Binormal Direction Research Articles

On the physical meaning of the power factor in acoustics

Following recent advancements in the study of time-averaged properties of energy propagation in linear acoustic fields, the well established concept of power factor known from the electric AC circuits analysis, is here extended to acoustics. This allows our outline of a complete acousto-electro-mechanic analogy, where the fundamental physical concept of energy trajectory is assimilated to a continuous line network of electric circuits, and the complex intensity vector field is defined by means of three special spatial directions: the tangent, the principal normal and the binormal direction at each point of any energy path. The notions of sound energy conductance and susceptance are then introduced and their relationship with complex intensity is highlighted. Finally, the frequency distributions of the defined quantities are measured in different acoustical contexts, thus illustrating their practical utility for advanced intensimetric metrology.

Vortex Ring Pinning for the Gross--Pitaevskii Equation in Three-Dimensional Space

We construct a stationary solution possessing two vortex rings to the nonlinear Schrodinger-type problem $ i u_t =-\varepsilon^2\triangle u +(V+|u|^2)u, $ where the unknown function $u$ is defined as $u:\mathbb {R}^3\times\mathbb{R}\rightarrow \mathbb{C}$, $\varepsilon$ is a small positive parameter, and $V$ is a real smooth potential. These two vortex rings will be pinned at a fixed site by the potential $V$. The two rings either lie in the same plane with neighboring interaction in the normal direction or they they lie in two parallel planes with interaction in the binormal direction, such that the neighboring interaction will be balanced by the effect of the potential.

Diffraction of a Gaussian beam in a three-dimensional smoothly inhomogeneous medium: an eikonal-based complex geometrical-optics approach

We present an ab initio account of the paraxial complex geometrical optics (CGO) in application to scalar Gaussian beam propagation and diffraction in a 3D smoothly inhomogeneous medium. The paraxial CGO deals with quadratic expansion of the complex eikonal and reduces the wave problem to the solution of ordinary differential equations of the Riccati type. This substantially simplifies the description of Gaussian beam diffraction as compared with full-wave or parabolic (quasi-optics) equations. For a Gaussian beam propagating in a homogeneous medium or along the symmetry axis in a lenslike medium, the CGO equations possess analytical solutions; otherwise, they can be readily solved numerically. As a nontrivial example we consider Gaussian beam propagation and diffraction along a helical ray in an axially symmetric waveguide medium. It is shown that the major axis of the beam's elliptical cross section grows unboundedly; it is oriented predominantly in the azimuthal (binormal) direction and does not obey the parallel-transport law.

Rigorous and Generalized Derivation of Vortex Line Dynamics in Superfluids and Superconductors

We prove rigorously the asymptotic motion equation of a vortex line in a superconductor and a superfluid at small coherence length $\epsilon$. In superconductors, the leading order term of the motion equation of a vortex line is dominated by the curvature and the normal direction of the vortex line. In superfluids, the leading order term of the motion equation of a vortex line is determined by the curvature and the binormal direction of the vortex line. Fortunately, the motion equation of a vortex line in a superfluid has the same leading order term as the motion equation of a vortex line in an incompressible fluid at high Reynolds numbers as $\epsilon = (Reynolds number)^{-\frac{1}{2}}$. The method of our proof is more rigorous and generalized than the formal asymptotic analysis in the dynamics of fluid dynamic vortices.

Motion of vortex lines in the Ginzburg-Landau model

Equations of motion for vortex lines in the Ginzburg-Landau theory are derived. We construct asymptotic approximations for the complex order parameter which are valid in the core region and in the far field. Using the method of matched asymptotic expansions we show that, to leading order, the line moves in the binormal direction with a curvature-dependent velocity. We also consider the contribution of remote parts of the line, interaction between several vortex lines and interaction with external fields.

An investigation of curvature variations over recursively generated B-spline surfaces

The continuity properties of recursively generated B-spline surfaces over an arbitrary topology have been related to the eigenproperties of the local subdivision tranformation, and conditions have been established on the subdivision weightings for tangent plane continuity at extraordinary points. In this paper, curves through an extradordinary point, which align in both the tagent and binormal direction, are identified, and their curvatures are compared either side of the point. Further restrictions on the subdivision weightings are derived to optimize the curvature properties of the surface. In general continuity of curvature is not attained.

Approximate asymptotic solution of surface field due to a magnetic dipole on a cylinder

A simple approximate expression for the surface magnetic field due to a magnetic dipole on a conducting circular cylinder is obtained. This solution is asymptotic for a large cylinder radius, and may be used everywhere on the cylindrical surface including the penumbra and the deep shadow. In the limit that the cylinder radius is infinite, it becomes identical to the known exact solution of a dipole on a conducting plane. For a surface ray propagating in parallel to the axis of the cylinder, the transverse surface magnetic field is found to vary asymptotically as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(ks)^{-1/2}</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</tex> is the distance from the source. This behavior is distinctively different from the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(ks)^{-1}</tex> variation of the surface ray on a plane, and is explained in terms of the dependence of the surface curvature in the binormal direction of the ray. Our solution is basically derived from the classical work of V. A. Fock, but contains a conjecture that has been partially verified. We apply our solution to calculate the mutual admittance between two slots on a cylinder, and obtain results which are in excellent agreement with those calculated from the exact modal solution. For a cylinder with a radius of one wavelength or larger, this agreement is within several percent in magnitude and a few degrees in phase, even for zero separation between the two slots. A comparison of the present solution with two other asymptotic (GTD) solutions is also given.