Analytical Series Solution Research Articles

Analytic solution for TE-mode radiation from a flanged parallel-plate waveguide

The problem of TE-mode radiation from a flanged parallel-plate waveguide is reexamined. The technique of the Fourier transform is used to represent the radiation field in a simple analytic series form. The series solution is compared with other results, thus clarifying the discrepancies between different numerical approaches. The analytic series solution is exact and more efficient than other existing numerical solutions.

Diffraction of electromagnetic waves by crossed gratings: a series solution

An analytical series solution is derived within the Rayleigh hypothesis to compute the efficiencies of a crossed grating illuminated by an arbitrary polarized beam. A comparison with previously reported numerical results for a dielectric crossed grating illuminated by a plane directly implemented on a personal computer.

Infinite elements for 3-dimensional wave — Structure interaction problems

The finite element technique incorporating infinite elements is applied to analysing the general three-dimensional wave — structure interaction problems within the limits of linear wave theory. The hydrodynamic forces are assumed to be inertially dominated, and viscous effects are neglected. In order to analyse the corresponding boundary value problems, efficiently, two types of elements are developed. One is the infinite element for modelling the radiation condition at infinity, and the other is the fictitious bottom boundary element for the case of deep water. The shape functions for infinite elements in the radial direction are derived from the asymptotic expressions for the progressive wave and the first evanescent wave components in the analytical boundary series solutions. To validate the infinite and the fictitious bottom boundary elements, numerical analyses are performed for several floating structures. Comparisons with the results from other available solution methods show that the present method incorporating the infinite and the fictitious bottom boundary elements gives good results.

Infinite elements for evaluation of hydrodynamic forces on offshore structures

In this paper, the finite element technique incorporating the infinite element is applied to the evaluation of hydrodynamic forces on offshore structures. The hydrodynamic forces are assumed to be inertially dominated, and the viscous effects are neglected. Two types of elements are developed to discretize the fluid domain efficiently. They are the infinite elements developed to model the radiation condition at infinity, and the fictitious bottom boundary elements introduced to avoid extensive fluid domain discretizations for the case of deep water. The shape functions of the infinite elements, in the radial direction, are derived from the asymptotic expressions for the progressive wave and the first evanescent mode components in the analytical boundary series solutions. Numerical analyses are performed for vertical axisymmetric bodies to validate the infinite elements and the fictitious bottom boundary elements developed in this study. Comparisons with the results from other available numerical solution methods show that the present method gives fairly good results. Numerical experiments are also carried out to determine the proper distance to the infinite elements and the fictitious bottom boundary elements from the solid body, which directly affect accuracy and efficiency of the solution.

Scattering of Plane Sh-Waves By A Semi-Parabolic Cylindrical Canyon In an Elastic Half-Space

The 2-D scattering and diffraction of plane SH-waves by a semi-parabolic cylindrical canyon in an elastic half-space is analysed for the cases of SH-waves coming from the side of the half-space towards the front of the canyon. the exact analytic series solution of the problem is presented. It shows that the surface topography can have significant effects on the incident waves, which become prominent at high frequencies, or when the wavelengths of the incident waves are short compared with the focal length of the canyon. Higher frequencies will also lead to greater complexity of the pattern of surface displacement amplitudes involving abrupt changes of amplification or deamplification from one point to another.

The static spherically symmetric interior case of the non-symmetric theory of gravitation

The field equations for a spherically symmetric perfect fluid in non-symmetric gravitational theory are cast as a set of first-order differential equations suitable for numerical integration. An analytic series solution is presented as an expansion around r=0. It is shows how interior solutions match with the exterior one and how, at the boundary, the Euler equation for the fluid becomes the equation of motion of a test particle in the exterior metric. An expression is derived from a conserved pseudotensor for the total mass-energy of a static body in terms of its interior matter parameters.

ITPE technique applications to time-varying two-dimensional ground-coupling problems

The interzone temperature profile estimation (ITPE) procedure is used to find two-dimensional analytical series solutions for the time-varying heat transfer between ground and slab-on-grade floors or basements. The undisturbed soil temperature is approximated as a sinusoidal function of time and the ITPE procedure is coupled with the complex temperature technique to derive the steady-periodic solutions for both configurations. The influence of insulation and of a fixed-temperature water table on the temporal behavior of slab-on-grade floors and basements is treated analytically for the first time.

Waves generated by an inclined‐plate wave generator

AbstractThis paper describes the characteristics of small‐amplitude waves generated by a sinusoidally oscillating, inclined paddle‐type wavemaker operating in a constant‐depth channel. Two‐dimensional, linearized potential flow is assumed. A semi‐analytical method, the boundary collocation method, is used to establish the relationship between wave amplitude and paddle stroke. The numerical results are compared with the numerical results of the boundary integral equation method. It is found that the boundary collocation method is simpler and more flexible to implement and faster to compute. In addition, the numerical results are in reasonably good agreement with the laboratory experimental data. For the vertical wavemaker, which is a special case of the inclined wavemaker, an analytical series solution can be found. By using the boundary collocation method and the boundary integral equation method to solve the vertical wavemaker problem and comparing the results with the analytical series solution, it is found that the boundary collocation method yields a solution which is much more accurate than that from the boundary integral equation method. Finally, the relationships between wave amplitude and paddle stroke are established for different inclinations of the paddle‐type wavemaker, based on the boundary collocation method.

Series solution of the Schroumldinger equation with the Morse potential.

An analytic series solution to the Schroedinger equation with the Morse potential has been obtained. In the present solution the Morse potential itself is also expanded in a series about the origin. For application to the problem of the density dependence of muon-catalyzed d-t fusion, the effective two-body molecular problem is also formulated in terms of the superimposed potential field inside a liquid cell due to nearest-neighbor molecules (atoms). In this approach, more detailed studies can be carried out of the density effect on the various matrix elements needed in the calculations of the rates of ..mu..-meson d-t fusion.

Mathematical modelling of rate-limiting mechanisms of pyritic oxidation in overburden dumps

The mining of valuable minerals often leads to major environmental pollution problems especially when pyritic material is associated with the orebody being mined. In this thesis mathematical equations are formulated which describe a dual region model of pyritic oxidation within a waste rock dump, where it is assumed that oxygen transport is the rate-limiting step in the oxidation process, and that oxygen transport to reaction sites occurs in two stages; bulk diffusion of oxygen from the atmosphere, at the top surface of the dump, through the pore space of the dump to particles within the dump, and then diffusional transport of oxygen from the surface of particles to oxidation sites within the particles. A moving boundary (shrinking-core) formulation, describing transport and reaction processes within a single particle of the dump, is solved by a new analytic series solution technique involving a change of coordinates leading to a non-linear partial differential equation with fixed boundaries. Based on this analysis a pseudo-steady state assumption is made with respect to diffusion within the particles giving a simplified system of equations for the dump model. The system of equations is solved both in an approximate analytic way and numerically. An approximate analytic result is obtained by solving a planar moving boundary formulation which in turn comes from approximation of the position of the moving reaction front within the particles of the dump. The position of the planar front within -vthe dump is given by a transcendental equation which is shown to I collapse on the (time)2 classical planar moving boundary result when the particle size tends to zero. A combined Crank-Nicolson and Backward Euler discretisation of the system of equations is adopted for the numerical scheme. The approximate and numerical solutions

Mie scattering into solid angles

A group-theory expansion of scalar spherical harmonics is used to obtain an expansion of vector spherical harmonics. This expansion is applied to the multipole-expansion treatment of Mie theory, and explicit expressions suitable for computation of the coefficients of the expansion are obtained. The result is a Mie-theory solution given in the variables and basis vectors of the spherical-coordinate system associated with a Cartesian system that is rotated with respect to the conventionally chosen coordinate system. This result is then used to obtain an analytical-series solution for the power scattered into a conical solid angle centered on any chosen direction from the scattering sphere.

Lens in the nearfield of a circular transducer: Gaussian–Laguerre formulation

The case of a thin lens located in the nearfield of a circular transducer is analyzed assuming circular symmetry for arbitrary transducer size and lens aperture, with or without absorption in the lens medium. Using the Fresnel approximation, the acoustic field is mathematically expressed as a series expansion of Gaussian–Laguerre functions. This technique is applied to the case of a circular-piston–lens system and an analytical series solution of the field obtained for arbitrary positions of the lens. Numerical results are presented corresponding to several positions of the lens in the nearfield of the circular piston. It is found that the field pattern behind the lens is strongly dependent on the distance between the transducer and the lens, and the ratio of the focal length to the parameter a2/λ. Here a is the radius of the transducer and λ is the acoustic wavelength. It is also shown that the theory presented here can be applied to study the field of finite Gaussian transducers, focused and unfocused, and transducer–iris systems.

Some considerations on the melting of reactor fuel plates and rods

Abstract Current practice generally leads to overdesigning with respect to catastrophic accidents involving gross melting of fuel elements. It is clear, however, that from both the standpoints of cost and safety it would be desirable to be able to base the design on calculations regarding the details of the melting process itself. This paper considers the gross melting of fuel plates and rods and seeks to estimate the extent of melting and the rapidity with which it occurs. Methods of analysis are presented in detail in the present exploratory study. For short times after the onset of melting, an analytical series solution is constructed by the ‘embedding technique’. The effect of the change in density which accompaies the change in phase is examined, and is found to be relatively unimportant as far as the melting front position is concerned. An exact steady-state solution is found, which makes possible a prediction of the maximum extent of melting in cases in which steady state can be expected. The actual progress of melting for all times is more difficult to calculate, and an approximate solution is therefore introduced. Its accuracy can, in some cases, be estimated by comparison with the known final melting extent. The approximate solution is applied to a class of melting problems, and the results are presented in the form of graphs depicting the progress of the melting front for a range of parameters representing various possible conditions.

The permeability of gases through reacting solutions: the carbon dioxide—bicarbonate membrane system

The transport of a gas across a stationary liquid film containing reactive species is investigated for the purpose of determining gas permeabilities or mass transfer coefficients in reacting solutions. Under limiting conditions when the reaction time constant far exceeds the diffusional time constant, the flux of the transported gas follows Fick's law of diffusion. Analytical series solution for the contribution of the chemical reaction to the transport process is obtained using the technique of perturbation analysis; criteria for the validity of various terms in the series solution are presented. The permeability of carbon dioxide in water and in 1N NaHCO 3−Na 2CO 3 solution is estimated. It is shown that a high degree of accuracy in the data is necessary for obtaining separate estimates of diffusivity and solubility by this technique.

Reduction of diffraction of use of a Lyot stop

The efficiency of a Lyot stop in reducing the effects of diffraction is analyzed for a Fourier optical system. The problem reduces to the evaluation of an integral whose integrand is the product of two Bessel functions of different order and argument. An analytic series solution is found for the reduction of diffracted light reaching the focal plane.

Steady Nonlinear Ekman-Hartmann Boundary Layers

The steady, axially symmetric Ekman-Hartmann boundary layer on a flat insulating surface is analyzed in its fully nonlinear form. Emphasis is placed on evaluation of Ekman pumping and induced Hartmann current as functions of Rossby number, magnetic interaction parameter, and magnetic Prandtl number. Significant quantitative departures from the estimates of linear theory are found. Yet, for some purposes (such as for predicting its influence on spin-up) the strength of this hydromagnetic pump is shown to be adequately described by an analytic series solution truncated after the terms quadratic in Rossby number.

Analytic Power Series Solution of the Schrödinger Equation for the Helium Atom and Integral Series Solution of the Schrödinger Equation

Analytic Power Series Solution of the Schrödinger Equation for the Helium Atom and Integral Series Solution of the Schrödinger Equation

Analytic Power-Series Solution of the Schrödinger Equation for the Helium Atom

Analytic power series solution of nonrelativistic Schroedinger equation for two-electron atom, assuming fixed nucleus and singlet and triplet S states

Stresses in spheres with concentric spherical cavities under diametral compression by three-dimensional photoelasticity

This paper presents the experimental solution of the problem of a sphere with a concentric spherical cavity under diametral compression using the three-dimensional photoelastic method and calculations of the complete stress distributions along lines of interest using Golecki's analytical series solution. The experimental procedure, the stress distributions along different lines as well as the comparison with the theoretical results and a discussion of Golecki's solution are included.