Showing papers on "Superposition principle published in 1974"
TL;DR: In this paper, the authors proposed a new iterative procedure for solving the equations, called the natural iteration method, which does not use differentiation nor matrix inversion and may be called the Natural Iterative Process (NIP) method.
Abstract: In the cluster‐variation method of cooperative phenomena and also in the quasichemical method, the bottleneck step has been to solve simultaneous equations. This paper proposes a new iterative procedure for solving the equations. This iteration does not use differentiation nor matrix inversion and may be called the natural iteration method. The free energy always decreases as the iteration proceeds, with a consequence that the iteration always converges to a stable solution (a local minimum of free energy) as long as the initial state is a physically acceptable one. The method derives in its introductory step a superposition approximation which writes the distribution variables of the basic cluster as a product of those of subclusters. The method is first explained with the pair approximation of the Ising ferromagnet, and then is applied to the fcc binary alloys to derive a phase diagram which is compared with the one reported recently by van Baal.
389 citations
TL;DR: In this paper, the authors derived the basic hydromagnetic wave equation and applied it to a cylindrical model of the inner magnetosphere, which offers a good representation of the spatially dependent field line frequencies present in a dipole field.
Abstract: The asymptotic temporal behavior of hydromagnetic waves in a model of the inner magnetosphere is shown to be characterized by guided modes. The basic hydromagnetic wave equation is derived and applied to a cylindrical model of the inner magnetosphere. This model offers a good representation of the spatially dependent field line frequencies present in a dipole field. The initial value problem for the symmetric toroidal mode is solved, and its singularities are treated by Fourier superposition. Singularities also are present in the asymmetric poloidal mode wave equation and are shown to be logarithmic. Fourier superposition leads to the solutions for the electric and magnetic fields, which are asymptotic in time. The results indicate decay of the poloidal modes and domination by the toroidal modes, i.e., field line control of the propagation. At the latitude at which the maximum amplitude of a particular frequency occurs, the wave becomes linearly polarized. The asymptotic micropulsation periods depend on the characteristic field line periods and are usually longer at higher latitudes. Undamped guided waves lead to some terms, such as change density and parallel current density, that increase linearly with time. The inclusion of loss mechanisms in the wave equation, e.g., conductivity, limits the guided modes and prevents such nonphysical effects from occurring.
100 citations
TL;DR: The superposition of several periodic or quasiperiodic patterns produces moire effects as discussed by the authors, and the combination of three grid structures adds great flexibility to the use of moire phenomena.
Abstract: The superposition of several periodic or quasiperiodic patterns produces moire effects. In particular, the combination of three grid structures adds great flexibility to the use of moire phenomena. Information about the fringe structure, given by vector addition in Fourier space, allows investigation of formations, interpretation, and expectation problems concerning moire-pattern parameters. Any frequency and orientation of the pattern is possible, using three-line gratings. On the other hand, three quasiperiodic gratings with variations of frequency and/or orientation result in a zone-plate shape of the moire pattern regardless of grating distortions. Both elliptical and hyperbolic zone plates occur. Several illustrations demonstrate these phenomena and indicate some potential applications.
79 citations
75 citations
TL;DR: This paper uses the statistics of the atmospheric impulse response (Green's function) to show that the atmospheric mode decomposition exhibits far-field and near-field regimes very similar to those of free-space propagation.
Abstract: Recent studies have used the superposition principle (extended Huygens-Fresnel principle) to characterize completely the statistics of a field that has propagated through a thick slab of turbulent air in terms of the statistics for spherical-wave sources. In this paper, we consider the normal-mode decomposition associated with this linear system propagation model. In particular, we use the statistics of the atmospheric impulse response (Green's function) to show that the atmospheric mode decomposition exhibits far-field and near-field regimes very similar to those of free-space propagation. The significance of these results for optical communication through the atmosphere is briefly discussed.
60 citations
TL;DR: In this article, the authors considered the Love mode as a superposition of propagating plane SH waves in a surface layer and an inhomogeneous plane SH wave falling off with depth in the half-space below.
Abstract: Love-wave transmission and reflection are studied numerically for a vertical interface between two layered models. By a new method of approximation, the transmission and reflection coefficients can be computed for propagation directions differing from normal incidence. We consider the Love mode as a superposition of propagating plane SH waves in a surface layer and an inhomogeneous plane SH wave falling off with depth in the half-space below. The method is based on satisfying all boundary conditions on the vertical interface and computing the coupling between the interface stress displacement pattern and the transmitted/reflected Love modes. The computations are done rapidly on a small computer. The transmission coefficients change rapidly with period. For propagation from ocean to continent, they increase with decreasing periods in the interval 60 to 20 sec. The opposite trend is found for propagation from continent to ocean. The amplitude transmission coefficient curves show almost no dependence on the angle of incidence except for large angles. The fine structure of the curves is similar to that for plane SV waves incident on an interface between two homogeneous elastic media. Love-wave amplitude observations at OBS on the ocean bottom northwest of San Francisco and at BKS in Berkeley also show these trends as a function of period and of angle of incidence.
46 citations
TL;DR: In this article, the problem of constructing the scattering field for plane-wave incidence on a sinusoidal surface is considered, and linear equations for the coefficients of these waves are derived for both soft and hard boundaries.
Abstract: The problem of constructing the scattering field for plane‐wave incidence on a sinusoidal surface is considered. Above the highest excursion of the surface the scattered field is written in the usual way as a superposition of upgoing and decaying waves. Linear equations are derived for the coefficients of these waves. Both soft and hard boundaries are considered, as well as arbitrary incidence angle. The equations for the soft boundary condition are similar to those of Uretsky, but the derivation is simpler. The full results for the hard boundary condition are apparently new.
44 citations
TL;DR: The general problems associated with multiple scan photon/X-ray transmission image reconstruction are formulated and a technique which leads to a highly accurate solution is presented.
Abstract: The general problems associated with multiple scan photon/X-ray transmission image reconstruction are formulated and a technique which leads to a highly accurate solution is presented. The method, 'Technique of Linear Superposition with Compensation', is detailed, together with the computer algorithm and simulation results.
41 citations
TL;DR: In this article, the superposition model was used to analyse ESR data for Gd3+ and Eu2+ in strained cubic crystals, and the quadrupolar (n=2) parameters derived from this data were shown to be consistent with the model proposed by Newman and Urban (1972).
Abstract: The superposition model is used to analyse ESR data for Gd3+ and Eu2+ in strained cubic crystals. The quadrupolar (n=2) parameters derived from this data are shown to be consistent with the model proposed by Newman and Urban (1972). It is also shown that the n=4 parameters are consistent with the hypothesis that they represent a simple superposition of contributions from the coordinated ligands.
31 citations
TL;DR: In this article, the authors introduced the concept of coherence width of the interaction between a molecule and light and proposed a mathematical formulation in terms of the reduced density operator for the molecule.
31 citations
TL;DR: In this article, a method for determining the stresses and displacements, under plane strain or plane stress conditions, in a linear elastic multizoned continuum having an irregular boundary geometry is presented.
Abstract: A method is presented for determining the stresses and displacements, under plane strain or plane stress conditions, in a linear elastic multizoned continuum having an irregular boundary geometry. Each zone is homogeneous and consists in general of a different orthotropic material, with one plane of elastic symmetry parallel to the problem plane, and the other two planes orientated arbitrarily. The singular solution used is that for a point load acting within a homogeneous orthotropic infinite lamina. The derivation of this new solution is given. Since the principle of superposition applies, the numerical integration of multiples of such singular solutions around zone boundaries produces a result satisfying both the governing equations and the zone boundary conditions on discrete boundary elements. The main advantage of the method is that, since only the boundary is discretized, the system of equations is small compared with other numerical methods.
01 Jan 1974
TL;DR: In this paper, the authors discuss conservation laws in quantum mechanics and show that the wave function completely determines the state of a physical system in quantum physics, and that the value of the derivative of a wave function with respect to time at any given instant must be determined by the value itself at that instant, and, by the principle of superposition, the relation between them must be linear.
Abstract: This chapter discusses conservation laws in quantum mechanics. It describes the Hamiltonian operator, the differentiation of operators with respect to time, stationary states, matrices of physical quantities, and different types of momentum. The wave function completely determines the state of a physical system in quantum mechanics. This means that, if this function is given at some instant, not only are all the properties of the system at that instant described, but its behavior at all subsequent instants is determined. The mathematical expression of this fact is that the value of the derivative of the wave function with respect to time at any given instant must be deter-mined by the value of the function itself at that instant, and, by the principle of superposition, the relation between them must be linear. Further, if the system is closed or is in a constant external field, its Hamiltonian cannot contain the time explicitly. This follows from the fact that all times are equivalent so far as the given physical system is concerned. Since any operator of course commutes with itself, a conclusion can drawn that Hamilton's function is conserved for systems which are not in a varying external field. As is well known, a Hamilton's function which is conserved is called the energy. The law of conservation of energy in quantum mechanics signifies that, if in a given state the energy has a definite value, this value remains constant in time. States in which the energy has definite values are called stationary states of a system.
TL;DR: In this article, a generalized form of the equations of linear viscoelasticity is presented which enables theoretical treatment of situations where a material is not at equilibrium with its environment and/or is subjected to a changing environment.
Abstract: A generalized form of the equations of linear viscoelasticity is presented which enables theoretical treatment of situations where a material is not at equilibrium with its environment and/or is subjected to a changing environment. These generalized equations are used successfully to predict the recovery behavior of wool and nylon fibers from bending deformations in experiments involving conditions of changing temperature and relative humidity.
TL;DR: In this article, a generalized formulation of the imposed rotation method for the inelastic analysis of multistory frames is presented. But this method is not applicable to the analysis of multi-dimensional structures.
Abstract: The imposed rotation method for the inelastic analysis of frames rests on the interpretation of the actual bending moment distribution as the superposition of linear elastic moment responses to loads and to unknown plastic rotations regarded as imposed strains. The method given herein is a generalized formulation, covering second-order geometric effects and moment-axial force interaction. Finite element models of frames and piecewise linear moment-rotation laws for critical sections are assumed. Recent algorithms for solving quadratic programming and linear complementarity problems are shown to be efficiently applicable to the analysis of multistory frames. The safety factor with respect to local failure because of “brittle” flexural behavior turns out to be attainable by a suitably modified linear programming procedure.
TL;DR: The inversion technique of Backus and Gilbert is applied to the problem of restoration of optical objects by superposition of images and curves for the trade off between resolution and noise are obtained.
Abstract: The inversion technique of Backus and Gilbert is applied to the problem of restoration of optical objects by superposition of images. The problem is formulated in the general case of space-variant systems. Curves for the trade off between resolution and noise in restoring objects degraded by simple diffraction and susceptible to uncorrelated noise are obtained. The technique is used to restore a test object whose image is sampled at the Nyquist rate and to which noise (simulated on a computer) is added. Restorations corresponding to several points on the trade-off curve are obtained.
TL;DR: In this article, it was shown that the (n −1)2 independent moduli and (2n − 1) unphysical phases completely specify all n-dimensional unitary matrices in two and three dimensions, but false in four or more.
Abstract: A conjecture that the (n −1)2 independent moduli and (2n −1) unphysical phases completely specify all n‐dimensional unitary matrices is shown to be true in two and three dimensions, but false in four or more. The implications for quantum theory are discussed.
TL;DR: In this article, the quantum characteristic functions, quasi-distributions and their moments for radiation passing through a random medium are obtained and some useful approximations are derived for the photon counting distribution and its factorial moments.
Abstract: The present paper uses results of the preceding paper (Czech. J. Phys.B 24 (1974), 374) to obtain the quantum characteristic functions, quasi-distributions and their moments for radiation passing through a random medium. It is shown that the quasi-distribution related to normal ordering does not exist as an ordinary function and the time behaviour of the quasi-distribution related to antinormal ordering is discussed. Exact photon counting distribution and its factorial moments are obtained and some useful approximations are derived. General theory is applied to the superposition of coherent and chaotic radiation. Finally some consequences are discussed such as self-radiation effects of the medium, the relation of the present active non-linear quantum description and the previous passive linear quantum description by Tatarski and passive linear quasi-classical description by Diament and Teich and the behaviour of the photon counting distribution for various levels of turbulence.
TL;DR: In this paper, the analysis of infinite arrays of waveguide apertures on cylinders covered by a dielectric is approached by enforcing the continuity of the EM fields at the air-dielectric and at the cylindric-cylinder interfaces.
Abstract: The analysis of infinite arrays of waveguide apertures on cylinders covered by a dielectric is approached by enforcing the continuity of the EM fields at the air-dielectric and at the dielectric-cylinder interfaces. The continuity of the EM fields at the first interface is enforced by representing the fields in the air and in the dielectric by an orthonormal set of modes LSE and LSM with respect to the radial direction. The matching of the fields at the second interface is performed by resorting to the "eigenexcitation" method [1]. The fields external to the cylinder are represented by a set of space harmonics matching the symmetry of the excitation of the array and the fields in the waveguides by a superposition of normal waveguide modes. The continuity of the fields is approximately enforced by using Galerkin's method. The rigorous analysis of these structures leads to rather involved expressions for the element driving point admittance and for the far fields. An approximate analysis is introduced to simplify the design of these structures for cylinders of large radius. Numerical examples illustrate the good approximation given by the simplified analysis. Numerical results are presented for the case of an array on a cylinder with radius approximately 100\lambda . The array element pattern shows the presence of resonance dips much more pronounced than the notches due to grating lobe phenomena.
TL;DR: In this paper, the authors applied the network theory developed by Yamamoto to explain at least qualitatively some of the nonlinear viscoelastic behavior of concentrated polymer systems in shearing flow, with the assumption that the probability of chain breakage is proportional to the square of the end-to-end distance in the chain.
Abstract: The network theory developed by Yamamoto (J. Phys. Soc. Jpn., 11, 413 (1956)) is applied to explain at least qualitatively some of the nonlinear viscoelastic behavior of concentrated polymer systems in shearing flow, with the assumption that the probability of chain-breakage is proportional to the square of the end-to-end distance in the chain. It is shown that the shear-rate dependence of the steady viscosity is similar to the frequency dependence of the absolute value of the complex viscosity. The so-called stress overshoot at the beginning of shearing flow, the stress relaxation after the stoppage of flow, the ordinary stress relaxation under large deformation, and the superposition of a small oscillation upon steady shearing flow are treated; the results are in good qualitative agreement with the experiment. The rate-dependent and the deformation-dependent relaxation spectra are derived from the time dependence of the stresses in the two kinds of stress relaxation.
TL;DR: In this article, the analysis of the element pattern in an array of waveguide elements on a cone is approached by systematically exploiting the geometrical symmetry of the structure, which consists of decomposing the arbitrary array excitation into fundamental excitations (eigenexcitations) for which the solution of the electromagnetic problem is simpler than in the general case.
Abstract: The analysis of the element pattern in an array of waveguide elements on a cone is approached by systematically exploiting the geometrical symmetry of the structure. The method consists of decomposing the arbitrary array excitation into fundamental excitations (eigenexcitations) for which the solution of the electromagnetic problem is simpler than in the general case. Following Bailin and Silver, the fields external to the cone are represented as a superposition of modes TE and TM to the radial direction. For each eigenexcitation the fields in the waveguide elements are represented as a superposition of normal waveguide modes. The enforcement of the continuity of the electromagnetic fields at the conical interface provides complete information about the radiation from the elements and the reflection coefficients of the modes in the waveguide. The array element pattern is obtained by the superposition, with suitable weights, of the patterns of the eigenexcitations. The communication presents the computational procedures followed to evaluate with high speed and precision the roots of the Legendre functions necessary to represent the fields external to the conical structure. Numerical results for the realized element patterns of a particular conical array are presented in the final section.
TL;DR: In this article, a generalized form of the equations of linear viscoelasticity is presented which describes the behaviour of a material whose inherent mechanical properties are changing with time as a result of ageing or of changing external environmental constraints.
Abstract: A generalized form of the equations of linear viscoelasticity is presented which describes the behaviour of a material whose inherent mechanical properties are changing with time as a result of ageing or of changing external environmental constraints. The equations are used to successfully predict recovery from stress relaxation measurements for a Lincoln wool fibre subjected to large step changes in relative humidity during the experiment.
TL;DR: The existence of steady state solutions to the linearized ion and electron Vlasov equation for longitudinal waves in an initially stable plasma is demonstrated in this article, where the evolution of an arbitrary initial perturbation can be described by superposition of these solutions.
Abstract: The existence of steady‐state solutions to the linearized ion and electron Vlasov equation is demonstrated for longitudinal waves in an initially stable plasma. The evolution of an arbitrary initial perturbation can be described by superposition of these solutions. Some common approximations to the full set of equations can be solved in the same way. In some special cases, relevant, for instance, for single‐ended Q machine experiments, a problem with given boundary conditions can be solved by superposition of normal modes.
TL;DR: The method of superposition of images used previously to restore coherent or incoherent degraded images, adapted to the restoration of partially coherent images, is adapted to restore partially coherent image as mentioned in this paper.
TL;DR: In this article, a generalization of the superposition principle of quantum mechanics is proposed introducing the concept of maximal state of a logic, where the maximal state is defined as the sum of all the maximal states of the logic.
Abstract: A generalization of the superposition principle of quantum mechanics is proposed introducing the concept of maximal state of a logic.
TL;DR: In this article, the Yvon Born-Green integral equation with the Kirkwood superposition approximation is modified via a truncation of dense fluid contribution to third-particle correlations at the boundary of the intermolecular potential wells of the interacting molecular pair.
Abstract: The Yvon‐Born‐Green integral equation with the Kirkwood superposition approximation is modified via a truncation of dense fluid contribution to third‐particle correlations at the boundary of the intermolecular potential wells of the interacting molecular pair. This de‐emphasis of third‐body contributions, manifesting itself in a ``truncated superposition approximation,'' improves a broad spectrum of equilibrium and transport property predictions significantly while simultaneously shortening the computational extent of the integral equation problem. The results are compared with earlier work performed by the authors using the Kirkwood superposition approximation and with experiment. The nature of both the Kirkwood and truncated superposition approximations is discussed in light of recent schemes used to shorten the computations required to obtain solutions to the Percus‐Yevick integral equation.
TL;DR: In this article, the authors pointed out the dangers of using the superposition technique when the data contain variations caused by several different periodic sources and proposed a delay filter representation of the process.
Abstract: Superposing and averaging of consecutive periods of data can be used to estimate variations in the data produced by a periodic source (Chapman and Bartels, 1940a). This paper points out the dangers in using the superposition technique when the data contain variations caused by several different periodic sources. By considering a delay filter representation of the superposition process, the optimum data handling procedure is found. The problem of separating weak lunar effects from strong solar effects in terrestrial data is considered as a specific example.
TL;DR: In this article, it was shown that the assumption of linearity is by no means unreasonable and the principle of superposition is only necessary for linearity, when used in this sense.
Abstract: A linear analysis of ship motions (for example, of directional stability and control) does not require that slow-motion derivatives be employed to represent fluid forces and moments. Indeed the notion of such a representation implies the use of a Taylor series in circumstances where it is not strictly tenable. For linearity, it is only necessary that the principle of superposition be employed. Experimental data show that, when used in this sense, the assumption of linearity is by no means unreasonable.
01 Jun 1974
TL;DR: In this paper, the geometrical theory of diffraction (GTD) was employed to analyze the radiation from a perfectlyconducting rectangular cylinder illuminated by an array of line sources, and a new diffraction coefficient valid in the transition regions of the shadow and reflection boundaries is employed to obtain a continuous total field.
Abstract: The geometrical theory of diffraction (GTD) is employed to analyze the radiation from a perfectly-conducting rectangular cylinder illuminated by an array of line sources. The excitation of the cylinder by a single electric or magnetic current line source is considered first, and a solution which includes contributions from the geometrical optics rays and all singly- and doubly-diffracted rays is obtained. A new diffraction coefficient valid in the transition regions of the shadow and reflection boundaries is employed to obtain a continuous total field, except for negligible discontinuities in the doubly-diffracted field at its shadow boundaries. Patterns calculated by the GTD method are found to be in excellent agreement with those calculated from an integral equation formulation. Using superposition the solution for array or aperture excitation of the rectangular cylinder is obtained. A computer program for this solution is included.
TL;DR: The application of Tobolsky's time-temperature superposition principle for the stress-relaxation behavior of amorphous polymers is reviewed, and its extension to crystalline polymers and the employment of X-ray and optical methods is discussed as mentioned in this paper.
Abstract: The application of Tobolsky's time-temperature superposition principle for the stress-relaxation behavior of amorphous polymers is reviewed, and its extension to crystalline polymers and the employment of X-ray and optical methods is discussed The desirability of applying the superposition principle to time-dependent crystal orientation functions is described The use of those techniques for the study of the relaxation of a low-density polyethylene sample is illustrated Spherulite orientation theory is used for analysis of the results