Showing papers on "Superposition principle published in 1988"
TL;DR: In this article, the Boltzmann superposition principle based on the general standard linear solid rheology is implemented in the equation of motion by the introduction of memory variables, and the propagation in time is done by a direct expansion of the evolution operator by a Chebycheff polynomial series.
Abstract: SUMMARY A new approach for viscoacoustic wave propagation is developed. The Boltzmann’s superposition principle based on the general standard linear solid rheology is implemented in the equation of motion by the introduction of memory variables. This approach replaces the conventional convolutional rheological relation, and thus the complete time history of the material is no longer required, and the equations of motion become a coupled first-order linear system in time. The propagation in time is done by a direct expansion of the evolution operator by a Chebycheff polynomial series. The resulting method is highly accurate and effects such as the numerical dispersion often encountered in time-stepping methods are avoided. The numerical algorithm is tested in the problem of wave propagation in a homogeneous viscoacoustic medium. For this purpose the l-D and 2-D viscoacoustic analytical solutions were derived using the correspondence principle.
381 citations
TL;DR: This is the first of two papers that develop the theory of oscillatory spectra, which is found that the absorption as a function of energy is a superposition of sinusoidal oscillations.
Abstract: This is the first of two papers that develop the theory of oscillatory spectra. When an atom is placed in a magnetic field, and the absorption spectrum into states close to the ionization threshold is measured at finite resolution, so that individual energy levels are not resolved, it is found that the absorption as a function of energy is a superposition of sinusoidal oscillations. These papers present a quantitative theory of this phenomenon. In this first paper, we describe the physical ideas underlying the theory in the simplest possible way, and we present our first calculations based upon the theory. In the second paper, the theory is developed in full detail, proofs of all of the assertions are given, and we describe the algorithm that was used to make the calculations.
279 citations
TL;DR: A solution of the bio-heat transfer equation for a 'step-function point source' is presented and from this basic solution one can, in principle, obtain the temperature field resulting from a general heat source distribution by superposition.
Abstract: A solution of the bio-heat transfer equation for a 'step-function point source' is presented and discussed. From this basic solution one can, in principle, obtain the temperature field resulting from a general heat source distribution by superposition. As an example, the method is used to calculate the temperature on the body surface at a point where therapeutic ultrasound is applied. Comparison is made with experimental results recently published by Williams and co-workers.
193 citations
TL;DR: In this article, a new technique is presented to measure the frequency dependent complex modulus simultaneously at several frequencies instead of consecutively as in a frequency sweep, where the strain in a dynamical mechanical experiment is prescribed as a superposition of several different modes (three in our case).
Abstract: A new technique is presented to measure the frequency dependent complex modulus simultaneously at several frequencies instead of consecutively as in a frequency sweep. For this purpose, the strain in a dynamical mechanical experiment is prescribed as a superposition of several different modes (three in our case). The resulting stress is decomposed into sinusoidal components, each of them characterized by their frequency, amplitude, and phase shift with respect to the corresponding strain component. Phase shift and amplitude are expressible in a frequency dependent complex modulus. A single experiment gives, therefore, values for the complex modulus at a set of prescribed frequencies. The method was demonstrated on three stable viscoelastic fluids and was applied to determine the instant of sol-gel transition (gel point) of a crosslinking polymer.
177 citations
TL;DR: In this paper, the quadratic residual (weighted sum of squares of coordinate differences) that results when one vector set is rotated relative to another is shown to be in a 4-dimensional form of order 4, and the stationary values of the residual are given by the eigenvalues of a matrix of 4.
Abstract: A rotation axis vector with magnitude tan (θ/2) for a rotation angle θ and a closely related unit vector of dimension 4 are used to show that : (i) the quadratic residual (weighted sum of squares of coordinate differences) that results when one vector set is rotated relative to another is a quadratic form of order 4, (ii) the stationary values of the residual are given by the eigenvalues of a matrix of order 4, (iii) the minimum residual is given by the largest eigenvalue, (iv) the rotations required to obtain such residuals are uniquely defined by the corresponding eigenvectors, and (v) the stationary values are related by the operations of 222 symmetry. No precautions against the generation of improper rotations are required. In addition, an equivalent solution based on a scalar iteration is presented, together with some relationships of general interest.
97 citations
TL;DR: In this paper, a molecular dynamics technique is introduced for the simulation of the adiabatic dynamics of an excess electron coupled to a classical many-body system, where the instantaneous ground state wave function of the electron is represented by a superposition of distributed Gaussian basis functions with equal amplitude.
Abstract: A molecular dynamics technique is introduced for the simulation of the adiabatic dynamics of an excess electron coupled to a classical many‐body system. The instantaneous ground state wave function of the electron is represented by a superposition of distributed Gaussian basis functions, each with equal amplitude. We present generalized equations of motion for the coupled system, which optimize the positions and widths of the Gaussians by simulated annealing. The condition of equal amplitude ensures the aggregation of the Gaussians in regions of finite electron probability density and hence yields a particularly efficient representation of localized ground states. The method is applied to an electron solvated in liquid ammonia and results for equilibrium properties are compared to quantum path integral calculations. New results for the dynamics are discussed in the light of mobility measurements.
78 citations
TL;DR: In this paper, a volume integral representation of the scattered wave field propagated backward in time into an arbitrary background medium is related via volume integral to perturbations in velocity about the background, which are expressed as a scattering potential.
Abstract: The scattered wave field propagated backward in time into an arbitrary background medium is related via a volume integral to perturbations in velocity about the background, which are expressed as a scattering potential. In general, there is no closed-form expression for the kernel of this integral representation, although it can be expressed asymptotically as a superposition of plane waves backpropagated from the receiver array. When the receiver array completely surrounds the scatterer, the kernel reduces to the imaginary part of the Green's function for the background medium.This integral representation is used to relate the images obtained by imaging algorithms to the actual scattering potential. Two such relations are given: (1) for the migrated image, obtained by deconvolving the extrapolated field with the incident field; and (2) for the reconstructed image, obtained by applying a one-way wave operator to the extrapolated field and then deconvolving by the incident field. The migrated image high-lights rapid changes in the scattering potential (interfaces), whereas the reconstructed image can, under ideal conditions, be a perfect reconstruction of the scattering potential. 'Ideal' conditions correspond to (1) weak scattering about a smoothly varying background medium, (2) a receiver array with full angular aperture, and (3) data of infinite bandwidth.Images obtained from a multioffset vertical seismic profile (VSP) illustrate some of the practical differences between the two imaging algorithms. The reconstructed image shows a much clearer picture of the target (a reef structure), in part because the one-way imaging operator eliminates artifacts caused by the limited aperture of the receiver array.
70 citations
TL;DR: In this paper, the authors used time-temperature superposition to extend the time scale of creep tests in polymers from the short times easily obtained in the laboratory to long times seen in actual use.
Abstract: In the past, time-temperature superposition has been used to extend the time scale of creep tests in polymers from the short times easily obtained in the laboratory to long times seen in actual use. A fundamental assumption of time-temperature superposition is, however, that the polymer does not change in structure as a function of time. Because ductile amorphous thermoplastics physically age below Tg, the structure of the polymer changes on a time scale comparable to the time duration of the creep test. Thus, the time-temperature superposition prediction greatly exaggerates the amount of creep in amorphous thermoplastics. For samples aged at the test temperature for one hour before testing, the difference between the time-temperature superposition prediction and the actual creep data after 10 days is greater than a factor of ten in time.
49 citations
TL;DR: In this article, the capillary instability of compound jets has been studied using a linear model derived from the two-dimensional equations of motion, where the flow was considered as a superposition of steady-state plug flow and travelling waves of small amplitude.
34 citations
TL;DR: In this article, a method is presented to evaluate quantitatively the reflection of elastic waves by a distribution of coplanar cracks, based on a novel calculation of crack-opening volumes.
Abstract: A method is presented to evaluate quantitatively the reflection of elastic waves by a distribution of coplanar cracks. The method is based on a novel calculation of crack‐opening volumes. To compute the crack‐opening volume of a specified crack, the effects of neighboring cracks are approximated by the effects of pairs of vector dipoles at the geometrical centers of those cracks. The strengths of the dipoles are proportional to the crack‐opening volumes of the neighboring cracks. Thus a crack‐opening volume is expressed as a superposition of solutions of two generic problems. One concerns the interaction of the incident wave with a single crack, while the other is concerned with the interaction of a vector dipole with a single crack. For a plane containing a large number of cracks, closed‐form expressions have been derived for the reflection and transmission coefficients in terms of the crack‐opening volumes. The results have been generalized to statistical distributions of cracks.
33 citations
TL;DR: In this paper, the authors show that in the particular case of floating breakwater, the superposition, which uncouples the wave diffraction and the body movement, neglects important effects of coupled phenomena and introduces errors into the simulation.
Abstract: Very often the linear models of floating caisson breakwaters simulate the complete hydrodynamic phenomenon by dividing it into elementary problems, such as wave diffraction around a fixed surface body, motion of the moored structure caused by the wave, and change of the wave field due to waves generated by the breakwater. The authors show that in the particular case of floating breakwater, the superposition, which uncouples the wave diffraction and the body movement, neglects important effects of coupled phenomena and introduces errors into the simulation. The present paper suggests some corrections, which, on the whole, can account for the nonlinearity of the coupled phenomenon. Furthermore, laboratory tests check the validity of the theoretical solutions of the elementary problems, the result obtained by working out these solutions, and the effectiveness of the proposed corrections. The tests also prove that, in some cases, the hydrodynamic coefficients of added mass, calculated on the usual assumption ...
TL;DR: In this article, a marked Poisson cluster process (PCP) is defined as a model for live loads in buildings and the outcrossing rate for this PCP and the superposition of such processes are derived for the determination of structural failure probabilities.
TL;DR: In this paper, the influence of iceberg shape on its wave-induced motion, the use of linear superposition to obtain estimates of iceberg response in irregular waves, and the application of linear diffraction theory to compute iceberg response amplitude operators (RAO's) were extensively studied utilizing data generated in a wave tank.
TL;DR: In this article, general mathematical expressions for a marine source array's (1) far-field pulse spectrum, (2) radiated energy density, and (3) directivity are developed for both a source in an infinite homogeneous medium and a source operating near the ocean surface.
Abstract: General mathematical expressions for a marine source array’s (1) far‐field pulse spectrum, (2) radiated energy density, and (3) directivity are developed for both a source in an infinite homogeneous medium and a source operating near the ocean surface. These results, intended to assist the analysis and design of marine source arrays, apply to any marine source array when (1) individual elements radiate isotropically, (2) their individual waveforms are specified, and (3) the array geometry is specified. Arbitrary geometry and arbitrary isotropic waveforms are allowed. The theory assumes linear superposition of the individually specified waveforms, and is consistent with the “square law effect” for identical elements. For an array of small elements, expended energy agrees with the array’s radiated energy found using far‐field methods. Also, the energy radiated from an array with large element spacing is equal to the sum of the independently radiated energies. Two closely spaced identical elements radiate fo...
TL;DR: In this article, an approach for the description of the time evolution of an arbitrary physical state by a superposition of generalized Bethe states is proposed, which does not contain the sum over the string configurations.
TL;DR: In this paper, the authors extended the first-order single scattering theory to study the scattering of surface waves as well as body waves by distributed point scatterers in a layered medium.
Abstract: Malin’s (1980) first-order single scattering theory has been extended to study the scattering of surface waves as well as body waves by distributed point scatterers in a layered medium. The scattered waveform itself is generated and examined instead of its energy envelope. The theory used allows 1) mode conversion, 2) wave type conversion, 3) finite scatterer distribution, and 4) the effect of attenuation from scattering as well as intrinsic absorption. The cases studied are for elastic or slightly attenuative media with any kind of source and receiver at any place in the layered structure. This direct calculation of coda waves provides us an immediate description of the relation of coda and scattering. The objectives are to find 1) the effect of layering on scattering, 2) the effect of scatterer distribution on recorded vertical and horizontal motion, 3) the relation of scattering Q to intrinsic Q, 4) the scattering behavior of surface and body waves, and 5) the superposition of scattering waves to form the coda. The generation of body waves by ‘locked mode’ approximation, which makes the body-wave scattering a subset of the ‘surface-wave’, scattering. Preliminary results explain some observed coda behavior surprisingly well. We find a larger geometrical spreading for near scatterers, which is caused by mode conversion or wave type conversion because of the wide angle scattering. This makes the spreading correction higher for early part of coda which may account for the low Q observed in early coda of regional earthquakes. This study is of practical value as an effort to understand the complicated coda phases.
TL;DR: In this paper, a ray-Kirchhoff method is developed for body-wave calculations which extends previous ray methods to rapidly varying media, based on a newly derived integral solution to wave equations which indicates that the wave field at a receiver point is given by a superposition of ray solutions determined by the transport and extended eikonal equations.
Abstract: SUMMARY A ray-Kirchhoff method is developed for body-wave calculations which extends previous ray methods to rapidly varying media. It is based on a newly derived integral solution to wave equations which indicates that the wave field at a receiver point is given by a superposition of ray solutions determined by the transport and extended eikonal equations. The latter is in turn solved by an asymptotic series. In a slowly varying medium, only the leading term of this series needs to be considered, and the extended eikonal equation reduces to the well-known eikonal equation. Wave fields in this case can be calculated using asymptotic ray theory. For a rapidly varying medium where velocity gradients are no longer small, the higher-order terms of the series must not be disregarded. These frequency-dependent higher-order terms represent the scattering effect of velocity gradients and provide a basis for avoiding caustics. The new method also includes a procedure for estimating the errors introduced by truncating higher-order terms from the asymptotic series. In particular, validity conditions for the ray-Kirchhoff method in elastic media are formulated which indicate that the new method is less restrictive than some previous ray methods such as the Gaussian-beam technique. For implementation, a perturbation scheme is developed for solving the ray and transport equations. In addition to computing the higher-order terms of the asymptotic series, this scheme avoids most of the ray tracing required for computing wave fields in median with weak lateral variations. Using this scheme, the ray-Kirchhoff method is extended to anelastic media. Approaches for removing singularities on an integral surface used in the ray-Kirchhoff method are proposed which not only prevent the infinite amplitude at a caustic, but also predict the phase shift caused by this singularity.
TL;DR: In this paper, a load dependent transformation vector is proposed for dynamic response analysis of large structures by vector superposition, which is an economic alternative to the usual mode superposition method.
TL;DR: In this paper, the authors developed a method of obtaining two-dimensional integrable evolutions corresponding to singular dispersion relations and applied it to the 2*2 linear first-order eigenvalue problem and the inverse spectral transform scheme is established.
Abstract: The authors develop a method of obtaining two-dimensional integrable evolutions corresponding to singular dispersion relations. The method is applied to the 2*2 linear first-order eigenvalue problem and the inverse spectral transform scheme is established. The Backlund transformation and a non-linear superposition formula are used to obtain the soliton solutions.
TL;DR: In this paper, the wave equation c2(x)uxx−utt=0 is solved for wave speeds c(x), corresponding to two-layered media with smooth transition from layer to layer.
Abstract: The wave equation c2(x)uxx−utt=0 is solved for wave speeds c(x) corresponding to two‐layered media with smooth transition from layer to layer. The wave speed c(x) has four free parameters to fit a given medium. Solutions are constructed from invariant solutions of a related system of first‐order partial differential equations that admit a four‐parameter symmetry group. These solutions are superposed to solve general initial value problems for data with compact support; the computation of the superposition coefficients uses elementary Fourier analysis. Solutions are illustrated for various initial conditions.
TL;DR: In this paper, the Backlund transformation for a fifth order KdV equation is presented in the bilinear form, and a nonlinear superposition formula related to the BT obtained above is proved rigorously.
Abstract: The Backlund transformation (BT) for a fifth order KdV equation is presented in the bilinear form. Furthermore, a nonlinear superposition formula related to the BT obtained above is proved rigorously. By the way, a nonlinear superposition formula of a modified fifth order KdV equation is also given.
TL;DR: In this paper, the Hartley transform is used to obtain the location of the centroid of a source in the Fourier transform plane, which is constructed by the superposition of two Fourier transforms, one rotated spatially by 180° and shifted in phase by 90°.
Abstract: We recently reported a method of optical phase measurement1 that involves the creation of a Hartley transform2 plane, which is constructed by the superposition of two Fourier transforms3, one of which was rotated spatially by 180° and shifted in phase by 90°. Because the Hartley transform, unlike the Fourier transform, is entirely real, it is able to extract more information when a phase-insensitive detector (for example, an optical detector) is used. In particular, the Hartley intensity encodes the location of the centroid of a source, whereas the intensity distribution of a Fraunhofer diffraction pattern does not change when the source shifts because source location is hidden in the phase information. Through the Hartley transform one gains access to the Fourier phase by amplitude measurements only. We have now extended this technique to the microwave range, and have also demonstrated how to obtain an image of an isophase source without using a phase-sensitive detector.
TL;DR: Gabor's expansion of an aperture field in a discrete two-dimensional superposition of Gaussian elementary beams is extended here to incorporate exponential beams as mentioned in this paper, which has an advantage over the commonly used Gaussian beam representation in situations involving an exponential aperture distribution.
Abstract: Gabor's expansion of an aperture field in a discrete two-dimensional superposition of Gaussian elementary beams is extended here to incorporate exponential beams. The resultant expansion scheme has an advantage over the commonly used Gaussian beam representation in situations involving an exponential aperture distribution such as the modal field in a thin slab laser amplifier.
TL;DR: It is shown that if three-level atoms in the V configuration are injected in a coherent superposition of the upper two states in a doubly resonant cavity, the diffusion coefficient for the relative phase vanishes under certain conditions.
Abstract: A nonlinear theory of correlated emission lasers is presented It is shown that if three-level atoms in the V configuration are injected in a coherent superposition of the upper two states in a doubly resonant cavity, the diffusion coefficient for the relative phase vanishes under certain conditions
TL;DR: In this article, the beam-series expansion is proposed to represent the initial field conditions, specified on a prescribed surface, by elementary beam constituents, leading to the beam series expansion, which constitutes a two-dimensional superposition of appropriately weighted Gaussian beams.
Abstract: The frequency-domain wave equation is shown to be asymptotically replaceable by a discrete collection of parabolic equations formulated along ray-centered coordinates. This approximate formulation replaces the original boundary-value problem with a considerably simpler set of initial-value problems whose exact solutions are the well-understood Gaussian beams. The basic question addressed herein pertains to the methodology by which the initial field conditions, specified on a prescribed surface, can be represented by elementary beam constituents. An exact procedure based on the Gabor representation is proposed, leading to the so-called beam-series expansion. The beam series constitutes a two-dimensional superposition of appropriately weighted Gaussian beams. A simple and numerically efficient method for evaluating the expansion coefficients, the beam spectrum, is a direct by-product of the proposed scheme. Three distinct advantages are notable. First, since the proposed representation is intrinsically discrete, no further discretization is required in the implementation stage. Second, the elementary beam-field components are nonsingular. Thus foci and caustic regions do not require special attention. Third, the truncation rules of the beam series are well established and sharply defined. The basic beam-series expansion is extended herein to encompass curved initial surfaces, basic Gaussian beam components whose waists are freely located (not necessarily confined to the initial surface), and inhomogeneous media. The relationship between the Gabor and Gauss–Hermite expansions is also discussed.
TL;DR: In this paper, the self energy of an extended moving material particle was associated with a proper photon with proper momentum making an angle α, such that cosα= v c, with the direction of the velocity v. The Compton effect was then obtained by superposition of velocity c waves with different directions of propagation and different frequencies.
TL;DR: In this paper, a two-level phase sensitive amplifier is proposed, where two level atoms are injected in pairs, initially prepared in a coherent superposition of upper and lower levels, and it is shown that the additive noise in such an amplifier is a broadband squeezed field.
TL;DR: In this article, superposition formulas are derived expressing the general solution of several different systems of nonlinear ordinary differential equations in terms of a fundamental set of particular solutions, which are induced by the action of the exceptional Lie group G2 (complex or real) on a homogeneous space G2/G, where G⊆G2 is a maximal subgroup of G2.
Abstract: Superposition formulas are derived expressing the general solution of several different systems of nonlinear ordinary differential equations in terms of a fundamental set of particular solutions. The equations, as well as the superposition formulas, are induced by the action of the exceptional Lie group G2 (complex or real) on a homogeneous space G2/G, where G⊆G2 is a maximal subgroup of G2. When G is either parabolic, or simple, three particular solutions are needed. When G is SL(2,C)×SL(2,C) (or one of its real forms), then two particular solutions suffice.
TL;DR: In this article, the authors show that the presence of two modes, respectively parallel and perpendicular to the direction of the forcing, may produce either a stable or a time-dependent superposition of the two modes.
Abstract: Surface waves instability has been experimentally studied in a cylindrical basin subjected to a horizontal oscillation. We show that the presence of two modes, respectively parallel and perpendicular to the direction of the forcing, may produce either a stable or a time-dependent superposition of the two modes. The phase diagram of the system, the surface deformation amplitude and the features of the time-dependent regimes are in satisfactory agreement with the available theoretical predictions.