First-order Correction Terms Research Articles

Diffraction by surfaces of variable curvature

To investigate the influence of variable curvature on the damping of diffracted waves (creeping waves) the integral equation method has been applied to an arbitrary convex cylinder. In addition to the well-known damping factor, depending on the radius of curvature <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</tex> , a first-order correction term yields an amplitude factor <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R^{1/6}</tex> while the second-order correction term results in a change in damping depending on the curvature and its first and second derivatives.

Limiting frequency-modulation spectra

The power spectrum of a carrier frequency-modulated by statistically stationary noise of any sort is studied in the limiting cases of large and small rms deviation. In the case of an rms deviation large compared to the frequencies present in the modulating wave, the spectrum is given by Woodward's theorem (the “adiabatic” theorem), which is derived together with a first-order correction term. The case of small rms deviation may be subdivided into (i) the case where the rms deviation is small compared to the frequencies present in the modulating wave and (ii) the case where the spectrum of modulating wave extends down to zero frequency. In case (i) the spectrum of the modulated carrier is the same as if it were amplitude modulated by the equivalent phase modulation, whose spectrum may be determined by dividing the spectrum of the frequency-modulating wave by the square of the frequency. In case (ii) the carrier, rather than appearing as a discrete spectral line, is spread into a bell-shaped Witch of Agnesi.

Curved Fluid Interfaces. II. The Generalized Neumann Formula

Following an analysis of the phenomenological concept of surface stress, the confluence properties of three fluid phases are examined from the molecular point of view. The detailed theory leads to a generalization of the Neumann surface tension triangle, which provides a boundary condition for the Laplace equations describing the surfaces of a fluid lens. It is found that the classical equation must be supplemented by a thermodynamic length parameter which, however, does not contribute to the Archimedean equilibrium of the lens. As in the preceding investigation, the first-order correction terms of the theory again provide criteria for the breakdown of thermodynamic concepts.

Higher Order Corrections to the Field Emission Current Formula

In this paper, higher order terms in expansions made to obtain the field emission current formula are retained. Analytic expressions for first-order correction terms to the current formula are derived.

Electron Flow Through Small Tubes With Magnetic Focusing

When a uniform finite magnetic field passing through the cathode is used for focusing a beam of electrons, random emission at the cathode results in the electrons traveling down the beam in a helical manner. Such a rotational motion severely limits the amount of current that can be made to travel through a small tube or between closely spaced parallel plates. The amount of current transmitted through such structures is calculated in this paper and presented in the form of normalized curves. It is assumed that the current initially fills the tube or the region between the parallel plates. If an alternating electric field having a frequency comparable to the rotational frequency of the electrons is applied normal to the beam, the average electron radius of motion is increased. Should the electron beam be made to flow through a tube after the application of such an alternating electric field, the current transmission through the tube will be reduced over that when no electric field exists. The first-order correction term for this case is calculated and presented in the form of normalized curves. The fact that the average electron radius is increased when an electric field is present implies added energy and leads to the formulation of a conductance expression relating the power in the rotational motion of the beam to the electric field normal to the beam. The current interception and the equivalent beam conductance are important in many electronic devices such as traveling-wave tubes. The beam conductance may be quite large in some cases—on the order of that present off the axis of a helix.