Showing papers on "Superposition principle published in 1971"
TL;DR: By the superposition of Bethe's wave functions, using the Lieb's solution for the system of identical bosons interacting in one dimension via a $\ensuremath{delta}$-function potential, this paper constructed the wave function of the corresponding system enclosed in a box by imposing the boundary condition that the wave functions must vanish at the two ends of an interval.
Abstract: By the superposition of Bethe's wave functions, using the Lieb's solution for the system of identical bosons interacting in one dimension via a $\ensuremath{\delta}$-function potential, we construct the wave function of the corresponding system enclosed in a box by imposing the boundary condition that the wave function must vanish at the two ends of an interval. Coupled equations for the energy levels are derived, and approximately solved in the thermodynamic limit in order to calculate the boundary energy of this Bose gas in its ground state. The method of superposition is also applied to the analogous problem of the Heisenberg-Ising chain (not the ring).
333 citations
TL;DR: In this paper, the authors deal with the application of homomorphic deconvolution to the recovery of the seismic wavelet from a time series formed by the convolution of this wavelet with an impulse train.
Abstract: Homomorphic systems (Oppenheim, 1965a and 1965b) are a class of nonlinear systems which satisfy a generalized principle of superposition. Such systems are particularly useful in separating signals which have been combined through convolution. This paper deals with the application of homomorphic deconvolution to the recovery of the seismic wavelet from a time series formed by the convolution of this wavelet with an impulse train. The unique point about this approach is that it does not require the usual assumptions of a minimum-phase wavelet and a random distribution of impulses.
231 citations
TL;DR: In this paper, a non-linear analysis is made of disturbances to a basic flow, where disturbances that are considered are represented as a superposition of many functions each of which is periodic in a coordinate ξ normal to η and is independent of the third co-ordinate direction.
Abstract: This paper deals with a system of equations which includes as special cases the equations governing such hydrodynamic stability problems as the Taylor problem, the Benard problem, and the stability of plane parallel flow. A non-linear analysis is made of disturbances to a basic flow. The basic flow depends on a single co-ordinate η. The disturbances that are considered are represented as a superposition of many functions each of which is periodic in a co-ordinate ξ normal to η and is independent of the third co-ordinate direction. The paper considers problems in which the disturbance energy is initially concentrated in a denumerable set of ‘most dangerous’ modes whose wave-numbers are close to the critical wave-number selected by linear stability theory. It is a major result of the analysis that this concentration persists as time passes. Because of this the problem can be reduced to the study of a single non-linear partial differential equation for a special Fourier transform of the modal amplitudes. It is a striking feature of the present work that the study of a wide class of problems reduces to the study of this single fundamental equation which does not essentially depend on the specific forms ofthe operators in the original system of governing equations. Certain general conclusions are drawn from this equation, for example for some problems there exist multi-modal steady solutions which are a combination of a number of modes with different spatial periods. (Whether any such solutions are stable remains an open question.) It is also shown in other circumstances that there are solutions (at least for some interval of time) which are non-linear travelling waves whose kinematic behaviour can be clarified by the concept of group speed.
105 citations
TL;DR: In this article, a solution by the Riemann-Hilbert method of the problem of the diffraction of a plane electromagnetic wave incident arbitrarily on a cylinder with a longitudinal slot is presented; this solution can be completely validated, and the scattered field for any cylinder with any slot width and with an arbitrary ratio of wave length to cylinder radius can be determined by the reduction method.
Abstract: A SOLUTION by the Riemann-Hilbert method of the problem of the diffraction of a plane electromagnetic wave incident arbitrarily on a cylinder with a longitudinal slot is presented; this solution can be completely validated, and the scattered field for a cylinder with any slot width and with an arbitrary ratio of wave length to cylinder radius can be determined by the reduction method. The study of open cylindrical structures can be carried out by two methods: by solving the homogeneous Maxwell equations (in which case the propagation of the natural modes of an open wave-guide is considered) or by a study of a problem with sources. In the first case the consideration of the energy radiation through the aperture into free space is described by the introduction of a complex propagation constant. This method was first used in [1] to analyze the properties of a cylinder with a longitudinal slot used as a slot antenna. The natural modes of slotted waveguides were studied in detail in [2–6] with the aid of equivalent circuits and the concept of transverse resonance. In the second approach the problem of the diffraction of a plane wave (or the superposition of plane waves) by an open structure is solved: for a circular cylinder with a longitudinal slot various methods for the formal solution of the problem were proposed in [7–10]; little attention was given to the consideration of the physical picture of the scattered wave in these papers. Only in 18] for the long-wave approximation and resonance cases is a qualitative analysis given of the solution obtained and important results on the structure of the diffracted field formulated. The first detailed investigation of this problem, based on the numerical solution of a system of algebraic equations, already obtained [7], was presented in [11] (in this paper there is also a series of formulae extending the limits of application of the solution of [7]). In all the papers enumerated above no basis for the methods used is given. In this paper we present a solution by the Riemann-Hilbert method [12] of the problem of the diffraction of a plane electromagnetic wave incident arbitrarily on a cylinder with a longitudinal slot; this solution can be completely justified, and the scattered field for a cylinder with any slot width and with an arbitrary ratio of wavelength to cylinder radius can be determined by the reduction method.
29 citations
01 Jan 1971
TL;DR: The influence of a deformable foundation on the response of buildings to earthquake motion is examined in this article, where the authors consider the forced horizontal, rocking and vertical harmonic oscillations of a rigid disc bonded to an elastic half-space, which is considered as a mathematical model for the soil.
Abstract: The influence of a deformable foundation on the response of buildings to earthquake motion is examined. The study is divided into two parts; the vibration of the base of the building on the foundation medium, and the response of the whole building-foundation system.
Studied first are the forced horizontal, rocking and vertical harmonic oscillations of a rigid disc bonded to an elastic half-space, which is considered as a mathematical model for the soil. The problem, formulated in terms of dual integral equations, is reduced to a system of Fredholm integral equations of the second kind. For the limiting static case these equations yield a closed form solution in agreement with that obtained by others.
Using the force-deflection relations for the base, the equations of motion of linear building-foundation systems are solved by both direct and transform methods. It is shown that, under assumptions which appear to be physically reasonable, the earthquake response of the interaction system reduces to the linear superposition of the responses of damped, linear one-degree-of-freedom oscillators subjected to modified excitations. This result is valid even for systems that do not possess classical normal modes. Explicit approximations in terms of the parameters of the system are obtained for the dynamic properties of the one-degree-of -freedom oscillator which is equivalent to a single story building -foundation system. For multi-story buildings it is shown that the effect of an elastic foundation, as measured by the change in the natural frequencies of the building, is negligible for modes higher than the first for many types of building structures.
24 citations
TL;DR: In this paper, the far field of a rectangular piston-like radiator is shown to be composed of four components of equal magnitude, each of which behaves as if generated at one of the corners.
Abstract: As an example of a previously described theory for plane and gently curved radiators, the farfield of a rectangular pistonlike radiator is shown to be composed of four components of equal magnitude, each of which behaves as if generated at one of the corners. The magnitude of each component varies monotonically with increasing angle from the acoustic axis and is independent of the dimensions of the radiator. For the steady state, the vectorial superposition of components, taking account of the separations between the four effective sources, yields the familiar CW field, containing lobe structure. In the two normal central planes parallel to the radiator sides, the four components reduce to two, and on the acoustic axis they reduce to one. If instead of a CW signal a pulse is applied to the radiator, the field components arrive sequentially, according to the travel times from each of the corners to the field point. If the applied pulse is sufficiently short, these field pulses may be fully resolved. Successful experimental verification of the theory is described in which resolution was obtained of the two pulses in a central plane of a rectangular transducer.
18 citations
TL;DR: In this article, the normal mode solution for the response of a point-excited cylindrical shell converges poorly for high frequencies, and a new alternative representation of the solution which converges with only very few terms in the high frequency regime is presented.
Abstract: The normal‐mode solution for the response of a point‐excited cylindrical shell converges poorly for high frequencies. Converting the normal‐mode series to an integral representation using a Watson transformation, we obtain a new alternate representation of the solution which converges with only very few terms in the high‐frequency regime. Furthermore, this new representation, when compared to the response of a point‐excited plate, can be interpreted as a superposition of propagating disturbances that circumnavigate the cylinder in helical paths from the drive path to the field point, with the phase velocity characteristic of flexural waves in a flat plate.
15 citations
TL;DR: In this paper, the motion of a polytropic gas is represented as the superposition of two small amplitude simple waves which interact only at the boundaries of the tube, and the problem reduces to solving a nonlinear difference equation, and is effected on the basis that the solution is in the neighborhood of a linear standing wave.
12 citations
TL;DR: In this paper, the edge effects in two arrays of uniform slits fed by parallel-plate waveguides terminated on a ground plane were investigated, and it was shown that the realized gain pattern of an element depends substantially upon its position in the array, particularly for the deep resonance notches in the patterns which are present for certain element spacings.
Abstract: The results obtained by modeling a linear array as an infinite periodic structure can be used for the analysis of finite arrays as the zero-order approximation of a perturbation technique. This idea is utilized to investigate the edge effects in two arrays of uniform slits fed by parallel-plate waveguides terminated on a ground plane. It is shown that the realized gain pattern of an element depends substantially upon its position in the array. This is true particularly for the deep resonance notches in the patterns which are present for certain element spacings. When the array is excited with uniform magnitude and linear phase, the aperture voltages are the superposition of a term, corresponding to the infinite array model, plus another correction term (a "spatial transient") representing the edge effect. The influence of this term is particularly relevant when the array is scanned at endfire. In such a case, the method introduced here allows the prediction of the element terminal admittances and the array pattern, while according to the infinite array model no radiation would be permitted.
12 citations
IBM1
Abstract: Solution of electronic problems involving plane surfaces on crystals requires solution of the band problem for real energy E in complex k space, and superposition of the generalized Bloch functions at the surface. A compact and general formulation of the problem of finding these Bloch functions and matching them across a plane makes use of a numerical matrix, the propagation matrix P, obtained from the Schrodinger equation. The eigenvectors of P are just the desired Bloch functions, and the eigenvalues give all k⊥ values at given E, k// (component parallel to the surface). Thus once P is found, the band problem is reduced to an ordinary eigenvalue problem; the bands can be followed along any line in k space parallel to k⊥ by varying k//; the potential may be complex (to describe inelastic scattering). A procedure for generating P by integration of a matrix equation has the advantage that a general anistropic potential can be used, but the disadvantage of a Fourier expansion parallel to the surface plane which does not hold well near the nucleus; hence it applies best for potentials that are weak or have a small number of Fourier coefficients. By generating P for a single layer by a two-dimensional version of KKR, this limitation is avoided for muffin-tin potentials.
12 citations
TL;DR: In this article, the authors derived formulae for the integrated intensity distribution, the photon-counting distribution and its factorial moments in the statistics of the superposition of coherent and chaotic multimode fields.
Abstract: Recently derived formulae for the integrated intensity distribution, the photon-counting distribution and its factorial moments in the statistics of the superposition of coherent and chaotic multimode fields are proposed as approximate formulae for light of arbitrary spectrum. It is shown by explicit calculations of the third factorial moment for the superposition of a one-mode coherent field with a Gaussian-Lorentzian field that the proposed formulae hold with good accuracy over a wide range of conditions. An application to the determination of spectral parameters of light is given.
TL;DR: In this article, the third factorial moment of the photon-counting distribution for the superposition of a one-mode coherent field with a Gaussian Lorentzian field is calculated.
Abstract: Formulae recently derived for the integrated intensity distribution, the photon-counting distribution and its factorial moments in the statistics of the superposition of multimode coherent and chaotic fields are analyzed in greater detail and their validity as approximate formulae for light of arbitrary spectrum is investigated It is shown by explicit calculation of the third factorial moment of the photon-counting distribution for the superposition of a one-mode coherent field with a Gaussian Lorentzian field that the proposed formulae hold with very good accuracy over a wide range of conditions
TL;DR: In this paper, approximate formulae for two-photon counting statistics for the superposition of coherent and chaotic fields are proposed and the exact second factorial moment is calculated, which is demonstrated by comparing the exact and approximate third factorial moments for the chaotic lorentzian field.
TL;DR: In this paper, a linear system model was developed to predict irradiance distributions of visible light below an idealized optically thick atmospheric cloud, which is illuminated from above in an arbitrary manner.
Abstract: A linear-system model has been developed to predict irradiance distributions of visible light below an idealized optically thick atmospheric cloud, which is illuminated from above in an arbitrary manner. The model offers elegant mathematical simplicity at the expense of some precision. As such, it is applicable to a broad class of problems in which correct functional forms are required, but levels of accuracy better than a factor of 2 are not necessary. Optical thicknesses can range from about 5 to 32. One example of a problem in this class, the design of a laser communication system to operate through clouds, provided the original motivation for development of the light-transmission model. The optical effects of the cloud are calculated by means of a four-dimensional linear superposition integral, which takes account of multiple scattering. Two illustrations of the method are given in detail, with incident illumination represented by a tightly collimated beam and by a sum of infinite plane waves, respectively.
TL;DR: In this paper, a method of resolving anomalous superposition of arbitrary numbers of dispersions is presented, assuming that the lowest-frequency dispersion follows the Debye equation, and simulated data suggests that a skewed-arc dispersion may be resolved, within the usual experimental error, into closely overlapping Debye-type dispersions.
Abstract: Assuming superposition of arbitrary numbers of dispersions, a method of resolving them is presented. A restriction of the method is that the lowest‐frequency dispersion follows the Debye equation. Apparent violations of superposition which have been reported may be due to errors in the analysis of data. The method is illustrated with simulated data which suggests that a skewed‐arc dispersion may be resolved, within the usual experimental error, into closely overlapping Debye‐type dispersions.
TL;DR: In this paper, the minimum thickness of a semiconductor plate that is able to support an approximate plane wave with the dispersion relation being practically equal to that of the one-dimensional theory is estimated.
Abstract: Starting from the two-dimensional carrier-wave theory of Kino and Robson the minimum thickness of a semiconductor plate that is able to support an approximate plane wave with the dispersion relation being practically equal to that of the one-dimensional theory is estimated. In the limit of small external permittivity such a plane wave is approximated by the fundamental lateral mode, whereas in the limit of large external permittivity a superposition of a few orders of the lateral modes is necessary. At small plate thickness the fundamental mode stays an approximate plane wave. It does the same for larger external permittivity, however, with a changed dispersion relation.
TL;DR: By using the superposition principle of fields, a combined field tensor is defined as mentioned in this paper, which is divergence-free and can be used to define any tensor tensor in the literature.
TL;DR: In this paper, the authors deal with three geometric operations that permit the determination of the fields of displacements and their time and space derivatives in any solid subjected to deformations, including acceleration, second-order space derivatives, combined space and time derivatives, and other related fields.
Abstract: This paper deals with three geometric operations that permit the determination of the fields of displacements and their time and space derivatives in any solid subjected to deformations. The first operation consists in the superposition of the image of a nondeformed on the image of a deformed network of parallel lines. The intersections of the two networks of lines give the loci of equal components of displacement (isothetics). The second operation gives the time derivatives of the components of displacement (isotachics) by superposition of two isothetics obtained at different times. The third operation gives the space derivatives of the components of displacement (isoparagogics) by superposing and shifting two images of the same isothetic. These three operations can also be applied to obtain acceleration, second-order space derivatives, combined space and time derivatives, and other related fields. For coarse networks, usually called grids, the results can be obtained by connecting the intersections by hand. For dense networks, usually called gratings, they can be obtained by the moire effect. Examples of applications are given.
TL;DR: The electroluminescence due to the superposition of sinusoidal voltages of two different or two equal frequencies has been investigated in this paper, where the voltage dependence of the light output shows deviation from the standard equation, B = B0 exp(-b/V 1 2 ), at lower voltages.
01 Jan 1971
TL;DR: In this article, a study of the free and forced oscillations in dynamic systems with hysteresis, on the basis of a piecewise -linear, nonlinear model proposed by Reid, is made.
Abstract: A study is made of the free and forced oscillations in dynamic systems with hysteresis, on the basis of a piecewise -linear, nonlinear model proposed by Reid. The existence, uniqueness, boundedness and periodicity of the solutions for a single degree of freedom system are established under appropriate conditions using topological methods and Brouwer's fixed-point theorem. Exact periodic solutions of a specified symmetry class are obtained and their stability is also examined. Approximate solutions have been derived by the Krylov-Bogoliubov-Van der Pol method and comparison is made with the exact solutions.
For dynamic systems with several degrees of freedom, consisting of "Reid oscillators", exact periodic solutions are derived under certain restricted forms of "modal excitation" and the stability of the periodic solutions has been studied. For a slightly more general form of sinusoidal excitation, a simple way of obtaining approximate solutions by "apparent superposition" has been indicated. Examples are presented on the exact and approximate periodic solutions in a dynamic system with two degrees of freedom.
01 May 1971
TL;DR: In this paper, the velocity fluctuation field is decomposed by modeling the velocity fluctuations as non-deterministic travelling waves with random phase angles, and the power spectral density function is identified as the appropriate frequency weighting function with which to synthesize the broad band (unfiltered) from the narrow band correlation functions.
Abstract: : The classical difficulties in synthesizing the velocity fluctuation field from turbulence data are discussed, including the closure problem, consequent non-uniqueness of the representation, and the loss of information due to averaging. In the present work the inverse approach is used, that is, several analytical models of the velocity fluctuation field, suggested by the flow visualization data, are compared with the existing space-time correlation function data. This approach effectively eliminates the closure problem, but due to its inductive nature, causes some difficulties in disclosing the underlying physics. Using the most promising model, the turbulent flow field has been decomposed, by modeling the velocity fluctations as non-deterministic travelling waves with random phase angles. It is found that this phenomenological model correctly represents the observed trends in the narrow band (frequency filtered) correlation function data. Next, the power spectral density function is identified as the appropriate frequency weighting function with which to synthesize the broad band (unfiltered) from the narrow band correlation functions. The functional form of the power spectral density function which agrees with the observed data is taken to be the superposition of a strong, unorganized background turbulence (Markoff noise) and an organized turbulent structure. The derived broad band correlation functions agree very well with a wide range of turbulent correlation function measurements.
TL;DR: The first and second N-photon factorial moments for the superposition of coherent and chaotic fields were given for arbitrary spectrum and counting time intervals in this paper, where the chaotic field was considered.
Abstract: The first and the second N-photon factorial moments for the superposition of coherent and chaotic fields and the third N-photon factorial moment for the chaotic field are given for arbitrary spectrum and counting time intervals.
01 Jan 1971
TL;DR: In this article, the question of whether a superposition X(Y) can have a continuous probability distribution was investigated, assuming that Y is a nonnegative random variable independent of the differential process X(t).
Abstract: Assuming that Y is a nonnegative random variable independent of the differential process X(t), attention is given to the question of whether or not the superposition X(Y) can have a continuous probability distribution. If the process has continuous distributions, then the superposition is continuous if and only if P/Y = 0/ = 0. If the process has discontinuous distributions and no trend, then no superposition can have continuous distribution. If the process has discontinuous distributions and nonzero trend, then the superposition onto a random epoch has continuous distribution if and only if Y has continuous distribution.
TL;DR: In this paper, a method is presented to calculate the output of a hard limiter if the input consists of a superposition of three phase-coded signals, and a detailed investigation is made of the case where the limiter input signals are phase reversal modulated according to a maximum length linear code.
Abstract: A method is presented to calculate the output of a hard limiter if the input consists of a superposition of three phase-coded signals. A detailed investigation is made of the case where the limiter input signals are phase reversal modulated according to a maximum length linear code. It is shown in this case that a false target signal is generated at the limiter output. The amplitude distribution of the false target signal is investigated along with the distribution of the captured true target signals. A digital computer simulation confirms the theoretically predicted effects.
TL;DR: In this paper, the authors developed a simple numerical procedure to determine the electromagnetic field penetration through a slab many hundreds of wavelengths in thickness of cold plasma with collisions, which leads to a differential equation of the Riccati type.
Abstract: One objective of this paper is to develop a simple numerical procedure to determine the electromagnetic field penetration through a slab many hundreds of wavelengths in thickness of cold plasma with collisions. The electron‐density‐collision frequency profiles may be arbitrary but must contain only a finite number of discontinuities. The solution of this problem leads to a differential equation of the Riccati type. Other objectives are to evaluate exactly the transmission loss through the aforementioned inhomogeneous plasma and to determine the field variation within the plasma slab when the incident field is of arbitrary polarization. This is accomplished by solving the E and H plane incidence problems separately and then employing superposition. The results are in the form of an integral involving the reflection coefficient. The differential equations for the reflection coefficient and the transmission integrals are easily evaluated to practically any desired degree of accuracy by computer. Analytical and numerical results are presented for a plasma half‐space with an exponential electron‐density distribution.
TL;DR: In this paper, the superposition principle for the tangential-velocity field of two parallel vortices is established for the interaction between two parallel whirling jets, and the results of an experimental study of the interaction are presented.
Abstract: The results of an experimental study of the interaction between two parallel whirling jets are presented. The superposition principle is established for the tangential-velocity field of two parallel vortices.
TL;DR: In this paper, a method for holographic reconstruction of diffusely illuminated objects which suppresses the known effect of granularity is described. But the experimental setup is modest, it consists of a source of polychromatic light.
Abstract: Described is a method for holographic reconstruction of diffusely illuminated objects which suppresses the known effect of granularity. The experimental setup is modest. It consists of a source of polychromatic light. Additionally there are inserted a simple ground glass and a diffraction grating in the reconstruction wave. Holographic reconstruction without granularity is achieved by incoherent superposition of longitudinal modes of the reconstruction wave.