Index of content:
Volume 23, Issue 2, 01 February 1952

On the Accumulation of Errors in the Numerical Solution of Differential Equations
View Description Hide DescriptionThe following gives a method for the evaluation of errors in the numerical solution of systems of differential equations.
Under certain assumptions the errors can be obtained by finding the difference r(x) between the computed solution y(x) passing through the given initial point, and the solution Y(x) passing through a neighboring point—that is, by solving the same equations repeatedly with different initial conditions.

The Thermal Conductivity of Metals at High Temperature
View Description Hide DescriptionThe Forbes bar method has been modified and adapted to the problem of measuring the thermal conductivity of metals and alloys in the temperature range from 25°C to 1000°C. A mathematical analysis is presented which indicates the error involved by assuming plane isothermals in the sample. The electrical and thermal conductivities of commercial ``A'' nickel, inconel, several stainless steel alloys, and 1010 steel are reported, and the results are compared, qualitatively at least, with the theory presented by Wilson^{1} and Makinson.^{2}

Cathodic Sputtering for Micro‐Diffusion Studies
View Description Hide DescriptionA method which has been developed to allow uniform removal of micro‐layers of metal from cylindrical specimens is based on sputtering of the metal from the cathode of a gas glow discharge tube. This technique is directly applicable to diffusion studies and offers many advantages over other more common methods.

Unsteady‐State Separation Performance of Cascades. I
View Description Hide DescriptionThis paper concerns the exact solution of the nonlinear differential equations which describe the time dependent behavior of multistage cascade separating processes for separation of two very similar molecular species. The Rayleigh separation law is postulated for each stage.
The main questions of interest in the nonstationary operation of cascades involve (a) their behavior while the concentration of the required component is being built up to the desired value; (b) the behavior of the cascade during the transition from one stationary mode of operation to another; and (c) the manner in which local concentration fluctuations are propagated through the cascade. We have derived analytical expressions that are suitable for the discussion of all of these points in square cascades. The cases of finite, half infinite, and infinite cascades are considered with and without product withdrawal.
A brief discussion of the linearization of a class of nonlinear second‐order partial differential equations is given in Appendix I.

An Iterative Method for the Solution of Eigenvalue Problems
View Description Hide DescriptionA simple iterative procedure is developed for the determination of eigenvalues and eigenfunctions associated with the solution of Sturm‐Liouville problems in a finite interval. Both the frequency and the displacement for a given mode may be determined with an accuracy which is independent of the accuracy involved in the calculation of other modes. Convergence of the iteration process is rapid, and the successive approximations to a given eigenvalue are shown to form a monotone sequence. The method is particularly useful when the coefficients of the differential equation are not expressed in analytical form.

Phenomena Associated with the Flight of Ultra‐Speed Pellets. Part II. Spectral Character of Luminosity
View Description Hide DescriptionThe spectroscopic character of the luminous trails generated by ultra‐speed pellets has been investigated. A total of seven spectrograms have been measured and reduced. Three of these were of 5.0 km/sec aluminum pellets; two were of 5.9 km/sec magnesium pellets; one was of a 6.0 km/sec magnesium‐lithium‐aluminum pellet; and one was of a titanium pellet whose velocity was estimated at about 4.5 km/sec.
The pronounced features of the spectrograms are: (1) AlO bands in the aluminumspectra, (2) Mg lines and MgO bands at the magnesiumspectra, (3) strong Li lines and a pronounced continuum in the magnesium‐lithium‐aluminum spectra, and (4) TiO bands in the titaniumspectra. Other features of the spectra are: (1) prominent AlI lines at λλ3944 and 3961 and a relatively weak continuum in the aluminumspectra, (2) a stronger continuum in the magnesiumspectra, and (3) relatively weak MgO bands in the Mg‐Li spectra.
The time resolvedspectra show that the AlO persists in the excited state twice as long as the aluminum. The aluminum oxidizes almost completely leaving AlO in an excited state which continues to emit light after the aluminum is burned.

The Reversion Method for Solving Nonlinear Differential Equations
View Description Hide DescriptionThe subject considered in this paper is the presentation of a method for the solution of ordinary nonlinear differential equations of a certain type. The method is based on the algebraic procedure used in reverting power series. By the use of this method, ordinary differential equations having certain type of nonlinearity are reduced to systems of linear differential equations. The resulting linear differential equations are then solved by the Laplace transform or operational method. The general procedure is illustrated by means of examples, and the method is applied to nonlinear electric circuits.

Space Charge and Transit Time Considerations in Planar Diodes for Relativistic Velocities
View Description Hide DescriptionThe usual Child‐Langmuir equation is extended to the case of relativistic velocities for a diode with parallel plane electrodes. The solutions obtained are valid for any value of the accelerating voltage. As the variation of mass with velocity becomes important, the exponent giving the dependence of current density on anode voltage becomes less than three‐halves, and for very large values of accelerating potential approaches unity. The variation with anode voltage of the transit time, both for the space‐charge‐free case and for the space‐charge‐limited case, has been calculated. The potential distribution across the diode is also discussed.

The Structure of Tetragonal Tungsten Trioxide
View Description Hide DescriptionA study of tungsten trioxide (WO_{3}) by differential thermal analysis and x‐ray diffraction at temperatures up to 980°C showed a new tetragonal form to exist at temperatures of 740°C and higher. This form has a distorted ReO_{3} structure. There are two molecules in a unit cell of dimensions a _{0}=5.25A, c _{0}=3.91A. All atoms lie in the 2‐ or 4‐fold special positions of the space group P4/nmm. The tungsten atoms are alternately displaced in opposite directions from the center of charge of the oxygen atoms so as to resemble the antiferroelectric structure proposed by Kittel.

Space‐Charge‐Wave Propagation in a Cylindrical Electron Beam of Finite Lateral Extension
View Description Hide DescriptionA method is outlined for obtaining the effect of geometry on the propagation of space‐charge waves in a cylindrical electron beam under arbitrary direct current conditions. In particular, the case of the complete‐space‐charge diode is discussed in detail. The results of this theory are verified by the experiments of Cutler and Quate.

The Dielectric Constant of Water Vapor in the Microwave Region
View Description Hide DescriptionThe dielectric constant of water vapor was determined at 9280 Mc by a cavity comparison method at 10 temperatures ranging from 32 to 103°C. In addition, observations were made at 24,800 Mc at the single temperature of 24.5°C. Although the vapor pressure was raised to within 10 percent of the saturation value at 24.5°C and 103°C, the variation of (ε′−1)/(ε′+2) with pressure remained linear, contrary to observations made at radiofrequencies. These data are discussed in relation to the questions of association and adsorption of the vapor. The present results are in agreement with those obtained at radiofrequencies (sufficiently below saturation pressures) and do not show the dispersion reported in an early microwave experiment. Results obtained here and those obtained by others are used to determine mean values for the constants in the Debye equation: namely, P=[3.96±0.32]+[(2.077±0.016)10^{4}/T]. This gives a dipole moment of (1.846±0.005)10^{−18} esu.

Generalized Ideal Filters
View Description Hide DescriptionOne of the basic concepts in the theory of electrical filters is that of an ideal filter. As commonly defined, an ideal filter is a network which passes without distortion all frequency components falling within a certain frequency range while rejecting all other frequency components. In the present paper a broader definition is formulated which extends the concept of ideal filter to both linear varying‐parameter and nonlinear types of systems. Briefly, a filter N is said to be ideal if it can extract a set of signals from the sum of and some other set . A direct consequence of this definition is that any ideal filter N (linear or nonlinear) is idempotent, that is, is equivalent to a tandem combination of two filters each of which is identical with N. The converse, however, is true only in the case of linear filters.
The basic properties of ideal filters are investigated by the use of function space techniques. It is shown, in particular, that by employing linear ideal filters one can separate two (or more) simultaneously transmitted sets of signals which occupy overlapping frequency bands, provided only the sets in question span disjoint manifolds in the signal space.

The Effect of Gas Pressure in the Electric Bulb on the Polarity of the Rectified Current
View Description Hide DescriptionIn sequel to the experiment by Ch'en and Liu (see reference 2) that a rectified current was derivable from the surface of the bulb when the bulb was coated with Chinese ink and was run by alternating current, the following points were observed with 220‐v bulbs: (1) When the nitrogen pressure inside the bulb was more than a hundred mm Hg, the polarity of the rectified current was always negative, and the current amounted to a few milliamperes. The current decreased with the decrease of pressure. At a pressure of a few mm Hg the current was decreased to a very small value, and then turned to positive polarity when the pressure was further decreased. The positive current increased to a weak maximum at a few tenths of a mm Hg and came down again as the pressure was further lowered. (2) The currentvs pressure curves for argon, helium, and neon were essentially similar to those for nitrogen. (3) When the inner surface of the bulb was coated with a layer of metallic sodium, the rectified current, which was positive while without the coating, turned out to be negative. (4) When the material used to paint the bulb contained no sodium compounds, a quick fatigue was observed.

Considerations Regarding the Distribution of Radiation Damage in Matter Bombarded by Light Ions and a Method of Making it Uniform
View Description Hide DescriptionIf a beam of high energy particles impinges on a solid, radiation damage is produced only in a very thin layer at the end of the range of the particles. Computations determining the extent of this inhomogeneity of damage are outlined and carried out for the specific case of 20‐Mev deuterons impinging on a copper sample. A simple method is discussed for producing uniform damage in a thickness corresponding to the range of the particles.

Studies in Newtonian Flow. III. The Dependence of the Viscosity of Liquids on Molecular Weight and Free Space (in Homologous Series)
View Description Hide DescriptionThe free‐space concept, previously applied to variation resulting from a change in temperature only, is here adapted to a case where both temperature and molecular weight vary. The molecular weight range of the n‐paraffins illustrative of this case is limited to m=100 through m=240. Over this range of molecular weights the family of lines represented by lnη=B(ν_{0}/ν_{ f })+lnA intersects the vertical axis at very nearly a common point. By assuming a common intercept and representing the slopes of this family of lines in terms of molecular weight, an expression defining viscosity as a function of molecular weight and free space is deduced. This expression reproduces the ``selected data'' satisfactorily over the molecular weight range mentioned. Its greater significance as a step in the development of a far more useful function will become apparent in the succeeding paper.

Space‐Charge‐Limited Currents between Inclined Plane Electrodes
View Description Hide DescriptionThe method of attack devised by Walker has been applied to the problem of current flow between inclined plane electrodes. It is assumed that the initial velocities of emission are negligible. Solutions for the potential distribution, the space‐charge characteristic (``perveance''), the particle trajectories, and the transit time in such diodes under conditions of complete space‐charge limitation are given. The trajectories and transit time have also been determined for the space‐charge‐free case. The results are presented in both graphical and tabular form. In most cases the solutions are given for angles between the electrodes as great as 180°.
For an angle between the electrodes of approximately 118° the transit time in the space‐charge‐free case is equal to that for complete space charge limitation. The effect of accidental electrode tilt on the perveance of a diode, which presumably has parallel electrodes, has also been investigated. A rectangular electrode whose width is 1.24 times the interelectrode spacing is relatively insensitive to tilt about a line passing through its center and parallel to its side. A disk‐shaped electrode is insensitive to tilt about any axis passing through its center if its diameter is 1.42 times the interelectrode spacing.

Diffraction Measurements at 1.25 Centimeters
View Description Hide DescriptionMeasurements have been made, using an image plane technique, of microwavediffraction near obstacles whose characteristic dimension is comparable to a wavelength (1.2668 cm). Detector, obstacle, and source of radiation are mounted on one side of a horizontal 4‐ft by 6‐ft conducting plane, and the operator and all auxiliary apparatus are located on the other. Measured distributions of both the phase and amplitude of the electric field near long cylinders of brass and polystyrene were obtained, and for the conducting cylinders these are compared directly with the results of a suitably modified theory. The technique has been extended to diffraction by a semi‐infinite conducting plane screen, and it was found in this case that considerable error is introduced by the point source. With the substitution of a line source the measurements are again in close agreement with the theory.

The Emission of Radiation from Diatomic Gases. III. Numerical Emissivity Calculations for Carbon Monoxide for Low Optical Densities at 300°K and Atmospheric Pressure
View Description Hide DescriptionNumerical emissivity calculations at 300°K and atmospheric pressure for nonoverlapping rotational lines have been carried out for CO using a dispersion formula for the line‐shape representation. Use of the best available experimental data on integrated absorption and rotational line‐width leads to calculated emissivities which are in excellent agreement with extrapolated empirical data published by Hottel and Ullrich. In particular, the theoretical dependence of emissivity on optical density, for small optical densities at 300°K, has been shown to follow experimental observations with satisfactory precision.
For small optical densities the calculated emissivity is found to be proportional to the square root of the assumed rotational line‐width, thus emphasizing the need for accurate line‐width determinations at elevated temperatures. The limits of validity of the treatment utilizing nonoverlapping rotational lines are defined by examining overlapping between adjacent weak and strong rotational lines.
The calculation of emissivities can be simplified by the use of approximate treatments. Thus absolute values of the emissivity can be predicted within 10 percent by utilizing a treatment for nonoverlapping, equally spaced, and equally intense lines, together with empirically determined values for the equivalent mean integrated absorption of the rotational lines of CO. A better analytic solution, which does not involve the assumptions of equal spacing and equal intensity of the rotational lines, has been obtained by utilizing asymptotic relations for large values of modified Bessel functions.

Secondary Electron Emission from Metals under Positive Ion Bombardment in High Extractive Fields
View Description Hide DescriptionIn the further study of breakdown mechanisms in high vacuum, experiments show that secondary electron emission from metals under positive hydrogen ion bombardment is small and increases only slowly with the electric field strength at the bombarded surface. Although more significant emission may yet be found with the heavier ions obtained in an actual vacuum gap, the particle interchange component of the breakdown process appears to be quantitatively inadequate on the basis of present measured electron and positive ion emission coefficients. The possible importance of negative ion emission deserves consideration in the particle exchange process.