Infinite Set Of Simultaneous Equations Research Articles

Analytical modeling of wave propagation in orthogonally rib-stiffened sandwich structures: Sound radiation

Wave propagation in an infinite sandwich panel reinforced by orthogonal rib-stiffeners is theoretically formulated for harmonic point force excitation. The motions of the equally spaced rib-stiffeners are handled by considering their tensional, bending vibrations and torsional movements. The response of the sandwich is determined by employing the Fourier transform technique and considering the periodical nature of the structure, which is numerically solved by truncating two infinite sets of simultaneous equations insofar as the solution converges. The far field radiated sound pressure is examined relative to that of an unstiffened panel with useful conclusions deduced.

Acoustic scattering by a finite elastic strip

A method for exact reduction of finite Wiener—Hopf type diffraction problems to an infinite set of simultaneous equations is presented. The problem considered is that of acoustic scattering by a finite elastic strip. The numerical implementation of the method is described and the truncation of the infinite set of equations is shown to converge to the full solution as the number of equations and unknowns goes to infinity. Results of the numerical computations are given and show effects of the leaky wave above the coincidence frequency of the plate and resonances below that frequency.

Electromagnetic wave excitation in a two‐dimensional impedance tunnel with a T‐junction

AbstractAs an example of tunnels with a branch and a bend, the one with a T‐junction is selected for analysis of electromagnetic wave propagation when it is excited with a line source. As the boundary condition on the tunnel wall, a surface impedance approximation has been introduced. The calculations have been performed with the introduction of a two‐dimensional model. In the calculation, the tunnel is divided into the source region (Tunnel I) and the remainder (Tunnel II). The field in the former is modal expanded in a series with unknown coefficients; in the latter a Fourier transform is introduced so that the field is treated in the transform domain. The field matching at the aperture is performed in the Fourier‐transformed domain so that the unknown expansion coefficients described above can be obtained as the solutions of an infinite set of simultaneous equations. Finally, the numerical results are compared with those of a model experiment and the effectiveness of the surface impedance approximation is discussed. It is found that the numerical and experimental results agree rather well for the electric field intensity distribution in Tunnel II.

An elastic contact problem for a neo-hookean half-space indented by a annular rigid stamp of arbitrary profiles

The present paper deals with an elastic contact problem for a neo-Hookean half-space indented by a annular rigid stamp of arbitrary profile, which constitutes a three part mixed boundary value problem. The problem is reduced into triple integral equations which are further reduced into an infinite set of simultaneous equations. Numerical results are illustrated graphically for the distributions of the normal displacement and stress in the case of parabolic concave punch for different values of a/b at z = 0

Some axially symmetric thermal stress distributions in an elastic solid containing an annular crack

The present paper seeks to solve the axisymmetric thermal stress distribution in an infinite, isotropic solid containing an annular crack. The problem is reduced to triple integral equations, which are ultimately reduced to the solution of an infinite set of simultaneous equations.

Sound radiation from fluid loaded orthogonally stiffened plates

The radiation of sound from infinite fluid loaded plates is examined when the plates are reinforced with two sets of orthogonal line stiffeners. The stiffeners are assumed to be equally spaced and exert only forces on the plate. The response to a convected harmonic pressure is found by using Fourier transforms and is given in terms of the harmonic amplitudes of the stiffener forces. These forces satisfy an infinite set of simultaneous equations to which a numerical solution must be found. An expression for the response to a general excitation is derived and from this the acoustic pressure in the far field is determined with particular reference to point force excitation.

Simultaneous Wiener–Hopf equations

In Noble's book The Wiener Hopf Technique, Pergamon, 1958, he considers the coupled system of Wiener–Hopf equations (§4.4, pp. 153–154)[Formula: see text]He shows that provided the functions L(α), M(α), Q(α), and R(α) have only simple pole singularities the solution can be reduced to two sets of infinite simultaneous linear algebraic equations. In this article a different approach is used which gives the solution in the form of a Fredholm integral equation of the second kind. This Fredholm integral equation can be reduced to infinite sets of simultaneous linear algebraic equations under the less restrictive conditions that either (i) L(α)/M(α) has no branch points in the lower α-half plane: Im(α) < τ+; or (ii) Q(α)/R(α) has no branch points in the upper α-half plane: Im(α) > τ−. In the special case considered by Noble if L(α)/M(α) (or Q(α)/R(α)) only have simple poles in the lower (upper) half plane then the Fredholm integral equation reduces to one infinite set of simultaneous equations. This extends the Wiener–Hopf technique to yet a larger class of boundary value problems, and simplifies the numerical computations.

Radiation of sound from an unflanged rigid cylindrical duct with an acoustically absorbing internal surface

A rigorous and exact solution is obtained for the problem of the radiation of sound from a semi-infinite unflanged rigid duct with an internal acoustically absorbent lining. The solution is obtained by a modification of the normal Wiener-Hopf technique. The solution is in terms of an infinite series of unknowns, which are determined from an infinite set of simultaneous equations. The infinite system converges rapidly enough to make the solution suitable for numerical computations. Some numerical results are given in graphical form for the propagation of the principal symmetric mode in the duct.

An elastic contact problem for a half-space indented by a flat annular rigid stamp

We consider an elastic contact problem for a half-space indented by a flat annular rigid stamp, which is a three-part mixed boundary value problem. If the distribution of pressure on the contact region is assumed, the contact problem will be reduced to the solution of an infinite set of simultaneous equations. Numerical results are also illustrated for the distributions of the displacements and stresses in the half-space.

Boundary-layer separation on a sphere in a rotating flow

The velocity just outside the boundary layer and upstream of the separation ring on a sphere moving along the axis of a slightly viscous, rotating fluid is calculated through a least-squares approximation on the hypothesis of no upstream influence. A reverse flow is found in the neighbourhood of the forward stagnation point fork≡ 2Ωa/U>k= 2·20 (Ω = angular velocity of fluid,U= translational velocity of sphere,a= radius of sphere) and is accompanied by a forwardseparation bubble, such as that observed by Maxworthy (1970) fork[gsim ] 1. Rotation also induces a downstream shift of the peak velocity; the estimated shift of the separation ring in the absence of forward separation increases withkto a maximum of 24°, in qualitative agreement with Maxworthy's observations.The least-squares formulation is compared with that given by Stewartson (1958) for unseparated flow (Stewartson did not consider separation). Both formulations require truncation of an infinite set of simultaneous equations, but Stewartson's formulation yields a non-positive-definite matrix that may exhibit spurious singularities. The least-squares formulation yields a positive-definite matrix, albeit at the expense of slower convergence for fixedk, and is especially well suited for automatic computation.Anad hocincorporation of a cylindrical wave of strength [Uscr ], such that the maximum upstream axial velocity is [Uscr ]U, is considered in an appendix. It is found thatkdecreases monotonically from 2·2 to 0 as [Uscr ] increases from 0 to 1.

Bottomless harbours

Does the harbour of an artificial island need a bottom? The excitation of waves inside a partially immersed open circular cylinder is considered. An incident plane wave is expanded in Bessel functions and for each mode the problem is formulated in terms of the radial displacement on the cylindrical interface below the cylinder. The solution is obtainable either from an infinite set of simultaneous equations or from an integral equation. It is shown that the phase of the solution is independent of depth and resonances are found at wave-numbers close to those of free oscillations in a cylinder extending to the bottom. If the resonances of the cylinder are made sharper (by increasing the depth of immersion) the peak response of the harbour increases, but the response to a continuous spectrum remains approximately constant. Numerical results are obtained by minimizing the least squares error of a finite numberNof simultaneous equations. Convergence is slow, but the error is roughly proportional to 1/Nand this is exploited. The solution obtained from a variational formulation using the incoming wave as a trial function is found to give a very good approximation for small wave-numbers, but is increasingly inaccurate for large wave-numbers. Away from resonance the amplitude of the harbour oscillation is less than 10% of the amplitude of the incoming wave provided the depth of the cylinder is greater than about ¼ wavelength, and it is argued that in practice at the resonant wave-number oscillations excited through the bottom of the harbour will leak out through the entrance before they can reach large amplitudes. In an appendix the excitation of harbour oscillations through the harbour entrance is discussed, and some results of Miles & Munk (1961) on an alleged harbour paradox are re-interpreted.

Surface Integral Formulae for Geomagnetic Studies

Many geomagnetic studies are concerned with time-varying magnetic fields at the Earth’s surface, and their relation to electric current systems in the upper atmosphere and in the Earth. There are, for example, many recent (as well as many earlier) studies of the Solar and Lunar diurnal variation fields, and of the ionospheric and earth current systems associated with them. These studies have led to the development of various methods and formulae for separating the surface field into parts of external and internal origin, and for determining the ionospheric currents that correspond to the external part. Further, the relationship found between the fields of internal and external origin is of the form that would be expected if the earth currents were due simply to electromagnetic induction by the moving and varying ionospheric current systems. A study of this relationship can lead to information about the conductivity of the Earth at various depths. This requires the solution of various mathematical problems on the electromagnetic induction of currents in concentric spheres and spherical shells of non-uniform conductivity, and involves the evaluation of self and mutual induction effects in these conductors. One of the important problems is to estimate the influence of currents induced in the relatively highly conducting oceans, and the screening effects of these currents on the conducting layers of the earth below. In the earlier studies the current systems and magnetic fields were expressed in terms of series of spherical harmonics and many valuable results were obtained in this way. When, however, it is desired to study detailed features of certain fields, such as the great enhancement of the Sq field near the dip equator, or, again, when one attempts to solve an induction problem which has, as one of the conductors, a thin shell with abrupt changes in conductivity, as at the surface of the Earth, then this method leads to difficulties owing to the slow convergence of the spherical harmonic expansions required. In the solution of such induction problems, the numerical work involved in solving the infinite sets of simultaneous equations, that are found for the coefficients of the required spherical harmonic expansions, is often prohibitive, and the solution obtained by truncating the series at a point that would make the numerical work possible very inaccurate. (See for example, the discussion by Ashour (1965a) of a problem considered by Rikitake & Yokoyama (1955)). Hence, alternative methods of treating these problems have been sought.

An Infinite Set of Simultaneous Equations

An Infinite Set of Simultaneous Equations

Simultaneous Wiener-Hopf Equations and Their Application to Diffraction Problems in Electromagnetic Theory. II

An infinite set of simultaneous equations of the Wiener-Hopf type with the kernel which is a Toeplitz matrix ∥ K m - n (ζ)∥ and their application of diffraction problems in electromagnetic theory are discussed. Owing to the special form of the kernel, it can be shown that the Fourier series expansion method transforms the infinite system into a single equation of the Wiener-Hopf type, so the standard Wiener-Hopf procedure can be applied. On the basis of this result, a treatment is made of the radiation of electromagnetic waves of TE type from an infinite set of staggered, equally spaced, semi-infinite plates.

An Infinite Set of Simultaneous Equations

An Infinite Set of Simultaneous Equations

Diffraction of Plane Electromagnetic Waves by a Perfectly Conducting Cylindrical Lamina

The diffraction of a plane electromagnetic wave by a perfectly conducting cylindrical lamina is solved for the case where the E vector is parallel to the generators of the lamina. Eigenfunction expansions for the interior and exterior diffracted field are employed. By application of the boundary conditions, the problem reduces to the solution of an infinite set of simultaneous equations for the diffracted field expansion coefficients. These equations are solved by numerical methods. Special attention has been paid to the long-wave scattering region. The scattered intensity in the far field and the total scattered power are studied.

BENDING OF A SQUARE PLATE WITH TWO ADJACENT EDGES FREE AND THE OTHERS CLAMPED OR SIMPLY SUPPORTED

T LITERATURE abounds with applications of classical thin elastic plate theory to rectangular plates having a multitude of combinations of boundary conditions and transverse loading. By far the greatest number of these papers deal with plates having only simply-supported and/or clamped boundary conditions, which requires restrictions upon the deflection and its first or second derivatives along the edges. Considerably fewer solutions have been obtained when a free edge is involved because of the relatively difficult combination of third derivatives of deflection which is encountered in the boundary conditions. When two adjacent free edges are involved, so that a free corner is created, the problem becomes exceedingly difficult. No closed form of solution is known to exist for any of this latter class of problems. Some solutions have been obtained for cantilevered rectangular plates—i.e., three edges free and the other clamped. Finite difference solutions for various aspect ratios and loadings were presented by Holl, Barton, MacNeal, Nash, and Livesly and Birchall. Nash also solved the problem of the uniformly loaded contilevered plate having a span-to-chord ratio of 1/2 by using a form of collocation known as point matching. This method depends upon choosing deflection functions which satisfy the partial differential equation of the continuum exactly, while matching the boundary conditions at only a finite number of discrete points. Nash solved the problem twice, once using an algebraic polynomial and, again, using a hyperbolic-trigonometric series. A recent paper by Leissa and Niedenfuhr presents the solution for the uniformly loaded cantilevered square plate using two other approaches: point matching, using an algebraic-trigonometric polynomial, and a Rayleigh-Ritz minimal-energy formulation. Doubtlessly, much of the foregoing work with cantilevered plates has been the result of the need for accurate information for the design of aircraft wings. However, approximate deflections and stresses may be obtained for these problems by using elementary beam theory. In marked contrast, the rectangular plate having two adjacent edges free and the others clamped or simply supported is in no way representable by beam theory. These problems are no more difficult than the cantilevered plate, but very little has been done with them up to the present. The problem of the uniformly loaded square plate with two adjacent edges free and the others clamped was solved by Huang and Conway. This involved a skillful superposition of five problems and the partial solution of an infinite set of simultaneous equations. Yeh generalized this same problem by including the reaction due to an elastic foundation. The Rayleigh-Ritz method was employed. In the present work the writers exhibit solutions for the four problems of a square plate with two adjacent edges free and the others clamped or simply supported subjected to either a uniform transverse loading or a concentrated force at the free corner. The method of point matching is used to solve the problems and the results are compared with other known solutions for two of the cases.

多重衝突法による炉の臨界条件

The criticality conditions of the multiplying medium are derived analytically by the multiple collision method based on the viewpoint of the Monte Carlo approach. The successive life cycles of a typical neutron, in which the collision with a nucleus in the medium is regarded as an end of each life cycle. are followed statistically in onder of time. The total number of neutrons in the medium is obtained by summing up those in every life-cycle and the criticality condition is derived from the convergence condition as time proceeds to infinity. The criticality of the one-dimensional homogeneous medium with isotropic scattering in the onevelocity theory is treated. The well-known criticality condition is obtained easily for the infinite medium. For the finite medium, the critical equation is reduced to an infinite set of simultaneous equations which are valid when the thickness of the slab is much smaller than the mean free path. The approximate solution is obtained and discussed (auth)

The finite range Wiener-Hopf integral equation and a boundary value problem in a waveguide

In this paper the boundary value problem of a finite bifurcation in a rectangular waveguide is formulated in terms of a finite range Wiener-Hopf integral equation, and the solution of the integral equation is presented. As a first step, an infinite set of simultaneous equations is obtained from the integral equation. The solution of the above set of equations is then obtained by analytic means. It is indicated that the method developed can be applied to certain other problems of mathematical physics and a number of examples of such problems are included.

POLYNOMIAL APPROXIMATIONS IN PLANE ELASTIC PROBLEMS

A general solution of plane elastic problems using the complex variable has been developed by Muschelisvili. His solution, for simply connected areas, requires the determination of two analytic functions, given by two integral equations, in which the loaded area in the z-plane is transformed to a unit circle in the ζ-plane. The integrations involved have been carried out only for cases in which the mapping function is rational. In this paper a method of solution is derived when the Taylor's series for the mapping function is known. This leads to solving an infinite set of simultaneous equations in which the unknowns are the coefficients in the Taylor's expansion of the analytic functions defined by Muschelisvili. These equations are readily solved by an iterative process, giving results of any required degree of accuracy. To illustrate the application of the method, the problem of a square with equal and opposite tensions directed along a diagonal, and applied at the extremities of that diagonal, is given.