. Therefore, contrary to method of moments, there is no need for weighting functions. Four examples are considered, namely, line source and plane wave incidence on a 2-D PEC circular cylinder and also plane wave incidence on 2-D elliptical and equilateral triangular PEC cylinders.

Multipole method for microstructured optical fibers I Formulation

We describe a multipole method for calculating the modes of microstructured optical fibers. The method uses a multipole expansion centered on each hole to enforce boundary conditions accurately and matches expansions with different origins by use of addition theorems. We also validate the method and give representative results.

Use of addition theorems in evaluation of multicenter nuclear-attraction and electron-repulsion integrals with integer and noninteger n Slater-type orbitals

Multicenter integrals appearing in the Hartree–Fock–Roothaan equations for molecules are calculated using different kinds of series expansion formulas obtained from the expansions of integer and noninteger n Slater-type orbitals, in terms of Ψα-exponential-type orbitals (where α=1, 0, –1, –2,...) at a displaced center, that form complete orthonormal sets and are represented by linear combinations of integer n Slater-type orbitals. The convergence of these series is tested by calculating concrete cases. The accuracy of the results is quite high for quantum numbers, screening constants, and location of orbitals.

Fast spheroidal multipole imaging of elementary magnetic sources on the axis

The multipole image (MI) is the set of coefficients in a harmonic expansion of the scalar magnetic potential of a magnetic source. Compared to the common spherical harmonics, spheroidal harmonics provide an improved description of the field near an elongated source. The total (spherical or spheroidal) MI can be retrieved using a superposition of elementary source images. This article presents fast two-term recursive formulae for multipole imaging of an elementary dipolar source on the axis. These results are compared to one-term recursions for the spherical MI. Another useful result is formulae linking the spherical and spheroidal MI by use of addition theorems and expansions of the Green function. Although it is possible to obtain the spheroidal MI from the spherical MI, perfect accuracy for a complex elongated source is not possible. In contrast, the spherical MI, which is appropriate for the magnetic field at the remote region, is accurately generated from the spheroidal MI, which provides the most precise description at the near zone.

Solution of a non-isotropic random flight problem in the case of a non-isotropic point source

In connection with the calculation of the perturbation caused by a thin circular detector in a thermal neutron flux, we have solved the following problem. Given is a point source whose strength is described by an isotropic distribution plus a non-isotropic cosine term. The single scattering law also contains one non-isotropic term. We have to calculate the neutron density and the distribution of the velocity directions in every point of space. Keeping the path probabilities arbitrary for the reason of generality, we have established exact recurrence relations between the defined probability distributions. Expanding these functions in series of spherical harmonics and making use of addition theorems involving Bessel functions and Legendre polynomials, it has been possible to transform the integral relations to simple linear equations. The final results are obtained in the form of integrals which can be calculated by numerical evaluation using computing machines. By introducing simplified functions under the integral signs, it is also possible to obtain good approximations for the neutron density and current. A simple example of this method is discussed in detail and the results are compared with those given by diffusion theory.