Curl Of Vector Research Articles

Vortex properties of the acoustic intensity vector in a shallow sea

We study the vortex properties of the field of the acoustic intensity vector as a function of distance and radiation frequency, determined using a combined acoustic system in Peter the Great Bay in the Sea of Japan. We present probability density histograms of the normalized intensity vector curl components at a frequency of 110 Hz for the distance interval between the source and receiver within the limits of 1200–1725 m. The vortex structure of the acoustic intensity vector was observed in the vertical plane at the full distance (4000 m) from the source to the receiver. The discovered vortex structures are of interest both for physical acoustics and for applied problems in underwater acoustics.

Comparison of the Vorticity of Acoustic Intensity Vector at 23 Hz and 110 Hz Frequencies in the Shallow Sea

This paper presents the results of vortex features research of the field of acoustic intensity vector depending on distance and frequency by the combined acoustic systems in Peter the Great Bay (Sea of Japan). The receiver was submerged to 70 m depth at the site depth of 120 m, sound velocity on the surface was higher than that of the bottom’s one. To determine the vorticity three components of curl of intensity vector were calculated. The work provides the distribution histograms of probability density in the normalized components of intensity vector curl at 110 Hz and 23 Hz frequencies for the distances interval between source and receiver within 1200 – 2000 m. The vortex structure of acoustic intensity vector was discovered in vertical plane along the entire researched distance (5000 m) between source and receiver. The described mechanism of acoustic energy transfer reveals new opportunities in the study of acoustic characteristics of the shallow sea. The disclosed vortex structures are of interest both for physical acoustics and for applied problems in underwater acoustics.

An efficient time‐domain method analysis of quasiperiodic structures by a finite‐element tearing and interconnecting algorithm

AbstractBased on the finite‐element approximation and nonoverlapping domain decomposition, an efficient tearing and interconnecting algorithm of the finite‐element time‐domain method is presented. Using edge‐based finite element, the method divides the computation domain into several nonoverlapping subdomains and computes electric fields in each subdomain by solving the vector curl–curl wave equation. Adjacent subdomains are related to each other by Dirichlet and Neumann boundary condition. By use of the repetition of the quasiperiod structures, the method further reduces the memory requirement and CPU time. The time‐dependent formulation employs an unconditional stable time‐integration method based on the Newmark‐beta method. An anisotropic perfectly matched layers absorbing material terminated by the first‐order boundary condition is applied for the truncation of lattices. Numerical results demonstrate that our proposed method is extremely efficient for the time‐domain analysis of the quasiperiodic structures, such as electromagnetic band gap structures.© 2010 Wiley Periodicals, Inc. Microwave Opt Technol Lett 52: 1072–1078, 2010; Published online in Wiley InterScience (www.interscience. wiley.com). DOI 10.1002/mop.25100

Normal forms for locally exact Poisson structures in [formula omitted

We consider smooth singular Poisson tensors in R 3 which admit a trivial curl vector field. We use normal forms for smooth functions in R 3 to produce normal forms for such Poisson tensors around the singular point.

Novel generalization of three-dimensional Yang–Mills theory

A class of new non-Abelian gauge theories for vector fields on three manifolds is presented. The theories describe a generalization of three-dimensional Yang–Mills theory featuring a novel nonlinear gauge symmetry and field equations for Lie-algebra-valued vector potential fields. The nonlinear form of the gauge symmetry and field equations relies on the vector cross-product and vector curl operator available only in three dimensions, and makes use of an auxiliary Lie bracket together with the Lie bracket used in Yang–Mills theory. A gauge covariant formulation of the new theories is given which utilizes the covariant derivative and curvature from the geometrical formulation of Yang–Mills theory. Further features of the new theories are discussed.

Reconstruction of fluid motion in acoustic diffraction tomography

This paper shows how the technique of diffraction tomography can be applied for reconstructing the magnitudes and the directions of the velocity vectors as well as stationary sound-speed perturbation in a fluid flow. The analysis is based on the wave equation describing the propagation of the sound in an inhomogeneous moving fluid medium. For plane-wave irradiation and scattered field detection by a linear array of receivers, the Fourier diffraction theorem for the fluid vorticity has been obtained. Similar to the scalar case, the theorem relates the Fourier transform of the measured forward scattered data to the spatial spectrum of the vorticity and leads to the values of the two-dimensional transform of the velocity vector curl along a semicircular arc in the frequency domain. Utilizing the nonreciprocity of sound scattering from moving fluid (as opposed to the reciprocity of the elastic scattering on the scalar potential), a procedure for the separation of scalar and moving components of the flow has been proposed. It has been shown that nonreciprocity of the sound scattering by flow offers additional information which can be used to separate two independent components of the velocity vector and thus to solve the vector reconstruction problem uniquely for an arbitrary 2-D vector field without any additional constraints concerned with the flow model, what was impossible in previous algorithms based on the time-of-flight measured data. The results offer independent confirmation of those derived by Rouseff and Winters [D. Rouseff and K. B. Winters, ‘‘Two-dimensional vector flow inversion by diffraction tomography,’’ Inverse Problems 10, 687–697 (1994)].

The Generation of Secondary Vorticity in an Incompressible Fluid

A general equation for the development of secondary vorticity in a homogeneous incompressible fluid is developed by extending the analysis of Hawthorne [1]. It is shown that secondary vorticity can be generated not only through the action of a nonuniform flow field on curved streamlines, but also whenever the flow kinetics are such that the vector curl (V × Ω) has a component in the flow-wise direction. Applied to the problem of the skewed boundary layer along a plane wall the general result partially explains the experimentally observed attenuation of skewing in the viscous sublayer region. The equation is applied to the problem of secondary flow generation in sharply curved passages of appreciable depth where the deep curved channel flow configuration dominates the secondary flow. Finally, one discusses the implication of secondary vorticity development from the vector curl (V × Ω) to the problem of laminar boundary-layer instability resulting in the occurrence of a system of longitudinal vortices.

Ampere's Law and the Vector Potential

Some general features of several elementary methods of determining the magnetic field of a current configuration are examined. Ampere's law, the circuital form of Maxwell's equation and a derived relation between the magnetic field and the vector curl of the current, are each applied to the simple problem of computing the magnetic field due to a thin current dipole in a conducting medium. The significance of closed current paths and the conditions under which the return currents in the medium must be included in the computation are discussed for the dc case. The conclusions concerning dc currents are then considered in relation to ac problems and the use of the vector potential in antenna problems. These considerations indicate why only antenna currents enter into the computation of vector potential and the nature of the “thick antenna” problem.

A New Proof of Lagrange's Theorem in Hydrodynamics

A new proof of Lagrange's theorem about the persistence of curl vector is given in the light of recent theories of linear partial differential equations. Velocity potential for an arbitrary pressure distribution can be constructed as the Extremal field.