Showing papers on "Superposition principle published in 1986"

Generating quantum mechanical superpositions of macroscopically distinguishable states via amplitude dispersion.

Abstract: It is pointed out here that a coherent state propagating through an amplitude dispersive medium will, under suitable conditions, evolve into a quantum superposition of two coherent states 180\ifmmode^\circ\else\textdegree\fi{} out of phase with each other. The response of a homodyne detector to this superposition of macroscopically distinguishable states is calculated. Signatures which an experimentalist might look for in the homodyne detector's output in order to verify the generation of such states are described.

Exact and Approximate Construction of Digital Phase Modulations by Superposition of Amplitude Modulated Pulses (AMP)

Abstract: Minimum shift keying and offset QPSK are two well-known modulations which can be interpreted as a set of time/phase-shifted AM pulses. We show in this paper that any constant amplitude binary phase modulation can also be expressed as a sum of a finite number of time limited amplitude modulated pulses (AMP decomposition). New methods for computing autocorrelation and power frequency spectrum are derived, which give very simple results for half-integer index modulations. We also show that the signal can be built with good accuracy using only one optimized pulse ("main pulse"). This synthesis is particularly satisfactory for modulations that have good spectral characteristics and/ or low index.

Critical Layers in Shear Flows

Abstract: The normal mode approach to investigating the stability of a parallel shear flow involves the superposition of a small wavelike perturbation on the basic flow. Its evolution in space and/or time is then determined. In the linear inviscid theory, if ū(y) is the basic velocity profile, then a singularity occurs at critical points yc, where ū = c, the perturbation phase speed. This is plausible intuitively because energy can be exchanged most efficiently where the wave and mean flow are travelling at the same speed. The problem is of the singular perturbation type; when viscosity or nonlinearity, for example, are restored to the governing equations, the singularity is removed. In this lecture, the classical viscous theory is first outlined before presenting a newer perturbation approach using a nonlinear critical layer (i.e., nonlinear terms are restored within a thin layer). The application to the case of a density stratified shear flow is discussed and, finally, the results are compared qualitatively with radar observations and also with recent numerical simulations of the full equations. ∗Address for correspondence: Department of Mathematics and Statistics, McGill University, Montreal, QC, H3A 2K6, Canada. e-mail: maslowe@math.mcgill.ca

Efficient global matrix approach to the computation of synthetic seismograms

Abstract: Summary. A numerically efficient global matrix approach to the solution of the wave equation in horizontally stratified environments is presented. The field in each layer is expressed as a superposition of the field produced by the sources within the layer and an unknown field satisfying the homogeneous wave equations, both expressed as integral representations in the horizontal wavenumber. The boundary conditions to be satisfied at each interface then yield a linear system of equations in the unknown wavefield amplitudes, to be satisfied at each horizontal wavenumber. As an alternative to the traditional propagator matrix approaches, the solution technique presented here yields both improved efficiency and versatility. Its global nature makes it well suited to problems involving many receivers in range as well as depth and to calculations of both stresses and particle velocities. The global solution technique is developed in close analogy to the finite element method, thereby reducing the number of arithmetic operations to a minimum and making the resulting computer code very efficient in terms of computation time. These features are illustrated by a number of numerical examples from both crustal and exploration seismology.

New Results in Dielectric-Loaded Resonators

Abstract: Analysis of cylindrical dielectric-loaded resonators is reviewed. The fields within the dielectric-loaded region are postulated as the superposition of hybrid, TE, or TM modes of the infinite dielectric-loaded waveguide, while the fields in the end regions of the resonators are described by the superposition of the normal modes of a homogeneously filled waveguide. Numerical results are presented which reveal that accurate representation of the fields in the resonant structure generally require several modes. Hence, the resonant modes cannot be correlated directly with single waveguide modes. A new method for mode identification is proposed. For a wide range of parameters, the resonant frequencies, mode charts, field expansion coefficients, field intensity, and distributions are presented. Excellent agreement of the mode charts with resonant frequency measurement results are obtained.

On the integrability of systems of nonlinear ordinary differential equations with superposition principles

Abstract: A new class of ‘‘solvable’’ nonlinear dynamical systems has been recently identified by the requirement that the ordinary differential equations (ODE’s) describing each member of this class possess nonlinear superposition principles. These systems of ODE’s are generally not derived from a Hamiltonian and are classified by associated pairs of Lie algebras of vector fields. In this paper, all such systems of n≤3 ODE’s are integrated in a unified way by finding explicit integrals for them and relating them all to a ‘‘pivotal’’ member of their class: the projective Riccati equations. Moreover, by perturbing two parametrically driven projective Riccati equations (thus making them nonsolvable in the above sense) evidence is discovered of chaotic behavior on the Poincare surface of section—in the form of sensitive dependence on initial conditions—near a boundary separating bounded from unbounded motion.

Mode-superposition methods in dynamic analysis of classically and non-classically damped linear systems

Abstract: Mode-superposition analysis is an efficient tool for the evaluation of the response of linear systems subjected to dynamic agencies. Two well-known mode-superposition methods are available in the literature, the mode-displacement method and the mode-acceleration method. Within this frame a method is proposed called a dynamic correction method which evaluates the structural response as the sum of a pseudostatic response, which is the particular solution of the differential equations, and a dynamic correction evaluated using a reduced number of natural modes. The greater accuracy of the proposed method with respect to the other methods is evidenced through extensive numerical tests, for classically and non-classically damped systems.

TE and TM Modes of Some Triangular Cross-Section Waveguides Using Superposition of Plane Waves (Short Paper)

Abstract: Exact transverse electric and magnetic mode solutions of four triangular cross-section waveguides have been found via a new general method using Snell's law and superposition of plane waves. This paper presents results for 1) equilateral, 2) 30°, 30°, 120°, 3) isosceles right, and 4) 30°, 60° right triangular waveguides. The electric and magnetic field solutions form finite sums of separable rectangular harmonics and are the only waveguides of triangular cross section for which such solutions have been found.

A comparison of time‐domain transmitting boundaries

Abstract: A non-linear interaction analysis with a (generalized) non-linear structure and a linear unbounded soil is analysed in the time domain, based either on the sub-structure method, which involves global convolution integrals, or on the direct method with local boundary conditions. Alternatively, the hybrid frequency–time-domain method of analysis, which is an iterative scheme, could be used. Approximate local boundary conditions to model the wave propagation towards infinity on the artificial boundary used in the direct method of non-linear soil–structure-interaction analysis to be performed in the time domain are examined. A semi-infinite rod supported elastically, which exhibits the same properties as certain unbounded soils such as dispersion and a cut-off frequency, is used for the investigation. For a transient excitation, the superposition boundary with frequent averaging, the well-known viscous damper and the extrapolation algorithm lead to good accuracy. Moving the artificial boundary further away from the structure (or more precisely, increasing the ratio of the distance of the artificial boundary to the wave length) improves the accuracy.

Gaussian beam synthetic seismograms

Abstract: This presentation is a tutorial on the Gaussian beam method used for the asymptotic synthesis of seismic and acoustic wave fields in inhomogeneous media. The method is based on the superposition of beam solutions, each of which is an approximate solution of the wave equation along particular rays. Smoothness conditions on the medium are required for the approximate propagation of beam solutions. Within a smoothly varying medium, various choices of the beam parameters can be used. Specifying broad planar beams at the source would result in a plane‐wave decomposition of the visible spectrum. The standard ray method would result by specifying narrow planar beams at the receiver. Another choice is to specify the minimum integral beam width along each ray. There are several advantages of the Gaussian beam method, including finite amplitudes at caustics, smoothing of endpoint errors, and the reduction of amplitude variability resulting from model parameterization. Several numerical examples will be given in 2‐D and 3‐D, and comparisons will be shown with other numerical methods.

Real spectra, complex spectra, compact spectra

Abstract: Plane-wave spectral and induced source representations of directly excited and (or) scattered fields constitute alternative approaches for analyzing wave propagation. Although the plane-wave spectra, on the one hand, and the source distributions, on the other, generally require continuous superpositions that lead to integral formulations in the spatial-spectral and the physical configuration domains, respectively, phenomena of constructive and destructive interference at high frequencies permit contraction of these distributed constituents around interference maxima represented by stationary points, end points, or other critical points in the integration interval. This leads to a representation of the high-frequency field in terms of physically meaningful compact spectral objects generated by local portions of the source distribution that radiate energy from the (actual or induced) source region to the observer by local plane waves traversing the ray trajectories of the geometrical theory of diffraction. If the critical points in the integrals are real, the compact representation identifies nonevanescent wave bundles emitted by source patches at a real physical location. However, for many wave phenomena involving beam-type initial source fields, concave and convex boundaries, leaky waveguides, etc., as well as damped resonances in the time domain, the spectral contraction occurs around damped complex constituents identifying bundles of evanescent plane waves that travel along complex ray trajectories. Thus the initial source configuration and propagation space must be extended by analytic continuation to complex values. Insisting on real spectral and configurational domains expresses in a smeared-out unnatural manner what is compact and natural in the complex domain. These concepts are illustrated here in various examples, with emphasis on the physical importance of compact representations.

Lower and Upper Bound Calculations on the Capacitance of Multiconductor Printed Transmission Line Using the Spectral-Domain Approach and Variational Method

Abstract: A unified spectral-domain method is developed for accurate evaluation of the parameters of single and coupled microstripline-type structures containing a number of additional conducting strips with induced and/or zero potentials, located on several interfaces of dielectric layers. The Green's function technique in the spectral domain and the superposition principle for solutions of simple Dirichlet's problems are applied for the first step of the analysis in which a set of algebraic equations is to be derived. Extreme values of two variational functional are found for estimation of the upper and lower bounds on the line capacitance. Specific computations, carried out for new coupled coplanar lines with additional tuning conductive septums, illustrate the validity and efficiency of the presented method. It has been shown that equalization of the even- and odd-mode phase velocities can be achieved in this structure.

Exact evaluation of the response of a layered elastic medium to an explosive point source using leaking modes

Abstract: A simple and rigorous theory of leaking modes for liquid layer or SH elastic wave propagation problems is presented. By taking the frequency ω and wave-number k simultaneously as complex variables and choosing appropriate paths of integration in the ω-plane, the integration with respect to k is performed exactly using Cauchy residue theory. The remaining integration with respect to ω is then carried out by use of the Fast Fourier transform. The method is simply to apply, accurate, and computationally efficient. There are no spurious arrivals, and provided the number of points in the Fast Fourier transform can be taken sufficiently large, there are no restrictions on distance. The theoretical results established in the paper show conclusively that complete seismograms, including all possible body waves, can be expressed simply as a superposition of modes. No branch line integrals are required, contrary to the widespread supposition in the literature. The application of the theory is illustrated by producing complete theoretical seismograms for a model consisting of six liquid layers.

Memory behaviour of the spin glass relaxation

Abstract: Isothermal magnetization measurements on a CuMn spin glass show that the memory behaviour of time relaxations in spin glasses is governed by the principle of superposition, which yields fundamental links between the time relaxations of the field-cooled, zero-field-cooled, isothermal-remanent and remanent magnetizations. The principle of superposition applies in spite of a continuous change of the response function at constant temperature, due to aging of the spin glass state.

Transient calculation of a transmission line using superposition law

Abstract: A method of calculation of transmission line transients using the superposition law is developed in the paper. In the method, impulse responses of a multiphase transmission line, in which all the frequency-dependent effects of the line are taken into account, are first obtained in the actual phase domain using a Laplace transform method. Then the impulse responses are included in a transient calculation using the superposition law. The frequency-dependent effect of a transformation matrix, which cannot be considered by the conventional method, is automatically taken into account. Results calculated by the method are compared with field test results and accurate solutions, and the accuracy of the method is proved to be good.

Interaction induced light scattering

Abstract: Several simplified models are discussed, more or less directly based upon Kirkwood's superposition approximation, currently employed in interpreting interaction induced light scattering spectra in high density simple fluids. Calculations are performed of the first two spectral moments for room temperature argon at moderate and high densities in Kirkwood's approximation using pair correlation functions from molecular dynamics and the dipole-induced-dipole pair polarizability model. Comparison with full molecular dynamics calculations indicates that at room temperature accurate zeroth and second spectral moments can be obtained using the superposition approximation and shows the limits of more simplified models in describing the density evolution of the spectral moments. The approximation is shown to work also for other induced polarizability models.

Discretization error for the superposition of Gaussian beams

Abstract: Summary. A high-frequency asymptotic integral expansion of a time-harmonic wavefield into Gaussian beams was derived in a previous paper by Klimes. The discretization error caused by replacing this integral superposition by a discrete summation of Gaussian beams is estimated in this paper.

Wave Propagation in Polyphase Transmission Lines a General Solution to Include Cases Where Ordinary Modal Theory Fails

Abstract: The solution of transmission line equations is usually written as a superposition of so called natural modes of exponential type. These are obtained through the use of a suitable transformation that decouples the original sets or N simultaneous 2nd order wave equations for voltages and currents into N independent equations. For such a transformation to exist the fundamental product matrix ZY must be diagonalizable. In a previous paper it has been shown that physically realizable transmission lines are possible for which ZY is not diagonalizable and to which ordinary modal theory does not apply. In the present paper a new generalized modal theory is developed for the purpose of including non-diagonalizing cases.

A Two-Dimensional Model of Inertial Oscillations Generated by a Propagating Wind Field

Abstract: A linear, two-dimensional, continuously stratified, viscous model has been developed to study the inertial oscillations generated by a propagating wind field. The model, an extension of that of Kundu and Thomson, includes the presence of a coast and superposition due to distributed forcing. These two effects generate a large subsurface oscillation, provided the wind spectrum has energy near the inertial frequency. The presence of the coast causes an additional blue shift of the frequency, and a downward flux from the surface-coast corner. The superposition of responses with random phases does not cancel out but initially increases the rms amplitude as (time)½. The model spectra have a blue shift that increases with depth and can also contain secondary peaks at higher frequencies if the speed of propagation is not too large. For a given propagation speed the blue shift, and hence the downward flux from the surface, is larger in the deep ocean where the gravity wave speeds cn are larger. A calculat...

Incoherent Flow Imaging

Abstract: A method for the multidimensional measurement of flow fields in liquids by nuclear magnetic resonance (NMR) is discussed. It is assumed that the flow field at a given point in space is given by the superposition of two components. The first component, characterizing the average bulk flow, is called the coherent velocity. The incoherent flow field super-imposed on top of the coherent one is assumed to be random as one may find in turbulent flow. Limiting cases are discussed. It is shown that coherent flow introduces a phase into NMR images whereas the incoherent one causes a reduction of signal intensity similar to the one given by T2 or diffusion processes.

Weight Functions of Radial Cracks Emanating from a Circular Hole in a Plate

Abstract: Explicit nodal weight functions for a pair of symmetric radial cracks emanating from a hole in a plate are presented with special emphasis on the load-independent characteristics of the explicit weight functions that can obviate repeated calculations of the Mode I stress intensity factors (K I ) under different loading conditions. An analytical expression, which relates the explicit crack-face weight function component for Mode I deformation to the radial distance from the crack tip along the crack face, is also provided to facilitate K I evaluation by combining the uncracked stress field and the explicit crack-face weight functions through the use of the linear superposition principle. The utilization of the explicit weight functions, which are obtained from a simple loading, for calculating K I under complex loading conditions such as biaxial loading and pin-joint loading with and without interference pressure, is uniquely demonstrated by combining the superposition principle and the weight function concept.