Volume 24, Issue 8, 01 August 1953

A Set of Principles to Interconnect the Solutions of Physical Systems
View Description Hide DescriptionA set of principles and a systematic procedure are presented to establish the exact solutions of very large and complicated physical systems, without solving a large number of simultaneous equations and without finding the inverse of large matrices. The procedure consists of tearing the system apart into several smaller component systems. After establishing and solving the equations of the component systems, the component solutions themselves are interconnected to obtain outright, by a set of transformations, the exact solution of the original system. The only work remaining is the elimination or solution of the comparatively few superfluous constraints appearing at the points of interconnection.
The component and resultant solutions may be either exact or approximate and may represent either linear or, with certain precautions, nonlinear physical systems. The component solutions may be expressed in numerical form or in terms of matrices having as their elements real or complex numbers, functions of time, or differential or other operators, etc. Boundary value, characteristic value, and time‐varying problems of partial differential equations, as well as problems in ordinary differential equations and algebraic equations may be solved in this manner.
The method shown may be extended and generalized so that one can tear apart and afterward reconstruct the solution of extremely large or highly intricate physical systems, often without calculating any inverse matrices at all and always without carrying along unduly large matrices. This extension and generalization of the method is analogous to building skyscrapers by erecting first a steel framework and only afterward filling the gaps between the girders as needed. Those versed in the science of tensorial analysis of interrelated physical systems on the one hand and of large electrical networks on the other, should thereby be able to solve, with the aid of already available digital computers, highly complex physical systems possessing tens of thousands and, in special cases, even hundreds of thousands of variables. The accuracy of machine calculations, of coding and even the correctness of the analytical procedure itself may be simultaneously checked by physical tests at various stages of the computation. The saving in computing time is considerable even in smaller problems; by tearing a physical system into n parts, the usual machine calculations are reduced, in matrix inversion for instance, to a fraction of about 2/n ^{2}.
The present paper develops in detail the solution of a simple boundary value problem of Poisson's equation. A numerical example of interconnecting the solutions of large electric‐power transmission systems appears in reference 3. Many simpler numerical examples are worked out in reference 1.

Contact and Rubbing of Flat Surfaces
View Description Hide DescriptionThe interpretation of certain phenomena occuring at nominally flat surfaces in stationary or sliding contact is dependent on the assumed distribution of the real area of contact between the surfaces. Since there is little direct evidence on which to base an estimate of this distribution, the approach used is to set up a simple model and compare the deduced theory (e.g., the deduced dependence of the experimental observables on the load) with the experimental evidence. The main conclusions are as follows. (a) The electrical contact resistance depends on the model used to represent the surfaces; the most realistic model is one in which increasing the load increases both the number and size of the contact areas. (b) In general, mechanical wear should also depend on the model. However, in wear experiments showing the simplest behavior, the wear rate is proportional to the load, and these results can be explained by assuming removal of lumps at contact areas formed by plastic deformation; moreover, this particular deduction is independent of the assumed model. This suggests that a basic assumption of previous theories, that increasing the load increases the number of contacts without affecting their average size, is redundant.

Measurement of Elastic Constants at Low Temperatures by Means of Ultrasonic Waves–Data for Silicon and Germanium Single Crystals, and for Fused Silica
View Description Hide DescriptionUltrasonicwaves (shear or longitudinal) in the 10–30 mc range are transmitted down a fused silica rod, through a polystyrene or silicone one‐quarter wavelength seal, and into the solid specimen. Measurement of reflections within the specimen yields values for velocities of propagation and elastic constants.
Data obtained over a temperature range of 78° to 300°K for silicon and germaniumsingle crystals, and 1.6° to 300°K for fused silica are listed. For the latter, a high loss is noted, with an indicated maximum near 30°K.

The Utilization of Degenerate Modes in a Spherical Cavity
View Description Hide DescriptionA general expression is given for calculation of the coupling coefficients between an arbitrary number of modes simultaneously excited in a cavity resonator. The mechanism of energy conversion between modes is discussed for several practical coupling devices. The technique is applied to the unlimited numbers of degeneracies available in spherical geometry and is shown to offer many interesting possibilities. An experimental model of a spherical cavity is described and shown to demonstrate the theory.

The Nature of Solutions of a Rayleigh‐Type Forced Vibration Equation with a Large Coefficient of Damping
View Description Hide DescriptionIn this paper, the author considers an approximation to the Rayleigh equation, , and its equivalent form under the transformation . A cubic function that appears in the altered equation is replaced by a piecewise linear function. For large ε, the existence of periodic solutions of the approximating equation is demonstrated. Strong orbital stability properties of neighboring solutions is also shown.

Diffusion of Barium in an Oxide‐Coated Cathode
View Description Hide DescriptionThe diffusion of Ba through activated BaO–SrO cathode coatings was studied as a function of temperature. Most of the radioactive tracer material diffused according to a dependence of log_{10} D vs 1/T similar to that obtained by Redington for diffusion of Ba in single crystals of BaO. The activation energy was 4.1±0.6 ev above 1280°K and 0.40±0.07 ev below that temperature.

Degradation of Polyisobutylenes on Shearing in Solution
View Description Hide DescriptionThe decrease of viscosity on shearing in a capillary at 37.8°C and a nominal rate of shear of 65 000 sec^{−1} has been measured for 5, 10, and 15 weight‐percent solutions in o‐dichlorobenzene of 10 polyisobutylenes having viscosity‐average molecular weights from 40 000 to 2 300 000. Very little, if any, viscosity decrease was observed for polymers having molecular weights below 500 000, but marked decreases were observed for polymers with molecular weights above 500 000. The observed decreases were not recovered on standing and correspond to appreciable decreases in the molecular weights of the polymers. The molecular weight reduction can be explained on the basis of polymer chain rupture if the mechanical energy developed in a volume less than that enveloping a single polymer molecule can be concentrated into a single bond.

Nonlinear Semiconductor Resistors
View Description Hide DescriptionA phenomenological theory for the nonlinear voltage‐current characteristic curve displayed by a granular aggregate of siliconcarbide is presented. The theory, which should apply to granular semiconductors other than siliconcarbide, is based on a very simple model of the aggregate and on the assumption that the essential resistance is located at the grain‐grain contacts, the impedance of the bulk material being considered negligibly small. The current i is related to the voltage V by the equation ,where A is the cross‐sectional area of the aggregate, t the thickness, P the applied pressure, and d the average particle diameter. The constants k, n, and m are structure‐sensitive, that is, they depend on the physical‐chemical nature of the particles forming the aggregate. Factors influencing the values of these constants include impurity concentration in the semiconductor, elastic constants, and particle shape. The validity of this equation is supported by careful measurements made on granular aggregates of siliconcarbide.

Filters for Detection of Small Radar Signals in Clutter
View Description Hide DescriptionRadar clutter is distinguished from thermal noise by being caused by random reflection of transmitted electromagnetic energy. This paper considers what may be done by means of linear filtering to improve the detection of small radar signals in the presence of this clutter. Defining the signal‐to‐clutter ratio as the ratio of the peak signal to the rms value of the clutter, the optimum linear filter is derived for enhancing this ratio. The optimum filter has a frequency characteristic given by the conjugate of the voltage spectrum of the transmitted pulse divided by the power spectrum of the clutter. The synthesis of the optimum filter and various approximations to it are considered. The influence of these filters upon the signal‐to‐thermal‐noise ratio is also investigated. Finally, the influence of the second detector upon the signal‐to‐clutter or signal‐to‐noise ratio is considered.

On the Interference of Pulse Trains
View Description Hide DescriptionA number of problems arising from the undesired overlapping or time coincidence of pulses from separate periodic pulse trains are considered. The coincidence time fraction is determined for two pulse trains with both a fixed initial phase and a randomly varying initial phase. The coincidence time fraction for these cases is then determined if only pulse coincidences equaling or exceeding a specified interval are considered. Finally, some problems involving more than two pulse trains are briefly discussed.

Beta‐Ray Spectrometer Line Shapes with Tilted Sources
View Description Hide DescriptionExperimental line shapes arising from tilted beta‐ray spectrometer sources are compared with analytically derived line shapes. Although the increased scattering in the source at higher tilting angles considerably alters the experimental line shape, the sharp leading edges predicted for the larger angles are largely retained. This is true for electron energies at least as low as 400 kev, even for quite dense sources (5 mg/cm^{2}). An application of the increased resolution resulting from this feature is given.

Measurement of Particle Size of Fine Ferromagnetic Powders
View Description Hide DescriptionThe particle sizes of four Fe and two Fe–Co powders with diameters ranging from 200 to 600 angstroms have been determined using electron microscope, x‐ray line broadening, and nitrogen adsorption techniques. Reasonable agreement was found between the electron microscope and the x‐ray methods, while the sizes determined through nitrogen adsorption were in every case larger. A method of dispersing single‐domain ferromagneticpowders for electron microscopy is described.

Determination of Reflection Coefficients and Insertion Loss of a Wave‐Guide Junction
View Description Hide DescriptionReflection and transmission coefficients are important parameters of a wave‐guide junction. It is shown that these parameters in phase as well as in magnitude may be read directly from a graph. A short‐circuit is moved in one of two wave guides that are connected by the junction, and the corresponding reflection coefficient is measured in the other wave guide as in a standard method. The positions of the short‐circuit, however, are taken equally spaced every ⅛th of a wavelength. When plotted in the complex plane, the corresponding reflection coefficients fall on a circle and the chords joining opposite points intersect at a point that is simply related to the desired parameters.
When very accurate measurements are desired, 8 or even 16 short‐circuit positions spaced at or wavelength can be used. The quality of the chord intersection then gives a control of the experiment by showing systematic or random errors. When the intersection is not perfect, because of random errors, a method of averaging is described that reduces these errors.

Erratum: Determination of the Drag on a Cylinder at Low Reynolds Numbers
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A Replica Technique in the Study of Chemical Precipitation Processes
View Description Hide DescriptionA replica technique has been developed for the electron microscope study of particles of precipitates formed by mixingsolutions of chemical reagents. The replica is formed thermally in a thin film of parlodion. A one‐step, shadowed replica has been found most satisfactory. The technique has been found suitable for studies of the effect of conditions of observation upon the specimen materials, studies of precipitates such as silver chloride which are known to be unstable during direct observation, studies of precipitates as they exist in the mother liquid; and studies which require detailed observation of surfaces.

Methods of Measuring the Properties of Ionized Gases at High Frequencies. IV. A Null Method of Measuring the Discharge Admittance
View Description Hide DescriptionThe admittance of a gas discharge may be obtained from measuring the ratio of the transmitted power through a microwave cavity to the incident power as a function of signal frequency near the cavity resonance. The method involves balancing the transmitted and incident signals to zero at the cavity resonance after they have passed through two separate receiving systems. When the signal frequency is changed, an attenuator is used to rebalance the two signals. The change in frequency from resonance and the corresponding change in attenuation gives the necessary data which will plot as a straight line whose slope yields the desired information.

Electrical Properties of Liquid Selenium I
View Description Hide DescriptionThe thermoelectric power of liquid selenium has been measured in the range 250–500°C. Values of activation energies calculated from the temperature dependences of thermoelectric powers and resistivities agree. This activation energy taken as the intrinsic activation energy (or the energy to break the Se–Se bond) is 2.31 ev calculated from data given by Henkels. The mobility of holes in the intrinsic region exceeds that of electrons and has a value (8.4×10^{6})/T ^{ <mml:math> <mml:mstyle> <mml:mfrac> <mml:mrow> <mml:mn>3</:mn> </:mrow> <mml:mrow> <mml:mn>2</:mn> </:mrow> </:mfrac> </:mstyle> </:math> } cm^{2}/volt sec, assuming electronic mass.
As a further check of the electronic nature of the conduction in the liquid and of theories of conduction in the hexagonal form, preliminary measurements of the thermoelectric power of pure selenium doped with small percentages of As, Sb, and Bi up to 4 percent were made. In the intrinsic region the specimens exhibited p type conduction, but at lower temperatures the sign of the thermoelectric powers reversed and values became quite large but negative. The crossover point depended qualitatively on the amount of impurity in the correct manner. The n type conduction is tentatively attributed to donor levels introduced by the normal trivalence of Group V_{b} elements. Two pelectrons are considered bound in the selenium chain, the third ionized at high temperature.
 LETTERS TO THE EDITOR


Schlieren Photographs of Sound Fields
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Effect of Cold Work and Anneal on Resistivity of Alpha Brass
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A Simpler Explanation for the Observed Shot Effect in Germanium Filaments
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