Terms Of Spheroidal Wave Functions Research Articles

A new practical approach to light scattering by spheroids with the use of spheroidal and spherical function bases

The approximation of real non-spherical scatterers by spheroids is used in various applications. It often requires fast massive calculations of the optical properties of spheroids. The most powerful approach to that is known to be a field expansion in terms of spheroidal wave functions, in particular, in the way suggested by Farafonov over 30 years ago. We improve the main shortcomings of such an approach. Our solution is formulated in terms of normalised spheroidal functions, and firstly for their definition given by Meixner & Schäfke, which is computationally favourable and is required by the unique subroutines recently created to compute these functions. By means of T-matrix transformations we solve a long-standing major problem of Farafonov’s version, namely the accuracy and time losses for one kind (TE mode) of the incident wave polarisation. Apart from this and other improvements of this solution, for the first time we relate its single particle spheroidal T-matrix to the standard spherical one which is widely employed for particle ensembles. The constructed algorithm has been extensively numerically tested. It is found to be very accurate for dielectric spheroids with the aspect ratio a/b reaching 100 and the diffraction (size) parameter xa=2πa/λ as large as 300, where a and b are the major and minor semi-axes, respectively, λ is the wavelength. The algorithm is supplied with a program interface to the free package CosTuuM to perform parallel computations of various optical properties for ensembles of spheroids with different distributions over orientation (alignment) and shape.

Electrostatic interaction between two spheroidal particles at large separations

Asymptotic expressions for the electrostatic interaction energy between two weakly charged spheroidal particles (prolate and oblate) with uniform surface potential or surface charge density at large separations in an electrolyte solution are derived on the basis of the linear superposition approximation. The electrostatic interaction between two spheroids is found to be the screened Coulomb interaction between two hypothetical point particles with orientation-dependent charges. Explicit interaction energy equations, which are expressed in terms of spheroidal wave functions, are given for four particular configurations of two similar spheroids, that is, side-to-side prolates, end-to-end prolates, side-to-side oblates, and end-to-end oblates. Comparison is also made with results obtained using Derjaguin’s approximation. Electrostatic interaction between two spheroids • Asymptotic expressions for the electrostatic interaction energy between two spheroidal particles are derived. • Constant surface potential and charge density cases are considered. • Side-to-side prolates, end-to-end prolates, side-to-side oblates, and end-to-end oblates are considered.

Acoustic scattering by two fluid confocal prolate spheroids

The exact spheroidal-function series solution for the time-harmonic acoustic scattering of a plane wave by two fluid confocal prolate spheroids is developed and a numerical implementation is formulated and validated by independent methods. The two spheroids define three regions in which the acoustic fields are expanded in terms of spheroidal wave functions multiplied by unknown coefficients. These expansions are forced to satisfy the boundary conditions and by using the orthogonality properties of the involved functions an infinite matricial system for the coefficients is obtained. The resulting system is then solved through a truncation procedure. The implementation has no limitations regarding the sound speed and density of the three media involved or in the incidence frequency.

Dynamically crowded solutions of infinitely thin Brownian needles.

We study the dynamics of solutions of infinitely thin needles up to densities deep in the semidilute regime by Brownian dynamics simulations. For high densities, these solutions become strongly entangled and the motion of a needle is essentially restricted to a one-dimensional sliding in a confining tube composed of neighboring needles. From the density-dependent behavior of the orientational and translational diffusion, we extract the long-time transport coefficients and the geometry of the confining tube. The sliding motion within the tube becomes visible in the non-Gaussian parameter of the translational motion as an extended plateau at intermediate times and in the intermediate scattering function as an algebraic decay. This transient dynamic arrest is also corroborated by the local exponent of the mean-square displacements perpendicular to the needle axis. Moreover, the probability distribution of the displacements perpendicular to the needle becomes strongly non-Gaussian; rather, it displays an exponential distribution for large displacements. On the other hand, based on the analysis of higher-order correlations of the orientation we find that the rotational motion becomes diffusive again for strong confinement. At coarse-grained time and length scales, the spatiotemporal dynamics of the needle for the high entanglement is captured by a single freely diffusing phantom needle with long-time transport coefficients obtained from the needle in solution. The time-dependent dynamics of the phantom needle is also assessed analytically in terms of spheroidal wave functions. The dynamic behavior of the needle in solution is found to be identical to needle Lorentz systems, where a tracer needle explores a quenched disordered array of other needles.

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.

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.

Spheroidal Phase Mode Processing for Antenna Arrays

This paper presents the theory and analysis of phase mode excitation of both prolate and oblate spheroidal antenna arrays in terms of spheroidal wave functions, which is an extension of the spherical counterparts reported in the literature. Theoretical and numerical results show that a spheroidal phase mode of the excitation function produces a far field radiation pattern with the same spheroidal phase mode form, and the elevation angle of that pattern increases (decreases) with the ratio of the interfocal distance of the prolate (oblate) spheroid antenna array to the wavenumber.

Microwave Specific Attenuation by Oblate Spheroidal Raindrops: an Exact Analysis of TCS's in Terms of Spheroidal Wave Functions

Microwave Specific Attenuation by Oblate Spheroidal Raindrops: an Exact Analysis of TCS's in Terms of Spheroidal Wave Functions

Microwave Specific Attenuation By Oblate Spheroidal Raindrops: an Exact Analysis of Tcs's in Terms of Spheroidal Wave Functions - Abstract

The electromagnetic plane wave scattering by oblate spheroidal raindrops is investigated in this paper using the full-wave eigen-expansion in terms of the spheroidal vector wave functions so as to obtain an exact solution of the total (or extinction) cross section of the spheroidal raindrops. In this analysis, both TE- and TM-mode incident plane waves (or parallel and perpendicular polarizations) are considered. As the permittivity of the raindrop water is a function of both the operating frequency and the outdoor temperature, the Ray's program is utilized to compute the lossy raindrop permittivity. A Mathematica program is developed based on the computation of the radial and angular spheroidal wave functions with complex arguments. Comparison between the spheroidal wave function values computed using this algorithm and the available data in the literature shows that the algorithm developed here has a better accuracy. Also, the total cross sections of the raindrops obtained using the full-wave oblate spheroidal eigenfunctions are compared with those using the T-Matrix approach; and a very good agreement between them is found.

CBR anisotropy from inflation-induced gravitational waves in mixed radiation and dust cosmology.

We examine stochastic temperature fluctuations of the cosmic background radiation (CBR) arising via the Sachs-Wolfe effect from gravitational wave perturbations produced in the early universe. We consider spatially flat, perturbed FRW models that begin with an inflationary phase, followed by a mixed phase containing both radiation and dust. The scale factor during the mixed phase takes the form a(\ensuremath{\eta})=${\mathit{c}}_{1}$${\mathrm{\ensuremath{\eta}}}^{2}$+${\mathit{c}}_{2}$\ensuremath{\eta}+${\mathit{c}}_{3}$, where ${\mathit{c}}_{\mathit{i}}$ are constants. During the mixed phase the universe smoothly transforms from being radiation to dust dominated. We find analytic expressions for the graviton mode function during the mixed phase in terms of spheroidal wave functions. This mode function is used to find an analytic expression for the multiple moments 〈${\mathit{a}}_{\mathit{l}}^{2}$〉 of the two-point angular correlation function C(\ensuremath{\gamma}) for the CBR anisotropy. The analytic expression for the multipole moments is written in terms of two integrals, which are evaluated numerically. The results are compared to multipoles calculated for models that are completely dust dominated at last scattering. We find that the multipoles 〈${\mathit{a}}_{\mathit{l}}^{2}$〉 of the CBR temperature perturbations for l>10 are significantly larger for a universe that contains both radiation and dust at last scattering. We compare our results with recent, similar numerical work and find good agreement. The spheroidal wave functions may have applications to other problems of cosmological interest.

The unsteady force on a body at low Reynolds number; the axisymmetric motion of a spheroid

In a recent paper by Lawrence & Weinbaum (1986) an unexpected new behaviour was discovered for a nearly spherical body executing harmonic oscillations in unsteady Stokes flow. The force was not a simple quadratic function in half-integer powers of the frequency parameter λ2 = −ia2ω/ν, as in the classical solution of Stokes (1851) for a sphere, and the force for an arbitrary velocity U(t) contained a new memory integral whose kernel differed from the classical t−½ behaviour derived by Basset (1888) for a sphere. A more general analysis of the unsteady Stokes equations is presented herein for the axisymmetric flow past a spheroidal body to elucidate the behaviour of the force at arbitrary aspect ratio. Perturbation solutions in the frequency parameter λ are first obtained for a spheroid in the limit of low- and high-frequency oscillations. These solutions show that in contrast to a sphere the first order corrections for the component of the drag force that is proportional to the first power of λ exhibit a different behaviour in the extreme cases of the steady Stokes flow and inviscid limits. Exact solutions are presented for the middle frequency range in terms of spheroidal wave functions and these results are interpreted in terms of the analytic solutions for the asymptotic behaviour. It is shown that the force on a body can be represented in terms of four contributions; the classical Stokes and virtual mass forces; a newly defined generalized Basset force proportional to λ whose coefficient is a function of body geometry derived from the perturbation solution for high frequency; and a fourth term which combines frequency and geometry in a more general way. In view of the complexity of this fourth term, a relatively simple correlation is proposed which provides good accuracy for all aspect ratios in the range 0.1 < b/a < 10 where exact solutions were calculated and for all values of λ. Furthermore, the correlation has a simple inverse Laplace transform, so that the force may be found for an arbitrary velocity U(t) of the spheroid. The new fourth term transforms to a memory integral whose kernel is either bounded or has a weaker singularity than the t−½ behaviour of the Basset memory integral. These results are used to propose an approximate functional form for the force on an arbitrary body in unsteady motion at low Reynolds number.

Rotary oscillations of a spheroid in an incompressible micropolar fluid

The paper discusses the flow generated by rotary oscillations of a spheroid (prolate and oblate) in incompressible micropolar fluid. The velocity and microrotation components are determined explicitly in terms of spheroidal wave functions and are expressed in infinite series form. The couple on the oscillating spheroid is evaluated and numerical studies are undertaken to examine the effects of the geometric parameter and material constant parameters of the fluid.

A General Addition Theorem for Spheroidal Wave Functions

A general addition theorem has been obtained for the spheroidal wave functions $R_{ml}^{(j)} (h,\xi )S_{ml}^{(1)} (h,\eta )\exp (im\varphi )$, $j = 1,2,3,4$. This theorem gives the expansion of a spheroidal wave function with reference to one coordinate frame in terms of spheroidal wave functions with reference to a second coordinate frame with arbitrary relative position and orientation. The expressions are applicable whether the two spheroidal coordinate frames are both prolate, both oblate, or one prolate and one oblate.

Self and mutual acoustic radiation impedances for two coplanar unbaffled disks

The self and mutual acoustic radiation impedances for two coplanar unbafed disks are calculated by using an eigenfunction expansion in terms of oblate spheroidal wave functions. Terms representing outgoing waves from both disks are included. One disk is assumed to be vibrating with a rotationally symmetrical normal velocity distribution, and the other disk is assumed to be stationary. The determination of the expansion coefficients from these boundary conditions is facilitated by the use of an addition theorem which expresses spheroidal wave functions with reference to one coordinate frame in terms of spheroidal wave functions with reference to a second coordinate frame. Numerical results for the special cases where the vibrating disk is either oscillating uniformly or vibrating uniformly on only one side are presented and discussed.

Self and Mutual Acoustic Radiation Impedances for Two Coplanar Unbaffled Disks

The self and mutual acoustic radiation impedances for two coplanar unbaffled disks are calculated using an eigenfunction expansion in terms of oblate spheroidal wave functions. Terms representing outgoing waves from both disks are included. One disk is assumed to be vibrating with a rotationally symmetrical normal velocity distribution, and the other disk is assumed to be stationary. The determination of the expansion coefficients from these boundary conditions is facilitated by the use of an addition theorem which expresses spheroidal wave functions with reference to one coordinate frame in terms of spheroidal wave functions with reference to a second coordinate frame. Numerical results for the special cases where the vibrating disk is either oscillating uniformly or vibrating uniformly on only one side are presented and discussed.

The Stokes-flow drag on prolate and oblate spheroids during axial translatory accelerations

Stokes's linearized equations of motion are used to calculate the flow field generated by a spheroid executing axial translatory oscillations in an infinite, otherwise still, incompressible, viscous fluid. The flow field, expressed in terms of spheroidal wave functions of order one, is used to develop general expressions for the drag on oscillating prolate and oblate spheroids. Formulae for the approximate drag, useful in making calculations, are obtained for small values of an oscillation parameter. These formulae reduce to the Stokes result in the limit when the spheroid becomes a sphere and the steady-state drag for a spheroid as the frequency of oscillation becomes zero. The fluid forces on spheroids of various shapes are compared graphically. The approximate formulae for the drag are integrated over all frequencies to obtain formulae for the drag on spheroids executing general axial translatory accelerations. The fluid resistance on the spheroid is expressed as the sum of an added mass effect, a steady-state drag and an effect due to the history of the motion. A table of added mass, viscous and history coefficients is given.

Acoustic Radiation from Two Spheroids

The acoustic radiation from two spheroids whose surface normal velocity distributions are specified is calculated using a Green's function approach. The necessary Green's function is expanded in spheroidal wavefunctions about both spheroids. The unknown expansion coefficients are determined from the boundary condition that the normal derivative vanishes over both surfaces. An addition theorem which expresses spheroidal wavefunctions in one system in terms of spheroidal wavefunctions in another system was developed to facilitate application of this boundary condition. The resulting Green's function is then used to obtain the acoustic fields for any desired velocity distribution over the two spheroidal surfaces. Numerical results for several different two-spheroid configurations and velocity distributions are presented and discussed. Extension to more than two spheroids is also discussed.

The couple on a rotating spheroid in a slow stream

AbstractThe problem considered here is to determine the couple required to maintain the steady rotation about its axis of symmetry of an oblate spheroid which is placed in a uniform stream of viscous fluid moving parallel to this axis. The conditions of flow are assumed to be such as to permit the use of Oseen's approximation in the equations of motion. An expression in terms of spheroidal wave functions is derived for the transverse component of velocity in the fluid, and from this the expression for the couple is deduced. The disk rotating about the normal through its centre is considered as a special case, and some numerical results are given for certain values of a Reynolds number. A comparison is made, and agreement found, with results obtained by Jeffery in a previous paper which treated the case in which the mainstream was at rest.