Showing papers on "Superposition principle published in 1995"

Handbook of radiation and scattering of waves

Abstract: Acoustic Waves in Fluids: Cartesian Tensors and Their Properties Integral-Transformation Methods Basic Equations of the Theory of Acoustic Waves in Fluids The Principle of Superposition and Its Application to Acoustic Wave Fields in Configurations with Geometrical Symmetry The Acoustic Wave Equations, Constitutive Relations, and Boundary Conditions in the Time Laplace-Transform Domain (Complex Frequency Domain) Acoustic Radiation from Sources in an Unbounded, Homogeneous, Isotropic Fluid Acoustic Reciprocity Theorems and Their Applications Plane-Wave Scattering by an Object in an Unbounded, Homogeneous, Isotropic, Lossless Embedding. Elastic Waves in Solids: Cartesian Tensors and Their Properties Integral-Transformation Methods Basic Equations of the Theory of Elastic Waves in Solids The Principle of Superposition and Its Application to Electromagnetic Fields in Configurations with Geometrical Symmetry The Elastic Wave Equations, Constitutive Relations, and Boundary Conditions in the Time Laplace-Transform Domain (Complex Frequency Domain) Elastodynamic Radiation from Sources in an Unbounded, Homogeneous, Isotropic Solid Plane Elastic Waves in Homogeneous Solids Elastodynamic Reciprocity Theorems and Their Applications Plane-Wave Scattering by an Object in an Unbounded, Homogeneous, Isotropic, Lossless Embedding. Electromagnetic Waves: Cartesian Tensors and Their Properties Integral-Transformation Methods The Electromagnetic Constitutive Relations The Electromagnetic Boundary Conditions Exchange of Energy in the Electromagnetic Field Vector Potentials, Point-Source Solutions, and Green's Function in the Theory of Electromagnetic Radiation from Sources The Principle of Superposition and Its Application to Electromagnetic Fields in Configuration with Geometrical Symmetry The Electromagnetic Field Equations, Constitutive Relations, and Boundary Conditions in the Time Laplace-Transform Domain (Complex Frequency Domain) Complex Frequency Domain Vector Potentials Point-Source Solutions and Green's Functions in the Theory of Electromagnetic Radiation from Sources Electromagnetic Radiation from Sources in an Unbounded, Homogeneous, Isotropic Medium Plane Electromagnetic Waves in Homogeneous Media Electromagnetic Reciprocity Theorems and Their Applications Plane-Wave Scattering by an Object in an Unbounded, Homogeneous, Isotropic, Lossless Embedding Interference and Shielding of Electromagnetic Systems Accessible via Low-Frequency Terminations Electromagnetic Compatibility (EMC).

Numerical solution of the hypernetted chain equation for a solute of arbitrary geometry in three dimensions

Abstract: The average solvent distribution near complex solid substrates of arbitrary geometry is calculated by solving the hypernetted chain (HNC) integral equation on a three‐dimensional discrete cubic grid. A numerical fast Fourier transform in three dimensions is used to calculate the spatial convolutions appearing in the HNC equation. The approach is illustrated by calculating the average solvent density in the neighborhood of small clusters of Lennard‐Jones particles and inside a periodic array of cavities representing a simplified model of a porous material such as a zeolite. Molecular dynamics simulations are performed to test the results obtained from the integral equation. It is generally observed that the average solvent density is described accurately by the integral equation. The results are compared with those obtained from a superposition approximation in terms of radial pair correlation functions, and the reference interaction site model (RISM) integral equations. The superposition approximation sig...

Constitutive model and wave equations for linear, viscoelastic, anisotropic media

Abstract: Rocks are far from being isotropic and elastic. Such simplifications in modeling the seismic response of real geological structures may lead to misinterpretations, or even worse, to overlooking useful information. It is useless to develop highly accurate modeling algorithms or to naively use amplitude information in inversion processes if the stress-strain relations are based on simplified rheologies. Thus, an accurate description of wave propagation requires a rheology that accounts for the anisotropic and anelastic behavior of rocks. This work presents a new constitutive relation and the corresponding time-domain wave equation to model wave propagation in inhomogeneous anisotropic and dissipative media. The rheological equation includes the generalized Hooke’s law and Boltzmann’s superposition principle to account for anelasticity. The attenuation properties in different directions, associated with the principal axes of the medium, are controlled by four relaxation functions of viscoelastic type. A dissipation model that is consistent with rock properties is the general standard linear solid. This is based on a spectrum of relaxation mechanisms and is suitable for wavefield calculations in the time domain. One relaxation function describes the anelastic properties of the quasi-dilatational mode and the other three model the anelastic properties of the shear modes. The convolutional relations are avoided by introducing memory variables, six for each dissipation mechanism in the 3-D case, two for the generalized SH-wave equation, and three for the qP - qSVwave equation. Two-dimensional wave equations apply to monoclinic and higher symmetries. A plane analysis derives expressions for the phase velocity, slowness, attenuation factor, quality factor and energy velocity (wavefront) for homogeneous viscoelastic waves. The analysis shows that the directional properties of the attenuation strongly depend on the values of the elasticities. In addition, the displacement formulation of the 3-D wave equation is solved in the time domain by a spectral technique based on the Fourier method. The examples show simulations in a transversely-isotropic clayshale and phenolic (orthorhombic symmetry).

Role of initial coherence in the generation of harmonics and sidebands from a strongly driven two-level atom.

Abstract: We investigate the coherent and incoherent contributions of the scattering spectrum of strongly driven two-level atoms as a function of the initial preparation of the atomic system. The initial ``phasing'' of the coherent superposition of the excited and ground states is shown to influence strongly the generation of both harmonics and hyper-Raman lines. In particular, we point out conditions under which harmonic generation can be inhibited at the expense of the hyper-Raman lines. Our numerical findings are supported by approximate analytical evaluation in the dressed state picture.

Quantum-state engineering via discrete coherent-state superpositions

Abstract: A representation of a Fock state \ensuremath{\Vert}n〉 is given by a superposition of n+1 coherent states. These discrete coherent-state superpositions provide experimental possibilities for generating an arbitrary quantum state of a single-mode electromagnetic field.

The inverse scattering transform: Tools for the nonlinear fourier analysis and filtering of ocean surface waves

Abstract: Surface wave data from the Adriatic Sea are analysed in the light of new data analysis techniques which may be viewed as a nonlinear generalization of the ordinary Fourier transform. Nonlinear Fourier analysis as applied herein arises from the exact spectral solution to large classes of nonlinear wave equations which are integrable by the inverse scattering transform (IST). Numerical methods are discussed which allow for implementation of the approach as a tool for the time series analysis of oceanic wave data. The case for unidirectional propagation in shallow water, where integrable nonlinear wave motion is governed by the Korteweg-deVries (KdV) equation with periodic/quasi-periodic boundary conditions, is considered. Numerical procedures given herein can be used to compute a nonlinear Fourier representation for a given measured time series. The nonlinear oscillation modes (the IST ‘basis functions’) of KdV obey a linear superposition law, just as do the sine waves of a linear Fourier series. However, the KdV basis functions themselves are highly nonlinear, undergo nonlinear interactions with each other and are distinctly non-sinusoidal. Numerical IST is used to analyse Adriatic Sea data and the concept of nonlinear filtering is applied to improve understanding of the dominant nonlinear interactions in the measured wavetrains.

Integrable Multi-Component Hybrid Nonlinear Schrödinger Equations

Abstract: Presented is a set of integrable multi-component hybrid nonlinear Schrodinger (MCHNS) equations. Each multi-component equation is a superposition of the nonlinear Schrodinger (NS) equation and the derivative nonlinear Schrodinger (DNS) equation. For the MCHNS equations, the inverse scattering formulation is given. The gauge transformation relating 2-component hybrid nonlinear Schrodinger equation with the Manakov equation is explicitly shown. This also confirms that the former is integrable since the latter is integrable.

Superpositions of handed wave functions

Abstract: Chiral molecules, as well as achiral racemic mixtures, are everywhere. Yet it is well known that the wave functions representing the true stationary states of molecules that are observed to be handed should in fact be invariant under inversion, and therefore not exhibit chirality (1, 2). Such stationary states are rarely seen, and the quantum dynamics of the chiral states has never been observed. However, recent developments in ultrashort-pulse laser methods may enable the preparation and experimental investigation of a large class of states of chiral molecules that resemble the achiral stationary states in being linear combinations of leftand right-handed states (3). In addition, such experiments could provide an opportunity to observe the microscopic processes by which the \"stationary\" achiral states are degraded, and handed states stabilized, by interactions with the surrounding medium. The quantum mechanical description of chiral molecules begins with a symmetric double-well potential (1, 2). That the two wells have equal depth is a reflection of parity conservation; the fact that, to a very good approximation, nature has no fundamental preference for either the leftor the right-handed isomer of a particular molecule. The left-handed isomer is represented by a wave function VL(X), localized in one well, while the wave function xVR(X) for the right-handed isomer is localized in the other. The \"chirality coordinate\" x in these states is some complicated combination of the actual coordinates of the atomic nuclei. The stationary states referred to in the preceding paragraph are symmetric and antisymmetric linear combinations of leftand right-handed states; that is, specific superpositions of the chiral wave functions having the forms

Method of fabricating semiconductor device including step of forming superposition error measuring patterns

Abstract: A method of fabricating a semiconductor device includes the steps of forming an inner circuit, a cell test pattern, and a superposition error measurement pattern. The inner circuit includes a plurality of recurring basic cells. The cell test pattern includes a test cell array having at least one test basic cell of the same design as the basic cells in the inner circuit and a plurality of test dummy cells disposed around the test cell array. The superposition error measurement pattern includes a first and a second pattern formed in the steps of a first and a second lithographic step, respectively, performed in the formation of the basic cells. The inner circuit, said cell test pattern and said superposition error measure pattern are integrated on the same semiconductor substrate. The method permits the formation of the test basic cell having the same proximity effect as that of the basic cells and further permits accurate monitoring of the correlation of the extent of superposition of semiconductor circuit patterns and superposition error.

Tomography of Atom Beams

Abstract: We investigate the accuracy of an experimental reconstruction of the Wigner function of the transverse motion in an atom beam using numerical wave packet simulations. For the example of a superposition of two Gaussians (the outcome of a double slit, for example) we study in detail the influence of experimental restraints on the quality of the reconstruction. For the potential candidate, metastable helium, we demonstrate that an accurate reconstruction of the Wigner function, especially of its negative parts, is well within experimental reach.

Improved genetic algorithm-based protein structure comparisons : pairwise and multiple superpositions

Abstract: Three major improvements to a previously described method for automatic protein structure comparison are described. First, a limit to translations for the rigid-body superposition is now assigned according to the dimensions of the structures being compared. Second, examination of the effect of the gap penalty on the derivation of a sequence alignment corresponding to a given structure superposition has led to a method to evaluate alternative structure-based sequence alignments. Third, the pairwise procedure has been generalized to multiple structure alignment. This implementation of rigid-body superposition can recognize well documented distant relationships which hitherto have required consideration of additional features and properties as well as those relationships between proteins of different sizes. A much larger common scaffold or framework between six globins can be extracted than that obtained using a standard algorithm for multiple structure superposition.

Stokes problems for moving half-planes

Abstract: New exact solutions of the Navier-Stokes equations are obtained for the unbounded and bounded oscillatory and impulsive tangential edgewise motion of touching half-infinite plates in their own plane. In contrast to Stokes classical solutions for the harmonic and impulsive motion of an infinite plane wall, where the solutions are separable or have a simple similarity form, the present solutions have a two-dimensional structure in the near region of the contact between the half-infinite plates. Nevertheless, it is possible to obtain relatively simple closed-form solutions for the flow field in each case by defining new variables which greatly simplify the r- and theta- dependence of the solutions in the vicinity of the contact region. These solutions for flow in a half-infinite space are then extended to bounded flows in a channel using an image superposition technique. The impulsive motion has application to the motion near geophysical faults, whereas the oscillatory motion has arisen in the design of a novel oscillating half-plate flow chamber for examining the effect of fluid shear stress on cultured cell monolayers.

Ripple Analysis in Ferret Primary Auditory Cortex. 3. Prediction of Unit Responses to Arbitrary Spectral Profiles

Abstract: : We examined whether Al responses to arbitrary spectral profiles can be explained by the superposition of responses to the individual ripple components that make up the spectral pattern. For each unit, the ripple transfer function was first measured using ripple stimuli consisting of broadband complexes with sinusoidally modulated spectral envelopes (Shamma et al. 1994). Unit responses to various combinations of ripples were compared to those predicted from the superposition of responses according to the transfer function. Spectral profiles included combinations of 2-5 ripples of equal amplitudes and random phases, and vowel-like profiles composed of 10 ripples with various amplitudes and phases. The results demonstrate that predicted and measured responses are reasonably well matched, and hence support the notion that Al analyzes the acoustic spectrum in a substantially linear manner.

Phase-sensitive reservoir modeled by beam splitters.

Abstract: The superposition of input fields in a lossless beam splitter is studied in the Schr\"odinger picture by using the convolution of the positive P representations, and the convolution law for these representations is extended to other quasiprobability functions such as the Wigner and Q functions. We show that the reservoir can be modeled by an infinite array of beam splitters, and we use the convolution law and this model to derive the Fokker-Planck equation for a system coupled with a phase-sensitive reservoir. Solving this equation shows that a phase-sensitive attenuation and amplification can be described by the superposition of two independent quantum fields, one of which is the initial signal field and the other the squeezed thermal noise field representing the reservoir.

Algorithm for the Generation of Non-diffracting Bessel Modes

Abstract: The iterative algorithm developed allows calculation of phase optical elements serving to transform incident coherent light into non-diffracting beams characterized by an unchanging transverse intensity distribution. Such non-diffracting light beams represent Bessel modes and are described as a superposition of a small number of Bessel functions with equal arguments. The results of numerical calculations are discussed.

Polymer Born–Green–Yvon equation with proper triplet superposition approximation. Results for hard‐sphere chains

Abstract: A site–site Born–Green–Yvon (BGY) equation is derived for polymeric fluids. This relates the pair and triplet site distribution functions, and superposition approximations for the latter are analyzed. It is shown that the pair functions to be superposed are uniquely determined by the exact normalization equations and asymptotic conditions. The Kirkwood superposition of pair distribution functions is shown to be valid only for the case of sites on three different polymers; for the cases of two or three sites on the same polymer different pair functions must be superposed. The polymer BGY equation is derived for a soft bonding potential between adjacent sites; the result for infinitely stiff bonds is given as a limiting case. Numerical results are obtained for soft and stiff tangent hard‐sphere chains, and comparison is made with simulations for packing fractions up to 0.4 and chains with up to 12 sites.

Comparison of ray and Fourier methods for modeling monostatic ground-penetrating radar profiles

Abstract: With increasing emphasis on shallow, high‐resolution geophysical techniques for environmental and engineering applications, it has become important to implement and evaluate tools for quantitative interpretation of ground‐penetrating radar (GPR) data Both ray and Fourier algorithms are viable for numerical simulation of 2-D monostatic GPR data, but they have different characteristics The ray algorithm uses geometrically complicated layers, where within each the dielectric permittivity and attenuation are constant The algorithm produces accurate amplitudes for reflection but does not include wave effects such as diffractions from layer truncations The Fourier algorithm uses a gridded parameterization in which reflections are constructed by superposition of diffractions in a background of constant dielectric permittivity and constant attenuation This technique includes all wave effects, but it does not contain the antenna directivities Both algorithms are able to simulate the main features in two repr

Decomposition of free-surface effects into wave and near-field components

Abstract: The classical Fourier representation of free-surface effects, as a two-dimensional superposition of elementary waves, is expressed in terms of a non-oscillatory near-field (local) flow component defined by a double integral and a wave component given by a single integral along the curve(s) defined in the Fourier plane by the dispersion relations. This Fourier representation of free-surface effects is valid for an arbitrary distribution of sources and/or dipoles and for a wide class of water waves including time-harmonic and steady flows, with or without forward speed, in deep water or in uniform finite water depth. An illustrative application to the Green function of wave diffraction/radiation at low forward speed is presented.

Dissipation and ordering in capillary waves at high aspect ratios

Abstract: We present an experimental study of high-aspect-ratio Faraday waves. We have measured the dispersion relation and the damping rate, together with the critical amplitude for the primary instability for a wide range of frequencies. We find that our results are well explained by the linear theory, if damping from the moving contact line is considered in addition to the bulk damping. Just above the primary instability a seemingly disordered stationary state is observed. We argue that this state is a superposition of normal modes. Approximately 5% above the primary instability this state breaks down in favour of a quasi-crystalline state. This result is discussed, partly in the light of the recent third-order nonlinear theory.

Phase properties of Kerr media via variance and entropy as measures of uncertainty

Abstract: The phase properties of an initially coherent field propagating through a Kerr medium are examined using two different measures of phase uncertainty: (i) variance of periodic functions and (ii) the entropy. The variance of a periodic function of phase (with period 2\ensuremath{\pi}/m) is found to be minimized only at the times when the field evolves into a discrete superposition of n distinguishable coherent states (Schr\"odinger-cat states) where n is a factor of m. The phase entropy is found to be more sensitive, as it is minimized at the times when the field evolves into a superposition of an arbitrary number of distinguishable coherent states. The sum of the photon number and phase entropies is found to display behavior very similar to the Wehrl entropy for the evolving field, i.e., the sum is minimal at the times the cat states occur, and at these times it has a value that indicates the number of component coherent states in the cat states.

Input impedance of a probe-excited semi-infinite rectangular waveguide with arbitrary multilayered loads. I. Dyadic Green's functions

Abstract: This paper presents both the electric and the magnetic types of dyadic Green's functions defined for electromagnetic fields due to electric and magnetic current sources in a semi-infinite rectangular waveguide filled with arbitrary multilayered media. Applying the principle of scattering superposition, the dyadic Green's functions in each of the multiple loads are constructed in general for such EM current sources located in an arbitrary layer of the waveguide. Analytical expressions of the scattering dyadic Green's functions' coefficients are obtained in terms of transmission matrices. To demonstrate how the method presented is used and how the results are obtained for some special cases, a semi-infinite rectangular waveguide with one load is considered. The dyadic Green's functions and their coefficients in such a case are derived in closed form by reducing the general formulae of the dyadic Green's functions for the arbitrary multiple case to those for the special case concerned. Further comparison of the dyadic Green's functions obtained here with previous publications shows good agreement, demonstrating the applicability of the results presented here. >

The correction of interferometric measurements of quadratic electrostriction for cross effects

Abstract: Electrostriction is the strain response of a dielectric material proportional to the square of an applied electric field. This effect is related to the strain-dependence of the dielectric function. The strain signal observed using optical interferometry is a superposition of true electrostriction and other effects like electrostatic forces and thermal stresses. Therefore, it is necessary to have a formula that allows the measured values to be corrected for these spurious effects. In this paper, a thermodynamic approach and the force law are used, combining continuum mechanics and Maxwell's electromagnetic equations, to derive a correction formula. The result can be applied to measurements of both the longitudinal and the transversal effect in single crystals of any symmetry.