Showing papers on "Superposition principle published in 1989"
TL;DR: The superposition model was originally developed to separate the geometrical and physical information in crystal field parameters as discussed by the authors, and its success in the analysis of lanthanide spectra has paralleled by the success of the related angular overlap model in analysis of d-electron spectra.
Abstract: The superposition model was originally developed to separate the geometrical and physical information in crystal field parameters. Its success in the analysis of lanthanide spectra has been paralleled by the success of the related angular overlap model in the analysis of d-electron spectra. The basic ideas, method of application and reliability of the superposition model are discussed and its relationship with the angular overlap model is clarified. Developments described are the application of the superposition model to the ground (L=0) multiplet splittings of d5 and f7 ions, orbit-lattice interactions, transition intensities and correlation crystal fields. Special attention is paid to work which has been claimed to support or disprove the postulates of the model.
1,007 citations
TL;DR: In this article, the long-term free evolution of wave packets formed by highly excited states of quantum systems performing regular periodic motion in the classical limit is considered, and the universal asymptotic scenario of the wave function temporal development is discovered which provides the generation of a certain sequence of initial wave packet fractional revivals.
412 citations
TL;DR: In this paper, a wave superposition integral is proposed for computing the acoustic fields of arbitrarily shaped radiators, which is shown to be equivalent to the Helmholtz integral, based on the idea that the combined fields of an array of sources interior to a radiator can be made to reproduce a velocity prescribed on the surface of the radiator.
Abstract: A method for computing the acoustic fields of arbitrarily shaped radiators is described that uses the principle of wave superposition. The superposition integral, which is shown to be equivalent to the Helmholtz integral, is based on the idea that the combined fields of an array of sources interior to a radiator can be made to reproduce a velocity prescribed on the surface of the radiator. The strengths of the sources that produce this condition can, in turn, be used to compute the corresponding surface pressures. The results of several numerical experiments are presented that demonstrate the simplicity of the method. Also, the advantages that the superposition method has over the more commonly used boundary‐element methods are discussed. These include simplicity of generating the matrix elements used in the numerical formulation and improved accuracy and speed, the latter two being due to the avoidance of uniqueness and singularity problems inherent in the boundary‐element formulation.
363 citations
Book•
01 Oct 1989
TL;DR: In this article, a single-degree-of-freedom (SDF) dynamic system is considered, and the effect of different degrees of freedom on the dynamics of the system is investigated.
Abstract: TABLE OF CONTENTS PREFACE 1 INTRODUCTION 1.1 Objectives of the Study of Structural Dynamics 1.2 Importance of Vibration Analysis 1.3 Nature of Exciting Forces 1.4 Mathematical Modeling of Dynamic Systems 1.5 Systems of Units 1.6 Organization of the Text PART I 2 FORMULATION OF THE EQUATIONS OF MOTION: SINGLE-DEGREE-OF-FREEDOM SYSTEMS 2.1 Introduction 2.2 Inertia Forces 2.3 Resultants of Inertia Forces on a Rigid Body 2.4 Spring Forces 2.5 Damping Forces 2.6 Principle of Virtual Displacement 2.7 Formulation of the Equations of Motion 2.8 Modeling of Multi Degree-of-Freedom Discrete Parameter System 2.9 Effect of Gravity Load 2.10 Axial Force Effect 2.11 Effect of Support Motion 3 FORMULATION OF THE EQUATIONS OF MOTION: MULTI-DEGREE-OF-FREEDOM SYSTEMS 3.1 Introduction 3.2 Principal Forces in Multi Degree-of-freedom Dynamic System 3.3 Formulation of the Equations of Motion 3.4 Transformation of Coordinates 3.5 Static Condensation of Stiffness matrix 3.6 Application of Ritz Method to Discrete Systems 4 PRINCIPLES OF ANALYTICAL MECHANICS 4.1 Introduction 4.2 Generalized coordinates 4.3 Constraints 4.4 Virtual Work 4.5 Generalized Forces 4.6 Conservative Forces and Potential Energy 4.7 Work Function 4.8 Lagrangian Multipliers 4.9 Virtual Work Equation For Dynamical Systems 4.10 Hamilton's Equation 4.11 Lagrange's Equation 4.12 Constraint Conditions and Lagrangian Multipliers 4.13 Lagrange's Equations for Discrete Multi-Degree-of-Freedom Systems 4.14 Rayleigh's Dissipation Function PART II 5 FREE VIBRATION RESPONSE: SINGLE-DEGREE-OF-FREEDOM SYSTEM 5.1 Introduction 5.2 Undamped Free Vibration 5.3 Free Vibrations with Viscous Damping 5.4 Damped Free vibration with Hysteretic Damping 5.5 Damped Free vibration with Coulomb Damping 6 FORCED HARMONIC VIBRATIONS: SINGLE-DEGREE-OF-FREEDOM SYSTEM 6.1 Introduction 6.2 Procedures for the Solution of Forced Vibration Equation 6.3 Undamped Harmonic Vibration 6.4 Resonant Response of an Undamped System 6.5 Damped Harmonic Vibration 6.6 Complex Frequency Response 6.7 Resonant Response of a Damped System 6.8 Rotating Unbalanced Force 6.9 Transmitted Motion due to Support Movement 6.10 Transmissibility and Vibration Isolation 6.11 Vibration Measuring Instruments 6.12 Energy Dissipated in Viscous Damping 6.13 Hysteretic Damping 6.14 Complex Stiffness 6.15 Coulomb Damping 6.16 Measurement of Damping 7 RESPONSE TO GENERAL DYNAMIC LOADING AND TRANSIENT RESPONSE 7.1 Introduction 7.2 Response to an Impulsive force 7.3 Response to General Dynamic Loading 7.4 Response to a Step Function Load 7.5 Response to a Ramp Function Load 7.6 Response to a Step Function Load With Rise Time 7.7 Response to Shock Loading 7.8 Response to a Ground Motion Pulse 7.9 Analysis of Response by the Phase Plane Diagram 8 ANALYSIS OF SINGLE-DEGREE-OF-FREEDOM SYSTEMS: APPROXIMATE AND NUMERICAL METHODS 8.1 Introduction 8.2 Conservation of Energy 8.3 Application of Rayleigh Method to Multi Degree of Freedom Systems 8.4 Improved Rayleigh Method 8.5 Selection of an Appropriate Vibration Shape 8.6 Systems with Distributed Mass and Stiffness: Analysis of Internal Forces 8.7 Numerical Evaluation of Duhamel's Integral 8.8 Direct Integration of the Equations of Motion 8.9 Integration Based on Piece-wise Linear Representation of the Excitation 8.10 Derivation of General Formulae 8.11 Constant Acceleration Method 8.12 Newmark's beta Method 8.13 Wilson-theta Method 8.14 Methods Based on Difference Expressions 8.15 Errors involved in Numerical Integration 8.16 Stability of the Integration Method 8.17 Selection of a Numerical Integration Method 8.18 Selection of Time Step 9 ANALYSIS OF RESPONSE IN THE FREQUENCY DOMAIN 9.1 Transform Methods of Analysis 9.2 Fourier Series Representation of a Periodic Function 9.3 Response to a Periodically Applied Load 9.4 Exponential Form of Fourier Series 9.5 Complex Frequency Response Function 9.6 Fourier Integral Representation of a Nonperiodic Load 9.7 Response to a Nonperiodic Load 9.8 Convolution Integral and Convolution Theorem 9.9 Discrete Fourier Transform 9.10 Discrete Convolution and Discrete Convolution Theorem 9.11 Comparison of Continuous and Discrete Fourier Transforms 9.12 Application of Discrete Inverse Transform 9.13 Comparison Between Continuous and Discrete Convolution 9.14 Discrete Convolution of an Infnite and a Finite duration Waveform 9.15 Corrective Response Superposition Methods 9.16 Exponential Window Method 9.17 The Fast Fourier Transform 9.18 Theoretical Background to Fast Fourier Transform 9.19 Computing Speed of FFT Convolution 9.16 Exponential Window Method 9.17 The Fast Fourier Transform 9.18 Theoretical Background to Fast Fourier Transform 9.19 Computing Speed of FFT Convolution PART III 10 FREE VIBRATION RESPONSE: MULTI-DEGREE-OF-FREEDOM SYSTEM 10.1 Introduction 10.2 Standard Eigenvalue Problem 10.3 Linearized Eigenvalue Problem and its Properties 10.4 Expansion Theorem 10.5 Rayleigh Quotient 10.6 Solution of the Undamped Free-Vibration Problem 10.7 Mode Superposition Analysis of Free-Vibration Response 10.8 Solution of the Damped Free-Vibration Problem 10.9 Additional Orthogonality Conditions 10.10 Damping Orthogonality 11 NUMERICAL SOLUTION OF THE EIGENPROBLEM 11.1 Introduction 11.2 Properties of Standard Eigenvalues and Eigenvectors 11.3 Transformation of a Linearized Eigenvalue Problem to the Standard Form 11.4 Transformation Methods 11.5 Iteration Methods 11.6 Determinant Search Method 11.7 Numerical Solution of Complex Eigenvalue Problem 11.8 Semi-definite or Unrestrained Systems 11.9 Selection of a Method for the Determination of Eigenvalues 12 FORCED DYNAMIC RESPONSE: MULTI-DEGREE-OF-FREEDOM SYSTEMS 12.1 Introduction 12.2 Normal Coordinate Transformation 12.3 Summary of Mode Superposition Method 12.4 Complex Frequency Response 12.5 Vibration Absorbers 12.6 Effect of Support Excitation 12.7 Forced Vibration of Unrestrained System 13 ANALYSIS OF MULTI-DEGREE-OF-FREEDOM SYSTEMS: APPROXIMATE AND NUMERICAL METHODS 13.1 Introduction 13.2 Rayleigh-Ritz Method 13.3 Application of Ritz Method to Forced Vibration Response 13.4 Direct Integration of the Equations of Motion 13.5 Analysis in the Frequency Domain PART IV 14 FORMULATION OF THE EQUATIONS OF MOTION: CONTINUOUS SYSTEMS 14.1 Introduction 14.2 Transverse Vibrations of a Beam 14.3 Transverse Vibrations of a Beam: Variational Formulation 14.4 Effect of Damping Resistance on Transverse Vibrations of a Beam 14.5 Effect of Shear Deformation and Rotatory Inertia on the Flexural Vibrations of a Beam 14.6 Axial Vibrations of a Bar 14.7 Torsional Vibrations of a Bar 14.8 Transverse Vibrations of a String 14.9 Transverse Vibration of a Shear Beam 14.10 Transverse Vibrations of a Beam Excited by Support Motion 14.11 Effect of Axial Force on Transverse Vibrations of a Beam 15 CONTINUOUS SYSTEMS: FREE VIBRATION RESPONSE 15.1 Introduction 15.2 Eigenvalue Problem for the Transverse Vibrations of a Beam 15.3 General Eigenvalue Problem for a Continuous System 15.4 Expansion Theorem 15.5 Frequencies and Mode Shapes for Lateral Vibrations of a Beam 15.6 Effect of Shear Deformation and Rotatory Inertia on the Frequencies of Flexural Vibrations 15.7 Frequencies and Mode Shapes for the Axial Vibrations of a Bar 15.8 Frequencies and Mode Shapes for the Transverse Vibration of a String 15.9 Boundary Conditions Containing the 15.10 Free-Vibration Response of a Continuous System 15.11 Undamped Free Transverse Vibrations of a Beam 15.12 Damped Free Transverse Vibrations of a Beam 16 CONTINUOUS SYSTEMS: FORCED-VIBRATION RESPONSE 16.1 Introduction 16.2 Normal Coordinate Transformation: General Case of an Undamped System 16.3 Forced Lateral Vibration of a Beam 16.4 Transverse Vibrations of a Beam Under Traveling Load 16.5 Forced Axial Vibrations of a Uniform Bar 16.6 Normal Coordinate Transformation, Damped Case 17 WAVE PROPAGATION ANALYSIS 17.1 Introduction 17.2 The Phenomenon of Wave Propagation 17.3 Harmonic Waves 17.4 One Dimensional Wave Equation and its Solution 17.5 Propagation of Waves in Systems of Finite Extent 17.6 Reection and Refraction of Waves at a Discontinuity in the System Properties 17.7 Characteristics of the Wave Equation 17.8 Wave Dispersion PART V 18 FINITE ELEMENT METHOD 18.1 Introduction 18.2 Formulation of the Finite Element Equations 18.3 Selection of Shape Functions 18.4 Advantages of the Finite Element Method 18.5 Element Shapes 18.6 One-dimensional Bar Element 18.7 Flexural Vibrations of a Beam 18.8 Stress-strain Relationship for a Continuum 18.9 Triangular Element in Plane Stress and Plane Strain 18.10 Natural Coordinates 19 COMPONENT MODE SYNTHESIS 19.1 Introduction 19.2 Fixed Interface Methods 19.3 Free Interface Method 19.4 Hybrid Method 20 ANALYSIS OF NONLINEAR RESPONSE 20.1 Introduction 20.2 Single-degree-of-freedom System 20.3 Errors involved in Numerical Integration of Nonlinear Systems 20.4 Multiple Degree-of-freedom System ANSWERS TO SELECTED PROBLEMS INDEX
248 citations
TL;DR: In this article, the authors considered each electron as a superposition of fictitious fractionally charged particles, and with the help of a natural ansatz, a systematic identification and characterization of the incompressible quantum Hall states at noninteger filling factors was performed.
Abstract: Considering each electron as a superposition of fictitious fractionally charged particles allows, with the help of a natural ansatz, a systematic identification and characterization of the incompressible quantum Hall states at noninteger filling factors. Explicit Laughlin-type wave functions are obtained for all the incompressible states and their quasiparticles. The order of stability of the various states predicted on the basis of physically plausible rules is in agreement with experiments. Although in principle all rational fractions are observable, these rules imply that the even-denominator fractions are in general much less stable than the odd-denominator ones.
215 citations
TL;DR: Numerical computations are based on the fast-Fourier-transform algorithm, and the practicality of this method is shown with several examples.
Abstract: Fourier decomposition of a given amplitude distribution into plane waves and the subsequent superposition of these waves after propagation is a powerful yet simple approach to diffraction problems. Many vector diffraction problems can be formulated in this way, and the classical results are usually the consequence of a stationary-phase approximation to the resulting integrals. For situations in which the approximation does not apply, a factorization technique is developed that substantially reduces the required computational resources. Numerical computations are based on the fast-Fourier-transform algorithm, and the practicality of this method is shown with several examples.
160 citations
TL;DR: In this paper, the authors considered the problem of converting the conventional causal impulsive-source Green's-function propagator into a noncausal propagator, which must be defined as an analytic signal because, owing to causality, the analytic continuation into the complex domain cannot be performed by direct substitution.
Abstract: Pulsed beams (PB’s) are time-dependent wave fields that are confined in beamlike fashion in transverse planes perpendicular to the propagation axis, whereas confinement along the axis is due to temporal windowing. Because they have these properties, pulsed beams are useful wave objects for generating and synthesizing highly focused transient fields. The PB problem is addressed here within the context of fundamental Green’s-function propagators for the time-dependent field equations. In a departure from known results in the frequency domain, by which beam solutions can be generated from point-source solutions by displacing the source coordinate location into a complex coordinate space, the complex extension is applied here as well to the source initiation time. This procedure converts the conventional causal impulsive-source Green’s-function propagator into a noncausal PB propagator, which must be defined as an analytic signal because, owing to causality, the analytic continuation into the complex domain cannot be performed by direct substitution. This being done, PB’s can be manipulated as conventional Green’s functions. Some previous results obtained by similar methods are viewed here from a sharper perspective, and new results, both analytical and numerical, are presented that grant basic insight into the PB behavior, including the ability to excite these fields by finite-causal-aperture-source distributions. Besides the basic (analytic Green’s-function) PB, examples include PB’s with frequency spectra of special interest. Particular attention is paid to the PB synthesis of focus-wave modes, which are source-free solutions of the time-dependent wave equation, and to the compact PB formulation of wave fields synthesized by focus-wave-mode spectral superposition.
134 citations
TL;DR: In this article, a new method is described for the analysis of long and complicated structures which are composed of spatially periodic units or sections of spatically periodic units, where the response of such a structure to external excitations is treated as a superposition of wave motions, with account taken of the effects of wave reflection due to change in the construction pattern along the structure and boundary conditions.
91 citations
TL;DR: In this article, the authors present a model which predicts seismological complexity even with no complexity in geometry or heterogeneity in material properties, using a Dieterich-type rate and state variable friction law at the planar interface of two infinitely long massless elastic slabs.
Abstract: We present a model which predicts seismological complexity even with no complexity in geometry or heterogeneity in material properties Fault slip is numerically modeled using a Dieterich-type rate and state variable friction law at the planar interface of two infinitely long massless elastic slabs A constant velocity boundary condition is imposed a distance H from each interface No geometrical, frictional, elastic, or remote loading variations are allowed in the direction parallel to the plane of the fault In the numerical solution a periodic boundary condition is imposed, and the fault surface is divided into N subregions each of which is represented by a mathematical point At these N points, all of which are mutually coupled by discretized two-dimensional elasticity solutions, the friction law differential equations are numerically solved The character of the solutions depends on model parameter values and initial conditions Solutions are found that are periodic, quasi-periodic, or aperiodic in time; and that are spatially homogeneous for all time, nearly homogeneous except during fast slip events, or essentially inhomogeneous for all time For given parameter values the solutions have a qualitative character which is nearly independent of initial conditions At any instant in time these solutions ultimately appear roughly as some superposition of those spatial sine waves which are unstable in a linearized calculation When spatially complex, the solutions can simultaneously exhibit regions that have steady sliding, large slip rates, and local propagating creep events Special initial conditions can generate other solutions such as steady propagating creep waves that span the whole fault The variety of simulated slip motions and long-term patterns of slip predicted by this spatially homogeneous nonlinear dynamical model suggests a possible role for dynamics, and not just complex geological structure, as a generator of temporal and spatial complexity in seismic phenomena
89 citations
TL;DR: In this paper, the modal expansion of the wave function in the discontinuity region based on the superposition principle together with a mode-matching technique was used to investigate the transmission characteristics of semiconductor quantum wire structures with discontinuities.
Abstract: We have used the modal expansion of the wave function in the discontinuity region based on the superposition principle together with a mode‐matching technique to investigate the transmission characteristics of semiconductor quantum wire structures with discontinuities. Our calculations compare quite well with published results for the theoretical transmission coefficient and experimental conductance of a T‐stub and split‐gate geometry, respectively. We apply this technique to analyze the effect of right‐angle bends in narrow quantum wires which show strong resonant behavior due to the presence of discontinuities in this geometry.
88 citations
TL;DR: In this article, a robust and efficient numerical algorithm is presented, which solves the coupled modal equations by iteration, and it is shown that the numerical integration algorithm always converges for proportional damping and for loading that varies linearly within an arbitrary time interval.
01 Mar 1989
TL;DR: The superposition principle of the wave function is defined in this article, which is the fundamental principle of quantum mechanics that the system of states forms a linear manifold, in which a unitary scalar product is defined.
Abstract: It is perhaps the most fundamental principle of Quantum Mechanics that the system of states forms a linear manifold,1 in which a unitary scalar product is defined.2 The states are generally represented by wave functions3 in such a way that φ and constant multiples of φ represent the same physical state. It is possible, therefore, to normalize the wave function, i.e., to multiply it by a constant factor such that its scalar product with itself becomes 1. Then, only a constant factor of modulus 1, the so-called phase, will be left undetermined in the wave function. The linear character of the wave function is called the superposition principle. The square of the modulus of the unitary scalar product (ψ,Φ) of two normalized wave functions ψ and Φ is called the transition probability from the state ψ into Φ, or conversely. This is supposed to give the probability that an experiment performed on a system in the state Φ, to see whether or not the state is ψ, gives the result that it is ψ. If there are two or more different experiments to decide this (e.g., essentially the same experiment, performed at different times) they are all supposed to give the same result, i.e., the transition probability has an invariant physical sense.
TL;DR: In this paper, the authors considered the problem of evaluating the scattered field at a finite distance from the edge of an impedance wedge which is illuminated by a line source, and derived an exact expression for the diffracted field and the surface wave contributions.
Abstract: The canonical problem of evaluating the scattered field at a finite distance from the edge of an impedance wedge which is illuminated by a line source is considered. The presentation of the results is divided into two parts. In this first part, reciprocity and superposition of plane wave spectra are applied to the left far-field response of the wedge to a plane wave, to obtain exact expression for the diffracted field and the surface wave contributions. In addition, a high-frequency solution is given for the diffracted field contribution. Its expression, derived via a rigorous asymptotic procedure, has the same structure as that of the uniform geometrical theory of diffraction (UTD) solution for the field diffracted by a perfectly conducting wedge. This solution for the diffracted field explicitly exhibits reciprocity with respect to the direction of incidence and scattering. >
TL;DR: It is shown that, for a certain choice of this phase, ``coherent trapping'' occurs in two-level atoms, and in the case of spectra, for the same choice of the phase, instead of a three-peaked symmetric spectrum, the authors have an asymmetric two- peaked spectrum.
Abstract: Considering a system consisting of a two-level atom, initially prepared in a coherent superposition of upper and lower levels, interacting with a coherent state of the field, we show that the dynamics of the atom as well as the spectrum of the field are sensitive to the relative phase between the atomic dipole and the cavity field. It is shown that, for a certain choice of this phase, ``coherent trapping'' occurs in two-level atoms. In the case of spectra, for the same choice of the phase, instead of a three-peaked symmetric spectrum, we have an asymmetric two-peaked spectrum.
TL;DR: In this article, a modified integral equation is obtained by the linear superposition of the classical Green equation and its normal derivative with respect to the field point, which removes the effects of all irregular frequencies in boundary-integral equations governing the interaction of regular waves with floating bodies of general geometry.
Abstract: The paper presents a method that removes the effects of all irregular frequencies in boundary-integral equations governing the interaction of regular waves with floating bodies of general geometry. A modified integral equation is obtained by the linear superposition of the classical Green equation and its normal derivative with respect to the field point.
TL;DR: In this paper, an aluminum hemicylindrical sample has been irradiated with an array of laser lines, with each line acting as a source of acoustic waves, and detection of the generated ultrasonic waves was performed using both a wide-band stabilized Michelson interferometer and a 20 MHz piezoelectric transducer.
TL;DR: In this article, the shape of a two-dimensional time-harmonic acoustic scatterer with Dirichlet boundary conditions from the knowledge of the far-field pattern is considered.
Abstract: A numerical method, developed by Kirsch and Kress (1988), for the determination of the shape of a two-dimensional time-harmonic acoustic scatterer with Dirichlet boundary conditions from the knowledge of the far-field pattern is considered. For the full-aperture problem with n incoming waves new convergence results are found. The corresponding computational part deals with the reduction of the computing time by a superposition of n incoming waves to a few 'total waves'. The method is extended to the limited-aperture problem; especially an estimation of its ill-posedness is given. Some numerical examples are presented with a measuring angle of 90 degrees and 180 degrees where it is shown that this method produces acceptable results for low wavenumbers.
TL;DR: The resulting photoionization signal, as a function of the delay between pulses, shows the classical periodicity of the classical Kepler orbit.
Abstract: An experiment is described in which a coherent superposition of the Rydberg states of atomic potassium is excited by a short optical pulse. The coherent superposition forms a wave packet localized in the radial coordinate. The radial motion of the wave packet is periodic with the period of the classical Kepler orbit. The time evolution is probed by a second short pulse. The resulting photoionization signal, as a function of the delay between pulses, shows the classical periodicity.
TL;DR: In this article, a general method that produces analytic distribution functions for gravitating systems is presented, where a typical model fits photometric and kinematical data and is a superposition of components.
Abstract: This paper presents a general method that produces analytic distribution functions for gravitating systems. A typical model fits photometric and kinematical data and is a superposition of components. The coefficients of the linear combination are determined by a quadratic programming technique. An explicit example for systems with spherical symmetry is given and the method is applied to the Coma cluster of galaxies. Statistically significant fits are obtained using only a small number of components. This technique is also a dynamical mass estimator. 49 refs.
TL;DR: In this article, a modified superposition and dan estimation technique for the parameters of this model is presented to model the magnetic recording channel including nonlinearities, and experiments were conducted on a 1300 Oe, 3*10/sup -3/ emu/cm/sup 2/ thin-film disk using a 3380E 18-turn thinfilm head at 70 mA pk-pk current and 23 m/s velocity.
Abstract: A channel model called modified superposition and dan estimation technique for the parameters of this model are presented to model the magnetic recording channel including nonlinearities. It is shown that this model and estimation technique can describe the magnetic recording channel including nonlinearities much more accurately than simple superposition with isolated transient response. Computer software was developed to extract automatically the three nonlinear parameters from a single measured waveform. As an example of the usefulness of this estimation technique, experiments were conducted on a 1300 Oe, 3*10/sup -3/ emu/cm/sup 2/ thin-film disk using a 3380E 18-turn thin-film head at 70 mA pk-pk current and 23 m/s velocity. >
TL;DR: In this paper, a mathematical model based on a detailed representation of soil impedance, an approximate identification of surface waves and a deconvolution of body waves in P and SV contributions is presented.
Abstract: The surface motion during an earthquake is different from point to point depending on the propagation properties of the seismic waves. Rocking and torsion are thus present in the free field, in proportion to the spatial derivatives of the surface motion with respect to a given direction. These derivatives are inversely proportional to the apparent wave velocity in that direction, so the smaller the wave apparent velocity, the more important its contribution to the rotations. In this respect, a marked contribution to surface rotations from surface waves is expected.
A mathematical model is presented, based on a detailed representation of soil impedance, an approximate identification of surface waves and a deconvolution of body waves in P and SV contributions. Through this model the surface motion obtained from the records of strong-motion accelerometers can be expressed as a superposition of plane waves of known wavelengths.
Rocking response spectra are computed and results are compared with previously published spectra. A sensitivity analysis is performed on some parameters of the model.
TL;DR: In this paper, the authors derived exact Levy-type solutions for the quasi-static problem and for the dynamic case, the governing partial differential equation is first reduced to a system of ordinary differential equations in time by means of the Galerkin method.
Abstract: The dynamic response of homogeneous, orthotropic plates having two parallel, simply supported edges and exposed to rapid surface heating is examined. Uncoupled, thin-plate theory is used to determine the inducedflexural vibrations. The solution is obtained as the superposition of two displacement fields, representing the quasi-static and the dynamic behaviors. An exact Levy-type solution is derived for the quasi-static problem. For the dynamic case, the governing partial differential equation is first reduced to a system of ordinary differential equations in time by means of the Galerkin method. For special situations in which the latter equations are uncoupled (e.g., plates having all edges simply supported), the solution is obtained using the Laplace transform. For cases involving more general support conditions, the equations are integrated numerically using the Runge-Kutta-Gill technique. Numerical results are presented for both isotropic and orthotropic plates having various combinations of edge cond...
TL;DR: The nonlinear oscillator generates a particular superposition of two squeezed vacuum states and the properties of this superposition state are discussed and contrasted with the other superposition states.
Abstract: The superposition of two squeezed vacuum states is analyzed by studying the photon-number probability distribution and the quadrature-phase-eigenstate marginal distributions. Interference fringes in the distributions are observed for some superposition states. The nonlinear oscillator generates a particular superposition of two squeezed vacuum states and the properties of this superposition state are discussed and contrasted with the other superposition states.
TL;DR: In this paper, the authors treat thermal-wave field diffraction as the extreme near-field approximation of a three-dimensional superposition integral that includes the generating optical aperture function.
Abstract: Thermal-wave field diffraction has been treated as the extreme near-field approximation of a three-dimensional superposition integral that includes the generating optical aperture function. This formalism is quite general and is convenient for applications with many experimental diffracting apertures. Specific examples of useful photothermal excitation apertures have been treated explicitly. These include the spatial impulse function, a Gaussian laser beam, a circular aperture, and an expression for the interference field generated by two Gaussian laser beams.
TL;DR: In this paper, the authors considered quantum-mechanically partially polarized light propagating through a Kerr-like medium and formulated the theory in terms of an effective Hamiltonian which is quartic with respect to the operators for two orthogonally polarized modes.
Abstract: We consider quantum-mechanically partially polarized light propagating through a Kerr-like medium. Using the usual form of the induced polarization P=A(E.E)E+B(E.E)E, the theory is formulated in terms of an effective Hamiltonian which is quartic in terms of the operators for two orthogonally polarized modes. Exact solutions in closed form for the Heisenberg equations of motion are obtained. These solutions are used to evaluate the physical behavior of various observables as the field propagates through a nonlinear medium. We also present explicit results for the time evolution of the input coherent and Fock states of the field. We show the generation of states that are macroscopic superposition of coherent states. We also find that if the input field is completely polarized, then due to quantum effects the output field becomes partially polarized. This is in contrast to the classical prediction and can have an important bearing on questions like topological phases of light propagating through a nonlinear medium. Numerical results for the energy in each mode, the correlation between two modes, and the higher-order correlations are presented. The input photon statistics is found to make a considerable difference in the dynamics.
TL;DR: In this paper, the authors considered a viscous fluid film flowing down a vertical plane wall and showed that there exist various kinds of solitary waves characterized by the number of their humps.
Abstract: Waves on a viscous fluid film flowing down a vertical plane wall are studied theoretically. Stationary solutions are sought for a nonlinear equation that describes the variation of the free surface, and a solitary wave is found as an exact solution. The corrugation of the surface concentrates in a small portion comparable with the film thickness and propagates with a constant speed. There exist various kinds of solitary waves characterized by the number of their humps. To compare these solitary waves with the so‐called single waves, solitary waves of the same kind are arranged at regular intervals on the laminar flow. When the nonlinearity is comparatively weak, individual solitary waves are so close to one another that this procedure of finding the periodic solution has to be replaced by the method of expanding the free surface as a Fourier sum and determining its coefficients. The periodic solution of arbitrary wavelength is obtained from these two procedures. There are as many kinds of periodic solutions as those of solitary waves, and a selection rule is proposed as to the problem of which kind of periodic solutions actually appears. Each kind of our periodic solutions is compared with the nearly sinusoidal wave as well as the single wave. It is seen that the wave that was observed experimentally is nothing but a superposition of solitary waves that are mistaken for the solitary waves with only one principal hump. Further, it is expected that solitary waves are still excited, even though there are no artificial perturbations.
TL;DR: In this paper, a new form of the general solution of the initial value problem for colliding gravitational plane waves with collinear polarization is obtained, which is expressed as a linear superposition ∫da(σ)ω(u,v,σ) of a one-parameter family of basic solutions of the form ω(U,V,σ), where u and v are arbitrary null coordinates and σ is the parameter.
Abstract: A new form of the general solution of the initial value problem for colliding gravitational plane waves with collinear polarization is obtained. The solution of the linear hyperbolic field equation for ψ(u,v) is expressed as a linear superposition ∫da g(σ)ω(u,v,σ) of a one‐parameter family of basic solutions of the form ω(u,v,σ)=ω1(u,σ)ω2(v,σ), where u and v are arbitrary null coordinates and σ is the parameter. The coefficients g(σ) in this superposition are expressed in terms of the initial data by using a generalization of an integral transform obtained by Abel in his solution of a tautochrone problem of classical particle mechanics.
TL;DR: In this article, the orthogonal superposition of small and large amplitude oscillations upon steady shear flow of elastic fluids has been considered and the theoretical results, obtained by numerical methods, are based on the Leonov viscoelastic constitutive equation.
Abstract: The orthogonal superposition of small and large amplitude oscillations upon steady shear flow of elastic fluids has been considered. Theoretical results, obtained by numerical methods, are based on the Leonov viscoelastic constitutive equation. Steady-state components, amplitudes and phase angles of the oscillatory components of the shear stress, the first and second normal stress differences as functions of shear rate, deformation amplitude and frequency have been calculated. These oscillatory components include the first and third harmonic of the shear stresses and the second harmonic of the normal stresses. In the case of small amplitude superposition, the effect of the steady shear flow upon the frequency-dependent storage modulus and dynamic viscosity has been determined and compared with experimental data available in literature for polymeric solutions. The predicted results have been found to be in fair agreement with the experimental data at low shear rates and only in qualitative agreement at high shear rates and low frequencies. A comparison of the present theoretical results has also been made with the predictions of other theories. In the case of large amplitude superposition, the effect of oscillations upon the steady shear flow characteristics has been determined, indicating that the orthogonal superposition has less influence on the steady state shear stresses and the first difference of normal stresses than the parallel superposition. However, in the orthogonal superposition a more pronounced influence has been observed for the second difference of normal stresses.
TL;DR: In this paper, a method for determining the vibratory response of continuous systems subjected to distributed excitation forces is presented, which is exact and yields closed from expressions for vibratory displacements, in contrast with the well known eigenfunction superposition method which requires expressing the distributed forcing functions and the displacement response functions as infinite sums of free vibration eigenfunctions.
TL;DR: Existing theory is extended to multichannel systems by applying results from point process theory to derive some distributional properties of the various types of sojourn time that occur when a given number of channels are open in a system containing a specified number of independent channels in equilibrium.
Abstract: Membrane patches usually contain several ion channels of a given type. However, most of the stochastic modelling on which data analysis (in particular, estimation of kinetic constants) is currently based, relates to a single channel rather than to multiple channels. Attempts to circumvent this problem experimentally by recording under conditions where channel activity is low are restrictive and can introduce bias; moreover, possibly important information on how multichannel systems behave will be missed. We have extended existing theory to multichannel systems by applying results from point process theory to derive some distributional properties of the various types of sojourn time that occur when a given number of channels are open in a system containing a specified number of independent channels in equilibrium. Separate development of properties of a single channel and the superposition of several such independent channels simplifies the presentation of known results and extensions. To illustrate the general theory, particular attention is given to the types of sojourn time that occur in a two channel system; detailed expressions are presented for a selection of models, both Markov and non-Markov.