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Experimental results are presented which confirm the diffraction effects predicted by the MacDonald (1915) form of the Green's function for the baffled field in the theory of trailing edge noise. Using this Green's function, the acoustic intensity of noise generated by flow over a trailing edge has been determined to be proportional to the 5th-power of the flow velocity.
A plug-in-and-grind formalism is derived for calculating the gravitational waves emitted by any system with weak internal gravitational fields. If the internal fields have negligible influence on the system's motions, then the formalism reduces to standard linearized theory. Whether or not gravity affects the motions, if the motions are slow and internal stresses are weak, then the new formalism reduces to the standard quadrupole-moment formalism. In the general case the new formalism expresses the radiation in terms of a retarded Green's function for slightly curved spacetime, and then breaks the Green's-function integral into five easily understood pieces: direct radiation, produced directly by the motions of the sources; whump radiation, produced by the the gravitational stresses of the source; transition radiation, produced by a time-changing time delay (Shapiro effect) in the propagation of the nonradiative, 1/r field of the source; focussing radiation produced when one portion of the source focusses, in a time-dependent way, the nonradiative field of another portion of the source, and tail radiation, produced by backscatter of the nonradiative field in regions of focussing.
We study the renormalization of four-quark operators in one-loop perturbation theory. We employ a coordinate-space gauge-invariant renormalization scheme (GIRS), which can be advantageous compared to other schemes, especially in nonperturbative lattice investigations. From our perturbative calculations, we extract the conversion factors between GIRS and the modified minimal subtraction scheme ( MS ¯ ) at the next-to-leading order. As a by-product, we also obtain the relevant anomalous dimensions in the GIRS scheme. A formidable issue in the study of the four-quark operators is that operators with different Dirac matrices mix among themselves upon renormalization. We focus on both parity-conserving and parity-violating four-quark operators, which change flavor numbers by two units ( Δ F = 2 ). The extraction of the conversion factors entails the calculation of two-point Green’s functions involving products of two four-quark operators, as well as three-point Green’s functions with one four-quark and two bilinear operators. The significance of our results lies in their potential to refine our understanding of QCD phenomena, offering insights into the precision of Cabibbo-Kobayashi-Maskawa (CKM) matrix elements and shedding light on the nonperturbative treatment of complex mixing patterns associated with four-quark operators. Published by the American Physical Society 2024
We test an innovative inversion scheme using Green’s functions from an array of pressure sources embedded in finite-element method (FEM) models to image, without assuming an a-priori geometry, the composite and complex shape of a volcano deformation source. We invert interferometric synthetic aperture radar (InSAR) data to estimate the pressurization and shape of the magma reservoir of Rabaul caldera, Papua New Guinea. The results image the extended shallow magmatic system responsible for a broad and long-term subsidence of the caldera between 2007 February and 2010 December. Elastic FEM solutions are integrated into the regularized linear inversion of InSAR data of volcano surface displacements in order to obtain a 3-D image of the source of deformation. The Green’s function matrix is constructed from a library of forward line-of-sight displacement solutions for a grid of cubic elementary deformation sources. Each source is sequentially generated by removing the corresponding cubic elements from a common meshed domain and simulating the injection of a fluid mass flux into the cavity, which results in a pressurization and volumetric change of the fluidfilled cavity. The use of a single mesh for the generation of all FEM models avoids the computationally expensive process of non-linear inversion and remeshing a variable geometry domain. Without assuming an a-priori source geometry other than the configuration of the 3-D grid that generates the library of Green’s functions, the geodetic data dictate the geometry of the magma reservoir as a 3-D distribution of pressure (or flux of magma) within the source array. The inversion of InSAR data of Rabaul caldera shows a distribution of interconnected sources forming an amorphous, shallow magmatic system elongated under two opposite sides of the caldera. The marginal areas at the sides of the imaged magmatic system are the possible feeding reservoirs of the ongoing Tavurvur volcano eruption of andesitic products on the east side and of the past Vulcan volcano eruptions of more evolved materials on the west side. The interconnection and spatial distributions of sources correspond to the petrography of the volcanic products described in the literature and to the dynamics of the single and twin eruptions that characterize the caldera. The ability to image the complex geometry of deformation sources in both space and time can improve our ability to monitor active volcanoes, widen our understanding of the dynamics of active volcanic systems and improve the predictions of eruptions.
We test an innovative inversion scheme using Green’s functions from an array of pressure sources embedded in finite-element method (FEM) models to image, without assuming an a-priori geometry, the composite and complex shape of a volcano deformation source. We invert interferometric synthetic aperture radar (InSAR) data to estimate the pressurization and shape of the magma reservoir of Rabaul caldera, Papua New Guinea. The results image the extended shallow magmatic system responsible for a broad and long-term subsidence of the caldera between 2007 February and 2010 December. Elastic FEM solutions are integrated into the regularized linear inversion of InSAR data of volcano surface displacements in order to obtain a 3-D image of the source of deformation. The Green’s function matrix is constructed from a library of forward line-of-sight displacement solutions for a grid of cubic elementary deformation sources. Each source is sequentially generated by removing the corresponding cubic elements from a common meshed domain and simulating the injection of a fluid mass flux into the cavity, which results in a pressurization and volumetric change of the fluid-filled cavity. The use of a single mesh for the generation of all FEM models avoids the computationally expensive process of non-linear inversion and remeshing a variable geometry domain. Without assuming an a-priori source geometry other than the configuration of the 3-D grid that generates the library of Green’s functions, the geodetic data dictate the geometry of the magma reservoir as a 3-D distribution of pressure (or flux of magma) within the source array. The inversion of InSAR data of Rabaul caldera shows a distribution of interconnected sources forming an amorphous, shallow magmatic system elongated under two opposite sides of the caldera. The marginal areas at the sides of the imaged magmatic system are the possible feeding reservoirs of the ongoing Tavurvur volcano eruption of andesitic products on the east side and of the past Vulcan volcano eruptions of more evolved materials on the west side. The interconnection and spatial distributions of sources correspond to the petrography of the volcanic products described in the literature and to the dynamics of the single and twin eruptions that characterize the caldera. The ability to image the complex geometry of deformation sources in both space and time can improve our ability to monitor active volcanoes, widen our understanding of the dynamics of active volcanic systems and improve the predictions of eruptions.
The scattering properties of perfectly conducting and resistive strips are predicted for strips which are located on a dielectric slab backed by a perfectly conducting ground plane. The spectral domain Green's function is used to relate the currents and fields on the strip, and the resulting integral equation is solved using the method of moments. Both TE and TM strips are examined using piecewise linear and pulse subdomain basis functions, respectively, to model the current on the strip. Calculated results are compared with results measured at the NASA Langley Research Center.
A development is presented for the Green's function for a point source in the vicinity of a rigid wedge. The diffraction contributions to the Green's function for arbitrary source and listener location is expressed in a form which can be readily evaluated using the Laguerre technique for numerical integration. The present approach offers the advantages of efficient numerical evaluation and of relatively straightforward reduction to well-known analytical approximations in limiting cases. Comparisons with previously obtained experimental and numerical results obtained by Ambaud and Bergassoli (1972) are presented. The comparison with the experimental results is excellent; the advantages of the present numerical technique, vis a vis that of Ambaud and Bergassoli, are pointed out.
The strong fluctuation theory is applied to the study of electromagnetic wave scattering from a layer of random discrete scatterers. The singularity of the dyadic Green's function is taken into account in the calculation of the effective permittivity functions. The correlation functions for the random medium with different scatterer constituents and size distributions are derived. Applying the dyadic Green's function for a two-layer medium and using the bilocal and distorted Born approximations, the first and the second moments of the fields are then calculated. Both the backscattering and bistatic scattering coefficients are obtained, and the former is shown to match favorably with experimental data obtained from snow fields.
The Green's function method and the computer program SOUSSA (Steady Oscillatory and Unsteady Subsonic and Supersonic Aerodynamics) are reviewed. The Green's function method is applied to the fully unsteady potential equation yielding an integro-differential-delay equation. This equation is approximated by a set of differential-delay equations in time using the finite element method. The Laplace transform is used to yield a matrix relating the velocity potential to the normal wash. The matrix of the generalized aerodynamic forces is obtained by premultiplying and postmultiplying the matrices relating generalized forces to the potential and the normal wash by the generalized coordinates. The program SOUSSA is compared with existing numerical results. Results indicate that the program is not only general, flexible, and easy to use, but also accurate and fast.
We are in the process of carrying out calculations of e-H total cross sections using the 'complex-correlation Kohn-T' (CCKT) method. In a later paper, we described the methodology more completely, but confined calculations to the elastic scattering region, with definitive, precision results for S-wave phase shifts. Here we extend the calculations to the (low) continuum (1 much less than k(exp 2) much less than 3) using a Green's function formulation. This avoids having to solve integro-differential equations; rather we evaluate indefinite integrals involving appropriate Green's functions and the (complex) optical potential to find the scattering function u(r). From the asymptotic form of u(r) we extract a T(sub L) which is a complex number. From T(sub L), elastic sigma(sub L)(elastic) = 4pi(2L+1)((absolute value of T(sub L))(exp 2)), and total sigma (sub L)(total) = 4pi/k(2L+1)Im(T(sub L)) cross sections follow.
The nonequilibrium Green's function gives access to one-body observables for quantum systems. Of particular interest are quantities such as density, currents, and absorption spectra which are important for interpreting experimental results in quantum transport and spectroscopy. We present an integration scheme for the Green's function's equations of motion, the Kadanoff-Baym equations (KBE), which is both adaptive in the time integrator step size and method order as well as the history integration order. We analyze the importance of solving the KBE self-consistently and show that adapting the order of history integral evaluation is important for obtaining accurate results. To examine the efficiency of our method, we compare runtimes to a state-of-the-art fixed time step integrator for several test systems and show an order of magnitude speedup at similar levels of accuracy. Published by the American Physical Society 2024
The paper examines the mathematical and physical foundations for the kinetic theory of reactive turbulent flows, discussing the differences and relation between the kinetic and averaged equations, and comparing some solutions of the kinetic equations obtained by the Green's function method with those obtained by the approximate bimodal method. The kinetic method described consists essentially in constructing the probability density functions of the chemical species on the basis of solutions of the Langevin stochastic equation for the influence of eddies on the behavior of fluid elements. When the kinetic equations are solved for the structure of the diffusion flame established in a shear layer by the bimodal method, discontinuities in gradients of the mean concentrations at the two flame edges appear. This is a consequence of the bimodal approximation of all distribution functions by two dissimilar half-Maxwellian functions, which is a very crude approximation. These discontinuities do not appear when the solutions are constructed by the Green's function method described here.
We outline a complete spectral formalism for computing high spatial resolution three-dimensional deformations arising from the surface mass loading of a spherically symmetric planet. The main advantages of the formalism are that all surface mass loads are always described using a consistent mathematical representation and that calculations of deformation fields for various spatial resolutions can be performed by simpley altering the spherical harmonic degree truncation level of the procedure. The latter may be important when incorporating improved observational constraints on a particular surface mass load, when considering potential errors in the computed field associated with mass loading having a spatial scale unresolved by the observational constraints, or when treating a number of global surface mass loads constrained with different spatial resolutions. The advantages do not extend to traditional 'Green's function' approaches which involve surface element discretizations of the global mass loads. Another advantage of the spectral formalism, over the Green's function approach, is that a posteriori analyses of the computed deformation fields are easily performed. In developing the spectral formalism, we consider specific cases where the Earth's mantle is assumed to respond as an elastic, slightly anelastic, or linear viscoelastic medium. In the case of an elastic or slightly anelastic mantle rheology the spectral response equations incorporate frequency dependent Love numbers. The formalism can therefore be used, for example, to compute the potentially resonant deformational response associated with the free core nutation and Chandler wobble eigenfunctions. For completeness, the spectral response equations include both body forces, as arise from the gravitational attraction of the Sun and the Moon, and surface mass loads. In either case, and for both elastic and anelastic mantle rheologies, we outline a pseudo-spectral technique for computing the ocean adjustment associated with the total gravitational perturbation induced by the external forcing. Three-dimensional deformations computed using the usual Love number approach are generally referenced to an origin at the center of mass of the undeformed planet. We derive a spectral technique for transforming the results to an origin located at the center of mass of the deformed planet.
We present a real-space Green's function theory and calculations of two-photon x-ray absorption (TPA). Our focus is on nonresonant 𝐾-shell TPA in metallic Cu, which has been observed experimentally at intense x-ray free electron laser (XFEL) sources. The theory is based on an independent particle Green's function treatment of the Kramers-Heisenberg equation and an approximation for the sum over nonresonant intermediate states in terms of a static quadrupole transition operator. XFEL effects are modeled by a partially depleted 𝑑 band. This approach is shown to give results for 𝐾-shell TPA in quantitative agreement with XFEL experiment and with a Bethe-Salpeter equation approach. Furthermore, we also briefly discuss many-body corrections and TPA sum rules.
Recent developments of the Green's function method and the computer program SOUSSA (Steady, Oscillatory, and Unsteady Subsonic and Supersonic Aerodynamics) are reviewed and summarized. Applying the Green's function method to the fully unsteady (transient) potential equation yields an integro-differential-delay equation. With spatial discretization by the finite-element method, this equation is approximated by a set of differential-delay equations in time. Time solution by Laplace transform yields a matrix relating the velocity potential to the normal wash. Premultiplying and postmultiplying by the matrices relating generalized forces to the potential and the normal wash to the generalized coordinates one obtains the matrix of the generalized aerodynamic forces. The frequency and mode-shape dependence of this matrix makes the program SOUSSA useful for multiple frequency and repeated mode-shape evaluations.
A new and very efficient numerical method for solving equations of the Helmholtz type is specialized for problems having axisymmetric geometry. It is then demonstrated by application to the classical problem of acoustic radiation from a vibrating piston set in a stationary infinite plane. The method utilizes 'Green's Function Discretization', to obtain an accurate resolution of the waves using only 2-3 points per wave. Locally valid free space Green's functions, used in the discretization step, are obtained by quadrature. Results are computed for a range of grid spacing/piston radius ratios at a frequency parameter, omega R/c(sub 0), of 2 pi. In this case, the minimum required grid resolution appears to be fixed by the need to resolve a step boundary condition at the piston edge rather than by the length scale imposed by the wave length of the acoustic radiation. It is also demonstrated that a local near-field radiation boundary procedure allows the domain to be truncated very near the radiating source with little effect on the solution.
A nonperturbative analytic solution of Green's function for heavy ion transport in a single medium is extended to multilayer transport media. This extension is implemented in the Green's function code (GRNTRN), which is being validated with laboratory experiments. Good agreement is achieved between an iron radiation beam experiment and GRNTRN only when the interactions of the iron radiation beam with several important layers of material located upstream of the target within the beamline are considered.
The dyadic Green's function for a current embedded in a grounded dielectric slab is used to analyze microstrip lines at millimeter wave frequencies. The dyadic Green's function accounts accurately for fringing fields and dielectric cover over the microstrip line. Using Rumsey's reaction concept, an expression for the characteristic impedance is obtained. The numerical results are compared with other reported results.