Showing papers on "Superposition principle published in 1973"
TL;DR: In this paper, the effects of building-soil interaction on the earthquake response and steady-state response to sinusoidal excitation are examined, assuming that the interaction system possesses n + 2 significant resonant frequencies, the response of the system is reduced to the superposition of the responses of damped linear oscillators subjected to modified excitations.
Abstract: In this study of the dynamics of building-soil interaction, the soil is modeled by a linear elastic half-space, and the building structure by an n-degree-of-freedom oscillator. Both earthquake response and steady-state response to sinusoidal excitation are examined. By assuming that the interaction system possesses n + 2 significant resonant frequencies, the response of the system is reduced to the superposition of the responses of damped linear oscillators subjected to modified excitations. The results are valid even though the interaction systems do not possess classical normal modes. For the special cases of single-story systems and the first modes of n-story systems, simplified approximate formulas are developed for the modified natural frequency and damping ratio, and for the modified excitation. Example calculations are carried out by the approximate and more exact analysis for one-story, two-story and ten-story interaction systems.
The results show that interaction tends to decrease all resonant frequencies, but that the effects are often significant only for the fundamental mode for many n-story structures and are more pronounced for rocking than for translation. If the fixed-base structure has damping, the effects of interaction on the earthquake responses are not always conservative, and an increase or decrease in the response can occur, depending on the parameters of the system.
304 citations
TL;DR: In this paper, an approximation to the Voigt function is described, which is valid over the entire domain of the independent variables that characterize it and which is accurate to the order of 0.0001 of the peak value of the function.
Abstract: An approximation to the Voigt function is described, which is valid over the entire domain of the independent variables that characterize it and which is accurate to the order of 0.0001 of the peak value of the function. Relations between the parameters of the function are also given. The approximation is used to develop a procedure for fitting observed lines with Voigt functions; the class of asymmetric lines that arise from the superposition of two Voigt functions is considered in some detail, and methods for extracting the Voigt parameters of the components from the observed contour of the envelope are given. The measurement of the width and shape of a single line with a Fabry–Perot interferometer is also discussed. All of the calculations described here can be handled by programmable calculators or small computers.
175 citations
TL;DR: In this article, the authors define the time response function of an optical system, that is the result given by any pupil receiving a very short impulse of light, which can be used to generalize the explanation of the channelled spectra and bring an argument to the parallelism between space and time domains.
Abstract: The concepts and experiments discussed in this article are scarcely used in physical optics although they can introduce new ways of processing the information content of any optical pupil. They are related to diffraction phenomena in polychromatic light. One aspect deals with the well-known channelled spectra. The other consists in the definition of the time response function of an optical system - that is the result given by any pupil receiving a very short impulse of light. The time response function allows to generalize the explanation of the channelled spectra. It is also deduced from Fourier techniques and brings an argument to the parallelism between space and time domains. If the entrance pupil of a spectrometer is set in the interference pattern of a two-beam device illuminated in white light, the coloured spectrum is crossed by dark bands that are closer and closer as the difference between the optical paths increases. Along a time frequency (or wave-number) scale, the dark bands are sinusoidal. In this manner, one can say that the time spectrum is the Fourier transform of the couple of the wave trains. The time spectrum of white light being broad, a beam of white light should be considered as juxtaposition of very short wave groups. So, one can consider two individual wave groups passing along each arm of the interferometer. The same reasoning holds as one deals with several well-arranged wave trains, issued for instance from a grating or a multiple beam interferometer, or even further, with a sequence of wave trains in complete disorder. The situation is summarized in the diagram (fig. i): an incident plane wave is diffracted when transmitted through various devices. The spectrometer slit is placed in the region where the parallel beams combine with each other, at infinity or not. One should note the Fourier relationships of the spectral modulation curves with each sequnce of wave trains drawn along a time axis. These considerations immediately open out into new possibilities of conveying and collecting information. While a familiar way of transmission is to let the signal modulate the carrier which is in optics a parallel beam of monochromatic light, here the carrier consists of a parallel beam of polychromatic light, the modulation law being of temporal type. In turn this law is determined by the configuration of the optical system. Therefore any given function can be transmitted to any receiver station having a spectrometer at disposal - the role of this spectrometer being the spectral analysis of the message (or of its Fourier Transform). As an example suppose the tansmission of a (sinX/X)2 function is desired. It is the intensity distribution in the far-field diffraction pattern of a rectangular aperture. Therefore one has to perform the diffraction of a beam of white light by a slit of suitable width, a. In a first approximation (fig. ii), in a direction θ, the sequence of wave trains is limited by a rectangle function of width a cosθ and the power spectrum is expected to be given by the square modulus of a sinc-function. Likewise another test signal for studying the transmission mechanism would be a pair of thin slits that yields a cos2 function, or else a pupil of gaussian transmittance would be nessary to build up a message described by the reciprocal gaussian law. The previous process suffers of limitations. It applies to real and positive function whose Fourier transforms are real and positive. For a wider class of functions, namely complex ones, one has to resort to an equivalent of holography. The transposition comes to send a reference wave-group before or after the information proper. The phase terms are thus kept, same as in conventional holography. Such a simple experiment was carried out by means of a narrow slit set in the same plane as that of the diffracting pupil. A temporal hologram was then recorded. It appears as an image of the time spectrum striped with thin dark fringes of sinusoidal profile, carrying the amplitude and phase temrs. The usual reconstruction process in monochromatic light applies to that temporal Fourier hologram. Information concerning the previous rectangular pupil has been transmitted in the form of a hologram recorded at the output of a spectrometer 10 meters away from the source. Incidentally a much more interesting and promising technique enables real-time operating. All one has to do is get the temporal autocorrelation function of the whole of the message with its reference waves group. The pair of antisymmetrical side band distribution represents the image and its conjugate. Practically such phenomena are observable as the amplitude and phase modulation of the visibility function of the fringes displayed at the output of a Michelson interferometer. An important requirement is the use of a spatially coherent source of high luminance, with a broad spectrum (its transform - the reference - would be narrow then). The input function of the Michelson interferometer is the sequence of wave trains. Therefore this represents the basic arrangement in Fourier transform spectroscopy, with a slight difference: instead of gathering a signal and making an interferogram whose spectrum would describe the source, one gets here the autocorrelation function of the information under test, that is of the time holographic signal coming from the pupil. As a matter of fact the time-hologram itself is never formed; no intermediate recording is needed as one directly reaches three terms: the central one being the superposition of the autocorrelation of the signal and the reference respectively, the other two the symetrical side-bands displaying, as said before, the cross correlations of the signal by the reference which was assumed as a narrow impulse along a time axis. A more rigourous treatment has proved necessary to give a full account of the mechanisms involved in the transmission of a time message. In particular a time impulse response and a time transfer function can be defined for any pupil. Looking at figure iii, one notes that the scale of the Fraunhofer spatial diffraction pattern of a rectangular aperture varies as the reciprocal of the time frequency ν. As ν of the incident plane wave is tuned continuously from 0 to ∞, an observer set in a fixed position at infinity will see the « breathing » of the pattern according to the law sin Kν/Kν To complete the determination of the amplitude H(ν) at the observation point, one must add an important fact: usually, in the calculation of the amplitude diffracted by any pupil by means of the Fresnel-Kirchhoff integral, one omits the multiplicative factor, - j/λ. Therefore: H(ν) = (j2πν sin Kν/Kν. The reciprocal of ν in the time domain is t. The time response of the system to a unit impulse h(u, t) in the u-direction is then the Fourier Transform of H(ν). After a well-known derivative theorem, the transform of H(ν) is the derivative of a rectangle function. The same procedure would show that if one considers any pupil, whatever its contour and amplitude distribution, the time impulse response is the derivative of the pupil function. A similar result is obtained with phase pupils. Generally, the time impulse response of a complex pupil in a given direction is described by the derivative of the complex pupil function projected on this direction. This is illustrated by various drawings of functions corresponding to one or several slit apertures. Applications are envisaged, namely in metrology - for instance in the assessment of the quality of surfaces and the measurement of thicknesses.
136 citations
TL;DR: In this paper, an analytic formulation for the total potential in atomic and molecular systems, based on the electrostatic approach from the Hellmann-Feynman theorem, is given.
79 citations
TL;DR: In this article, the theory of wave fields in terms of superpositions of monochromatic plane waves is presented for fields satisfying the inhomogeneous scalar wave equation.
Abstract: The theory of the representation of wave fields in terms of superpositions of monochromatic plane waves is presented for fields satisfying the inhomogeneous scalar wave equation The discussion includes expansions of the type originally used by E T Whittaker involving only homogeneous plane waves, and of the type introduced by H Weyl involving both homogeneous and inhomogeneous plane waves Expressions for the plane-wave amplitudes for both types of representations are obtained in terms of the source function, and precise conditions under which each expansion is valid are given It is shown that when both types of expansions are valid, the superposition of inhomogeneous plane waves in the Weyl-type representation is equal to the superposition of the homogeneous plane waves that propagate into a specific half-space in the Whittaker-type representation It is shown also that in restricted space-time regions only a certain subset of the plane waves in the Whittaker-type expansion contribute to the field This result leads to a simple expression for the field valid at large distances from the source
72 citations
TL;DR: In this article, an investigation into the evolution from a sinusoidal initial wave train, of long periodic waves of small but finite amplitude propagating in one direction over water in a uniform channel is made, and the spatially periodic surface displacement is expanded in a Fourier series with time-dependent coefficients.
Abstract: An investigation is made into the evolution, from a sinusoidal initial wave train, of long periodic waves of small but finite amplitude propagating in one direction over water in a uniform channel. The spatially periodic surface displacement is expanded in a Fourier series with time-dependent coefficients. Equations for the Fourier coefficients are derived from three sources, namely the Korteweg–de Vries equation, the regularized long-wave equation proposed by Benjamin, Bona & Mahony (1972) and the relevant nonlinear boundary-value problem for Laplace's equation. Solutions are found by analytical and by numerical methods, and the three models of the system are compared. The surface displacement is found to take the form of an almost linear superposition of wave trains of the same wavelength as the initial wave train.
68 citations
IBM1
TL;DR: A precise nondestructive optical contour mapping method with adjustable sensitivity for noncontact testing of surface deformations up to 30 microm/cm is described and applications for surface flatness testing of semiconductor wafers and photomasks are presented.
Abstract: A precise nondestructive optical contour mapping method with adjustable sensitivity for noncontact testing of surface deformations up to 30 μm/cm is described. The method employs an optical grating that is placed in front of the test surface. Illumination of the grating by a monochromatic plane wave generates an interference pattern between the beam components of two different diffraction orders. Reflection at the test surface and superposition with the fixed grating generate a fringe pattern that resembles the surface contours of the test object. This fringe pattern consists of a superposition of an interference line system and a moire line system. Whereas the distance between two adjacent contour lines of the interference pattern corresponds to a surface deformation of a half-wavelength of the illuminating light the distance between two lines of the moire system is determined by the grid constant and the direction of the grid illumination. Therefore, the scale of measurement can be chosen according to the problem. Applications for surface flatness testing of semiconductor wafers and photomasks are presented.
40 citations
36 citations
TL;DR: In this paper, an analogous superposition principle is constructed for turbulent plasmas, which shows that the turbulent fluctuations may suppress the growth of the instabilities that give rise to them.
Abstract: It is well known that the properties of stable quiescent plasma may be calculated by a method known as the ‘superposition of dressed particles’. In this paper an analogous superposition principle is constructed for turbulent plasmas. This shows that the turbulent fluctuations may suppress the growth of the instabilities that give rise to them. The fluctuation spectrum, diffusion coefficient and dielectric constant of turbulent plasma are calculated for the guiding centre model. The relationship of this theory to that of Dupree and Weinstock is discussed.
25 citations
TL;DR: The conception of hybrid modulation is conceived as the superposition of two z -plane zero-singularity (ZS) patterns associated with amplitude- and angle-modulating signals, respectively, and important spectral properties of the modulated signal are explicit in the resultant pattern.
Abstract: Certain properties of periodic signals are defined in terms of the zeros and singularities of associated analytic functions of a complex time variable z . This algebraic approach is a generalization of analytic signal theory, and leads to the conception of hybrid modulation as the superposition of two z -plane zero-singularity (ZS) patterns associated with amplitude- and angle-modulating signals, respectively. It is shown that important spectral properties of the modulated signal, such as band limitation, are explicit in the resultant pattern. Signal design is then interpreted in terms of ZS manipulation and placement. The theory is applied in a unified approach to compatible singlesideband (CSSB) modulation systems. It is shown that two types of proposed CSSB systems give rise to essentially nonband-limited output signals. The relation between conventional and single-sideband (SSB) angle modulation is also discussed in terms of their characteristic ZS patterns.
TL;DR: In this paper, a treatment based on the Preisach model and ignoring the demagnetizing field is presented, and it is shown that superposition is inconsistent with this model when more than two transitions are involved.
Abstract: Reasons for, and the character of, deviations from the "superposition" principle which is customarily used in considerations about digital magnetic recording are analyzed. First, a treatment is presented which is based on the Preisach model and ignores the demagnetizing field. It is shown that superposition is inconsistent with the Preisach model but that, when more than two transitions are involved, a weaker form of the superposition principle can be derived from the Preisach model. Next, the effects produced by the demagnetizing field in the medium are illustrated by results of a "self-consistent" computation for a thin layer. Finally, some experimental results obtained from a large-scale simulation model are presented.
Patent•
14 Jun 1973
TL;DR: In this paper, the factors are b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17, b18, b19, b20, b21, b22, b23, b24, b25, b26, b27, b28, b29, b30, b31, b32, b33, b34, b35, b
Abstract: The factors are b0, b1, b2....bi-1, where b is a base, e.g. 2, 3, 4.... and i the number of input signals; the products are then added and the sum is transmitted through a transmission channel; it is decoded at the receiving end by an amplitude filter and the transmission line is a multiplex line. Input signals are D.C. pulses. Addition is carried out by voltage superposition, and the whole dynamic range of the transmission channel (negative and positive amplitudes) is used for transmission. The input signals may be constant frequency A.C. pulses.
TL;DR: In this paper, a gas filled tube has a pressure sensitive heat source at the center, while energy is allowed to radiate from the ends, and the small amplitude disturbance in the pipe is represented as the superposition of two simple waves traveling in opposite directions.
Abstract: : A gas filled tube has a pressure sensitive heat source at the center, while energy is allowed to radiate from the ends. For a sufficiently large feedback coefficient the system is linearly unstable. Within the nonlinear theory, the self-excited oscillation initially grows until a shock forms. The shock acts as a dissipative mechanism so that a balance is achieved and ultimately a time periodic state is reached. The small amplitude disturbance in the pipe is represented as the superposition of two simple waves traveling in opposite directions, and without interaction. Based on this representation the problem reduces to solving a nonlinear difference equation. (Author)
TL;DR: In this article, a superposition model is proposed to describe the electronic density of states curves of binary alloys with the aid of those of the component metals, based on a charge transfer between the different atomic cells.
Abstract: A superposition model is suggested to describe the electronic density of states curves of binary alloys with the aid of those of the component metals. It is based on a charge transfer between the different atomic cells. The application of this model to different spectroscopic methods studying the density of states brings up the question of the localizing and averaging properties of the corresponding experiments.
TL;DR: In this paper, a superposition model is proposed to describe the electronic density of states curves of binary alloys with the aid of those of the component metals, based on a charge transfer between the different atomic cells.
Abstract: A superposition model is suggested to describe the electronic density of states curves of binary alloys with the aid of those of the component metals. It is based on a charge transfer between the different atomic cells. The application of this model to different spectroscopic methods studying the density of states brings up the question of the localizing and averaging properties of the corresponding experiments.
TL;DR: In this paper, a new method for reducing the pattern effect in PCM current modulation of DH-GaAlAs lasers without superposition of dc bias current was demonstrated at 200 Mb/s.
Abstract: A new method for reducing the pattern effect in PCM current modulation of DH-GaAlAs lasers without superposition of dc bias current is demonstrated at 200 Mb/s. Also PCM current-modulation characteristics are analyzed by simulation using nonlinear rate equations.
TL;DR: In this paper, the atomic exchange reaction A + BC → AB + C was investigated quantum mechanically employing a coupled differential equations approach and the relative motion in reactant and product channels was described in the common coordinate R 3 (the AC nuclear separation) and developed in three-dimensional space.
Abstract: The atomic exchange reaction A + BC → AB + C is investigated quantum mechanically employing a coupled differential equations approach. The relative motion in reactant and product channels is described in the common coordinate R 3 (the AC nuclear separation) and is developed in three-dimensional space. The total wave functions of the system are expressed as a superposition of valence bond electronic states of the initial (A, BC) and final (AB, C) configurations, with the coefficients describing the relative and internal (vibrational, rotational) nuclear motions. Choosing convenient trial functions with the appropriate boundary conditions and using the Kohn variational principle, a set of differential (rather than the usual integro-differential) equations is obtained for the relative motion wave functions in R 3. The potential matrix elements turn out to be dynamical in that they depend on the initial k 1 and final k 2 wave vectors. Two-state coupled channel calculations of the differential and integral cro...
CSAV1
TL;DR: In this paper, the wave function describing the electron interference produced by an electrostatic biprism derived from the perturbation theory involves an integral which is not easy to evaluate numerically.
Abstract: The wave function describing the electron interference produced by an electrostatic biprism derived from the perturbation theory involves an integral which is not easy to evaluate numerically. Using the method of stationary phase it is shown in the paper that the expression of this integral in terms of the Fresnel integrals considerably facilitates the calculations. Physically this expression corresponds to the superposition of two waves emerging from two virtual images of the source diffracted by two opaque half-planes of opposite orientation. In such a way the model widely used for discussions of the interference of this type is justified.
TL;DR: In this article, a simple classical model is used to analyze reactive processes in H + 2 -He collisions at relative energies above a few eV. Detailed structure of the cross section occurs and is a consequence of the superposition of probabilities due to distinct binary collision mechanisms which depend on impact parameter and initial orientation.
TL;DR: A statistical analysis is presented that predicts the ambiguity for both the mask and the Gaussian beam velocimeters and the results are shown to be in close agreement with experiment.
Abstract: In a laser Doppler velocimeter the signal arises from the superposition of several pulses at random phase, and the instantaneous frequency recorded at the output, which is a measure of the velocity, contains an ambiguity component due to this effect. A statistical analysis is presented that predicts the ambiguity for both the mask and the Gaussian beam velocimeters and the results are shown to be in close agreement with experiment. Based on this theory, a computer simulation model has been developed that is a useful tool for analyzing and optimizing the parameters in a particular system. Finally, the specific applications of turbulence measurement in fluid flow is considered in more detail.
TL;DR: In this paper, the authors investigated the dependence of s-photon absorption probability on the statistical properties of the incident pulsed light and found that when the two pulses have the same spectral distribution and the same energy, the probability induced by the incoherent pulse is s! times higher than that induced by a coherent pulse.
Abstract: We deal with the models of a coherent pulse and of an incoherent one. The former pulse is described by using a polychromatic pure coherent state and the latter is obtained by an incoherent superposition of identical polychromatic elementary pulses shifted in time by a statistically distributed delay. For the incoherent pulse the ratio between its duration and its coherence time can assume any possible value. By using these pulse models the dependence ofs-photon absorption probability on the statistical properties of the incident pulsed light is investigated. It is found that, when the two pulses have the same spectral distribution and the same energy, the ratio between the probabilities fors-photon absorption induced by the incoherent pulse and that induced by the coherent pulse is given by\(s!\left[ {\frac{{I_{inc} }}{{I_{coh} }}} \right]^{s - 1} \), where Iinc and Icoh are the intensities of the incoherent pulse and of the coherent pulse, respectively, at the time in which the field interacts with the atomic system. This ratio shows that, when the two pulses have also the same duration and, therefore, the same intensity, the probability induced by the incoherent pulse is s! times higher than that induced by the coherent pulse. Explicit calculations on the multiphoton absorption probability are carried out for the pulses when their time shapes are Gaussian.
Abstract: This work is the computation of the shift, parallel to the incident plane, by total reflection, of a light beam (Goos and Hanchen effect). The incident wave is choosen so as to satisfy the Maxwell equations. It is expanded as a superposition of plane waves, part of which are inhomogeneous (or evanescent). The corresponding reflected plane waves are recomposed, which allows to obtain the energy flux of the reflected beam. The results are compared with those obtained by the approximate known formulas, and also with Goos and Hanchen's measurements.
Abstract: Model stars whose convection zones had been prepared in accordance with the standard mixing-length theory were used as a basis for the computation of unstable convective modes. It was found that no superposition of statistically independent, nonviscous, adiabatic, convective modes can reproduce the radial dependence of the convective flux of the model. This implies that the representation of a stellar convection zone as a superposition of unstable adiabatic linear modes is inconsistent with the mixing-length theory, and that conclusions based upon such a representation should be regarded with caution. It is also shown that if the linear scale of convective motions is greater than (or of the same order as) the pressure scale height, then the fractional deviation of the pressure from equilibrium will generally not be negligible, as assumed in the mixing-length theory, but will be at least of the same order as the fractional deviation of the density from equilibrium.
TL;DR: In this article, the validity of the use of superposition in high-density recording systems was investigated for thin oxide, thick oxide, and metallic disks, both as regards pulse shape and validity of using superposition.
Abstract: In high-density recording systems it is important to know the actual pulse shape as well as the validity of the use of superposition. Experimental results for these phenomena were obtained for thin oxide, thick oxide, and metallic disks, both as regards pulse shape and the validity of superposition.
TL;DR: In this article, a general formula for diffraction and interference of a light wave passing through two superposed sound waves has been derived and two-beam interference is considered as a special case.
Abstract: The diffraction and interference of a light wave passing through two superposed sound waves have been studied theoretically on the basis of Huygens' principle. As a result of frequency shifts in every deflected light ray caused by the sound waves, the instantaneous intensity resulting from linear superposition of the deflected light rays consists of the d.c. component as well as the a.c. component as a sum of beating signals among those rays, provided that the coherence time of the light is long enough compared with the period of a beating signal under consideration. A general formula for diffraction is derived and two-beam interference is considered as a special case. The resultant formula illustrates that the complex degree of coherence is modulated, after traversing the sound field, with a form of the Bessel function of zeroth-order which varies with Raman-Nath parameters as well as with the separation of two points on the wavefront of the light. This validity will hold so long as the inclination facto...
TL;DR: In this article, upper and lower bounds for the optimal weight of a beam are derived, for single as well as for alternative loading conditions, and a superposition principle is established to establish the optimal design.
TL;DR: In this paper, a classroom demonstration has been developed which allows the correlation functions of a wide variety of pre-selected signals to be displayed on an oscilloscope and all the steps involved in generating a given correlation function can be displayed in a logical step-by-step sequence from the input signals to the output function.
Abstract: A classroom demonstration has been developed which allows the correlation functions of a wide variety of pre-selected signals to be displayed on an oscilloscope. The demonstration utilizes standard equipment found in most undergraduate electrical engineering laboratories and requires no special construction of new equipment or extensive modification of existing instruments. All the steps involved in generating a given correlation function can be displayed in a logical step-by-step sequence from the input signals to the output function. Either autocorrelation or cross-correlation can be performed and the effects of varying different parameters illustrated. Additionally the same two signals can be convolved so that the validity of the superposition integral can be demonstrated. The application of correlation techniques to the detection of signals in the presence of noise can also be demonstrated. Examples of all these possibilities are given.
TL;DR: In this paper, the photon counting distribution for the M-mode superposition of chaotic and periodically modulated coherent fields is studied and the average numbers of photons of chaotic field are assumed to be the same in all modes.
Abstract: The photon counting distribution for the M-mode superposition of chaotic and periodically modulated coherent fields is studied. The average numbers of photons of chaotic field are assumed to be the same in all modes. Three types of modulation are considered: square-wave, triangular and cosinusoidal. The modulation of the coherent field changes considerably the photon counting distribution.