Volume 24, Issue 7, 01 July 1953

Nonequilibrium Thermodynamics
View Description Hide DescriptionThe foundations and implications of the concepts essential to the thermodynamicanalysis of nonequilibrium systems are investigated. Following a review of the background developments in irreversible thermodynamics the concept of thermostatic isolation is introduced. This provides a conceptual device whereby a thermostatic state may be established for any nonequilibrium system and a means of justifying extended use of basic relations of classical thermodynamics or thermostatics.
The conventional development of the form of entropy production is outlined. This in general is the sum of extrinsic entropy production and intrinsic entropy production. The former occurs as the sum of products of generalized diffusion effects and property gradients and is the result of interactions with discrete adjacent systems. The latter occurs as the sum of products of generalized affinities and generalized reaction rates and is the result of interaction between geometrically coincident overlapping systems. Between the diffusion effect and gradients of extrinsic entropy production generally occur linear relations with phenomenological coefficients subject to the Onsager reciprocal relations. The intrinsic entropy production occurs as a relaxation phenomenon developed from the first‐order relaxation between the affinity and the displacement of the reaction from equilibrium.
The relation of the ultimate limits of applicability of thermodynamics to the uncertainty in establishing thermostatic state under conditions of microscopic fluctuations is discussed.

Experiments on Radiation by Fast Electron Beams
View Description Hide DescriptionThe results of some experiments on millimeter wave and light generation by means of an undulator are described. After a brief survey of the theoretical background the design of a magnet system is discussed. An experiment is described in which a 100‐Mev electron beam from the Stanford linear accelerator passed through the undulator. Light radiated by the beam was observed and the plane of polarization determined. A small linear accelerator with good bunching action was used for an experiment on millimeter wave generation. At a beam energy of 3 Mev, radiation in a wavelength band below 1.9 millimeters was observed. A peak power output of the order of one watt was obtained. Millimeter waves generated in the accelerator tube were also observed.

A Theorem on the Impedance‐Transforming Properties of Reactive Networks
View Description Hide DescriptionComplex impedance‐matching problems are simplified substantially when they are studied in terms of appropriate circle transformations in the R‐X impedance plane. The geometric approach yields valuable insight into the behavior of specific networks and leads to the interesting Circle‐Locus Theorem which applies to any arbitrary, linear, reactive, two‐terminal‐pair network with lumped or distributed parameters (or a combination of both).

Experimental Verification of the Metal‐Strip Delay‐Lens Theory
View Description Hide DescriptionIn this paper a comparison is made between the index‐of‐refraction values measured on five metal‐strip test samples and values computed from four solutions that have appeared in the literature. Three of the samples are wave‐guide equivalents, and two are free‐space arrays. The techniques of measurement for the two types of structures are described briefly. It has been found that all four solutions have certain ranges of validity, but that only one is reliable over the full practical range of the design parameters. This solution, derived previously by the writer, is based on a transmission‐line equivalent circuit in which proximity effects are taken into account.

The Equivalence of Optimum Transducers and Sufficient and Most Efficient Statistics
View Description Hide DescriptionThe problem of the design of transducers for the purpose of restoring the fidelity of a signal (which has been corrupted by noise), while preserving information in the sense of Shannon is shown to be a particular though unusual one of statistical inference and to be amendable to the methods of mathematical statistics.
Adopting the view that the inference of the approximate form of a function s(t) (the signal) from the available mixture of the signal and the noisetime seriesn(t) is the natural limiting case of inference from discrete series of random variables, it is seen that the transducer under discussion is the physical analog of a ``statistic''—in the sense of Fisher—and that the properties of the optimum transducer are equivalent to the statistical properties of sufficiency (preservation of information) and efficiency (maximization of fidelity).
The case of a Gaussian signal to which Gaussian noise has been added is considered in detail. Using the probability density functional of the Gaussian time series, the maximum likelihood estimate (which in this case is efficient, hence sufficient) of the signal is derived, and it's physical analog is identified as the smoothing filter of Wiener with infinite lag.

Transmission of Electromagnetic Waves through Pairs of Parallel Wire Grids
View Description Hide DescriptionThe control and direction of electromagnetic waves is accomplished, in general, by the use of boundary materials and structures which produce desired effects on such waves when irradiated by them. An important class of boundary structure for this purpose is a surface having alternate areas of conducting and nonconducting material. Grids formed of a large number of parallel cylindrical conductors of circular cross section, uniformly spaced and lying in a common surface, are among the most generally useful of this class of boundary structure. Single layer parallel wire grids have received considerable study in the past. However, little information is available concerning the characteristics of multiple layers of such grids
An analytical solution is obtained for the power transmission coefficient of a double grid system. Power transmission coefficients for various combinations of system parameters are computed from the analytical solution and tested experimentally. The solution is found to yield excellent correlation with experiment until the interwire spacing approaches the wavelength.

Starting Currents in the Backward‐Wave Oscillator
View Description Hide DescriptionThe starting current of a simple model of the backward‐wave oscillator described by Kompfner and Williams has been calculated. The effect of space charge is included. The starting current I _{0} may be written in the form

New Developments in the Production and Measurement of Ultra High Vacuum
View Description Hide DescriptionA new vacuum technology is described which makes it possible to achieve working pressures as low as 10^{−10} mm Hg in a routine and straightforward fashion. This is accomplished without the use of chemical getters, special traps, or refrigerants of any kind. Essential to this technology are several new instruments which have been especially developed for the production or measurement of ultra high vacua. These new tools have permitted us to carry out a number of researches concerning the phenomena which occur at very low pressures. The ultra high vacuum technology has also made possible a new experimental approach in the fields of gaseous and physical electronics.

Convection Currents in Porous Media. V. Variational Form of the Theory
View Description Hide DescriptionA theory for the mean thermal gradient at the onset of convective flow of a fluid in a porous medium and heated from below, is given in variational form. Thermal equilibrium is not assumed, nor is viscosity assumed to be constant. An approximate solution is given for the theory, and is shown to agree quite well with published data. No attempt is made to maintain complete mathematical rigor in the derivations.

Probe Studies of Energy Distributions and Radial Potential Variations in a Low Pressure Mercury Arc
View Description Hide DescriptionProbe measurements were made in the plasma of a low pressuremercury arc. The electron‐energy distributions showed depletions from a Maxwellian distribution in the high energy range. Coupling effects between adjacent probes were investigated and were found to be quite small but in the proper direction to agree with the Langmuir‐Tonks theory. Drift‐current distortion of the random electron‐energy distributions was measured with a bidirectional probe and compared with theory. A multisection probe extending from tube axis to tube wall allowed a determination of radial potential and density variations. Results over a pressure range from 3.4 microns to 35 microns showed good agreement with the ambipolar diffusion theory based on cumulative ionization. A direct calculation of ionization rate in the plasma was made from the ionization probability for a one‐step ionizing process; comparison of this calculation with the observed ionization rate at 1.7 microns indicated that at that pressure the ionization is half direct, half cumulative. For higher arc pressures cumulative ionization evidently predominates.

A Certain Problem in Heat Conduction
View Description Hide DescriptionA homogeneous sphere is surrounded by an infinite homogeneous medium. Heat is generated inside the sphere at a prescribed rate. The temperature distribution inside and outside the sphere is computed.

Potential Analog Network Synthesis for Arbitrary Loss Functions
View Description Hide DescriptionA general method is developed for designing networks with arbitrary loss functions based on the potential analogy. An appropriate potential problem is formed on the basis of the given loss function by introducing continuous charge distribution on the complex frequency plane. After the potential problem is solved, the technique of quantization of charge is used to find the natural modes of the network function.

Matrix Solution of Equations of the Mathieu‐Hill Type
View Description Hide DescriptionThis paper presents a method for the solution of a class of linear, second‐order differential equations with periodic coefficients of the Mathieu‐Hill type. The method is based on a procedure involving powers of matrices and is adequate for the study of a large class of physical problems. The stability of the solution is considered and the general method is illustrated by its application to typical special cases.

The Relaxation Distribution Function of Polyisobutylene in the Transition from Rubber‐Like to Glass‐Like Behavior
View Description Hide DescriptionThe steady flow viscosity of a sample of polyisobutylene of viscosity‐average molecular weight 1.35 million, distributed by the National Bureau of Standards, has been measured from 15° to 100°C. Its logarithm is a linear function of 1/T ^{2}. Application of the method of reduced variables to dynamic mechanical data from −45° to 100°, previously reported for this polyisobutylene, yields composite curves reduced to 25°C for the real and imaginary parts of the complex compliance and complex shear modulus; the real part of the complex dynamic viscosity; and the mechanical loss tangent. The latter exhibits a broad and peculiarly asymmetric maximum. The reduced time scale extends from 1 to 10^{−9} sec. The reduction factors a_{T} obtained in this way are slightly higher than those derived either from the viscosity or from stress relaxationmeasurements of Tobolsky and associates. The distribution functions of relaxation and retardation times have been calculated by second approximation methods and their detailed shapes are defined in the transition region between rubber‐like and glass‐like behavior. The relaxation distribution function is compared with the idealized distribution of Tobolsky.

Helix Impedance Measurements Using an Electron Beam
View Description Hide DescriptionImpedance measurements made on a tape helix with an electron beam are described. Curves showing the measured impedances of the fundamental and first backward space‐harmonic components of the mode commonly used in traveling‐wave tubes are presented for values of ka (circumference to free‐space wave‐length ratio) ranging from 0.15 to 0.6. The impedance of the fundamental is found to be less than that calculated from the sheath model by a factor ranging from 0.8 to 0.3. Phase velocities of other modes and components were observed for ka from 0.1 to 1.1; these agree with an analysis by Sensiper which predicts that certain values of the phase constant are not allowed for a single wire helix. A relationship for the impedance of one space‐harmonic component in terms of the impedance of its fundamental is presented in approximate agreement with the experimental data. In addition to providing impedance data over a wide frequency range, the helix tester performed as a continuously voltage tunable backward‐wave oscillator from 1500 to 4300 mc at a beam current of 1 ma.

Consolidation Around Pore Pressure Meters
View Description Hide DescriptionResponse of pore pressure meter on variations in loading conditions of surrounding soil is retarded by the necessity that pore water has to enter the instrument. This property is introduced as an instruments coefficient influencing the boundary conditions. With regard to the surface of the instrument, two types are considered: a rigid type and a cavernous type. At t=0 for unit step loading, consolidation has not yet started and the response of the instrument depends on shear modulus of soil only. Calculation of response as a function of time involves three‐dimensional consolidation theory and is established with the aid of spherical solutions for simple harmonic and unit step loading conditions.

Reflection of Sound in the Ocean from Temperature Changes
View Description Hide DescriptionWhen the temperature of the ocean changes with depth within a horizontal layer, it is accompanied by a change in the velocity of sound. The reflection of sound from such a transition layer depends on the incident angle and wavelength of the sound, the thickness of the layer, the manner in which the velocity changes, and the total amount of the change. The usual formula for reflection assumes a transition layer of negligible thickness. The reflection coefficient then depends only on the incident angle and the total velocity change. This case is treated and numerical values are given for the reflection coefficient under conditions of practical interest. It is then shown that for a gradual velocity change which corresponds to certain temperature data, the reflection coefficient for small glancing angles of incidence changes gradually with increasing layer thickness and may be approximated very closely by the usual formula. Quantitative comparison of the cases is made and the general applicability of the approximation is discussed.

On the Statistical Theory of Detection of a Randomly Modulated Carrier
View Description Hide DescriptionA method is outlined for estimating the probability of detection for a pulsed radar, assuming a randomly modulated carrier. For a square law detector closed forms for the moments of the distribution of the envelope are presented in terms of three different choices of the distribution of carrier amplitude, thus leading to an Edgeworth series representation of the desired probability. At least one choice of distribution of the carrier amplitude leads to closed forms for the moments for the case of a linear detector. Curves of probability of detection vs signal‐to‐noise power ratio are constructed and cross checked by a method of numerical integration.

A New Method for Continuous Viscosity Measurement. General Theory of the Ultra‐Viscoson
View Description Hide DescriptionA new continuous ultrasonic viscometer employing a pulsed resonant exponentially damped magnetostrictive strip is described and treated theoretically.Properties of viscoelastic materials affecting the propagation of transverse elastic waves are derived and are related to the response of the viscometer. Typical applications of the use of the ultrasonic viscometer are presented.
 LETTERS TO THE EDITOR


Comments on ``The Dirac Delta Function and the Summation of Fourier Series''
View Description Hide Description
