Showing papers on "Superposition principle published in 1969"

High-resolution frequency-wavenumber spectrum analysis

Abstract: The output of an array of sansors is considered to be a homogeneous random field. In this case there is a spectral representation for this field, similar to that for stationary random processes, which consists of a superposition of traveling waves. The frequency-wavenumber power spectral density provides the mean-square value for the amplitudes of these waves and is of considerable importance in the analysis of propagating waves by means of an array of sensors. The conventional method of frequency-wavenumber power spectral density estimation uses a fixed-wavenumber window and its resolution is determined essentially by the beam pattern of the array of sensors. A high-resolution method of estimation is introduced which employs a wavenumber window whose shape changes and is a function of the wavenumber at which an estimate is obtained. It is shown that the wavenumber resolution of this method is considerably better than that of the conventional method. Application of these results is given to seismic data obtained from the large aperture seismic array located in eastern Montana. In addition, the application of the high-resolution method to other areas, such as radar, sonar, and radio astronomy, is indicated.

On the characterization of nonlinear viscoelastic materials

Abstract: Starting with specific constitutive equations, methods of evaluating material properties from experimental data are outlined and then illustrated for some polymeric materials; these equations have been derived from thermodynamic principles, and are very similar to the Boltzmann superposition integral form of linear theory. The experimental basis for two equations under uniaxial loading and the influence of environmental factors on the properties are first examined. It is then shown that creep and recovery data can be conveiently used to evaluate properties in one equation, while two-step relaxation data serve the same purpose for the second equation. Methods of reducing data to accomplish this characterization and to determine the accuracy of the theory are illustrated using existing data on nitrocellulose film, fiber-reinforced phenolic resin, and polyisobutylene. Finally, a set of three-dimensional constitutive equations is proposed which is consistent with nonlinear behavior of some metals and plastics, and which enables all properties to be evaluated from uniaxial creep and recovery data.

Light Scattering from Binary Solutions

Abstract: The spectrum of the light scattered by a binary solution is calculated from thermodynamic fluctuation theory and the linearized hydrodynamic equations appropriate to a two‐component fluid The spectrum consists of three peaks Expressions are obtained for the positions and widths of the two‐side, Brillouin peaks In general the central, unshifted Rayleigh peak is found to consist of a superposition of two Lorentzians that involve the combined dynamical effects of heat conduction and diffusion The condition is stated under which it is possible to separate the central peak simply into two contributions, one arising from diffusion and one from thermal conduction For many binary systems this separation is justified In these cases measurement of the spectrum of the scattered light should prove to be an attractive alternative means of measuring the diffusion coefficient of binary solutions

Echo removal by discrete generalized linear filtering.

Abstract: : A new approach to separating convolved signals, referred to as homomorphic deconvolution, is presented. The class of systems considered in this report is a member of a larger class called homomorphic systems, which are characterized by a generalized principle of superposition that is analogous to the principle of superposition for linear systems. A detailed analysis based on the z-transform is given for discrete-time systems of this class. The realization of such systems using a digital computer is also discussed in detail. Such conputational realizations are made possible through the application of high-speed Fourier analysis techniques. As a particular example, the method is applied to the separation of the components of a convolution in which one of the components is an impulse train. This class of signals is representative of many interesting signal-analysis and signal-processing problems such as speech analysis and echo removal and detection. It is shown that homomorphic deconvolution is a useful approach to either removal or detection of echoes.

Imperfectly mode-locked laser emission and its effects on nonlinear optics

Abstract: The output of an ideally mode-locked laser, namely, one having equal phase angles of the modes, consists of a train of bandwidth-limited short pulses. The influence of deviations from equal phase angles on the laser output is investigated. Random distributions of the phases of the modes around equal phase angles introduce only limited fluctuating background, whereas the duration of the pulses remains bandwidth-limited. A systematic deviation of the phase angles at least quadratic with mode number is necessary for lengthening of the pulses. The results obtained by the superposition of these two effects are consistent with the experiments published so far. The efficiency of second- and higher-harmonic generation is discussed by help of the moments of the intensity probability distributions. It is shown how these distributions change as a result of harmonic generation and optical mixing. From intensity-correlation measurements, the moments of the intensity probability distribution can be obtained directly, as well as information on the time behavior of the light field. Experimental arrangements used to measure intensity correlations are described. © 1969 The American Physical Society.

Quantum Statistics of Three Interacting Boson Field Modes

Abstract: The quantum theory of the stimulated Raman and Brillouin effects, the parametric amplifier, and the frequency converter is developed, starting with a simple Hamiltonian for a system of three coupled field modes. The intense incident beam is treated classically. The evolution in time of the statistical properties of the quantized Stokes and anti-Stokes waves are analyzed by means of appropriate time-dependent phasespace distributions under various assumptions for the initial fields. Expansions of the field variables in terms of complete sets of orthogonal operators are discussed. Several regimes of operation are considered. It is shown that an initially coherent state develops, in general, into a superposition of a coherent state and a chaotic state. The amplitudes of the coherent components follow the same equations of motion as the mode operators. The chaotic components stem from the amplification of the vacuum fluctuations.

Analysis of unbalanced polyphase networks by the method of phase co-ordinates. Part 2: Fault analysis

Abstract: A general analysis of power-system polyphase networks under fault conditions is developed in terms of the phase-co-ordinate representation. It is shown that the method requires only the assembly of the nodal phase admittance or impedance matrices from connection tables, and either the subsequent solution of sets of linear algebraic equations subject to constraints, or the use of the superposition principles of the generalised Norton theorem. By these techniques, the problems of multiple unbalanced faults on unbalanced systems can be analysed with no more difficulty than balanced 3-phase faults on balanced systems.

Use of superposition in digital computers to obtain wind tunnel interference factors for arbitrary configurations, with particular reference to V/STOL models

Abstract: Computer program for calculating wall interference factors of V/STOL aircraft and conventional airplanes

On the non-linear Lamb-Taylor instability

Abstract: A non-linear analysis of the inviscid stability of the common surface of two superposed fluids is presented. One of the fluids is a liquid layer with finite thickness having one surface adjacent to a solid boundary whereas the second surface is in contact with a semi-infinite gas of negligible density. The system is accelerated by a force normal to the interface and directed from the liquid to the gas. A second-order expansion is obtained using the method of multiple time scales. It is found that standing as well as travelling disturbances with wave-numbers greater thanwhere a is the disturbance amplitude and kc is the linear cut-off wave-number, oscillate and are stable. However, the frequency in the case of standing waves and the wave velocity in the case of travelling waves are amplitude dependent. Below this cut-off wave-number disturbances grow in amplitude. The cut-off wave-number is independent of the layer thickness although decreasing the layer thickness decreases the growth rate. Although standing waves can be obtained by the superposition of travelling waves in the linear case, this is not true in the non-linear case because the amplitude dependences of the wave speed and frequency are different. A mechanism is proposed to explain the overstability behaviour observed by Emmons, Chang & Watson (1960).

Analytical solutions of the neutron transport equation in arbitrary convex geometry.

Abstract: The integral equation describing the transport of monoenergetic, isotropically scattered neutrons in a one‐, two‐, or three‐dimensional body of arbitrary convex shape, containing distributed sources, is considered. An exact representation of the neutron density ρ(r) is obtained, involving a superposition of functions belonging to the null space of a simple differential operator. In general, when a countable basis is chosen to span the null space, the coefficients in the expansion of ρ(r) satisfy a coupled system of singular integral equations which is reducible to a system of Fredholm equations. If no sources are present, an exact criticality condition is also obtained. Some techniques for evaluating the expansion coefficients are given and several examples are considered.

Theory of linear superposition in tape recording

Abstract: The conditions under which linear superposition (LSP) of isolated pulses is valid for the synthesis of multibit waveforms are examined theoretically. LSP is found to be valid subject to three conditions; first, that all the processes following the write process be linear operations on the tape magnetization; second, that the write field rise time be less than the bit interval; and third, that each change in magnetization, occurring during the write process, be a function only of the field causing that change. Additionally, the validity of LSP at 15 000 bit/in is demonstrated for a high resolution tape recorder using standard γ-Fe 2 O 3 tape.

On the quantum statistics of the superposition of coherent and chaotic fields

Abstract: The generating function, the corresponding integrated intensity distribution, and the photon-counting distribution and its fractional moments are derived for the superposition of coherent and chaotic M-mode fields on the basis of a formalism of arbitrary ordering of field operators in quantum optics An alternative description is also given based on a recent analysis of the superposition of coherent and chaotic fields in terms of the correlation functions which makes it possible to derive a number of results valid for arbitrary mean occupation numbers per mode from which the model considered follows as a special case Many earlier results obtained for coherent and chaotic fields and their superposition are included as special cases The present results generally describe a superposition experiment for an N-mode coherent field (eg generated by an N-mode laser operating far above threshold) and an M-mode chaotic field (M>or=N) as well as a field generated by an M-mode laser operating above threshold

Rigorous field expressions for piecewise-sinusoidal line sources

Abstract: Superposition is used to obtain expressions for the field of a line source with piecewise-sinusoidal current distribution which are rigorous even in the near zone, despite their simplicity

I Multiple-Beam Interference and Natural Modes in Open Resonators

Abstract: Publisher Summary This chapter discusses multiple beam interference and natural modes in open resonators. Interferograms from multiple-beam interferometers with and without phase objects can always be expressed quantitatively as the superposition of several simultaneously excited modes of the interferometer system. The classical interference fringe and ring systems result from the superposition of the fields of many modes. The modes become prominent only when diffraction effects begin to play a major part. Application of the mode concept shows that the simple connection between object and interferogram predicted by the classical theory of multiple-beam interference holds a certain “resolution limit”. The relation is confirmed by measurements with centimeter waves and by experiments on resolution of small phase object structures in the optical region. The phase object in a wave field changes the emerging wave producing an exact phase image of the object. These effects are not noticed if only the Kirchhoff diffraction theory in the simple form is applied.

Zur Überlagerung von Erneuerungsprozessen

Abstract: If a finite sequence of independent (not necessarily stationary) renewal processes is given, a superposition process can easily be defined as the union of all point sequences represented by the given processes. The properties of such superposition processes are investigated. First, a necessary and sufficient condition for a superposition process to be a renewal process is given. Essentially, this condition reads thus: the given processes must be Poisson processes. The main result given in this paper is a limit theorem for superposition processes which shows that, even with largely arbitrary renewal processes superimposed, the superposition process has local properties which approach the properties of the Poisson process as the number of given processes increases. The theorem contains some well-known special theorems of this type [e.g. Khintchine, 1960; Franken, 1963].

Signal Strength in Two-beam Interferometers with Laser Illumination

Abstract: The depth of modulation of the interference pattern in a two-beam interferometer is considered for systems in which the superposed wavefronts are nominally parallel. The effects are considered, for laser illumination with a Gaussian distribution, of imperfect superposition and parallelism of the wavefronts.

Nonlinear Systems with a Restricted Additivity Property

Abstract: A special but nontrivial class of time-invariant systems is described, which is neither linear nor memoryless. This class of systems is characterized by the "semiadditive" property that the response to the sum of two nonoverlapping input time functions is equal to the sum of the responses to each input acting alone. These semiadditive systems are found to be amenable to characterization, analysis, and synthesis with a simplicity almost comparable to that of linear systems. In particular, it is shown that a semiadditive system is completely specified by an amplitude dependent step response function, a "needle pulse" response function of time and amplitude, or an amplitude dependent frequency response function. Explicit input-output relationships are presented, which include 1) an additivity integral that generalizes the superposition integral of linear systems, 2) a relation that yields the output spectrum directly from the input time function, and 3) a relation that yields the spectral density of the response to a stationary Gaussian input process directly from the autocorrelation function of the input. An analogy exists between semiadditive nonlinear systems and linear time-varying systems by interchanging the roles of amplitude and time.

External thermal resistance of three buried cables in trefoil-touching formation. Restricted application of superposition

Abstract: An approximate formula, with error bounds, is derived for the thermal resistance between three touching circular cylinders and a plane. The formula is based on a restricted application of the principle of superposition. The distance between the cylinder configuration and the plane is assumed to be much greater than the cylinder radius, which is the case for the evaluation of the external thermal resistance of three equally loaded identical cables laid direct in the ground in trefoil-touching formation. Numerical calculation indicates that formulas now in use for this application may give values that are noticeably on the high side. The method may be used to obtain approximations to the solution of other thermal resistance problems involving a plane as one of the boundaries.

External thermal resistance of two buried cables. Restricted application of superposition

Abstract: New formulas are derived for the external thermal resistance of one of two equally loaded identical cables laid direct in the ground at the same depth. The formulas are based on a restricted application of the principle of superposition, and remain valid when the cables are close or touching. Two assumptions made are that the depth of the buried cables is much greater than the cable radius, and that the cable surfaces are isothermal at a common temperature. The former assumption is justified as the cable depth is generally equal to at least ten times the cable radius. The latter assumption is recognised to be an approximation, but the present approach may prove to be more acceptable than alternative techniques. Comparison is made with previously published formulas.

Underwater Acoustic Propagation Prediction by the Alternating‐Direction Implicit‐Explicit Computational Method

Abstract: A computer solution of the wave‐difference equations is found by using an interlacing implicit‐explicit scheme. In the computation, the entire two‐dimensional field is found as a function of time. The examples considered involve propagation in a homogeneous shallow‐water channel, where the effect of superposition of discrete spectra produces characteristic modal patterns. The spatial sound‐pressure fluctuations are represented as plot‐density variations along the two‐dimensional channel. Examples of discrete propagated modes as well as evanescent modes are presented. The size of the field that can be presented is limited by the size, accuracy, and speed of the computer.

The Extension of a Superposition Principle to Wave Propagation in Nonhomogeneous Elastic Media

Abstract: In this paper a superposition principle is extended to examine the wave propagation problem in composite materials, which can be phenomenologically characterized as being nonhomogeneous- isotropic. The method is particularly useful for the analysis of stresses and displacements in such media with finite geometrical bounds.The existence of a superposition principle is first established and the resolution of the wave propagation problem in nonhomogeneous media into the solution of a static and dynamic eigen-value problem demonstrated. Finally, for purposes of illustration, a one-dimensional problem is solved and data are presented.

Theorie der unelastischen Streuung von Elektronen in Kristallen

Abstract: A general solution of the basic equations of electron diffraction in crystals is found which shows that each state of a diffracted electron can be described as a linear superposition of Bloch waves with amplitudes varying in the direction normal to the entrance surface of the crystal. For each state the Bloch waves composing it belong to the same wave vector but are eigenfunctions of different energy bands. The absorption effect can be described by the variation of the amplitudes which can be calculated by solution of a system of linear homogeneous differential equations of first order. Since this variation is different for every state, a dispersion effect is coupled with the absorption. The mathematical structure of this solution is the same for elastic and inelastic scattering, and thus both kinds of transitions may contribute to the dispersion and absorption mechanism.

Fourier Analysis and Nonlinear Optical Systems

Abstract: The modulation transfer function is based on Fourier techniques which imply the validity of the principle of superposition. This is clearly violated in the nonlinear film development process. A pseudotransfer function is defined to represent an artificial linear filter which, if it replaced the actual optical and photographic process, would result in precisely the same image as would be produced by the real system. The significance of such a function in a statistical approach and the results of a typical calculation are presented for purpose of illustration.

Analysis of an Integral Equation Arising from the Investigation of Vertical Directivity of Sea Noise

Abstract: An integral equation relates the single frequency vertical distribution of ambient ocean noise to the measured array output when the noise power at the array arises from the superposition of uncorrelated plane waves. This integral equation has an infinite number of solutions. However, from the physical viewpoint, there is one solution, referred to as the “principal solution,” which is the most plausible representation of the true noise field. The “principal solution,” is precisely defined, and then classical Hilbert space theory is used to determine this solution from a finite number of array measurements. The number of measurements required depends only on the interelement spacings of the array. In selecting this finite number of steering angles to make the array measurements, any choice that yields independent samples for a certain finite sequence of trigometric functions is sufficient. Not surprisingly, these measurements contain all the information available in the observed distribution, the array out...