Showing papers on "Superposition principle published in 1968"

Integrals of Nonlinear Equations of Evolution and Solitary Waves

Abstract: In Section 1 we present a general principle for associating nonlinear equations evolutions with linear operators so that the eigenvalues of the linear operator integrals of the nonlinear equation. A striking instance of such a procedure discovery by Gardner, Miura and Kruskal that the eigenvalues of the Schrodinger operator are integrals of the Korteweg-de Vries equation. In Section 2 we prove the simplest case of a conjecture of Kruskal and Zabusky concerning the existence of double wave solutions of the Korteweg-de Vries equation, i.e., of solutions which for |I| large behave as the superposition of two solitary waves travelling at different speeds. The main tool used is the first of remarkable series of integrals discovered by Kruskal and Zabusky.

Nonlinear filtering of multiplied and convolved signals

Abstract: An approach to some nonlinear filtering problems through a generalized notion of superposition has proven useful. In this paper this approach is investigated for the nonlinear filtering of signals which can be expressed as products or as convolutions of components. The applications of this approach in audio dynamic range compression and expansion, image enhancement with applications to bandwidth reduction, echo removal, and speech waveform processing are presented.

Nonlinear filtering of multiplied and convolved signals

Abstract: An approach to some nonlinear filtering problems through a generalized notion of superposition has proven useful In this paper this approach is investigated for the nonlinear filtering of signals which can be expressed as products or as convolutions of components. The applications of this approach in audio dynamic range compression and expansion, image enhancement with applications to bandwidth reduction, echo removal, and speech waveform processing are presented.

Photon, spin, and Raman echoes

Abstract: The general technique of the excitation of a system into a linear superposition of states is reviewed. Recent results in ruby are presented together with a simple theory which predicts Raman echoes.

Dynamic modulus of polyisobutylene solutions in superposed steady shear flow

Abstract: Measurements are reported for the first time of the effect on the dynamic modulus of a viscoelastic liquid of the superposition of a steady shearing orthogonal to the oscillatory shearing. The author believes that the study of this flow, together with the case of parallel superposition of a steady and an oscillatory shearing, will clarify the importance of the second invariant of the rate of deformation tensor used in a number of rheological equations of state for viscoelastic liquids. 8.54, 6.86 and 5.39% solutions of polyisobutylene in cetane were studied using a recently developed electromagnetic transducer over the frequency range 0.3 to 150 c.p.s. Preliminary experiments on the orthogonal superposition of two oscillatory shear flows are also reported.

Distribution Function of Classical Fluids of Hard Spheres. II

Abstract: The pair‐ and triplet‐correlation functions g12[≡ g(r12)] and g123[≡ g(r12, r13, r23)] are used to make a superposition approximation g123g124g134g234 / (g12g13g14g23g24g34) to the quadruplet correlation function g1234. Introduction of this approximation into the second equation of the Born–Green–Yvon (BGY) hierarchy makes it possible to truncate the hierarchy. The resulting equations are a pair of simultaneous integro‐differential equations involving g12 and g123, which hereafter are referred to as the BGY2 equations. The BGY2 theory was further elucidated by the use of the hard‐sphere potential to solve the corresponding BGY2 equations at two selected values (10 and 15) of λ, which is used as the independent variable and is equal to 4πρg(σ) (ρ = density, σ = sphere diameter). The densities {ρ = λ / [4πg(σ)]} corresponding to these values of λ can be established from the BGY2 solutions of g(r) at contact distance of two spheres. They are found to be, respectively, 0.298ρ0 and 0.372ρ0 (ρ0 = the close‐pack...

Finite Elements in Linear Viscoelasticity

Abstract: : A stress analysis method is presented that includes the capability for transient, nonhomogeneous temperature distribution. The method is based on the following assumptions: linear viscoelasticity with hereditary integral form of stress-strain relation; validity of the reduced time hypothesis; bulk modulus constant in time; and a homogeneous, isotropic material. The associated, finite element computer program, called TAP (Thermoviscoelastic Analysis Program), is a plane strain formulation with uniform strain and linear temperature distribution assumed in each element. The element matrices involve superposition integrals which are approximated numerically for stepping out the time-varying solution. TAP solutions for simple problems are compared with exact and other approximate solutions and with other approximate methods to more complex problems. Different numerical procedures are considered, in particular one that avoids integration back to the time origin, thus minimizing computer storage and solution times.

A Generalized Approach to a Solution of Aperiodic Arrays and Modulated Surfaces

Abstract: A generalized approach to a solution of the aperiodic phased-array problem, and derived periodically modulated surfaces, is possible by an extension, using linear superposition, of the solutions for the related periodic phased array. This approach can be applied to any aperiodic array of elements such as planar arrays of waveguides, or dipoles, or thick walled parallel-plate arrays. It also leads to the solution for the dispersion relation, in terms of the periodic phased-array solution, for the surface of short-circuited waveguide clements; the short circuit depth being periodically modulated along the surface.

Normal and antinormal correlations for the superposition of thermal and coherent fields

Abstract: The sixth-order normal correlation function for the superposition of thermal and coherent fields is calculated in coherent state formalism with the help of a method which enables us to understand why the semiclassical and quantum descriptions of light are equivalent. An experiment for measuring this sixth-order correlation function, which provides the possibility of examining the coherent state aspects of a laser, is proposed. Antinormal correlation functions for the superposition of thermal and coherent fields are also calculated and general relations between normal and antinormal correlation functions, which generalize the relations obtained in recent papers, are derived. A graphic method for computing the moments of the integrated intensity of arbitrary order for the superposition of thermal and coherent fields is introduced.

Classical fluids and the superposition approximation

Abstract: The statistical theory of fluids in equilibrium is developed in that form which is based on the definition of correlation functions and which invokes a superposition closure approximation. Because of the importance of pair interactions between the constituent molecules of a simple fluid, the theory is particularly concerned with the calculation of the pair correlation function. The arguments are set against alternative developments which do not involve a closure approximation. Microscopic conditions in the fluid are first briefly reviewed in a form which allows the known formulae of statistical mechanics to be adapted to conditions appropriate to fluids. The equations of the theory are developed and a superposition approximation introduced. Ways of improving this approximation are investigated. The article ends with the consideration of some numerical results derived from the theory.

Transmission and Reflection of Electromagnetic Waves at the Boundary of a Relativistic Collisionless Plasma

Abstract: The exact solution of the transmission and reflection problem for transverse electromagnetic waves incident on a bounded plasma has been discussed to some extent by several authors. Shure considered the special cases of perpendicular incidence on nonrelativistic half‐space and slab plasmas and made use of van Kampen‐Case modes to construct the solution. For both half‐space and slab plasmas, we generalize his results to (i) arbitrary temperatures (relativistic) and (ii) arbitrary angles of incidence. For simplicity, we consider the special case where the incident electric field is perpendicular to the plane of incidence and assume that particles are reflected specularly at the interface. We proceed somewhat differently from Shure, and use a Laplace transformation in obtaining our solution. We also show that present solutions can be expressed as a superposition of van Kampen‐Case modes appropriate to a relativistic plasma.

Symbolic Approach to Dynamical Problems in Plates

Abstract: A formal approach is used to obtain two‐dimensional differential equations (of infinite order) for dynamical problems in plates. It is assumed that the displacements may be expanded in power series in z, the thickness coordinate. These power series are substituted into the three‐dimensional dynamical equations of linear elasticity. The coefficients of powers of z are equated to zero leading to an infinite sequence of differential equations which by formal manipulation are reduced to three differential equations of infinite order in which the midplane displacements are the dependent variables and x, y, t are the independent variables. It is shown how various special theories including the classical theories may be obtained from the general equations by making certain assumptions on the frequency and wavelength of the expected solutions. A short discussion of the solution of an initial value problem by means of the superposition of solutions of the various special theories is also given.

Photostorage and retrieval of multiple images by diffraction processes with cross-talk suppression

Abstract: This application depicts methods and structures for suppressing crosstalk in the photostorage and selective retrieval of one or more of a plurality of images recorded in additive superposition and in respective multiplication with a unique spatial carrier function. A first method disclosed involves the formation of a density record which is processed to substantially gamma - 2. A second method involves the use of low duty-cycle carrier functions.

Evolution of Reduced Distribution Functions. II. The Classical Two‐Body Function

Abstract: A nonequilibrium BBGYK hierarchy is truncated at the two‐body equation by a dynamical superposition approximation equivalent to neglecting three‐body deviations from equilibrium in the triplet distribution function. This truncated hierarchy is solved by iteration in the case of a fluid system near equilibrium. The solutions are partially summed to series that are nearly series in the strength of the potential.

On the Action of Concentrated Loads in the Case of a Cosserat Continuum

Abstract: As it is known [4], all concentrated loads can be constructed starting from a single concentrated load, considered as a fundamental load; usually the concentrated force is considered as such a load. The result is valid for all deformable continua with mechanical symmetry for which the principle of the local superposition of effects can be used.

A synthesis method for bidimensional apertures

Abstract: A generalization of the Woodward-Lawson synthesis method is introduced for a wide class of apertures (including as particular cases the rectangular, the rhombic, and the hexagonal). By resorting to a rather general type of bidimensional Fourier series expansion, the aperture distribution is obtained as a superposition of orthogonal constant amplitude linearly phased components, whose complex amplitude coefficients are the values of the radiation pattern in a regular lattice of "cardinal points" on which the desired pattern function is exactly matched As a numerical application of the method, two examples of pattern synthesis with a hexagonal aperture are treated in detail.

Information in Noisy Photon Beams

Abstract: The fluctuations in a thermal beam due to the superposition of two similar thermal beams are computed. The information loss due to superimposing a quasi-monochromatic thermal-noise beam on a similar signal beam is investigated for two types of noise beams, sample noise, and path noise. In the Wien and Rayleigh–Jeans limits, the information loss j23 per macrocell is found to depend only on the signal-to-noise ratio. The relative loss in maximum obtainable information, I3,max/I2,max, for given signal and noise beams is given as well as some numerical examples.

Are the Optical Luminosity Fluctuations of 3C 273 Random

Abstract: WE have shown1 that a light curve similar to that of the quasi-stellar source 3C 273 can be generated by a superposition of unit light curves occurring at random times. Gudzenko, Ozernoy and Chertoprud2 have argued (1) that the distribution of the observed luminosities should be expected to be Gaussian on our model, and (2) that the observed distribution is not Gaussian, so that the luminosity curve cannot be regarded as the superposition of random events. The purpose of this communication is to question the validity of both arguments and to show that the luminosity record may still be considered to be of random origin.

On the superposition approximation and the Kirkwood-Salsburg integral equation

Abstract: Under the superposition approximation, the Kirkwood-Salsburg integral equation for the radial distribution function is expressed as a power series in the density, and the coefficients gk (x) in this series are compared with the exact quantities. It is found that, unlike the Kirkwood and the Born-Green-Yvon theories, this formalism provides g 1(x) and g 2(x) (and consequently the first four virial coefficients) correctly. Numerical values of g 3(x) and of the fifth virial coefficient for molecules interacting according to the Gaussian, rigid spherical, and square-mound potentials are presented. Comparisons between our results and those of other theoretical treatments are also given.

Strain-Gradient Effects on Force Transfer Between Embedded Microflakes:

Abstract: In this paper the plane strain problem concerned with shearing stresses along the fiber-matrix interface due to a fracture of the adjacent fiber is solved anew based on Mindlin's strain-gradient theory of linear elasticity. Classical method of selecting proper stress functions and determining superposition constants to satisfy all homogeneous boundary conditions is used to obtain the auxiliary solution for the problem. Applying a Fourier integral transformation to the auxiliary solution both homogeneous and non-homogeneous boundary conditions are simultaneously satisfied. The exact solu tion is thus obtained in integral form. Numerical results of the solution for some selected elastic constants have been worked out and compared with results previously obtained based on classical theory and couple-stress theory.

On uniform approximations to the solutions of the partial-wave Lippman-Schwinger equation

Abstract: The partial-wave Lippman-Schwinger equation for a superposition of Yukawa potentials for two specific forms of the weight function is reduced to infinite systems of linear (algebraic) equations for the two cases considered It is shown that the linear transformations which these systems of linear equations define are compact in Hilbert space for physical values of the energy

Moments of the Input, Output, and Impulse Response Functions of Linear Systems About Arbitrary Points

Abstract: A general expression is derived for the relation between the moments of the output, input, and impulse response functions of lumped linear time invariant systems about any three arbitrary points along the time axis. The derivation is given both by a method based on integration and summation operations and by a method using Laplace transforms. The latter method also gives the influence on their output of initial time lags in the input and in the impulse response function of linear systems.

Dissipative magnetogasdynamic flow

Abstract: An analysis of the structure of the wakes and waves in steady compressible magnetohydrodynamics is presented. No restriction is made on the equation of state of the gas or on the ratios of the various dissipative parameters. An asymptotic solution is obtained which furnishes directly the flow far from a body and which may be used in the construction of the entire flow field. The non-dissipative solutions are obtained as a non-uniform limit for vanishing dissipation; no matter how small the dissipation, one can go far enough from the origin that the flow is essentially dissipative. For non-aligned fields the wave pattern consists of a downstream wake and either two or four standing waves, depending on the flow regime. For aligned fields, two of these waves become wakes, so that the wake is a superposition of three structured layers, with either all downstream or two downstream and one up-stream. It is found that the nondissipative limit of the wake is non-unique for the aligned fields case. Different limits are obtained depending on how the various dissipative parameters vanish. In this paper we consider steady magnetohydrodynamic flow in the Oseen approximation. No restriction is placed on the equation of state of the gas or on the various dissipative parameters (viscosity, thermal and electrical conductivity). The method of approach is to obtain the fundamental solution. It was shown in an earlier paper (Salathe & Sirovich 1967) how this could be used to obtain the solution for arbitrary boundary-value problems. In 5 2 of the present paper we demonstrate that the fundamental solutions themselves provide the far field flow past a finite body. This is obtained simply in terms of such quantities as the total force on the body, heat added, etc. In 53 we obtain the fundamental solutions for the non-aligned fields case. An asymptotic solution is obtained applicable for distance from the origin large compared to the mean free path. It is well known that there exist two distinct flow regimes in magnetohydrodynamics, the doubly hyperbolic and the hyperliptic. In each of these we obtain a downstream wake which is a pure entropy wake. That is, it carries only density and temperature disturbances and is structured by thermal conductivity. In the doubly hyperbolic regime, the flow exhibits, in addition, four structured waves, while in the hyperliptic case, t Present address: Center for the Application of Mathematics, Lehigh University, Bethlehem, Pennsylvania.