Showing papers on "Superposition principle published in 1975"

Dynamics Of Structures

Abstract: Part 1 Single-degree-of-freedom systems: overview of structural dynamics analysis of free vibrations response to harmonic loading response to periodic loading response to impulse loading responses to general dynamic loading - step by step methods, superposition methods generalized single degree-of-freedom systems. Part 2 Multi-degree-of-freedom systems: formulation of the MDOF equations of motion evaluation of structural-property matrices undamped free vibrations analysis of dynamic response using superposition vibration analysis by matrix iteration selection of dynamic degrees of freedom analysis of MDOF dynamic response - step by step methods variational formulation of the equations of motion. Part 3 Distributed parameter systems: partial differential equations of motion analysis of undamped free vibrations analysis if dynamic response. Part 4 Random vibrations: probability theory random processes stochastic response of linear SDOF systems stochastic response of non-linear MDOF systems. Part 5 Earthquake engineering: seismological background free-field surface ground motions deterministic structural response - including soil-structure interaction stochastic structural response.

Time‐dependent approach to semiclassical dynamics

Abstract: In this paper we develop a new approach to semiclassical dynamics which exploits the fact that extended wavefunctions for heavy particles (or particles in harmonic potentials) may be decomposed into time−dependent wave packets, which spread minimally and which execute classical or nearly classical trajectories. A Gaussian form for the wave packets is assumed and equations of motion are derived for the parameters characterizing the Gaussians. If the potential (which may be nonseparable in many coordinates) is expanded in a Taylor series about the instantaneous center of the (many−particle) wave packet, and up to quadratic terms are kept, we find the classical parameters of the wave packet (positions, momenta) obey Hamilton’s equation of motion. Quantum parameters (wave packet spread, phase factor, correlation terms, etc.) obey similar first order quantum equations. The center of the wave packet is shown to acquire a phase equal to the action integral along the classical path. State−specific quantum information is obtained from the wave packet trajectories by use of the superposition principle and projection techniques. Successful numerical application is made to the collinear He + H2 system widely used as a test case. Classically forbidden transitions are accounted for and obtained in the same manner as the classically allowed transitions; turning points present no difficulties and flux is very nearly conserved.

On the dynamics of internal waves in the deep ocean

Abstract: The dynamical processes affecting the evolution of a random internal wave field are considered. If the statistical properties of the internal wave field vary slowly with space and time, these dynamical processes can be treated as small perturbations about the local steady state of the free linear field. The time evolution of the wave field is then governed by a radiative transfer equation describing changes in the action density spectrum of the wave field along wave group trajectories. The source function describing these changes is determined by the superposition of all processes governing the generation, transfer, and dissipation of wave energy. Some terms of the source function, those corresponding to expansible processes, can be derived rigorously by using weak interaction concepts. Other terms, corresponding to nonexpansible processes which are governed locally by strongly nonlinear dynamics, cannot be determined completely. For the case where the internal wave field can adequately be described in the WKBJ approximation, a rather complete list of source terms is presented. The evaluation of these source terms is difficult partly because of their complicated functional structure and partly because the geophysical fields involved are not sufficiently known. Those source terms which have been evaluated in detail are briefly reviewed, and their implications on the energy balance of the internal wave field are discussed.

Anelastic damping of acoustic and seismic pulses

Abstract: The basic question of the linearity or non-linearity of the seismic attenuation mechanism has not yet been conclusively answered. Alternative linear theories are compared with recent measurements of the degeneration of pulse shape in attenuating media. Only a theory due to Azimiet al., (1968) survives the test. The assumption of linearity conflicts with observations of stress-strain hysteresis loops which always have cusped (pointed) ends, but no successful hysteretic model of pulse attenuation has emerged. Observations of the independence of attenuations of superimposed waves are critical to the linearity assumption, but all superposition observations so far are defective in that they have considered the superposition of phase-uncorrelated waves, whereas only the superposition of phase-correlated waves is a true test of superposition.

The mechanism of visual pattern fixation in the walking fly,Drosophila melanogaster

Abstract: In freely walking flies,Drosophila melanogaster, stationary stripes induce fixation processes. If two vertical black stripes (width 6°) are presented against an illuminated white background and if the two stripes are separated by an angular distance ofαν 75°, the angular distributions of the flies' positions always show two maxima directed towards the two stripes (Fig. 3a). This result is also true for 6°-wide vertical stripes which are illuminated from behind in an otherwise dark drum (Fig. 4a). In experiments with a two- and three-stripesenvironment, the variability of the flies' reactions to the stripes has been measured (Figs. 3–6). From these results, a functiontr (μ) can be deduced which describes the strength of the turning tendencies induced in the flies by a stripe as a function of the walking directionμ of the fly (turning tendency function). This function is characterized (1) by a sharp increase oftr (μ) with increasing angular distance from the direction of the object to the maximum response atμ=25–30°, (2) by a sharp decrease oftr (μ) from the maximum toμ=60°, and (3) by a very slow approach oftr (μ) to the labile zero position of the turning tendency function atμ=±180° (Fig. 7). Additive superposition of two or three of these turning tendency functions, which are phase-shifted for different anglesαν, allows one sufficiently to describe the walking directions of the flies. This is possible because the stable zero positions of the summation turning tendency functionstrs (μ) coincide with the maxima of all the frequency distributionsf (β) of the flies' positionsβ (Fig. 8).

Normal‐mode program with both the discrete and branch line contributions

Abstract: In a stratified ocean model, in which the sound speed and density become constant as the depth coordinate becomes infinite, the pressure field can be represented as a finite sum of modes plus an integral superposition of modes. This later contribution is given by an integration around a branch cut. In this analysis the sound‐speed and density profiles are partitioned in layers such that in each layer the square of the index of refraction can be approximated by a straight line and the density by a constant. The profile is terminated in a high‐speed isovelocity half space. Using this profile, it is possible to express the depth‐dependent portion of the pressure field in terms of Airy functions. Evaluation of the modal and branch line contributions shows that the branch integral can contribute significantly to the pressure field over a range equal to at least one water depth and in some cases to many water depths.Subject Classification: 30.20, 30.25.

Band-structure effects in field-emission energy distributions in tungsten

Abstract: A calculation of the total energy distribution of field-emitted electrons from the (100) and (110) planes of tungsten is presented. The potential inside the metal is represented by a superposition of atomic potentials in the muffin-tin approximation as evaluated by Mattheiss. The crystal is terminated abruptly at the surface and the potential barrier outside the surface is represented by a straight line. Using the mathematical apparatus developed by low-energy-electron-diffraction theorists we reproduce the band structure of tungsten and calculate the tunneling probability by matching the wave function outside the metal to a superposition of Bloch waves, including a number of evanescent waves, inside the metal. The total energy distribution from the (100) plane exhibits a sharp peak at 0.29 eV below the Fermi level which suggests that the experimentally observed hump at about the same energy is due to band-structure effects. We show that this peak in the total-energy-distribution curve is due to electron states associated with certain sections of the constant-energy surface and we argue that spin-orbit interaction is not likely to remove this peak. The calculated distribution from the (110) plane does not show any marked deviation from the free-electron behavior in agreement with the experimental observations. For the (100) plane the ratio of the exact total energy distribution to its corresponding value evaluated in the free-electron approximation for the (100) plane is considerably smaller than unity away from the peak in the total-energy-distribution curve. For the (110) plane it is of the order of unity.

Numerical Analysis of Distortion of a Vortex Filament

Abstract: Motion of a vortex filament with a narrow core in an inviscid and incompressible fluid otherwise at rest is investigated numerically. Two types of initial conditions, a sinusoidal form with a single Fourier component and that of a superposition of the first and the third components, are assumed. In both cases, interaction between different components occurs owing to the nonlinear property of the governing equation, but the filament does not show random behavior and after a certain time it recurs nearly to the initial state. Analytical treatment is made and the numerical results are confirmed. Next, the case where an external flow field with an axisymmetric potential flow exists is examined, and an onset of randomness is observed. This is considered as due to a computational error amplified through the interaction between the filament and the external flow.

Dependence of reproducing gap null on head geometry

Abstract: The spatial frequency response of reproduce heads with infinite depth, but finite pole length, is determined to very good accuracy by superposition of conformal map solutions for simpler geometries. This approach yields closed form approximations of the frequency response which are accurate to better than 7% for any head length-to-gap length ratio and any spatial frequency. In particular, the effect of this ratio on the location of the first gap null in the frequency response spectrum is explored for narrow pole length heads. The results lead to design considerations for extending the useful frequency range of thin film reproduce heads.

Linear and non-linear performance of transducer and pupil in Calliphora retinula cells.

Abstract: 1. Intracellular recordings have been made from the blowfly (Calliphora erythrocephala) retinula cell; apart from the transducer mechanism, these cells also feature a pupil mechanism. 2. At several mean intensity levels, within the apparently linear range of response, frequency characteristics of amplitude and phase and responses to 'delta'-flashes and 'delta'-flash pairs have been obtained. 3. Fourier methods have shown these responses to be mutually compatible, confirming linearity in these circumstances. 4. Non-linear behaviour can be made to appear at the lower frequencies when the modulation depth is increased. 5. Non-linearities can also appear through application of the superposition test: a low frequency sine wave, modulated so as to elicit an apparently linear response, and a high frequence sine wave which does not give rise to non-linearity even at the highest modulation depths can, when superimposed, yield a greater response to the latter when situated at the minima of the former than at its maxima. 6. At frequencies above approximately 1 Hz these superposition non-linearities are attributed to the transducer mechanism gain control. Below this frequency the pupil mechanism takes part considerably in the retinula cell's total observed gain control: its characteristics remain yet to be cleared up. 7. The transducer's linear and non-linear properties fit in closely with those of the Fuortes-Hodgkin model which couples increases in gain and time constants. 8. The Fuortes-Hodgkin model will probably require some quantitative modifications in the originally treated case of Limulus, on account of its pupil. 9. Finally, the merits of Veringa's diffusion model, and the possibility of eventually joining this model with the Fuortes-Hodgkin one are pointed out briefly.

Quantum statistics of light after one-photon interaction with matter

Abstract: The time development of the normally ordered generating functional of a light beam interacting linearly with a collection of two-level atoms is determined exactly. Rigorous conclusions on changes of the statistical properties of the beam are drawn. It is shown that the field after the interaction appears to be the superposition of a thermal field and of a field statistically similar to the initial one.

A digital technique for analyzing a class of multicomponent signals

Abstract: There are numerous occasions in applied sciences where there is need to analyze multicomponent signals formed by a linear superposition of functions having the same shape and location but different widths and amplitudes. A fast digital technique capable of producing the spectrum, i.e., the distribution of component amplitudes versus component widths, of such experimental data is presented in this paper. The method described does not require any a priori numerical information regarding the composition of the signal, can be easily and efficiently implemented on digital computers, can be automated, and is essential for an automated interpretation of these types of laboratory data. The technique is based upon a nonlinear change of variables followed by a deconvolution. A low-pass filtering is necessary in the final stage of data processing to reduce the effect of computational and experimental noise. As a main example of the practical implementation of this technique within a laboratory environment, the paper details its usage in connection with pulsed NMR measurements on malignant tissues, where the data have the form of a superposition of exponential decays.

A unified view of high density digital recording theory (invited)

Abstract: In an attempt to unify digital recording theory, the principal ideas concerning pulse response, linear superposition, medium properties, head geometry, channel codes, signal-to-noise ratio, post-equalization and achievable bit densities are discussed.

Theory of diffraction effects in electron emission from solid surfaces

Abstract: Electron emission is represented in terms of an initial state that is a superposition of outgoing partial waves centred on an emitting ion core. A theory of the multiple scattering of this initial state by the ion cores of the solid is given using the angular momentum representation, and it is shown how to calculate the intensity of emission as a function of the direction relative to the surface. The theory is strictly confined to clean surfaces or surfaces with ordered adlayers. The calculations employ the Reverse Scattering Perturbation Theory previously developed for LEED, involving an iterative procedure which enables one readily to establish convergence. Given translational symmetry, surfaces of arbitrary complexity can be treated within the same framework. The theory can be applied to cases where several or many ion cores emit coherently, by summing the amplitudes arising from each before calculating the intensity. Therefore not only Auger emission and photoemission from core states can be described, but also photoemission from valence states of the surface.

Multifrequent Microwave Holography at Close Range

Abstract: By recording and reconstructing several microwave holograms of the same object at different frequencies, and superposition of the complex amplitude of the reconstruced images in an appropriate manner, the resolving power can considerably be improved. Experiments in the 8-12 GHz region are presented with simple objects. The reconstruction is done numerically on a digital computer.

Holographic Aperture Synthesis via a Transmitter Array

Abstract: The problem of the synthesis of a large holographic aperture by means of a smaller aperture and an array of sources is addressed. Emphasis is placed on the problem of parallel acquisition of the contributions associated with the different sources and on a comparison between alternative superposition approaches (either superposition of holograms prior to reconstruction or superposition of reconstructed image fields). Experimental results are presented in which synthetic-aperture images due to a 10 × 10 hologram array and a 2 × 2 transmitter array are compared with images due to (a) a 10 × 10 array and a single transmitter, and (b) a 20 × 20 array and a single transmitter.

Finite element calculation of stress intensity factors using superposition

Abstract: A new approach to the finite element technique for calculating stress intensity factors of cracks is presented with the aid of analytical functions in the formulation, in which the total displacements of the conventional finite element method and the unknown constants of the analytical solution are determined in the modified principle of virtual work. The simple problem of a square plate with a central crack subjected to in-plane tension is solved and compared with the available exact solutions.

. A Superposition Principle for Siegert Resonant-States

Abstract: The utilization of the Siegert resonant states as a natural basis for the scattering process has been constrained by the lack of a convergent superposition formula. This paper presents such an expansion of the exact scattering wavefunction over the s‐wave Siegert resonant states for finite range potentials. Any integral property of the scattering wavefunction can then be calculated from the resonances.

Perturbation theory using the Yvon–Born–Green equation of state for square‐well fluids

Abstract: Using the Yvon−Born−Green integral equation with superposition approximation for the square−well potential with R=σ2/σ1=1.5, values of the coefficients of the reciprocal temperature expansion of the Helmholtz energy are computed. The results compare favorably up to values of n*=0.7 with the machine computations of Alder, Young, and Mark through A4/NkT despite obvious differences in the hard−sphere reference pair correlation functions between the two approaches.