Showing papers on "Superposition principle published in 1977"

Computational Solution of Linear Two-Point Boundary Value Problems via Orthonormalization

Abstract: We discuss algorithmic matters of a computer code for solving linear two-point boundary-value problems. The method of solution uses superposition coupled with an orthonormalization procedure and a variable-step Runge–Kutta–Fehlberg integration scheme. Each time the linearly independent solutions start to lose their numerical independence, the vectors are reorthonormalized before integration proceeds. The underlying principle of the algorithm is then to piece together the intermediate (orthogonalized) solutions, defined on the various subintervals, to obtain the desired solution.

Large-time behavior of solutions of initial and initial-boundary value problems of a general system of hyperbolic conservation laws

Abstract: We study the asymptotic behavior of the solution of the initial and initial-boundary value problem of hyperbolic conservation laws when the initial and boundary data have bounded total variation. It is shown that the solution converges to the linear superposition of traveling waves, shock waves and rarefaction waves. The strength and speed of these waves depend only on the values of the data at infinity.

A fluid in contact with a wall

Abstract: The density profiles for a hard-sphere fluid in front of an ideal wall obtained either with the Percus-Yevick or the superposition approximation are compared. In an expansion of the profile in powers of the bulk density, the coefficient of the third power is better in the superposition approximation. The density at the wall is given exactly by the superposition approximation, whereas the Percus-Yevick result is too low. For the superposition approximation a calculated density profile and the excess surface density as function of the bulk density are also given.

An antenna for limited scan in one plane: Design criteria and numerical simulation

Abstract: An analytical and numerical study has been performed on a novel design scheme for an antenna system for limited one-dimensional scan. The system has a number of control elements approximately equal to the minimum theoretically compatible with the aperture size and field of view (FOV). The radiating structure consists of a "bootlace" lens with linear outer and circular inner profiles. This geometry plays a basic role in determining excellent scan performance over a moderate frequency band. A linear array whose size depends critically upon the scan requirements and the lens focal length is located on the focal plane and is focussed onto the inner lens profile. The array is fed by a hybrid network (HN) performing a spatial Fourier transformation. The input ports of the HN are fed by the output ports of a beam forming network (BFN) through a set of variable phase shifters. The BFN has separate input ports for the sum and difference patterns, controlled independently, The system works as follows. The antenna illumination is synthesized as the weighted superposition of components illuminations or "overlapping subarrays" each of which is due to the excitation of one of the ports of the HN. The amplitudes of the subarray excitations are fixed and determined by the power divisions provided by the BFN. Their phases are controlled by the set of variable phase shifters. A desirable feature of the scheme is that for a fixed phase shifter setting neither the beam scan nor its width changes for a moderate frequency variation. Through a suitable design of the BFN, ultralow sidelobes outside the FOV can be achieved at the expense of a slight reduction of the illumination efficiency, which is, however, always high, since the aperture is fully used. Extensive numerical computations for an antenna having a half-power beamwidth of 1.2\deg show that the sum beam can be scanned in a sector greater than \pm3 beams, on a band of 20 percent with excellent performance form the viewpoint of gain and sidelobes-the scan sector being slightly less for the difference beam.

Superposition of the radiation from N independent sources and the problem of random flights

Abstract: It is often stated that the intensity of the signal produced by superposition of N equally intense, but randomly phased, monochromatic coherent waves tends to N as N becomes large. An examination (first made by Rayleigh) of the distribution of intensities obtained by superposing N independent monochromatic waves shows that the mean intensity is N and the variance is N2−N. Exploiting the analogy of this problem to random walk in two dimensions (’’random flight’’), we have recalculated the distribution for N=2 to 6 and compared it with the results from a computer experiment.

Maximization of lineal recording density

Abstract: A brief outline is given for design and analysis of the analog signal processing portion of peak sensing digital magnetic recording systems for which lineal recording density is maximized. This is a linear system analysis in which superposition is applicable to saturation recording if the media is indeed fully saturated at the termination of each transition. Results are stated in terms of parameters of run bounded codes because of their practical importance. However, the concept applies in general to the maximization of the information rate through a channel of finite bandwidth under pulse-position modulation.

A wave-envelope technique for wave propagation in non-uniform ducts

Abstract: The method of variation of parameters is used to analyze wave propagation in variable area, plane ducts with no mean flow. The method has an explicit representation of the fast axial variation of the acoustic modes, and numerical integration is required only for the slower axial variations of the mode amplitudes and phases. Results are presented which demonstrate the numerical advantages of the method. Comparison of the results with those of a small perturbation theory are given. The relationship between this method and the method of weighted- residuals is discussed. 7 among others. Each approach has unique characteristics and advantages, as well as obvious limitations, either of a numerical or physical nature. For example, no tests have been made to establish the range of geometry variations for which the multiple-scale solutions remain valid; direct numerical analyses require small step sizes and large computation times for high frequencies and high-order modes because of the rapid axial and transverse oscillations; and weighted-residual analysis7 also requires a numerical integration over each axial oscillation of the signal although the transverse variations of high-order modes present no difficulty. The computation of these axial variations has been sim- plified in a direct numerical analysis of wave propagation in constant-area ducts by using an estimate of the harmonic axial variations of the fundamental mode.8 This procedure was reportedly advantageous even with estimates that were only moderately accurate. In the work reported here, acoustic propagation in variable- area ducts without mean flow is analyzed by the method of variation of parameters 9 in a manner that incorporates features of several previous investigations. To facilitate the study of high-order modes and multimodal interactions, the acoustic disturbance is represented as a superposition of parallel-duct eigenf unctions. Moreover, the fast axial variation is given explicitly, and numerical integration is required only for the slower axial variations of the amplitudes and the phases of the modes. We have borrowed the term 'wave-envelope method' as a description of this aspect of the method, although in most respects it bears little resemblance to the procedure used by Baumeister.8 Finally, the representation of the acoustic wave is required to satisfy an integrability constraint derived from the wave equation; this feature is similar to that used in the multiple-scale analysis by Nayfeh and Telionis,3 but does not have the small per- turbation limitation.

Propagation in Slowly-Varying Wave-Guides

Abstract: It is shown that a slight modification to the ray ansatz of short wave asymptotics can lead to a considerable improvement in the uniformity of the solution. A superposition of ray solutions is used to obtain a description of the sound waves generated by a multi-pole point source in a weakly stratified ocean.

Coupled Modes Approach to Non-Linear Dispersive Wave. I. Recurrence of Soliton

Abstract: The K-dV equation is numerically solved by means of a mode coupling method for periodic waves launched at a certain boundary. The coupled modes equations for slowly varying complex amplitudes are first derived and are solved. The superposition of N modes yields almost periodic solutions in space which represent the recurrence of an initial state. The recurrence length is given explicitly as a function of a frequency and an initial amplitude at the boundary. The recurrence is, in general, imperfect and the waveform differs from each other at different recurrence points. The reason of the imperfect recurrence is discussed from the view points of wave-wave interaction and of soliton-soliton interaction.

Harmonic analysis over finite commutative groups in linearization problems for systems of logical functions

Abstract: In this paper we consider the linearization problems for systems of two- and many-valued logical functions by methods of abstract harmonic analysis. By an optimal linearization we mean a representation of the original system as a superposition of linear and nonlinear vectorfunctions, such that the complexity of the nonlinear part is minimized. The problems are solved for the three most simply computed criteria of the complexity of systems of logical functions. Logical functions are treated as functions defined on finite commutative groups. The solutions of the linearization problems involve the use of Fourier expansions of these functions in terms of the group characters. The spectral characteristics thus arising, as well as the correlation characteristics obtained from the original function by double spectral transforms, are used as a working tool in solving linearization problems. The solutions are exact and convenient from the computational standpoint. The paper illustrates the effectiveness of the methods of abstract harmonic analysis in problems of synthesis and optimization of digital devices.

Coherent optical implementation of 1-D Mellin transforms.

Abstract: Peter Kellman and Joseph W Goodman Stanford University, Department of Electrical Engineering, Stanford, California 94305 Received 20 May 1977 Recent interest in Mellin power spectra, complex filtering, and correlation is due to the scale invariance of the Mellin transform Mellin correlation has been cited for application to 2-D target detection and 1-D signal processing where parameters of scale and Doppler are unknown In such applications, Fourier domain matched filtering requires either sequential hypothesis testing or multichannel processing Loss of correlation in the Fourier implementation due to scale difference between received signal and reference is exchanged, however, for an equally severe loss in the Mellin correlation due to time or positional displacement A coherent optical technique for Mellin transforms is described here in which signals may be processed in real time to provide scale invariant correlations The parameterization on time uncertainty is provided for automatically by utilizing an acoustooptic spatial light modulator Methods for performing general linear integral transforms using parallel processing in both spatial dimensions allow real time operation Several of these methods have been outlined recently; one configuration, well suited for implementing Mellin spectrum analysis, is reviewed here The general linear superposition integral may be written as

On the g-shift of S-state ions

Abstract: A superposition model of the anisotropic g-shifts of S-state ions substituted in crystals is developed. Its relationship to the existing theoretical and experimental situation is explored.

On the thermodynamics of nonlinear single integral representations for thermoviscoelastic materials with applications to one-dimensional wave propagation

Abstract: Thermodynamic theory is used to develop single integral constitutive relations for the nonlinear thermoviscoelastic response to arbitrary stress and temperature histories; the thermomechanically coupled energy equation is also obtained. The thermorheologically simple material, modified superposition and the isotropic stress power law are discussed in detail. A modified Fourier heat conduction law is employed to ensure that the propagation of thermal disturbances takes place at a finite velocity. Using the nonlinear thermoviscoelastic stress power law along with the linearized energy equation and modified Fourier law, one-dimensional wave front solutions are obtained.

A Posteriori Compensation for Rigid Body Motion in Holographic Interferometry by Means of a Moire-technique

Abstract: A Moire-technique is introduced that regenerates fringe systems due to some deformation, which are distorted by rigid body motion of the object under test. The kind of rigid body motions that can be eliminated is determined by the condition, that one has to be able to simulate the fringe system the motion would generate by itself. Theory is discussed starting from the Sampling Theorem of the Information Theory. The superposition is realized experimentally by a T.V.-technique.

Geometric Structure of Charged Fields

Abstract: The concept of parallel transport is generalized. It leads to a geometric structure of charged fields. A quantization principle of charge is put forward and the Aharonov-Bohm effect is interpreted within the framework of a one-valued field representation and superposition principle. No explicit equation is demanded to describe the electron field: The interpretation is a pure geometric one.

Quantification du champ électromagnétique au voisinage d'un miroir plan métallique

Abstract: We quantify the electromagnetic field in a half-space bounded by a perfect mirror. By analogy with the quantization near a dielectric, by Carniglia and Mandel, we use doublet modes obtained by superposition of incident and reflected waves satisfying Fresnel's conditions. These functions form a complete basis in the space of the field states. By expanding the electromagnetic field in terms of these doublet modes, the energy and the impulsion parallel to the mirror reduce to the sum of the contributions of independent harmonic oscillators. It is therefore possible to quantify as in a free field. We use successively Maxwell–Minkowski's tensor and de Broglie's. With the latter, the calculus is more straightforward and easier than with the former. The interest in quantum formalism of doublet modes is that interactions with the electromagnetic field near a mirror can be studied as in a free field.

Wave Diffraction by a Crack: Finite Element Simulations

Abstract: The two-dimensional problem of a plane longitudinal stress wave diffracted by a straight crack is simulated by the finite element method. Time-dependent stress-intensity factors are computed with the aid of a crack-tip singularity element. Numerical results are obtained from two finite-element models using both consistent-mass and lumped-mass matrices and two different sizes of time step in the time-integration algorithm. The problem is simulated by application of two different superposition schemes. In one case the finite element model is loaded on a boundary remote from the crack; in the other case the model is excited through loading of the crack face. Computed stress-intensity factors are in good agreement with an existing analytical solution.

Coherent Transient Excitation and Superposition of Adiabatic States

Abstract: Double resonance, resonance Raman and resonance fluorescence are discussed in terms of a superposition of adiabatic states. The adiabatic states are the instantaneous eigenstates of a two-level Hamiltonian which includes the interaction with a near-resonant coherent pulse and which has been transformed by a unitary transformation so that its only time-dependence is that of the field envelope of the pulse. Resonance Raman results from spontaneous transitions to a third, lower state from the adiabatic following state which reduces to the initial (lower) state of the two-level system in the absence of radiation. Resonance fluorescence is produced by spontaneous transition to the lower third state from the upper adiabatic state which reduces to the upper state of the two-level system in the absence of radiation. Differential equations allowing the calculation of the probability amplitudes of the adiabatic states are given and a method of solving these equations by successive approximation is proposed. Saturation effects such as dynamic Stark shift and splitting (Autler—Townes effect) and optical nutation are interpreted in terms of perturbation of adiabatic energy levels and interference effects between adiabatic states. The possibility of observing optical nutation in a two-level system by using time-resolved double resonance experiments is suggested. Decay of the states of the two-level system is taken into account leading to extended adiabaticity conditions which show that adiabatic following becomes possible even at resonance. For weak fields it is shown that adiabatic following at resonance can produce light scattering narrower than the linewidth as predicted by Heitler. The adiabatic states theory has also been extended to near-resonance multiphoton interaction for systems that can be described by an effective two-level Hamiltonian.

Quantum statistical properties of higher-order non-linear optical processes

Abstract: We study the quantum statistical properties of radiation in higher-order parametric processes and higher harmonic generation. Adopting the Heisenberg picture, we use the short-time approximation to solve the Heisenberg equations and to calculate the quantum characteristic functions and quasi-distributions. The lossy mechanism is included with or without rotating terms. We show that the statistical properties of radiation involved in these non-linear optical processes are described by the model of the superposition of coherent and chaotic fields with correlated components in this approximation. Generalizations of the well-known conservation laws for the number operators are derived. We show for parametric processes that the pumping mode has tendency to be coherent while the signal modes are obtaining a noise. In the higher-harmonic generation, the basic radiation is losing its coherence proportionally to its intensity while the generated higher-harmonic radiation has tendency to be coherent.

On the boundary conditions at surface current distributions described by the first derivative of Dirac's impulse

Abstract: Boundary conditions at surface distributions of doublets of electric current are considered in this work It is shown that such current distributions can be treated as the superposition of a double plus a single layer of electric current, the latter being equivalent to a simple layer of magnetic current Accordingly, it is shown that proper distributions of purely electric current can be specified over any given closed surface so that no fields are excited inside the volume bounded by the surface

Spinor Soliton as an Elementary Particle

Abstract: Classical and quantized spinor soliton are investigated from the viewpoint that the ele­ mentary particles are regarded as a kind of soliton. The non-linear spinor equation proposed by Heisenberg is studied in two-dimensional case which is usually called the Thirring modeL The results obtained are as follows: The principle of superposition is realized in a certain sense. A wave packet and a soliton are the same content in this special model. The collision of two solitons are illustrated in detail. 2> Almost all of the known solitons are classical, spinless and two-dimensional (one­ dimensional space and time). Many particle physicists want to regard the solitons as the elementary particles. To do this the equation should be solved and quan­ tized in four-dimensional space-time. However a simple dimensional analysis tells us that the stability of soliton is usually lost in larger dimensions. An equation with higher derivatives might give a stable soliton in four dimensions, but it would cause other problems. On the quantization of soliton there are some inter­ esting approaches, 3> which, however, seem not to be successful completely, be­ cause of the complication of non-linearity. In this connection we are interested in whether the exact solution of a quantized non-linear field equation may allow a soliton. When we consider that the solitons represent the elementary particles, we encounter various problems. For example, the soliton should describe free waves as well as scattering waves. As to the free waves the soliton solution should have some freedom to choose its wave form such as plane wave or wave packet arbitrarily. In addition not only one-particle state but also many-particle state should be described by a soliton solution of an equation. If the particles are separated enough to each other, the wave forms of the particles should be chosen arbitrarily, because they are independent of each other. In other words the prin­ ciple of superposition should be satisfied by free waves. It is, however, generally believed that because of the non-linearity of equation the wave forms of solitons are not free but restricted or determined by the equation and the principle of superposition should never be realized in non-linear equations. In this note we want to show that it is not true for a special non-linear equations.

Resolution limit of the near-field scanning technique

Abstract: The near-field intensity distribution in a fully-excited fibre is calculated by superposition of propagating modes. It is shown that the resolution of the near-field scanning technique is limited by the number of modes supported by the fibre. Experimental confirmation is given.

Response of periodic beam to supersonic boundary-layer pressure fluctuations

Abstract: The response of a periodic beam (modeling a periodic fuselage) to supersonic boundary-layer pressure fluctuations is analyzed on the basis of a scheme in which a decaying turbulence is treated as a superposition of frozen-pattern components, thus allowing the structural response to be similarly superposed and the advantage of frozen-pattern analysis to be maximally utilized. The fundamental solution required for the construction of the total response is one corresponding to the excitation of a frozen-pattern sinusoid. To obtain this fundamental solution, the formulation follows Mead's wave-propagation method (1971), but also takes into account the effect of freestream velocity on the same side of the turbulence excitation and the effect of a cavity on the opposite side of the excitation. As a numerical example, the spectral density of the structural response is computed and the results are compared with experimental data.

Some superposition theorems for second-order fluids

Abstract: Theorems are derived, within the framework of the second-order Rivlin—Ericksen theory, relating to the superposition of longitudinal flows on plane flows in an incompressible isotropic viscoelastic fluid. Also, some theorems are obtained concerning the effects of inertia on the solution of plane flow problems in such viscoelastic fluids.