Use Of Boundary Conditions Research Articles

Scattering of sound waves by a periodic array of slotted waveguides

The problem of scattering of plane acoustic waves obliquely incident upon a barrier consisting of superposed rectangular slotted waveguides is treated by taking a waveguide approach. The wave equation is solved for each interior region, bounded by parallel walls, and for the half-spaces on both sides of the scattering arrangement. The use of boundary conditions then leads to two infinite systems of algebraic equations, the solution of which generates the complex transmission and reflection coefficients. The scattering effects of comblike structures show sharp variations over the frequency spectrum. They also vary strongly at certain particular frequencies with the angle of incidence of the incoming wave. Two geometries of slotted waveguides are studied and their behavior, regarding the amplitude and phase of the transmitted sound, is compared with that of a simple grating of rectangular bars having similar dimensions. Theoretical results compare favorably with experimental values which are presented in another article forming the second part of this study.

On laminar small reynolds-number flow over wavy walls

In the present paper a perturbation method is developed in order to study viscous, laminar flow over a small amplitude wavy solid boundary. The wall is transformed into a straight solid boundary in order to simplify the use of boundary conditions of the problem on the wall. The flow field is calculated numerically up to first perturbation order. separation of the flow is included.

Accuracy of determination of stress concentrations by empirical-numerical methods

1. The combined use of experimental methods and the finite elements method to study stress concentrations in individual zones of actual objects makes it possible to improve the accuracy of determination of the maximum stresses compared to the accuracy of a purely experimental determination. 2. The use of displacements as the boundary conditions for calculating the zones by means of the finite elements method places more stringent requirements on the accuracy with which they are assigned compared to the use of boundary conditions in stresses. 3. In several cases, the error of the calculation due to the error of assignment of the boundary conditions is of the same order as that due to the error of the numerical method. Thus, it is important that the examination of individual zones of objects makes it possible to subdivide them finely enough into finite elements and thus makes the problem of selecting an optimum grid less urgent. 4. Approximation of the experimental data on the contour leads to a substantial decrease in the number of points at which it is necessary to determine the stresses or displacements, which in turn reduces the volume of experimental work necessary and simplifies it as well.

The Slot Gap Model of Pulsars

A general picture of pulsar magnetospheres is outlined which suggests polar current flow should proceed within a magnetic flux tube with conducting boundaries. The dynamics of this current flow is considered, including the effect of formation of pair plasma on the potential. A slot gap surrounding the stream of pair plasma is described, and the electric fields and J·E work done in the gap are calculated. The inapplicability of curvature emission to the formation of γ-ray pulses is discussed. The dynamics of pair plasma flow is described, including electrostatic trapping of low energy positrons, polar cap bombardment and heating, and the emission of soft X-rays from the cap. The relation of this current flow and polar cap heating to the formation of fluctuations on the radio subpulse time scale is also described. The importance of the boundary layer between the pair plasma and the slot gap is illustrated by calculations of the boundary layer structure and microscopic instability, which show that this region may be the source of pulsar emission.I will describe a number of quantitative results and qualitative aspects of work in progress on the “slot gap” model for the emission regions of a pulsar, proposed by Arons and Scharlemann (1979, hereafter AS) and by Arons (1979b, hereafter I). The results and ideas concern the dynamics and emission physics of relativistic plasma and current flow along the polar field lines of an isolated, rotating magnetized neutron star with the axis of its dipole moment oblique to the rotation axis - a pulsar. The quantitative modelling is confined to the physics within a polar flux tube and is therefore “local”. Full detail will be given in papers now being written. However, development of this type of model requires the use of boundary conditions which follow from a qualitative view of the whole magnetosphere. Because this view involves some qualitative ideas not widely discussed in the literature, I describe it briefly.

Optimum shape design of shoulder fillets in tension bars and T-heads

Optimum shape design of shoulder fillets in flat tension bars, round tension bars and at T-heads are considered by a mathematical programming method. The curved shape is defined by a general third degree function in the coordinates x and y which is completely determined by making use of boundary conditions and four design variables. To have a better shape control during optimization, the problem is formulated as minimization of volume with limitation on stress concentration factor. The successive optimum shapes are obtained by changing the limitation on the stress concentration factor (SCF), until minimum possible SCF limitation is reached. Design charts are presented for a few cases. It is observed that stress concentration factor can be reduced from 5 to 10% in comparison with circular fillets.

On boundary conditions for hyperbolic difference schemes

We describe an investigation, largely experimental, to determine stable approximations for the boundary conditions of hyperbolic systems. We are concerned with the fully discrete leapfrog scheme, and also with the method of lines which uses a finite difference approximation in space and an ODE solver in time. We are especially interested in schemes which have fourth order accuracy. Our discussion concerns problems in only one space dimension. We describe a modification of the scheme developed by Oliger [7] to stabilize the leapfrog scheme when a fourth order spatial difference approximation is used. We analyze boundary approximations for the method of lines by a numerical study of the eigenvalues and norm of the matrix for the semi-discrete system. These are checked by integrations of the system using an ODE solver. We also study the use of boundary conditions which are differentiated in time. This is done in order to obtain a system to which an ODE solver can be conveniently applied.

Seismic analysis and design of hyperbolic cooling towers

The safety of hyperbolic cooling towers is important to the continuous operation of a power plant. Depending upon the site, earthquake may govern the design of the tower. Methods of seismic analysis have been presented. It is concluded that the response spectrum method of analysis is of maximum practical use. A method to construct the design response spectra for various earthquake zones is presented. An earthquake motion consists of three components; however, it is shown that designing for one horizontal component only is adequate. The use of boundary conditions and the effects of inelastic action on analysis and design are discussed.

The Excessive Functions of a Continuous Time Markov Chain

Boundary conditions in the form of equalities have been used by Feller, Dynkin, and others to characterize the range of the resolvent operator for certain continuous time Markov chains. Along similar lines Denzel, Kemeny, and Snell were able to establish a characterization and Riesz decomposition for the excessive functions of a more restricted class of Markov chains through the use of boundary conditions in the form of inequalities. The present paper sets out to clarify and build upon this work by reanalyzing these excessive functions in a more general setting. Here the boundary theory developed by Chung is brought to bear on the problem so that the results can be derived in canonical form for probabilistic interpretation.

Electroacoustic Sensitivity of Pressure-Equalized Spherical and Cylindrical Hydrophones

A simple method for the prediction of the stiffness-controlled voltage response of a hydrophone configured for deep-sea operation is described. The approach utilizes previous work in this area, enlarged to include the effects of various degrees of decoupling of the acoustic field from the back face of the hydrophone. Two configurations of importance, the spherical and the cylindrical, have been analyzed, and the results presented graphically. For both cases, the equations of equilibrium and the equation of compatibility are solved and evaluated by the use of boundary conditions. These stress equations, thus generated, and the electromechanical constants are then related to yield the voltage/pressure sensitivity equations. The graphs show the effects of thickness to diameter, comparison of the air-backed to the pressure-equalized hydrophone, and the various degrees of decoupling of the acoustic field from the back face of the hydrophone. For all cases, both barium titanate and lead zirconate titanate are considered.

An Exact Solution of the Acoustical Field Near a Circular Transducer

An exact solution for the acoustical field of an extended source is obtained by the use of boundary conditions differing from those of the simple piston.

Optical Properties and Electronic Structure of Solid Silicon

The existence of an empty (conduction) band 1.5 or 2.0 volts above the highest filled band in metallic-looking semi-conductors such as silicon and certain sulfides is inferred from the optical properties of these substances. Reflectivity data of silicon are examined with the aid of the expression for the complex index of refraction as given by classical electromagnetic theory. Through use of this expression, which is checked for internal consistency, an oscillator strength of 1.6 electrons is calculated for the absorption, and a dielectric constant at low frequencies of 12.5 is computed. The Wigner-Seitz-Slater method of computing electronic energy bands in crystals is used to determine the band structure of silicon. A combination of one $s$, three $p$, and three $d$ functions for each of the two atoms in the unit cell is made, and through use of boundary conditions of continuity of value, of normal, and of tangential derivatives, a solution is obtained for the plane $x=y$. The resulting band structure closely resembles that obtained by Kimball for the diamond, except that silicon is more nearly metallic than is the diamond. The $3s$ level of the silicon atom splits into two bands, the $3p$ level into six bands, and the $3d$ level into six bands; the two $3s$ bands and the lowest two $3p$ bands are completely filled by electrons. At the observed half-distance between nearest neighbors, the gap between the uppermost filled band and the lowest empty band is much greater than that expected from consideration of the optical data. The form of the optical absorption as expected from band structure considerations is proposed. The width of the filled bands as observed in soft x-ray emission spectra is about equal to the width of the computed bands.