Showing papers on "Superposition principle published in 1976"
TL;DR: In this paper, the Boltzmann after-effect equation is solved for a linear anelastic solid having a linear relationship between stress and strain and their first time derivatives, the ‘standard linear solid’, and having a distribution of relaxation times.
Abstract: The concept of a relaxation spectrum is used to compute the absorption and dispersion of a linear anelastic solid. The Boltzmann after-effect equation is solved for a solid having a linear relationship between stress and strain and their first time derivatives, the ‘standard linear solid’, and having a distribution of relaxation times. The distribution function is chosen to give a nearly constant Q over the seismic frequency range. Both discrete and continuous relaxation spectra are considered. The resulting linear solid has a broad absorption band which can be interpreted in terms of a superposition of absorption peaks of individual relaxation mechanisms.
The accompanying phase and group velocity dispersion imply that one cannot directly compare body wave, surface wave, and free oscillation data or laboratory and seismic data without correcting for absorption. The necessary formalism for making these corrections is given. In the constant Q regions the correction is the same as that implied in the theories of Futterman, Lomnitz, Strick and Kolsky.
736 citations
TL;DR: In this paper, a nonlinear term was added to the Schr\"odinger equation, which rapidly drove the amplitude of one or another of the state vectors in such a superposition to one, and the rest to zero.
Abstract: It is hypothesized that the state vector describes the physical state of a single system in nature. Then it is necessary that the state vector of a macroscopic apparatus not assume the form of a superposition of macroscopically distinguishable state vectors. To prevent this, it is suggested that a nonlinear term be added to the Schr\"odinger equation, which rapidly drives the amplitude of one or another of the state vectors in such a superposition to one, and the rest to zero. It is proposed that it is the phase angles of the amplitudes immediately after a measurement which determine which amplitude is driven to one. A diffusion equation is arrived at to describe the reduction of an ensemble of state vectors corresponding to an ensemble of macroscopically identically prepared experiments. Then a nonlinear term to add to the Schr\"odinger equation is presented, and it is shown that this leads to the diffusion equation in a weak-coupling approximation.
366 citations
TL;DR: In this article, the coupling efficiency of a phased multi-waveguide structure (the "Grill") designed to launch HF-waves at the lower hybrid resonance to heat large toroidal plasmas, while satisfying the accessibility condition, is studied.
Abstract: The coupling efficiency of a phased multi-waveguide structure (the "Grill") designed to launch HF-waves at the lower hybrid resonance to heat large toroidal plasmas, while satisfying the accessibility condition, is studied. To find the reflection and transmission coefficients, as well as the k||-spectrum of the excited field, the waveguide field, represented as a superposition of eigenmodes, is matched to the field in the plasma, which is evaluated on the assumption of a linear density profile near the plasma edge. It is found that the reflection coefficient can be made acceptably low and is not sensitively dependent on the plasma parameters. It is concluded that it is possible to design a Grill capable to launch lower hybrid waves at the power level required for the ignition of a reactor plasma.
283 citations
TL;DR: In this article, the nuclear relaxation mechanism in tumbling paramagnetic molecules is a superposition of two processes, the usual S-mechanism and an additional χ mechanism, which is conveniently described in terms of the molecular susceptibility χ.
Abstract: The nuclear relaxation mechanism in tumbling paramagnetic molecules is a superposition of two processes, the usual ‘S-mechanism’ and an additional ‘χ-mechanism’. The latter arises from the thermal average of the electron spin polarization and is conveniently described in terms of the molecular susceptibility χ. Expressions for T 1 and T 2 due to χ-relaxation are derived. The χ-contribution is appreciable in cases where the rotational correlation time is four orders of magnitude longer than the electron spin relaxation time. The χ-relaxation rates increase quadratically with the external field. The effects of anisotropic susceptibility and of chemical exchange are considered. It is shown that in certain important cases the N.M.R. line width is not determined by the chemical exchange rate, even when it is much faster than the tumbling rate.
190 citations
TL;DR: In this article, a mode superposition technique for approximately solving nonlinear initial-boundary-value problems of structural dynamics is discussed, and results for examples involving large deformation are compared to those obtained with implicit direct integration methods such as the Newmark generalized acceleration and Houbolt backward difference operators.
146 citations
01 Jan 1976
TL;DR: The theory of functionals is used to reformulate the notions of aerodynamic indicial functions and superposition, and integral forms for the aerodynamic response to arbitrary motions are derived that are free of dependence on a linearity assumption as discussed by the authors.
Abstract: The theory of functionals is used to reformulate the notions of aerodynamic indicial functions and superposition. Integral forms for the aerodynamic response to arbitrary motions are derived that are free of dependence on a linearity assumption. Simplifications of the integral forms lead to practicable nonlinear generalizations of the linear superpositions and stability derivative formulations. Applied to arbitrary nonplanar motions, the generalization yields a form for the aerodynamic response that can be compounded of the contributions from a limited number of well-defined characteristic motions, in principle reproducible in the wind tunnel. Further generalizations that would enable the consideration of random fluctuations and multivalued aerodynamic responses are indicated.
87 citations
TL;DR: In this article, the authors apply the superposition model to the 6S5/2 ground state of Mn2+ and Fe3+ and give a consistent description of experimentally determined parameters in several systems.
Abstract: Recent work has shown it possible to derive single-ligand contributions from the observed spin-Hamiltonian parameters for the 8S7/2 ground state of f7 lanthanide ions using the superposition model Here the authors apply this model to the 6S5/2 ground state of Mn2+ and Fe3+ and show it to give a consistent description of experimentally determined parameters in several systems The significance of this result is understanding the splitting of the 6S5/2 ground state is discussed
81 citations
TL;DR: In this article, the angular spectrum of plane waves is represented as a sum of two double integrals, one of which is a superposition of homogeneous plane waves and the other (ui) is an inhomogeneous plane wave superposition.
Abstract: Under rather general conditiosn a time‐harmonic wave field u (x,y,z) can be represented in a half‐space z≳0 by a double integral known as the angular spectrum of plane waves. The representation divides naturally into the sum of two double integrals, one of which (uH) is a superposition of homogeneous plane waves and the other (ui) is a superposition of inhomogeneous plane waves. We obtain asymptotic approximations to u (x,y,z), uH, and UI valid when the point of observation of the field recedes towards infinity in a fixed direction through a fixed point. The results apply when the spectral amplitude of the plane waves belongs to a specific class which arises frequently in applications. Our approach is based on the method of stationary phase, which we extend in order to permit the presence of inhomogeneous waves in the integrand. Although the analysis of u requires that we distinguish the directions that are perpendicular to the z axis from the directions pointing into the half‐space z≳0, the results for t...
65 citations
TL;DR: In this article, the first symmetric and antisymmetric free vibration modes of a cantilever plate for a wide range of aspect ratios were analyzed using the method of superposition.
64 citations
01 Jan 1976
TL;DR: In this article, the amplitude and phase spectra in the vicinity of the 60 GHz molecular oxygen resonant line are interpreted as a superposition of an absorption mechanism plus a scattering mechanism.
Abstract: Amplitude spectra in the vicinity of the 60-GHz molecular oxygen resonant line are interpreted as a superposition of an absorption mechanism plus a scattering mechanism. Also, a simple derivation for the amplitude and phase spectra in the limit of small fluctuation frequencies is given.
40 citations
TL;DR: By using the principle of superposition and the results of an earlier paper, the problem is reduced to that of solving a pair of singular integral equations as discussed by the authors, and a number of special cases are considered and numerical results given.
TL;DR: In this article, a series of models for the prediction of the viscoelastic behavior of isotropic soft polymers was developed for the time dependent stress through the Boltzmann superposition integral incorporating into it an appropriately chosen nonlinear measure of strain.
Abstract: Restricting consideration to deformations in which time shift invariance is preserved, a series of models was developed for the prediction of the viscoelastic behavior of isotropic soft polymers. The models describe the time dependent stress through the Boltzmann superposition integral incorporating into it an appropriately chosen nonlinear measure of strain. The theory was developed in its most general threedimensional form. It requires a single time function, the relaxation modulus of linear viscoelastic theory. In its simplest form the strain measure contains a single material parameter.
TL;DR: For a multilevel atom or molecule irradiated by a finite superposition of monochromatic fields, the time-dependent rotating-wave-approximation Schroedinger equation can be reduced to an eigenvalue problem when the detunings satisfy a consistency condition as discussed by the authors.
Abstract: For a multilevel atom or molecule irradiated by a finite superposition of monochromatic fields, the time-dependent rotating-wave-approximation Schroedinger equation can be reduced to an eigenvalue problem when the detunings satisfy a consistency condition. This condition is analyzed and the reduction performed using a graph associated with the interaction Hamiltonian. (AIP)
TL;DR: Using the paraxial ray approximation, simple formulas for the cross polarization introduced by curved reflectors are developed andoretical cross-polarization patterns are shown to be in accurate agreement with measurements on a symmetrical dual-reflector system.
Abstract: Using the paraxial ray approximation, simple formulas for the cross polarization introduced by curved reflectors are developed. In particular, when the reflectors are quadric surfaces of revolution with the center ray of the beam passing through the foci, the maximum cross-polarized field amplitude throughout a gaussian beam, relative to the on-axis copolarized field, is $C = {2\xi\kappa\bot\over \sqrt{e}}\sin \theta_{i},$ where e is the base of the natural logarithm, ξ is the 1/e power radius of the beam, k i is the curvature of the reflector perpendicular to the plane of incidence, and θ i is the angle of incidence. For such reflectors, the beam fields are accurately represented by a superposition of just two gaussian modes for each plane of polarization: the fundamental mode, which corresponds to the co-polarized gaussian beam, and a higher-order mode, which accounts for the cross-polarized field and the amplitude “space” taper. Transformation of a beam through a general sequence of such reflectors is influenced by three factors: the curved reflectors, longitudinal propagation lengths, and rotations of the plane of incidence. The effect of each factor is described by a 4 × 4 matrix relating the input and output gaussian modes. Several typical beam-reflector systems are analyzed by this method. Theoretical cross-polarization patterns are shown to be in accurate agreement with measurements on a symmetrical dual-reflector system.
TL;DR: In this article, the authors studied the statistical properties of parametric processes (parametric amplification, frequency conversion, splitting of frequency, etc.) described by the trilinear Hamiltonian and of the second harmonic generation including the lossy mechanism and rotating terms.
Abstract: We study the statistical properties of parametric processes (parametric amplification, frequency conversion, splitting of frequency, etc.) described by the trilinear Hamiltonian and of the second harmonic generation including the lossy mechanism and rotating terms. In a short-time approximation some conservation laws are derived and the existence of the Glauber-Sudarshan weighting function is discussed. It is shown in the second harmonic generation case that good coherence of the incident radiation is lost proportionally to the intensity in interaction while the second harmonic radiation has tendency to be coherent. The effects of the spontaneous emission of the medium and of the physical vacuum are also discussed. In this approximation the statistics is generally described by the superposition of coherent and chaotic fields with correlated components.
TL;DR: The wave aberration of an objective lens may be determined from superposition diffractograms of bright-field images with different tilt direction of the primary beam and single-sideband transfer.
TL;DR: In this article, a new EPR determination of the spin-Hamiltonian parameters for Gd3+:YAG has been made and a direct comparison of garnet crystal-field and spin- Hamiltonian parameters, as well as the success of the superposition model analysis, enables to eliminate the possibility of large non-linear (e.g. quadratic) crystal field contributions to the ground state splitting.
Abstract: A new EPR determination of the spin-Hamiltonian parameters for Gd3+:YAG has been made. A direct comparison of garnet crystal-field and spin-Hamiltonian parameters, as well as the success of the superposition model analysis, enables us to eliminate the possibility of large non-linear (e.g. quadratic) crystal-field contributions to the Gd3+ ground-state splitting. Spin-Hamiltonian superposition model parameters are derived and an analysis of the n=2 crystal-field and spin-Hamiltonian parameters suggests that the coordinate angles derived from X-ray data are in error by about 1 degrees .
TL;DR: In this article, a method of excitation which enhances quantum beats of certain frequencies and diminishes those at other frequencies is presented, which can even produce an effect when the original quantum beats appear to be washed out in large molecules because of the average over many pairs of interfering levels.
TL;DR: In this article, the authors determined the motion of an elastic, homogeneous and isotropic half-space, excited by a normal point load traveling uniformly over the free surface.
TL;DR: In this article, the principle of superposition for stress intensity factors for different types of loading was discussed and it was shown that stress intensity factor for different loading types are additive provided all loading types tend to open the crack.
TL;DR: In this article, the ground state of 58Ni was described as a superposition of Hartree-Fock-Bogolyubov wave functions of many different shapes and the corresponding charge density was calculated.
TL;DR: The test particle superposition principle of plasma kinetic theory is discussed in this article in relation to the recent theory of two-time fluctuations in plasma given by Williams and Oberman, and a new deductive and inductive proof of the principle are presented; the deductive approach appears here for the first time in the literature.
Abstract: The test particle superposition principle of plasma kinetic theory is discussed in relation to the recent theory of two‐time fluctuations in plasma given by Williams and Oberman. Both a new deductive and a new inductive proof of the principle are presented; the deductive approach appears here for the first time in the literature. The fundamental observation is that two‐time expectations of one‐body operators are determined completely in terms of the (x,v) phase space density autocorrelation, which to lowest order in the discreteness parameter obeys the linearized Vlasov equation with singular initial condition. For the deductive proof, this equation is solved formally using time‐ordered operators, and the solution is then re‐arranged into the superposition principle. The inductive proof is simpler than Rostoker’s although similar in some ways; it differs in that first‐order equations for pair correlation functions need not be invoked. It is pointed out that the superposition principle is also applicable to the short‐time theory of neutral fluids.
TL;DR: In this paper, a computer program was developed to investigate the elasto-plastic behavior of structures and to discuss the problems of nonlinear response of structures in nuclear power plants.
TL;DR: In this article, the authors presented a method of optimum synthesis of planar function generators, where the dimensions of an optimum mechanism are determined by minimizing the error in Freudenstein's input-output displacement equation of the mechanism at N design positions, where N is not limited by the number of unknowns of the system.
TL;DR: In this paper, a formal solution for traveling waves in isotropic linear elastic bars of rectangular cross section which have traction-free lateral surfaces is presented by crosswise superposition of two single series, each term of which gives zero shearing stresses on lateral surfaces.
Abstract: A formal solution for travelling waves in isotropic linear elastic bars of rectangular cross section which have traction-free lateral surfaces is presented. The solution is obtained by crosswise superposition of two single series, each term of which gives zero shearing stresses on the lateral surfaces. Frequency equations, which define the dispersion relations of elastic waves, are given by infinite determinants. Based on the approximate frequency equations, dispersion relations for a longitudinal mode, a torsional mode, and bending modes were calculated with the aid of a digital computer. Numerical results were compared with those given by other authers.
TL;DR: In this article, the determination of mechanical response of a non-linear viscoelastic solid based on a Frechet expansion was investigated from a slightly different point of view, and the effect of such a principle on the kernel functions and a geometric interpretation were discussed.
Abstract: This paper deals with the determination of mechanical response of a non-linear viscoelastic solid based on a Frechet expansion In the first portion, a modified superposition principle, suggested earlier by various authors, is arrived at from a slightly different point of view The effect of such a principle on the kernel functions and a geometric interpretation are discussed In the later portion, the tests required for the experimental determination of the kernel functions are discussed It is seen that besides the tests suggested by Onaran and Findley for the two-dimensional case, three additional tests are needed for the three-dimensional case to compute the response under constant combined stress
TL;DR: In this article, the Boltzmann superposition principle is applied to linearly viscoelastic materials with an imposition of changes in both the strain and the density of elastically effective chains in discrete increments.
Abstract: Methods reported by Moacanin et al. (1975) and Moacanin and Aklonis (1971) are generalized with the objective to include strains (or stress) applied in an arbitrary manner to linearly viscoelastic materials. An imposition of changes in both the strain and the density of elastically effective chains in discrete increments is considered. In accordance with the Boltzmann superposition principle, each strain increment may be treated as a new independent experiment which adds linearly to the total response of the system.
TL;DR: In this article, a generalized Fourier series is used to predict the forced vibration of membranes, beams, plates or shells when they have attached to them at a single point a linear lumped parameter element or assembly of elements.
01 Jan 1976
TL;DR: In this paper, it is shown that the collective electronic excitation, the exciton which is optically prepared in a molecular crystal, with a vector k ≃ 0, undergoes rapid subsequent localization: Bloch functions are solutions of the wave equation, and k can be used in the description of the crystal stationary states, only as long as perfect translational symmetry is present.
Abstract: The collective electronic excitation, the exciton, which is optically prepared in a molecular crystal, with a vector k ≃ 0, undergoes rapid subsequent localization: Bloch functions are solutions of the wave equation, and k can be used in the description of the crystal stationary states, only as long as perfect translational symmetry is present. At any non-zero temperature, phonons destroy partially this symmetry, and states of the system are now described by a superposition of Bloch functions which may span the whole accessible range Δk of k values, depending on the strength of exciton-phonon coupling. By the uncertainty principle, this may amount to a localization, say in l-dimension Δx such that ΔxΔk ~ 2π,and a wave packet has been built. This is a process common to all electronic states in solids, and the same line of thought is followed if Wannier functions are constructed.
TL;DR: Time-temperature superposition in the\(\mathop \varepsilon \limits^.\) = const regime can be applied to partially crystalline fluoroplastics with different specific volumes as mentioned in this paper.
Abstract: Time-temperature superposition in the\(\mathop \varepsilon \limits^.\) = const regime can be applied to partially crystalline fluoroplastics with different specific volumes. There is a correlation between the variation of the specific volume and the relaxation processes in fluoroplastics. Time-temperature superposition is valid only in the temperature regions between transitions. The master curves are theoretically described by means of a nonlinear equation.