Linear random operator equations in mathematical physics. II
Stochastic, random and ordinary Green function relations for two point correlation functions in mathematical physics
Adomian, G.↗
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The acoustic response of rooms with open windows to airborne sounds.
The objective of the work described in this and the companion paper was to establish a theory for predicting the sound field generated in a room by a sonic boom incident on an open window. In this paper, some basic theoretical results are presented. First, the case of a normally incident harmonic wave was considered. Expressions for the pressure field were obtained by viewing the room as a terminated duct and by using a Green function method. The concept of mode excitation distribution functions was formulated and used to match the boundary conditions. This concept has been extended for oblique incidence. A modified form of Laplace transform technique was used to obtain expressions in the time domain for transient signals.
Vaidya, P. G.↗
Subsonic potential aerodynamics for complex configurations - A general theory
A general theory of subsonic potential aerodynamic flow around a lifting body having arbitrary shape and motion is presented. By using the Green function method, an integral representation for the velocity potential is obtained for both supersonic and subsonic flow. Under the small perturbation assumption, the potential at any point in the field depends only upon the values of the potential and its normal derivative on the surface of the body. On the surface of the body, this representation reduces to an integro-differential equation relating the potential and its normal derivative (which is known from the boundary conditions) on the surface. The theory is applied to finite-thickness wings in subsonic steady and oscillatory flows.
Morino, L.↗
A General Theory of Unsteady Compressible Potential Aerodynamics
The general theory of potential aerodynamic flow around a lifting body having arbitrary shape and motion is presented. By using the Green function method, an integral representation for the potential is obtained for both supersonic and subsonic flow. Under small perturbation assumption, the potential at any point, P, in the field depends only upon the values of the potential and its normal derivative on the surface, sigma, of the body. Hence, if the point P approaches the surface of the body, the representation reduces to an integro-differential equation relating the potential and its normal derivative (which is known from the boundary conditions) on the surface sigma. For the important practical case of small harmonic oscillation around a rest position, the equation reduces to a two-dimensional Fredholm integral equation of second-type. It is shown that this equation reduces properly to the lifting surface theories as well as other classical mathematical formulas. The question of uniqueness is examined and it is shown that, for thin wings, the operator becomes singular as the thickness approaches zero. This fact may yield numerical problems for very thin wings.
Morino, L.↗
Steady and Oscillatory, Subsonic and Supersonic, Aerodynamic Pressure and Generalized Forces for Complex Aircraft Configurations and Applications to Flutter
A general method for analyzing aerodynamic flows around complex configurations is presented. By applying the Green function method, a linear integral equation relating the unknown, small perturbation potential on the surface of the body, to the known downwash is obtained. The surfaces of the aircraft, wake and diaphragm (if necessary) are divided into small quadrilateral elements which are approximated with hyperboloidal surfaces. The potential and its normal derivative are assumed to be constant within each element. This yields a set of linear algebraic equations and the coefficients are evaluated analytically. By using Gaussian elimination method, equations are solved for the potentials at the centroids of elements. The pressure coefficient is evaluated by the finite different method; the lift and moment coefficients are evaluated by numerical integration. Numerical results are presented, and applications to flutter are also included.
Chen, L. T.↗
Turbulent Chemically Reacting Flows According to a Kinetic Theory
A review of various methods of calculating turbulent chemically reacting flow such as the Green Function, Navier-Stokes equation, and others is presented. Nonequilibrium degrees of freedom were employed to study the mixing behavior of a multiscale turbulence field. Classical and modern theories are discussed.
Hong, Z. C.↗
Radiation from highly relativistic geodesics
A number of recent works are reviewed concerning the generation and emission of gravitational waves. It is shown that at high frequencies, the generation of gravitational radiation is a local phenomenon. Two examples are described illustrating this generation when a high-energy particle collides against the space-time curvature. One, after Matzner and Nutku, uses a method of virtual photons; the other, after Chrzanowski and Misner, is based on the W.K.B. approximation, corresponding to geometric optics, for the inhomogeneous wave equation. This method uses a factorized integral representation of the Green function which is valid asymptotically to infinity in space.
Misner, C. W.↗
Steady subsonic flow around finite-thickness wings
The general method for analyzing steady subsonic potential aerodynamic flow around a lifting body having arbitrary shape is presented. By using the Green function method, an integral representation for the potential is obtained. Under small perturbation assumption, the potential at any point, P, in the field depends only upon the values of the potential and its normal derivative on the surface of the body. Hence if the point P approaches the surface of the body, the representation reduces to an integral equation relating the potential and its normal derivative (which is known from the boundary conditions) on the surface. The question of uniqueness is examined and it is shown that, for thin wings, the operator becomes singular as the thickness approaches zero. This fact may yield numerical problems for very thin wings. However, numerical results obtained for a rectangular wing in subsonic flow show that these problems do not appear even for thickness ratio tau = .001. Comparison with existing results shows that the proposed method is at least as fast and accurate as the lifting surface theories.
Kuo, C. C.↗
Plasmon and exciton superconductivity mechanisms in layered structures
Plasmon and exciton superconductivity mechanisms are discussed. Superconductivity in a three layer metal semiconductor metal and insulator semimetal insulator sandwich structure was described in terms of the temperature dependent Green function of the longitudinal (Coulomb) field. The dependences of the superconducting transition temperature on structure parameters were obtained. In a semiconducting film, as a result of interactions of degenerate free carriers with excitons, superconductivity exists only in a certain range of parameter values, and the corresponding critical temperature is much lower than in the plasmon mechanism of superconductivity.
Gabovich, A. M.↗
Acoustic loading in planar networks
The acoustic loading in a complex planar network of ducts is determined by a method in which Green function surface elements are used. The network consists of straight ducts, elbows and branched ducts. A transfer matrix technique is developed in which each duct is treated separately and the matrix of the influence coefficients is transformed to tri-diagonal form allowing efficient inversion.
El-Raheb, M.↗
Geoid anomalies in a dynamic earth
Potential and surface deformation Love numbers for internal loading have been calculated in order to obtain a dynamically consistent relationship between the geoid and the earth's response to internal buoyancy forces. These quantities depend on the depth and harmonic degree of loading, and can be integrated as Green functions to obtain the dynamic response due to an arbitrary distribution of internal density contrasts. Constructing a series of spherically symmetric, self-gravitating flow models for a variety of radial Newtonian viscosity variations and flow configurations, and calculating relaxation times for spherically symmetric viscous earth models, it is demonstrated that boundary deformation due to internal loading reaches its equilibrium value on the same time scale as postglacial rebound; this is much less time than the time scale for significant change in the convective flow pattern.
Richards, M. A.↗
Inversion of chalcogen defect levels in silicon - An MNDO study
MNDO (modified neglect of diatomic overlap) calculations have been carried out for substitutional oxygen and sulfur impurities in silicon. The calculations of the gap levels reveal a reversal of trend with atomic ionization energies in agreement with self-consistent Green function results, and analysis of the MNDO charge distribution supports the view that the electronegativity difference between oxygen and sulfur gives rise to this shallower energy level.
Singh, R. K.↗
Biharmonic spline interpolation of GEOS-3 and Seasat altimeter data
An algorithm is presented for determining the minimum curvature surface passing through a set of nonuniformly spaced data points. The curve is generated as a linear combination of Green functions for the biharmonic operator at each data point, with the amplitudes of the functions adjusted so that the interpolating surfaces passes through each point. The function passing through the points can be regarded as a spline to which point forces are applied, defining the minimum curvature between the points. The technique was used to combine the along track slopes of the GEOS-3 and Seasat altimeter data into a consistent geoid height map of the Caribbean area, covering 0.5 million data points in the process. Sample images are provided and new topographic features that are revealed are discussed.
Sandwell, David T.↗
Design of magnets inside cylindrical superconducting shields
The design of magnets inside closed, cylindrical, superconducting shields is discussed. The Green function is given for the magnetic vector potential for cylindrically symmetric currents inside such a shield. The magnetic field everywhere inside the shield can be obtained from this function, which includes the effects of the induced shield currents exactly. The field is given for a thin solenoid as an example and the convergence of the series solution for this case is discussed. The shield can significantly reduce the strength and improve the homogeneity of a magnet. The improvement in homogeneity is of particular importance in the design of correction coils. These effects, and the maximum field on the shield, are examined for a typical solenoid. The results given are also useful, although not exact, for long shields with one or two open ends.
Rigby, K. W.↗
On the integration of image sources in exact image method of field analysis
Convergence conditions of image integration in the exact image method of field calculation have been investigated, and the method is extended to include more general media than previously considered. It is demonstrated that the integral is well behaved and the method works best when the field is calculated in a medium with less loss. If the medium has more loss, it is shown that the image line may enter the improper half-space and still produce valid results. Means of correctly selecting the integration path branch for the case of crossing branch cuts of the Green function in complex integration planes are proposed.
Lindell, I. V.↗
The use of the FFT for the efficient solution of the problem of electromagnetic scattering by a body of revolution
The enhancement of the computational efficiency of the body of revolution (BOR) scattering problem is discused with a view to making it practical for solving large-body problems. The problem of EM scattering by a perfectly conducting BOR is considered, although the methods can be extended to multilayered dielectric bodies as well. Typically, the generation of the elements of the moment method matrix consumes a major portion of the computational time. It is shown how this time can be significantly reduced by manipulating the expression for the matrix elements to permit efficient FFT computation. A technique for extracting the singularity of the Green function that appears within the integrands of the matrix diagonal is also presented, further enhancing the usefulness of the FFT. The computation time can thus be improved by at least an order of magnitude for large bodies in comparison to that for previous algorithms.
Gedney, Stephen D.↗
A boundary integral approach to the scattering of nonplanar acoustic waves by rigid bodies
The acoustic scattering of an incident wave by a rigid body can be described by a singular Fredholm integral equation of the second kind. This equation is derived by solving the wave equation using generalized function theory, Green's function for the wave equation in unbounded space, and the acoustic boundary condition for a perfectly rigid body. This paper will discuss the derivation of the wave equation, its reformulation as a boundary integral equation, and the solution of the integral equation by the Galerkin method. The accuracy of the Galerkin method can be assessed by applying the technique outlined in the paper to reproduce the known pressure fields that are due to various point sources. From the analysis of these simpler cases, the accuracy of the Galerkin solution can be inferred for the scattered pressure field caused by the incidence of a dipole field on a rigid sphere. The solution by the Galerkin technique can then be applied to such problems as a dipole model of a propeller whose pressure field is incident on a rigid cylinder. This is the groundwork for modeling the scattering of rotating blade noise by airplane fuselages.
Gallman, Judith M.↗
Energetics of molecular-beam epitaxy models
The removal energies of constituent atoms from various unreconstructed semiconductor surfaces are calculated using a Green function method. An efficient difference-equation approach within the second-neighbor tight-binding model is employed. For a compound AB, binding energies for the A and B atoms on the (111), (-1 -1 -1), (100), and (110) surfaces are calculated. Analyses are made of the energy to remove an atom from the nearly full surface and from the nearly empty surface. Results are presented for Si, GaAs, CdTe, and HgTe; and the surface sublimation energies are found to depend on surface coverage and do not display a simple linear relationship to the number of bonds broken, as is often assumed in modeling growth by MBE.
Krishnamurthy, Srinivasan↗