Volume 24, Issue 4, 01 April 1953

Statistical Criteria for the Detection of Pulsed Carriers in Noise. I
View Description Hide DescriptionA theory for the detection of pulsed carriers (of constant amplitude) in narrow‐band random noise is described, based on several types of optimum tests of a statistical hypothesis against an alternative (i.e., signal and noisevsnoise). Siegert's concept of the betting curve is introduced, whereby, for a finite integration time, the minimum detectable signal is uniquely defined. Three types of observer are next considered: the Neyman‐Pearson, the Ideal, and the Sequential observer, whose properties are determined by the manner in which the test is carried out. For each observer it is verified that the best second detector is a logI _{0}‐rectifier, which in practice is closely approximated by the usual half‐wave linear envelope‐tracer. In Part II specific betting curves are calculated and the performance of the three observers is analyzed for both the weak (threshold) and strong‐signal cases.

Statistical Criteria for the Detection of Pulsed Carriers in Noise. II
View Description Hide DescriptionIn Part I a statistical theory for the detection of a pulsed carrier in noise has been described, where specifically three types of observer are defined: (1) the Neyman‐Pearson, (2) the Ideal, and (3) the Sequential observer. In Part II specific betting curves are determined for the weak‐signal cases, and some approximate curves for strong signals are also included. The input signal‐to‐noise (power) ratio (a _{0} ^{2})_{min} corresponding to the minimum detectable signal at the output (with an arbitrary probability of a correct decision) is found, as expected, to vary as n ^{−½} for weak signals and as n ^{−1} for strong, where n=number of pulses integrated. Furthermore, comparison of observers on the basis of the minimum detectable signal reveals the Neyman‐Pearson to be better than the Ideal, by an amount 0 (0.5 db). On the basis of the assumed approximate distribution w_{s} (n_{s} ) of sample size, the Neyman‐Pearson also appears superior to the Sequential observer, but as indicated in Part I, the latter must always be as good as, if not better than the former. The discrepancy is the result of the approximate nature of the assumed distributions. A lower bound on the errors thus introduced in the calculations of the (a _{0} ^{2})_{min} is accordingly provided by the extent to which the Sequential appears worse than the Neyman‐Pearson observer for equal probabilities of type I and II errors α and β. For w_{s} (n_{s} ) =δ(n_{s} −<n>) this error is reasonably small 0 (1.5 db), but for a Gaussian distribution of n_{s} the effect is 0 (3.0 db) at least. Included also are an analysis of performance and some of the factors affecting it. The paper terminates with a brief account of a statistical theory of the observation process and its relation to the various observers considered here.

Use of Tschebyscheff‐Polynomial Operators in the Numerical Solution of Boundary‐Value Problems
View Description Hide DescriptionThis paper concerns the numerical solution, in two or more dimensions, of boundary‐value problems arising from linear partial differential equations, of which Laplace's and Poisson's equations furnish simple examples. Only techniques suitable for automatic computing machines are considered. The only method previously applicable to automatic machines is a straightforward iteration of an appropriate difference operator applied to an assumed trial solution on a network of points; as has been repeatedly pointed out, the rate of convergence of this iterative procedure is inordinately slow when the number of net points is large. The present paper shows how the employment of a Tschebyscheff polynomial of this same difference operator can cut the amount of work required in obtaining a solution by a very large factor—a factor of the order of magnitude of √N in the two‐dimensional case, in the three‐dimensional case, where N is the number of net points. This method is an outgrowth of that developed by Flanders and Shortley [J. Appl. Phys. 21, 1326 (1950)] for employment of such Tschebyscheff‐polynomial operators in the numerical solution of eigenvalue problems. The method is illustrated by a simple example involving Poisson's equation.

Optimum Nonlinear Filters
View Description Hide DescriptionThe theory of optimum nonlinear filters outlined in this paper is based on the consideration of a sequence of classes of nonlinear filters, designated as N_{1}, N_{2}, N_{3}, …, such that each class in the sequence includes all the preceding classes and, furthermore, the class of linear filters is a subclass of every class in the sequence. A filter of class N_{ m } is described in terms of a characteristic function which involves m age variables and m values of the input time‐function. The input‐output relationship for a filter of class N_{ m } has the form of an m‐fold integral of the characteristic function with respect to the m age variables. It is shown that the characteristic function of the optimum filter (in the least squares sense) within the class N_{ m } satisfies a linear integral equation of 2mth order. The optimization of filters of class N_{1} is treated in detail, and methods of approximate realization of such filters in the form of nonlinear delay line filters and power series filters are indicated. The results are extended to the case of nonstationary time series.

Effects of Very High Pressures on Glass
View Description Hide DescriptionIn contrast to other inorganic solids, glasses may be permanently compacted by application of pressures of the order of 10^{4} to 10^{5} atmospheres. This effect was studied on two simple oxide glasses (SiO_{2} and B_{2}O_{3}) and several silicate glasses. The effect of compacting was studied by measuring the densities, dimensions, and x‐ray diffraction patterns. A definite threshold pressure is observed in vitreous silica and silicate glasses, under which no effect takes place and above which the collapse takes place readily. Vitreous boric oxide behaves in a different manner, collapsing gradually, starting at the lowest pressures. Vitreous boric oxide exhibits also plastic flow with subsequent strain hardening. X‐ray diffraction measurements performed on vitreous silica and boric oxide indicate that the compacting proceeds on the atomic scale, leaving, however, the short‐range order of the basic structural units unchanged. The density of compacted glass can be restored to the original value by annealing to sufficiently high temperatures. Activation energy of this process was determined in the case of vitreous boric oxide.

The Susceptance of a Circular Obstacle to an Incident Dominant Circular‐Electric Wave
View Description Hide DescriptionThe problem of the susceptance of a circular obstacle in a circular wave guide with an incident TE _{01} mode is solved by a variational method. Theoretical expressions are obtained which are in good agreement with experimental results. Curves of normalized susceptance as a function of relative aperture, guide wavelength, and free‐space wavelength are included.

Bulk Viscosity of the System Polystyrene—Diethyl Benzene
View Description Hide DescriptionThe bulk viscosity of the system polystyrene‐diethyl benzene has been measured over the concentration range from 8 to 80 weight percent polymer and a temperature range from 30 to 130°C. A single sample of polymer was used. The activation energy for viscousflow is found to vary from about 2 to 100 kcal/mole as one goes from pure solvent to pure polymer. Using a series of 15 fractions of polystyrene ranging from 50 000 to 10^{6}, the molecular weight dependence for the viscosity of this system was measured at two polymer concentrations, 14 and 44 percent. The results are compared in a qualitative way with what one would expect from existing theory.

Positive and Negative Ions in Cathode‐Ray Tubes
View Description Hide DescriptionThe positive and negative ions accompanying the operation of a cathode‐ray tube contribute in a number of ways to tube failure. These ions have been studied in standard sealed‐off tubes. The identities of the most persistent ions have been determined and ion currents measured. Some attempts to limit or control the ions at their sources have been made.
The greatest offender is hydrogen which is present as both negative and positive ions. It is shown that aluminized screens offer little protection to the ion bombarded phosphor. Uncertainty in the literature as to whether negative m/e, 26 is C_{2}H_{2} or CN seems resolved in favor of C_{2}H_{2}.

The Effective Length of an Electrostatic Analyzer
View Description Hide DescriptionThe general formula for the effective length of an electrostatic analyzer with ``rounded'' corners is derived using a modified Schwartz transformation. For the dimensions considered the rounded corner effect introduces a correction of about 0.1 percent in the evaluation of the effective length of the analyzer.

Electrical Phenomena in Adhesion. I. Electron Atmospheres in Dielectrics
View Description Hide DescriptionRapid breaks of metal‐polymer‐metal adhesive specimens have shown the presence of a charge density on the metallic surface, provided the break occurs at the metal‐polymer interface, and a much smaller charge if it occurs in the interior of the polymer. This is analyzed in terms of the electron atmosphere existing external to the metal in a dielectric region of low barrier. The barrier values in certain cases are otherwise known to be of appropriate order of magnitude. Measured experimental surface charge densities permit estimation of the maximum limit of barrier height. Because of the presence of the charge distribution in the polymer, there is an electrostatic force on the adherend metal directed toward the dielectricpolymer, which represents a contribution to the total adhesive forces opposing break. Such electrostatic contributions to adhesion have not previously been considered; their order of magnitude and their relation to the thickness of the adhesive are discussed. The qualitative agreement between the theory and a number of previously unexplained experimental results in the literature is shown.

The Effect of Turbulence on Heat Transfer in a Pile with a Liquid Reactant
View Description Hide DescriptionThe spacing of cooling surfaces in a liquid reactant pile is considered in an elementary fashion. The resulting physical picture is that the eddydiffusion of thermal energy, which is enhanced by large free volumes of fluid, does not increase as fast as linearly with the spacing of cooling surfaces. Thus, for a given temperature difference between reactant and coolant, the total heat transfer decreases with increased spacing of the cooling surfaces. This disadvantage of large spacings is to be set off against the advantage of decreased parasitic neutron losses to the coolant.

A Retarding Potential Method for Measuring Electrical Conductivity of Oxide‐Coated Cathodes
View Description Hide DescriptionA retarding potential method is developed for measuring the electrical conductivity of normal oxide cathode coatings. The method is limited by normal currentmeasuring devices and can not be used for coatings which have a conductivity to thermionic emission ratio greater than 2 cm/volt. Advantages of the method are: (1) the conductivity of coatings which are in a normal state for thermionic emission may be measured without the use of probe wires or other devices which might impair the thermionic emission of the sample, and (2) conductivity and thermionic emissionmeasurements may be made simultaneously on the same coating sample. The theory of the method is discussed in detail and experimental results obtained using this method on both BaO and (BaSr)O coatings are given.

Engravement of Transient Stress Wave Particle Velocities
View Description Hide DescriptionQuantitative information on the high intensity transient waves set up by impulsive loading is usually required in order to design rationally against failure under such loads. To obtain reliable data is often difficult when the loads last only a few microseconds and reach magnitudes of several hundred thousand pounds per square inch. A technique is described which enables quantitative determinations to be made either of particle velocity within the transient wave or of velocity of propagation of the wave. The scheme consists basically of measuring the depth of the permanent impression or engravement that is left on a surface when a pellet that has been previously affixed to that surface flies off. The engravement is a direct consequence of the impingement of a transient wave against the surface. Considerable numerical data are presented which establish the validity of the scheme. The simplicity of the technique suggests that it may find extensive application in impulsive loading investigations.

Photoelastic Studies of Quenched Cylinders and Spheres
View Description Hide DescriptionA mathematical theory of residual stresses of quenching origin based on an idealized model of the quenching process is investigated. Photoelastic curves are computed from the theory and compared with those determined with a Babinet Compensator. Ten glass cylinders and spheres quenched in molten lead are employed. Although the qualitative agreement between theory and experiment is good, a quantitative comparison shows that the theory needs improvement. Major factors in the disagreement are probably the neglect of the effects of plastic flow and the temperature dependence of the physical constants.

Work Functions of Gas‐Coated Tungsten and Silver Surfaces
View Description Hide DescriptionThe average electronic work functions of tungsten and silversurfaces, subjected to glow discharges in various gases (He, A, H_{2}, N_{2}, O_{2}, freon‐12) and to water vapor, have been determined by Oatley's magnetron contact potential method. Oxygen and freon‐12 produced increases in work functions in excess of 1 ev, whereas nitrogen and particularly hydrogen caused decreases of the order of 0.5 ev. The inert gases and water vapor produced relatively little changes. The history and treatment of the surfaces is described, and the data presented have been found to be reproducible.

On the Zeros of Polynomials and the Degree of Stability of Linear Systems
View Description Hide DescriptionA method is presented for finding the zeros of any nth degree real or complex polynomial as its coefficients (or their parameters) are varied. Several new theorems are presented. The method is based on new theorems, some concerning determinants, and on known theorems. It is applied to characteristic equations of linear physical systems with or without feedback. Root trajectories in the complex plane can be visualized without plotting them from ``root trajectory diagrams'' which show curve families for real or complex roots of a polynomialequation in a plane having two coefficients of the polynomialequation as coordinates. The diagrams can be used in the converse manner to determine the coefficient values corresponding to a desired dominant complex pair or real root.

Approximate Elastic Spectrum of Acoustic Waves in AgCl from X‐Ray Scattering
View Description Hide DescriptionThe temperature diffuse scattering of x‐rays from a large single crystal of AgCl has been measured at room temperature using a Geiger counter spectrometer and crystal monochromated CuK‐α radiation. Measurements were made throughout the hk0 plane in reciprocal space, and in certain other regions. The data are analyzed on the basis of the theory developed by Laval and James, in which temperature vibrations are expressed in terms of traveling elastic waves. An assumed set of vibrational frequencies is used for the optical branch. Dispersion curves for waves in the acoustical branch traveling in the [100], [110], and [111] directions are obtained. The velocities at infinite wavelength are shown to agree fairly well with those given by the static elastic constants. An approximate frequency spectrum is then calculated, and suggestions are made as to how the effects of the optical and acoustical branches may be separated.

The Limiting Negative Pressure of Mercury in Pyrex Glass
View Description Hide DescriptionIt is generally considered that mercury does not ``wet'' glass, as evidenced by the convex surface of the mercury meniscus in a manometer. If this were true under all conditions, the development of a negative pressure in a mercury column would be impossible. But puremercury in a clean evacuated degassed tube shows a flat meniscus. In other words, when the tube is degassed,mercury does adhere to the glass wall.
A U‐tube manometer of 5‐mm bore with one leg sealed off was evacuated and degassed at 500°C with a mercury‐vapor pump, and then filled with pure mercury by distillation. The vertical height of the mercury column from the meniscus in the open tube to the top of the closed leg was 52 cm. When the manometer was again evacuated, the mercury column remained suspended from the top of the tube, withstanding a negative pressure of two‐thirds of an atmosphere.
In extending the measurements, short mercurial thermometers were first used. Each thermometer was cemented to the spinner of a variable‐speed motor, with the spin axis bisecting the distance from the end of the bulb to the meniscus. The maximum centrifugal stress on the free capillary column was consequently at the spin axis. Measurement of nine thermometers gave negative pressures ranging from 2 to 17 bars.
Finally, freshly drawn capillary tubes with fine bores were evacuated, degassed by heating, sealed off and filled by breaking one end under pure mercury. The negative pressure which these mercury columns could withstand under centrifugal stress increased rapidly as the degassing improved. The highest negative pressure observed for mercury was 425 bars at 28°C. In this case, the furnace temperature during degassing was raised to the point where the capillary tube started to deform.

On Unsteady Flow of Compressible Viscous Fluids
View Description Hide DescriptionIn order to solve the nonlinear equations of fluid dynamics, classical hydrodynamics introduced the concept of perfect incompressible fluids, viz., the perfect inviscid fluid and the perfect viscous fluid. The practical importance of considering compressibility nowadays has resulted in two new approaches from two distinct points of view. In the one case, compressibility is included in the discussion of a perfect inviscid fluid. As is well known, however, compressibile effects include shock‐wave phenomena, which are regarded in this instance as mathematical discontinuities. The goal then is to study the basic equations, allowing discontinuous solutions. The other approach retains the viscous terms together with the compressible ones. It is based upon the possibility of small coefficients producing large changes in the behavior of the solutions. The present paper is written from this point of view. The consequent mathematical difficulties of solving the equations are avoided by the use of a particle model of the continuum; the authors are indebted to Professor R. von Mises of Harvard University for this idea. The resulting equations are put into finite difference form for solution on a digital calculator.
 LETTERS TO THE EDITOR


A Simple Method for Measuring Interface Impedance
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