Showing papers on "Linear approximation published in 1973"

A note on tsunamis: their generation and propagation in an ocean of uniform depth

Abstract: The waves generated in a two-dimensional fluid domain of infinite lateral extent and uniform depth by a deformation of the bounding solid boundary are investigated both theoretically and experimentally. An integral solution is developed for an arbitrary bed displacement (in space and time) on the basis of a linear approximation of the complete (nonlinear) description of wave motion. Experimental and theoretical results are presented for two specific deformations of the bed; the spatial variation of each bed displacement consists of a block section of the bed moving vertically either up or down while the time-displacement history of the block section is varied. The presentation of results is divided into two sections based on two regions of the fluid domain: a generation region in which the bed deformation occurs and a downstream region where the bed position remains stationary for all time. The applicability of the linear approximation in the generation region is investigated both theoretically and experimentally; results are presented which enable certain gross features of the primary wave leaving this region to be determined when the magnitudes of parameters which characterize the bed displacement are known. The results indicate that the primary restriction on the applicability of the linear theory during the bed deformation is that the total amplitude of the bed displacement must remain small compared with the uniform water depth; even this restriction can be relaxed for one type of bed motion. Wave behaviour in the downstream region of the fluid domain is discussed with emphasis on the gradual growth of nonlinear effects relative to frequency dispersion during propagation and the subsequent breakdown of the linear theory. A method is presented for finding the wave behaviour in the far field of the downstream region, where the effects of nonlinearities and frequency dispersion have become about equal. This method is based on the use of a model equation in the far field (which includes both linear and nonlinear effects in an approximate manner) first used by Peregrine (1966) and more recently advocated by Benjamin, Bona & Mahony (1972) as a preferable model to the more commonly used equation of Korteweg & de Vries (1895). An input-output approach is illustrated for the numerical solution of this equation where the input is computed from the linear theory in its region of applicability. Computations are presented and compared with experiment for the case of a positive bed displacement where the net volume of the generated wave is finite and positive; the results demonstrate the evolution of a train of solitary waves (solitons) ordered by amplitude followed by a dispersive train of oscillatory waves. The case of a negative bed displacement in which the net wave volume is finite and negative (and the initial wave is negative almost everywhere) is also investigated; the results suggest that only a dispersive train of waves evolves (no solitons) for this case.

Waveform Segmentation Through Functional Approximation

Abstract: Waveform segmentation is treated as a problem of piecewise linear uniform (minmax) approximation. Various algorithms are reviewed and a new one is proposed based on discrete optimization. Examples of its applications are shown on terrain profiles, scanning electron microscope data, and electrocardiograms. The processing is sufficiently fast to allow its use on-line. The results of the segmentation can be used for pattern recognition, data compression, and nonlinear filtering not only for waveforms but also for pictures and maps. In the latter case some additional preprocessing is required and it is described in [19].

Adiabatic-Following Approximation

Abstract: The nonlinear response of an atom to a near-resonant light pulse is studied using a novel approximation scheme. In first order, the approximate solution reduces to the well-known rate equations. The second-order approximation contains Grischkowsky's adiabatic-following approximation. In each order, the approximate solution of the Bloch equations is presented with a closed-form expression for the error that can be used to investigate its range of validity.

On the dynamics of a system of rigid bodies

Abstract: The existent treatises on rational mechanics do not pay attention to the dynamics of a system of rigid bodies, confining their interest to the general theorems of mechanics, the theory of rotation of a rigid body about a fixed point, some problems of analytical mechanics and the theory of stability. At the same time, the problems whose formulation is reduced precisely to the dynamics of a system of rigid bodies are quite common nowadays, at least during the last 30 years. An attempt to solve such problems on the basis of Lagrange’s second method leads, as a rule, to cumbersome differential equations with a large number of terms; the mechanical interpretation of those is not at all easy. Even computers are then of little help. A solution of the problem of motion of a system of rigid bodies in terms of a linear approximation is a superposition of harmonic motions, and is, therefore, sufficiently transparent. However, in order to get the important qualitative characteristics of the motion, it is necessary to take into account the non-linear terms of initial differential equations and to integrate these equations over a time interval considerably longer than the period of partial oscillations of the associated linear system. However, such integration leads to a rapid accumulation of errors because the actual calculations by the method of finite differences involve a large number of steps.

An appraisal of the computation of statistical parameters in grain size analysis

Abstract: A computer program has been written to overcome the limitations of two published programs that compute statistical parameters in grain size analysis. This program uses all available data, obtained from standard sedimentological analyses, regardless of phi interval, and computes statistical parameters by direct integration of a linear approximation to the cumulative curve. This has the advantage that the distribution curve is known and hence the limitations of the parameters obtained can be assessed. Furthermore, it is argued, with examples, that this approximation introduces no greater degree of inaccuracy than is already present in standard sampling and analytical procedures.

Calculation of linear bestL p -approximations

Abstract: In this paper we are concerned with finding theL p -solution (i.e. minimizing theL p -norm of the residual vector) to a linear approximation problem or, equivalenty, to an overdetermined system of linear equations. An embedding method is described in which the damped Newton iteration is applied to a series of “perturbed problems” in order to guarantee convergence and also increase the convergence rate.

Convergence of Approximation Methods to Compute Eigenelements of Linear Operations

Abstract: The object of this paper is a theoretical study of the convergence of approximation methods (Galerkin and finite difference methods) to compute eigenelements of a closed linear operator T in a Banach space. The stability and strong stability of the approximation method are defined, and they reduce to very simple conditions when T is self-adjoint, either compact, or bounded from below with compact resolvent.

Linear approximations to √x 2 +y 2 having equiripple error characteristics

Abstract: An equiripple error constraint was used to design linear approximations to \sqrt{x^{2} + y^{2}} . Several examples are presented including one having a peak error of less than one percent, which is significantly less than the error obtained by using other design criteria. The mean and standard deviation of the relative error are also tabulated as are earlier results obtained by other authors.

Fault isolation with insufficient measurements

Abstract: A technique is given for approximating the "most likely" values of the internal components of a system when insufficient measurements have made their exact determination impossible. It is limited to linear systems, all of whose measurements are made at a single frequency and whose connections can be represented by sets of linear algebraic equations. The technique is based on the Moore-Penrose generalized inverse of a linear approximation of the system function and is formulated in terms of the component-connection model of a largescale dynamical system.

Stability of Planar Oscillations of a Satellite in an Elliptic Orbit

Abstract: A problem of stability of odd 2π-periodic oscillations of a satellite in the plane of an elliptic orbit of arbitrary eccentricity is considered. The motion is supposed to be only under the influence of gravitational torques. Stability of plane oscillations was investigated earlier (Zlatoustovet al., 1964) in linear approximation. In the present paper a problem of stability is solved in the non-linear mode. Terms up to the forth order inclusive are taken into consideration in expansion of Hamiltonian in a series. It is shown that necessary conditions of stability obtained in linear approximation coincide with sufficient conditions for almost all values of parameters α ande (inertial characteristics of the satellite and eccentricity of the orbit). Exceptions represent either values of the parameters α,e when a problem of stability cannot be solved in a strict manner by non-linear approximation under consideration, or values of the parameters which correspond to resonances of the third and fourth orders. At the resonance of the third order oscillations are unstable, but at the resonance of the fourth order both unstability and stability of the satellite's oscillations take place depending on the values of the parameters α,e.

An approximation method for time domain synthesis of linear networks

Abstract: Time domain synthesis of linear networks requires approximation of functions with a finite sum of weighted exponentials. In this paper an error function, obtained by orthonormal filtering, suitable to numeric optimization, is evaluated. The error function is the integrated squared difference between the obtained function and the desired one. The procedure can be used to form an optimum set of basis exponentials to an ensemble of functions. Finally, the procedure is illustrated with some simple examples.

Best discrete mean rational approximation

Abstract: The approximation problem is: given a function f on X to find an element r*eR to minimize [ I f r }1 over elements r of R. Any such element r* is called a best approximation tof . z'norms' were introduced by Motzkin and Walsh for linear approximation on an interval [5]. The case of Lp approximation is just a special case with z ( t ) = ttl". This paper makes extensive use of the techniques of [1] and [2].

Optical designs for the inverse regression method of calibration

Abstract: Optical experimental designs are developed for the inverse estitator in the Linear Calibration problem using the crieterion of minimum intetrated or average mean squared error. Designs are developed for a linear approximation when the true model is Linear and when it is quadratic. In both cases, the optimal designs depend on unknown model parameters and are not realistically usable. Eowever, designs are shown to exist which are near optimal and do not depend on unknown model parameters

Equation for nonsteady-state combustion velocity of a powder

Abstract: An integral equation is obtained for the nonsteady-state combustion velocity of a powder. It is shown that the effect of a variable tangential stream of gases on the rate of burning (nonsteady-state erosion) can be calculated in a similar way as for the change of pressure. The solution of the equation in linear approximation is considered (rate of burning differs slightly from steady-state).

Equations of motion of viscoelastic systems as derived from the conservation laws and the phenomenological theory of nonequilibrium processes

Abstract: The system of equations of motion for a liquid and a solid with internal parameters — scalars and second-order tensors — is written out in the linear approximation. From the system of equations for the class of motions with velocity gradients independent of the coordinates there follows the known equation of the linear theory of viscoelasticity. It is shown that the correct description of the motions of polymer systems requires a quadratic approximation. A nonlinear variant of the theory of viscoelasticity, in which spectral functions that depend on the internal parameters are introduced, is proposed.

Some optimal considerations in attitude control systems

Abstract: The conventional six-engine reaction control jet relay attitude control law with deadband is shown to be a good linear approximation to a weighted time-fuel optimal control law. Techniques for evaluating the value of the relative weighting between time and fuel for a particular relay control law is studied along with techniques to interrelate other parameters for the two control laws. Vehicle attitude control laws employing control moment gyros are then investigated. Steering laws obtained from the expression for the reaction torque of the gyro configuration are compared to a total optimal attitude control law that is derived from optimal linear regulator theory. This total optimal attitude control law has computational disadvantages in the solving of the matrix Riccati equation. Several computational algorithms for solving the matrix Riccati equation are investigated with respect to accuracy, computational storage requirements, and computational speed.

New Difference Equation for Diffusion Calculation

Abstract: A new difference equation for the numerical solution of the one-dimensional diffusion equation was obtained by using a semi-analytic method, in which the only approximation employed is that the source distribution within a mesh-region is represented by a linear function. A test program, EXX-1, was prepared and run for BWR calculations, and comparisons were made with the conventional method. The results show that by using the new difference equation, a very coarse mesh model (1 mesh point per material region) can be applied without seriously impairing the computational accuracy. It is also shown that the conventional difference equation becomes identical to the new expression if group-constants are multiplied by correction factors and the treatment of the source term appropriately modified. New difference equations for spherical and cylindrical geometries are also given, in an appendix.

Linear programming method for rational approximation

Abstract: Development of a new algorithm, based on linear programming, for the computation of the best rational approximation of a continuous function.

Magnetic hydrodynamics of two-dimensional flows in a plasma with an anisotropic pressure

Abstract: The hydrodynamic equations of Chew, Goldberger, and Low [1] are used to analyze certain types of two-dimensional flows of a plasma with an anisotropic pressure (the pressure along the magnetic field p∥ differs from the pressure across it p⊥). In Sec. 1 the relationships derived in [2] for the transition of plasma state across surfaces of strong discontinuity are invoked to investigate the variation of the hydrodynamic parameters in weak shock waves in the linear approximation. The flow around bodies which only slightly perturb the main flow is investigated in Sec. 2 in the linear approximation. Similar problems for the case of an isotropic pressure are studied in detail in [3–5], for example.

Normal Oscillating Modes of a One-Dimensional Maxwellian Beam in the Linear Space Charge Approximation

Abstract: Following a method used in plasma physics, a Fourier-Hermite transform is attempted to solve the linearized Vlasov-Poisson equation for small fluctuations of a stationary beam. This expansion method is well fitted to the case of a maxwellian beam confined by linear external forces. As a first approximation, if one retains only the linear part of the space charge forces, the eigenmodes for the small beam oscillations are found in a straightforward way. The associated eigenfrequencies are the roots of a 3-diagonal determinant. The spectrum involves only real and discrete eigenfrequencies, and is similar to the spectrum of a beam with uniform density in real space or in phase space.

Blowing coefficient with arbitrary variation of the mass flow rate of the blown gas past a body around which flows a stream with an exponential change in the velocity

Abstract: Formulas are derived permitting calculation of the linear corrections to the friction and heat-transfer coefficients with the blowing into the boundary layer of different gases, in small amounts but with a mass flow rate varying arbitrarily along the body. The case of a Mach number equal to zero and a temperature factor equal to unity was studied. Here it is postulated that bringing the relative heat-transfer coefficient down to a dependence on the dimensionless blowing renders possible, as with blowing which permits a self-similar solution, the use of the results obtained for arbitrary values of these parameters [1]. The proposed method of solution is based on the application, in the linear approximation, of a Duhamel integral for an arbitrary law of change in the mass flow rate along the body, if a solution is known with a discontinuous change in the mass flow rate. For a discontinuous change in the mass flow rate, the solution is sought using a Laplace transform; in this sense, the proposed method is similar to the method of [5].