Showing papers on "Superposition principle published in 1970"

Transient Radiation from Pistons in an Infinite Planar Baffle

Abstract: An approach is presented to compute the near‐ and farfield transient radiation resulting from a specified velocity motion of a piston or array of pistons in a rigid infinite baffle. The approach, which is based on a Green's function development, utilizes a transformation of coordinates to simplify the evaluation of the resultant surface integrals. A simple expression is developed for an impulse response function, which is the time‐dependent velocity potential at a spatial point resulting from an impulse velocity of a piston of any shape. The time‐dependent velocity potential and pressure for any piston velocity motion may then be computed by a convolution of the piston velocity with the appropriate impulse response. The response of an array may be computed using superposition. Several examples illustrating the usefulness of the approach are presented. The farfield time‐dependent radiation from a rectangular piston is discussed for both continuous and pulsed velocity conditions. For a pulsed velocity of time duration T it is shown that the pressure at several of the field points can consist of two separate pulses of the same duration, when T is less than the travel time across the piston.

Surface motion of a layered medium having an irregular interface due to incident plane SH waves

Abstract: A practical method is devised to calculate the elastic wave field in a layer-over-half-space medium with an irregular interface, when plane waves are incident from below. This method may be used for studying the interface shape of the M discontinuity, for example, using the observed spectral amplitude and phase-delay anomalies due to teleseismic body waves. The method is also useful for the engineering-seismological study of earthquake motions of soft superficial layers of various cross sections. The scattered field is described as a superposition of plane waves, and application of the continuity conditions at the interface yields coupled integral equations in the spectral coefficients. The equations are satisfied in the wave-number domain when the interface shape is made periodic and the equations are Fourier transformed and truncated. Frequency smoothing by using complex frequencies reduces lateral interferences associated with the periodic interface shape and permits comparison of computed results with those obtained from finite bandwidth observations. Analyses of the residuals in the interface stress and displacement, performed for each computed solution, provided estimates of the errors. The relative root-mean-square residual errors were generally less than 5% and often less than 1% for problems in which the amplitude of the interface irregularity and the shortest wavelength were comparable. The method is applied to several models of ‘soft basins’ ‘dented M discontinuity’ and ‘stepped M discontinuity’ The results are compared with those derived from the flat-layer theory and from the ray theory. In addition to vertical interference effects familiar in the flat-layer theory, we observe the effects of lateral interference as well as those of ray geometry on the motion at the surface.

Colliding gravitational waves.

Abstract: NOWHERE should the nonlinear features of general relativity show up more clearly than in the collisional interaction of two gravitational waves. One of the direct consequences of the linearity of Maxwell's equations is that electromagnetic waves pass straight through each other, and this is probably one of the best attested facts of physics. It may readily be demonstrated1 that such a principle of superposition can never apply to gravitational waves travelling in nonparallel directions, but the precise way in which the waves will diffuse through each other has not hitherto been understood. The problem may have a distinct bearing on observational phenomena, because the gravitational fluxes observed by Weber2 appear to be large enough to contribute significantly to the curvature of the universe3,4, hence large enough to make the linearized approximation invalid (a high frequency approximation has, however, been successfully applied to the cosmological problem by Isaacson5). We shall describe here a coordinate system in which the problem of colliding plane waves may be discussed in general relativity; a particular solution expressible in elementary functions and representing at least a portion of such a collisional situation will be given.

Alternating Current Distribution Between Planar Conductors

Abstract: The general theory for the vector potential produced by a sinusoidal current of finite cross section has been developed in a preceding paper.1 In the present work, this theory is applied to a coil between a number of plates of finite electrical conductivities and permeabilities. Linear, isotropic, and homogeneous media are assumed throughout. The Green's function solution is first obtained, and the principle of superposition is then used to effect the desired result. A number of physically observable phenomena, such as the impedance, the power loss due to the presence of the conductors, and the time‐averaged electromagnetic force on the conductors are subsequently calculated from the vector potential. The treatment is sufficiently general to allow the solution of a number of difficult electromagnetic induction problems, while each of the expressions obtained for the physically observable phenomena is in terms of an infinite integral that may readily be evaluated on a computer. In all cases for which exper...

Mutual coupling analysis of a conformal array of elements on a cylindrical surface

Abstract: The analysis of multiport antennas may be approached by resorting to the powerful concept of eigenexcitations of the structure. They may be defined as the eigenvectors of the admittance matrix, i.e., those sets of voltages and currents at the antenna terminals which are related simply by a proportionality factor (eigenvalue of the admittance matrix). Similarly, the eigenexcitations may be alternatively defined as the eigenvectors of the scattering matrix. The analysis for this special kind of excitation is sometimes relatively easy. The general case may then be treated by utilizing superposition principle. An array of infinitely long longitudinal slots equispaced on a conducting cylinder is analyzed in detail. The eigenvectors of the admittance and scattering matrices are easily found because of the symmetry of the structure. Expressions for the eigenvalues of these matrices, and for the radiation patterns due to the array eigenexcitations, are found in the form of infinite series rapidly convergent (even for cylinders of large radius). By using linearity an expression for the radiation pattern of the single excited element in the array environment (the main result of this paper) is then established. Expressions are also given for the coupling (or scattering) coefficients. A number of numerical examples with different cylinder radii are discussed in detail.

Analysis of the Experimental Meaning of Coherent Superposition and the Nonexistence of Superselection Rules

Abstract: The experimental meaning of coherent superposition of state vectors is analyzed. It is shown that the involvement of the measuring instrument is of basic importance and that it is erroneous to study superposition without it. From this analysis it is found that there are no such things as superselection rules. The meaning of the sign change of spinors under $2\ensuremath{\pi}$ rotation is considered, and it is found to have experimental consequences. The nonmeasurability of the relative parity of states with different $z$ components of angular momentum is shown. An appendix discusses the application of some of the considerations of the paper to decaying states and elementary particles.

Model for Nonlinear Dynamics of Offshore Towers

Abstract: It is possible to model the responses of deep-ocean tower structures subjected to wave forces. Interactions of four systems—the wave surface heights, the ocean, the hydraulic force, and the structure—lead to a problem formulation involving weakly nonlinear random vibrations. Analytical solution of the problem is achieved by the use of a discrete-system semilinear technique in which an equivalent linear system is studied. Certain coefficient parameters of the equivalent system differential equations of motion are selected in a manner minimizing the error of the equivalent system approximation. Any resulting linear system is nonclassical since normal mode superposition techniques do not apply. Nonclassical mode superposition, which involves complex-valued mode shapes and coordinates, supplies the solution. Principal conclusions are that a simplified analysis (using one nonclassical mode) becomes possible for increasing wind velocities, that amplitude and phasing effects in the various systems require design analyses at many wind velocities, and that form drag forces become noticeable at high wind velocities.

Normal modes of carrier waves in semiconductor plates

Abstract: A dispersion relation for waves in an extrinsic semiconductor plate or film with an applied electric field is derived. It is shown that there exists an infinite number of modes. An arbitrary perturbation can be expressed as a superposition of modes. Expressions for the expansion coefficients are derived. Finally, it is shown that a perturbation which is initially a plane wave in a thick plate, remains an approximate plane wave in a time interval corresponding to the decay time, except near the boundaries. Thus, an apparent discrepancy between the theories of infinite- and finite-dimension semiconductors is resolved.

Axially Symmetric Eddy Currents in a Spherical Conductor

Abstract: The general theory for the vector potential produced by a sinusoidal current of finite cross section is developed and applied to a coil encircling two concentric, spherical shells of finite electrical conductivities and permeabilities. Linear, isotropic, and homogeneous media are assumed throughout the analysis. First, the Green's function solution is obtained, and then the principle of superposition is used to effect the desired result. A number of physically observable phenomena—such as the impedance, the power loss due to the presence of the conductors, and the time‐averaged electromagnetic force on the conductors—are subsequently calculated from the vector potential. The treatment is sufficiently general to allow the solution of a number of difficult problems of electromagnetic induction, while each of the expressions obtained for the physically observable phenomena is in terms of an infinite sum that may be readily evaluated on a computer.

Hologram generator using superposition of plane waves

Abstract: A SYSTEM WHICH GENERATES HOLOGRAM FROM ELECTRICAL SIGNALS SUPERIMPOSES A SERIES OF LIGHT WAVES IN A THICK HOLOGRAM RECORDING MEDIUM. A PROCESSOR DETERMINES THE AMPLITUDE, PHASE, AND ANGLE OF INCIDENCE OF THE SERIES OF LIGHT WAVES NEEDED TO GENERATE THE HOLOGRAM FROM THE ELECTRICALL SIGNAL. THESE LIGHT WAVES ARE THEN GENERATED WITH AMPLITUDE MODULATION, PHASE MODULATION, AND DE- FLECTION DEVICES, AND ARE MADE TO INTERFERE WITH REFERENCE LIGHT WAVES. THIS CAUSES A SERIES OF INTERFERENCE PATTERNS TO BE SUPERIMPOSED IN THE RECORDING MEDIUM, THEREBY PRODUCING THE HOLOGRAM.

Analysis of a thin circular loop antenna over a homogeneous earth

Abstract: In this paper, the current distribution on a bare conducting loop, situated in free space over a semi-infinite medium, is obtained for arbitrary time harmonic excitations. The loop is assumed to be thin, perfectly conducting and the standard one-dimensional integral equation and its Fourier series solution are used as the starting points. The field due to the current in the loop, where the semi-infinite medium is absent, is expressed as a superposition of plane waves. The tangential component of the field reflected by the interface, of the semi-infinite medium, is evaluated using appropriate Fresnel reflection coefficients. This reflected field serves as a new source for the loop and induces a current on the loop. The field due to the induced current is treated in the same manner, and this process is repeated indefinitely. The summation of the original current and all the induced currents gives the steady-state current on the loop.

Mutually interacting instabilities in the magnetosphere.

Abstract: It may happen that a magnetospheric plasma can admit two or more instabilities of different types, for example, a high frequency instability (electrostatic or cyclotron) and a low frequency instability (hydromagnetic or drift wave). In many such cases, the plasma wave dynamics cannot even approximately be described by a simple linear superposition of the various instabilities which are possible, because slight changes in the plasma produced by one instability may cause major changes in the growth rate of another. It is always possible to describe this interaction mathematically by saving enough terms in the Vlasov-Maxwell equations (e.g., mode-mode coupling). To some extent a physically concise and useful description is supplied by following the analysis of Dupree (1966, 1967, 1968), certain aspects of which were rediscovered independently by the author in the course of the present work. In this analysis, one takes into account the influence of the instability wave fields on particle orbits which are used to calculate the dispersion relation. In this fashion, growth rates taken from the dispersion relation depend explicitly on wave intensities, in contrast to the usual linear stability calculations. In certain cases, this explicit dependence on wave intensities is essential for description of equilibrium, since the equilibrium conditions of quasi-linear theory (Kennel and Engelmann, 1966) cannot be applied.

Radiative transport of Lorentz lines in nonisothermal gases

Abstract: A modified Ladenburg-Reiche analysis is developed for predicting nonisothermal curves of growth for isolated, unshifted Lorentz lines. The superposition of Lorentz lines along nonisothermal paths is approximated by an equivalent Lorentz line, whose two fitting parameters are internally defined. The formulation is exact in the weak- and strong-line limits and is a valid approximation for all optical depths. Its accuracy is investigated for a family of linear variations in the line strength and half-half-width, for which exact results can be obtained analytically. The error in the nonisothermal equivalent width is strongly dependent on variations in the half-half-width and only weakly dependent on variations in the line strength. For a ten-fold change in the half-half-width over the optical path, a maximum error of only three per cent is observed for the equivalent width.

Test particle method in kinetic theory of a plasma

Abstract: A general relationship has been found between many body correlation functions and the conditional probability functions which describe the shield clouds surrounding test particles. This is the generalization of Rostoker's superposition principle. The relationship is useful because the problem of kinetic theory is reduced to determining the conditional probability functions, which essentially involves only the Vlasov equation.

Use of superposition in writing state equations for networks with excess elements

Abstract: When one tries to write state equations, in normal form, for an RLC network with excess elements employing principle of superposition, one encounters several difficulties. The method proposed in this correspondence avoids such difficulties and broadens the class of networks that could be formulated in normal form using the principle of superposition.

Analysis of spectra with nonlinear superposition

Abstract: A numerical method is described that analyses spectra with nonlinear superposition. Examples of the superposition of up to five species with up to five pair-interference spectra are presented. The effect of noise on the accuracy of the determination of species concentrations is discussed. The numerical method is based on optimization (nonlinear least squares fit). A FORTRAN listing is appended.

VI Coherence Theory of Source-Size Compensation in Interference Microscopy

Abstract: Publisher Summary This chapter reviews the general theory of two-beam interferometer and the unified theory of image-formation and interference in terms of coherence. Its applications to the theory of localized fringes and to the theory of source-size compensation are described. The chapter presents the classification of practical methods of source-size compensation which is achieved. The description of the images produced by a compensated interference microscope is also given. It is formulated in terms of four coherence functions between the radiation from the beams i and j at two points. This representation applies to both the localization of fringes and the source-size compensation in any plane of interest, whereas the latter is described formerly as the superposition of two systems of localized fringes. These studies yielded simple formulae, which serve as an analytic tool for devising new designs.

Coupling of Electrostatic Waves in a Nongyrotropic Plasma

Abstract: The dispersion relation for small‐amplitude electrostatic waves in a nongyrotropic plasma is derived. The equilibrium distribution function is not symmetric in the plane perpendicular to the magnetic field and is necessarily periodic in time. A single electrostatic mode contains a linear superposition of potentials with the same wavelength and damping rate but with different frequencies separated by the cyclotron frequency. The dispersion relation is solved for the case of small asymmetry. The asymmetry then leads to a linear coupling of known modes, with small changes in known dispersion relations.

Incoherent light pulses

Abstract: We study a light pulse for which the ratio between its time length and its coherence time can assume any possible value This light pulse is obtained by an incoherent superposition of identical polychromatic photons Its density matrix, evaluated by making use of the convolution law of theP representation, is analysed A new definition of coherence time that is a generalization of previous definitions is proposed; this new definition does not require the field to be stationary and consequently may apply to some light pulses

A Solution Using the Superposition Technique for Externally Pressurized Multi-Recess Journal Bearings Including Hydrodynamic Effects

Abstract: The paper presents details of a computer solution to the problem of a rotating multi-recess externally pressurized bearing. It is shown that the bearing number Nη/P8(cD/D)2correlates the results and permits the superposition of hydrodynamic and hydrostatic results for a given geometry. This significantly reduces the computer time necessary for a complete survey of results since only one solution for each of the Laplace equations and the Reynolds equation is required at any eccentricity. The problem of the control circuit is solved separately and is negligible in computer time taken in comparison with the main subroutines of the programme for finding the hydrodynamic and hydrostatic coefficients.

Decomposition of Converging Branch Multistage Systems

Abstract: This article considers an alternate dynamic-programming technique for analysis of converging branch systems. This technique, like superposition, is based on decomposition of the system considered. However, it is not restricted to linear systems. The validity of the approach is shown, and an example problem is presented.

Evolution of Reduced Distribution Functions. III. General Truncation and Solution of the Linearized Classical BBGYK Hierarchy

Abstract: A linearized version of the nonequilibrium BBGYK hierarchy applicable to simple fluid systems near equilibrium is derived. A sequence of successive extended dynamical superposition approximations is defined to truncate the hierarchy at any arbitrary level. A formal solution, applicable either to the truncated or the complete hierarchy, is obtained by iteration of the Laplace transformed hierarchy followed by inversion of the transform. For the lowest truncation, which produces a single kinetic equation for the one‐body reduced distribution function, the Fourier–Laplace transform of the solution is summed to a closed form.

Ordering ofM-mode field operators in quantum optics

Abstract: The general formalism of an arbitrary ordering ofM-mode field operators is developed and this formalism is applied to a model of the superposition of coherent and chaoticM-mode fields having importance for a description of laser light. Corresponding probability distributions and their moments as well as their interrelationships are derived. Physically important relations for the normal, symmetric and antinormal orderings are deduced as special cases and a criterion of the existence of the weighting function of the Sudarshan-Glauber diagonal representation of the density matrix inL2-space given.