Showing papers on "Superposition principle published in 1984"

Guided wave propagation in laterally varying media — I. Theoretical development

Abstract: Summary. The propagation of surface waves in a laterally varying medium can be described by representing the wavetrain as a superposition of modal contributions for a reference structure. As the guided waves propagate through a heterogeneous zone the modal coefficients needed to describe the wavetrain vary with position, leading to interconversions between modes and reflection into backward travelling modes. The evolution of the modal terms may be described by a set of first-order differential equations which allow for coupling to both forward and backward travelling waves; the coefficients in these equations depend on the differences between the actual structure and the reference structure. This system is established using the orthogonality properties of the modal eigenfunctions and is valid for SH-waves, P-SV-waves and full anisotropy. The reflected and transmitted wavefields for a region of heterogeneity can be related to the incident wave by introducing reflection and transmission matrices which connect the modal coefficients in these fields to those in the incident wavetrain. By considering a sequence of models with increasing width of heterogeneity we are able to derive a set of Ricatti equations for the reflection and transmission matrices which may be solved by initial value techniques. This avoids an awkward two-point boundary value problem for a large number of coupled equations. The method is demonstrated for 1 Hz Lg- and Sn-waves in a multilayered model for which there are 19 coupled modes. The method is applicable to three-dimensional heterogeneity, and we are able to show that the interconversion between Love and Rayleigh waves, in the presence of gradients in seismic properties transverse to the propagation path, leads to a net rate of increase of the transverse components of the seismogram at the expense of the other components.

The principle of superposition and its application in ground-water hydraulics

Abstract: The principle of superposition, a powerful mathematical technique for analyzing certain types of complex problems in many areas of science and technology, has important applications in ground-water hydraulics and modeling of ground-water systems. The principle of superposition states that problem solutions can be added together to obtain composite solutions. This principle applies to linear systems governed by linear differential equations. This report introduces the principle of superposition as it applies to ground-water hydrology and provides background information, discussion, illustrative problems with solutions, and problems to be solved by the reader.

The two-dimensional Gaussian beam synthetic method: Testing and application

Abstract: The Gaussian beam method of Cervený et al. (1982) is an asymptotic method for the computation of wave fields in inhomogeneous media. The method consists of tracing rays and then solving the wave equation in “ray-centered coordinates.” The parabolic approximation is applied to find the asymptotic local solution in the neighborhod of each ray. The approximate global solution for a given source is then constructed by a superposition of Gaussian beams along nearby rays. The Gaussian beam method is tested in a two-dimensional inhomogeneous medium using two approaches. One is the application of the reciprocal theorem for Green's functions in an arbitrarily heterogeneous medium. The discrepancy between synthetic seismograms for reciprocal cases is considered as a measure of the error. The other approach is to apply Gaussian beam synthesis to cases for which solutions are known by other approximate methods. This includes the soft basin problem that has been studied by finite difference, finite element, discrete wavenumber, and glorified optics. We found that the results of these tests were in general satisfactory. We have used the Gaussian beam method for two applications. First, the method is used to study volcanic earthquakes at Mount Saint Helens. The observed large differences in amplitude and arrival time between a station inside the crater and stations on the flanks can be explained by the combined effects of an anomalous velocity structure and a shallow focal depth. The method is also applied to scattering of teleseismic P waves by a lithosphere with randomly fluctuating velocities.

Geometrical theory of diffraction, evanescent waves, complex rays and Gaussian beams

Abstract: Summary. The Gaussian beam method has recently been introduced into synthetic seismology to overcome shortcomings of the ray method, especially in transition regions due to focusing or diffraction where ray theory fails. One proceeds by discretizing the initial data as a superposition of paraxial Gaussian beams, each of which is then traced through the seismic environment. Since Gaussian beam fields do not diverge in ray transition regions, they are ‘uniformly regular’ although the quality of this regularity depends on the beam parameters and on the ‘numerical distance’ which defines the extent of the transitional domain. However, when Gaussian beam patches are used to simulate non-Gaussian initial data, there arise ambiguities due to choice of patch size and location, beam width, etc., which are at the user's disposal. The effects of this arbitrariness have customarily been explored by trial and error numerical experiment but no quantitative recommendations have emerged as yet. As a step toward a priori predictive capability, it is proposed here to perform a systematic study on analytically tractable prototype models of how the parameters and location of a single beam affect the quality of the observed seismic field, especially in ray transition regions. The conversion of ordinary ray fields into beam fields in canonical configurations can be accomplished conveniently by displacing a real source point into a complex coordinate space. Thus, the desired beam solutions can be obtained directly from available ray, and even paraxial ray, fields. Complex ray theory and its implications are reviewed here, with an emphasis on improvements of beam tracking schemes employed at present.

Singularity expansion representations of fields and currents in transient scattering

Abstract: A unified treatment of the natural mode representations for induced currents and scattered fields is obtained by use of fundamental concepts regarding causality and superposition. The transient scattered field response is shown to have the form of a constant coefficient complex exponential sum only in the "late-time," after the last driven reponse is received from the object. Prior to this, the "early.time" response is found to be due to direct physical optics fields as well as a sum of temporally modulated natural modes produced by the progressive illumination of the incident wavefront. Alternate representations and s -plane behaviors are considered. The implications of these results on natural resonance target identification schemes are discussed.

The force between two planar electrical double layers

Abstract: An expression to calculate density profiles of ions between two charged plates is obtained through the hypernetted chain theory for pure electrolytes. In deriving this expression no superposition approximation is assumed. Separately, a contact theorem is derived from the Born–Green–Yvon equation for pure electrolytes. The combination of these two results allows the evaluation of the pressure which drives two charged plates apart, without assuming a superposition approximation. It is shown that for point ions, this theory reduces to the Deryaguin–Landau–Verwey–Overbeek theory.

Expansion of a high-frequency time-harmonic wavefield given on an initial surface into Gaussian beams

Abstract: Summary. A high-frequency asymptotic expansion of a time-harmonic wavefield given on a curved initial surface into Gaussian beams is determined. The time-harmonic wavefield is assumed to be specified on the initial surface in terms of a complex-valued amplitude and a phase. The asymptotic expansion has the form of a two-parametric integral superposition of Gaussian beams. The expansion corresponds to the relevant ray approximation in all regions, where the ray solution is sufficiently regular (smooth) in effective regions of the beams under consideration.

Linear longitudinal oscillations in collisionless plasma diodes with thin sheaths. Part II. Application to an extended Pierce-type problem

Abstract: The integral‐equation method developed in Part I is applied to a Pierce‐type diode [J. Appl. Phys. 15, 721 (1944)] whose external circuit involves a resistor, an inductor, and a signal generator. The general linear perturbational problem is solved analytically for the small‐amplitude quantities je(t) (external‐circuit current density) and E(x, t) (electrostatic field). Each of these quantities can be constructed from a ‘‘spatial’’ Green’s function (accounting for initial perturbations of the plasma), a ‘‘temporal’’ Green’s function (accounting for external‐generator signals), and two functions associated with the initial state of the external circuit. The solutions generally exhibit an initial transient and an asymptotic part, the latter being a superposition of eigenmodes only. Systematic numerical results for eigenfrequencies and eigenmode profiles in some typical parameter regions demonstrate that the linear response and stability behavior of the diode system may substantially depend on the propertie...

The Ray Method and the Theory of Edge Waves

Abstract: Summary. A modification of the ray method including diffraction is outlined. The study is designed for the computation of the wavefields in 3-D inhomogeneous media containing such structural elements as pinch-outs, vertical and oblique contacts. faults and so on. The approach is based on the theory of edge waves. The total wavefield is considered as the superposition of two parts. The first part is described by the ray method. It has discontinuities because of its shadow boundaries. The second part is a superposition of two types of diffracted waves, caused by the edges and vertices of interfaces. This part smooths away the above-mentioned discontinuities, so that the total wave-field is a continuous one. The effects of multiple diffraction are considered. Of special importance is a mathematical form of amplitudes of diffracted waves, described with unified functions of eikonals. In fact, it allows all additional computations to be considered by finding the eikonals of diffracted waves.

Time Domain Response of a Sphere in the Field of a Coil: Theory And Experiment

Abstract: The time-harmonic solution for the anomalous vector potential due to a conducting permeable sphere in the field of a current-carrying loop is used to derive the corresponding step response. The step response is then used to obtain analytical expressions for the voltage induced in a second loop due to a chosen exciting current pulse train. The voltage induced in an actual system of coils is obtained by superposition. The effect of the measurement system is included in the analysis in order to experimentally verify the model. Measured responses of a number of aluminum and steel spheres at various distances from the coils are compared with theoretical predictions. The agreement between the two is generally good.

Vortex-Induced Vibrations of a Flexible Cylinder Near a Plane Boundary Exposed to Steady and Wave-Induced Currents

Abstract: Model tests were carried out in a wave tank to determine the effect of combined steady and wave-induced currents and/or the proximity of a plane boundary (seabottom) on the vortex-induced vibrations of a flexible pipe. The response of the center of the pipe span was measured using a biaxial accelerometer. The results show that the proximity of the plane boundary and/or superposition of waves on the steady flow have a pronounced effect on the amplitude and frequency response in both the transverse and in-line directions.

Statistics of the Stokes parameters in speckle fields

Abstract: A certain kind of speckle field can be described as a superposition of two fully developed and linearly polarized speckle patterns with an additional elliptically polarized coherent background intensity. The first-order statistics of the Stokes parameters have been studied analytically for the case in which the background intensity is zero and for which the probability density function of the moduli of the electric field follows a bivariate Rayleigh distribution. The probability density functions of the Stokes parameters have been further evaluated when the speckle field is superimposed upon a coherent background intensity.

The relation between Gaussian beams and Maslov asymptotic theory

Abstract: The application of Maslov asymptotic theory in a general 3-D mixed subspace of 6-D complex phase space is proposed to obtain the integral superpositions of Gaussian packets and beams. The ray method and the superposition of plane waves (Maslov method of Chapman and Drummond [7]) are special limiting cases of the above mentioned approach. The same high-frequency asymptotic expansion formulae for seismic body waves were derived previously in [8] using the Gaussian beam method.

Superposition of solutions to Bäcklund transformations for the SU(n) principal σ-model

Abstract: We show that the Backlund transformations for the SU(n) principal σ‐model may be linearized using a geometrical interpretation of these equations involving the minimal orbit of SU(n, n) in the Grassmann manifold Gn (C2n). Linearization puts the equations in Zakharov–Mikhailov–Shabat (ZMS) form. Using this form of the equations, we prove inductively a nonlinear superposition law and a permutability theorem for iterated Backlund transformations analogous to known results in the theory of the sine–Gordon and KdV equations. From the superposition law we get an explicit form for multisoliton solutions to the σ‐model.

Analysis of active control by surface heating

Abstract: The excitation of boundary-layer disturbances by active, localized periodic heating of the flow surfaces is investigated analytically. A triple-deck model is used in the matched-asymptotics approach, and the incompressible case with small temperature variations, a linearized 3-deck problem, is considered. This case corresponds to the experimental (water-tunnel) conditions of Liepmann et al. (1982), who demonstrated that the induced disturbances can be used to enhance or suppress the Tollmien-Schlichting waves. Here the energy equation is uncoupled from the momentum and continuity equations to permit definition of the thermal-sublayer temperature profile by superposition of canonical solutions, which are presented in an appendix. A diagram of the 3-deck structure and graphs of calculated results are included.

Exact Formulation of Multiple Scattering in a Three-Dimensional Cylindrical Geometry

Abstract: Exact expressions for the source function, flux, and scattered intensity normal to the surface are developed in cylindrical coordinates for a three-dimensional, absorbing, emitting, isotropically scattering medium exposed to both diffuse and collimated radiation. Simplifications of these expressions for certain important geometries and uniform loading are presented. Also, superposition of these equations and radiative equilibrium are discussed. The generalized three-dimensional equations are shown to reduce to the familiar one-dimensional results. Also, the equations for a strongly anisotropic phase function which is made up of a spike in the forward direction superimposed on an otherwise isotropic phase function are expressed in terms of the isotropic expressions.

Spectral Analysis of Random Shrinkage Stresses in Concrete

Abstract: Random variation of environmental humidity is characterized by its power spectrum, and random variation of pore humidity and stresses in a halfspace is analyzed under the assumption that the problem is linear (or linearized). The diffusion equation and the superposition integral for stress relaxation are used. The dependence of both the creep properties and the drying diffusivity on age is taken into account. Complex variable expressions for the frequency response functions of humidity and stress are obtained, and are evaluated numerically. In contrast to nonaging structures, these functions depend on both the current age of concrete and the age when drying starts. The standard deviations of pore humidity and of stress exhibit oscillations about a drifting mean. For typical diffusivities of concrete, the solution is non-stationary for at least 50 yrs, and for environmental fluctuations whose period does not exceed one year, the fluctuations are not felt at depths over 20 cm below the surface. Since, even for aging structures, the spectral densities of input and response are related algebraically, the spectral method is computationally more efficient than the impulse response function method, in which the autocorrelation functions of the input and the response are related by integrals.

Waves in Plasmas

Abstract: Any periodic motion of a fluid can be decomposed by Fourier analysis into a superposition of sinusoidal oscillations with different frequencies ω and wavelengths λ. A simple wave is any one of these components. When the oscillation amplitude is small, the waveform is generally sinusoidal; and there is only one component. This is the situation we shall consider.

Symmetry constraints on the Hartree-Fock model: II. Basis set superposition errors and the ‘Algebraic Approximation’

Abstract: It is shown that the removal of the ‘spherical approximation’ from the Atomic Hartree-Fock equations generates a set of atomic orbitals which are not always eigenfunctions of L 2. In particular, the s-type and d-type orbitals ‘mix’. The result of this mixing is to generate a set of ‘valence’ orbitals for these atoms which solves the old problem of why, if the 3d lies below the 4s, does the 3d not always fill up first.

Representation of static three-body correlations in dense fluids

Abstract: This paper reports upon new studies of the error involved in the superposition approximation to the triplet distribution function g 3 in atomic fluids. The analysis is performed by first expressing g 3 in spherical perimetric coordinates (ρ, θ, φ) in which the ‘size’ of a triangle, ρ, is measured independently of its ‘shape’ θ and φ. Then g 3 is expanded in spherical harmonics of the triangular shape parameters. The expansion coefficients were evaluated by analysing molecular dynamics simulation data for a Lennard-Jones fluid and the results were compared with the corresponding coefficients appearing in an explicit expansion of the superposition approximation. We show how these two sets of coefficients may be combined to give a precise representation of the difference between the simulation and the superposition correlation functions. This formulation is then tested by evaluating the Axilrod-Teller triple-dipole energy directly from simulation and from the superposition approximation plus correction terms.

Line sources of instability waves in a Blasius boundary layer

Abstract: Numerical solutions of the instability wave pattern behind a harmonic point source in a Blasius boundary layer are used to form line sources by superposition. For infinite-length spanwise line sources of constant amplitude and phase, the result is just the two-dimensional normal mode of the same frequency; for a sinusoidal amplitude or linear phase distribution, the result is an oblique normal mode of the same spanwise wavenumber. A finite-length spanwise source simulates a vibrating ribbon. In a study of the influence of the source length on the downstream amplitude, the effective field of view is found to have a half-angle of about 16 deg. If the source tips are within this field, the amplitude may be either greater than or less than the comparable normal-mode amplitude, depending on the distance from the source and the spanwise location. For an oblique line source, the downstream wave development at each spanwise location is found to be close to, but not identical with, that of an oblique normal mode which originates at the source with the initial wave angle of the source and satisfies the irrotationality condition on the wavenumber vector.

A Three-Dimensional Numerical Method Based on the Superposition Principle

Abstract: A three-dimensional numerical method based on the superposition principle for the solution of the heat diffusion equation is derived for Cartesian coordinates and tested for three different boundary conditions: a constant heat flux density, a convective-type surface heat flux, and a sudden cooling of the surface to a constant temperature, In addition, this three-dimensional numerical method is compared with the popular three-dimensional Brian's alternating direction implicit (ADI) method. The method based on the superposition principle has the same degree of accuracy in most cases as the method normally used for these types of calculations. In addition, its algorithm is considerably simpler to formulate and easier to program, and it requires about half the computing time needed to solve the problem when using Brian's ADI method. On the other hand, Brian's ADI method is unconditionally stable, but the method based on the superposition principle is not.