Volume 24, Issue 3, 01 March 1953
Index of content:
24(1953); http://dx.doi.org/10.1063/1.1721257View Description Hide Description
A number of methods of numerical solution of systems of partial differential equations are surveyed. The size and speed of computing machines required to solve problems of varying complexity is discussed. Finally, a number of typical problems which have been solved numerically by various authors are described.
24(1953); http://dx.doi.org/10.1063/1.1721258View Description Hide Description
The authors present the design and construction of an analog computer unit which is particularly well adapted for solving nonlinear differential equations. The units are readily combined to solve simultaneous nonlinear differential equations. The unit is carefully checked by using it to solve some nonlinear differential equations for which known solutions exist. Finally the unit is used to find the solution to some nonlinear differential equations whose solutions are not given in the literature.
24(1953); http://dx.doi.org/10.1063/1.1721259View Description Hide Description
The effect of electron encounters on the velocity distribution in an electron beam after passage through an accelerating grid is calculated by direct integration of the Boltzmann equation. It is found, with electron densities and velocities typical of traveling wave tubes and diodes, that the electron encounters have a negligible effect, that the formulas of the usual kinetic theory for heat conduction do not hold, and that the longitudinal component of the electron gas pressure is negligible.
24(1953); http://dx.doi.org/10.1063/1.1721260View Description Hide Description
The drift velocity of electrons in an electric field in argon and alcohol mixtures has been measured. The measurement has been taken observing on the screen of a triggered sweep oscillograph the rise time of pulses as given by α‐particles in an ionization chamber. The results give, for the 10 percent mixture, a behavior of the w(E/p) function, of the w=aE/p type up to E/p=2 v cm−1 mm Hg−1 values, and of the w=const type for larger values of E/p. Measurements at 0.5 and 5 percent alcohol concentrations are also reported.
24(1953); http://dx.doi.org/10.1063/1.1721261View Description Hide Description
Artificial dielectrics have been constructed with indices of refraction as high as 7.2 at a frequency of 9364 mc/sec by imbedding microscopic conducting flakes in paraffin wax. When good conductors such as Cu and Al are used, the resulting permeability is observed to be complex while the permittivity is substantially real. This phenomenon is in agreement with the theoretical results of Lewin. It is indicated that a low loss, high refractive index, artificial dielectric can be constructed by using powdered conductors if the particles are very small and possess a high electric polarizability.
24(1953); http://dx.doi.org/10.1063/1.1721262View Description Hide Description
It is shown that the electromagnetic field in a loss‐free cavity excited through holes cannot be completely expressed in terms of the short‐circuit modes of the cavity satisfying the condition that the tangential component of the electric field is zero on the boundary of the cavity including the openings. For a complete expansion it is necessary to add an irrotational magnetic field, which contributes a term inversely proportional to the frequency, to the usual admittance matrix. If the cavity is presumed to include a reasonable portion of the guides feeding the openings, this irrotational component becomes almost diagonal.
24(1953); http://dx.doi.org/10.1063/1.1721263View Description Hide Description
A few formal observations are made concerning wave propagation through an inhomogeneous medium. A modification of Kirchoff's formula is suggested and an integral equation derived which, if certain integrals can be evaluated, gives an estimate of the error made in the usual approximate methods. Applications are indicated to the equivalent earth's radius model and to the flat earth, modified index model.
24(1953); http://dx.doi.org/10.1063/1.1721264View Description Hide Description
An extension of the Emmons numerical method for temperature fields is developed for application in cases of solid‐fluid heat transfer in which a gas or fluid passes through a porous wall generating nonuniform heat. The numerical approach is first presented in general form and then applied to the two specific cases of a porous power‐producing cylindrical pipe and porous power‐producing plain wall.
24(1953); http://dx.doi.org/10.1063/1.1721265View Description Hide Description
The dynamics of a line of traffic composed of n vehicles is studied mathematically. It is postulated that the movements of the several vehicles are controlled by an idealized ``law of separation.'' The law considered in the analysis specifies that each vehicle must maintain a certain prescribed ``following distance'' from the preceding vehicle. This distance is the sum of a distance proportional to the velocity of the following vehicle and a certain given minimum distance of separation when the vehicles are at rest. By the application of this postulated law to the motion of the column of vehicles, the differential equations governing the dynamic state of the system are obtained.
The solution of the dynamical equations for several assumed types of motion of the leading vehicle is effected by the operational or Laplace transform method and the velocities and accelerations of the various vehicles are thus obtained. Consideration is given to the use of an electrical analog computer for studying the dynamical equations of the system.
Surface Effects and the Creep of Zinc Single Crystals: I. Creep Characteristics of Clean and Oxide‐Coated Crystals24(1953); http://dx.doi.org/10.1063/1.1721266View Description Hide Description
Quantitative studies of the influence of the surface composition on the creep of zincsingle crystals under constant loads have been carried out. The creep curves of crystals with clean and oxidizedsurfaces were obtained in air and in acid. It is shown that a continuous layer of oxide strengthens the crystal only at low loads, and that the presence of cracks in this layer allows the metal to creep at its normal rate. These facts support the theory that the oxide film acts to relieve the metal of part of the load. An increase in creep rate is observed when the pure metalsurface is treated with dilute mineral acids. The heat liberated by the reaction is considered as a possible explanation of the sudden increase in creep rate when the surface layer of the crystal is removed. The normal creep curves of unoxidized crystals in air may be described by an empirical equation of a new form, and its application to other types of curves obtained is discussed.
24(1953); http://dx.doi.org/10.1063/1.1721267View Description Hide Description
This paper deals with the problem of finding the ``optimum'' method of detecting a sine wave of known frequency and amplitude in the presence of noise. The type of noise considered is the so‐called Gaussian process, which is obtained when thermal noise is passed through an arbitrary linear passive device.
The analysis takes into account the fact that in practice only a finite sample of observed signal is available.
The optimum detection method is defined as that which maximizes the probability of recognizing the presence of a sine wave if one has actually appeared; while the probability of falsely announcing the presence of a sine wave, if none has actually appeared, does not exceed some prechosen value.
It is shown that when the noise has a flat spectrum, all the relevant information is contained in the amplitude and phase of the Fourier transform of the received sample at the frequency of the sine wave. Almost the same result holds in the case where the noise has an exponentially decaying autocorrelation, except that in this case the values of the observed sample at the end points of the sample also play a role.
24(1953); http://dx.doi.org/10.1063/1.1721268View Description Hide Description
An exact expression for the temperature difference between the inside and outside of an infinitely long cylinder heated inductively has been derived in terms of Bessel functions. A numerical comparison shows that for small cylinders (radius about 0.3 cm) an assumed parabolic distribution of the magnetic intensity agrees with the exact solution within two percent.
24(1953); http://dx.doi.org/10.1063/1.1721269View Description Hide Description
A method is demonstrated for synthesizing an unbalanced RLC network with lossy coils, i.e., each inductance has an associated series resistance. The network realizes a minimum phase transfer function whose numerator and denominator are of degree not higher than the third and fourth, respectively, and whose poles and zeros may lie anywhere in the left half of the complex frequency plane. It is shown that in many cases higher degree functions may be realized as one network. For a transfer function of arbitrary degree, factoring into component transfer functions is carried out, each component having the above characteristics; the components are then realized separately and isolated from each other by vacuum tubes supplying gain.
The method makes use of the following new features:
1. A breakdown of a Hurwitz polynomial into two Hurwitz polynomials so that a division by one of them yields a sum of unity plus a positive real RLC function,each of whose partial‐fraction terms (complex poles taken in pairs, or course) is positive real.
2. An application of a network theorem which divides the desired network into two networks, thus simplifying the synthesis problem.
3. A method of zero shifting to any point in the left half‐plane with at least one pair of complex poles.
A further advantage of the procedure is that the network may be designed to possess input and output shunt capacitances so that tube capacitances may be automatically provided for.
24(1953); http://dx.doi.org/10.1063/1.1721270View Description Hide Description
In the following, the author explains the principle of the double resonant circuit method as used in measuring the electrical characteristics of extremely poor insulating materials such as wood or textile containing much moisture. The characteristics of the measuring circuit and the results obtained thereby are studied.
24(1953); http://dx.doi.org/10.1063/1.1721271View Description Hide Description
Series circuits of ordinary light bulbs and iron‐core transformer coils are found to have properties of the different types of multivibrators and ring circuits. Besides their intrinsic interest value these circuits have application to a diversity of processes, e.g, switching, counting, and time delaying. They can all indicate their action and immediate state by switching on or off large incandescent lamps.
24(1953); http://dx.doi.org/10.1063/1.1721272View Description Hide Description
The amplifying properties of electron streams passing near nonpropagating or lossy walls have been calculated. The method of field analysis is used to obtain values of gain and phase velocity for walls of arbitrary complex admittance. It is shown how one may find easily and quickly these values for actual tube design through use of two contour maps of the complex functions encountered. The gain is found to be zero for open or short circuit, or capacitive wall impedance, low for resistive‐capacitive, higher for resistive and resistive‐inductive, and highest for inductive walls. The behavior of a physically realizable model of the resistive‐capacitive wall is determined. In conclusion, some comment is offered concerning the usefulness of these structures in the realm of microwave tubes.
24(1953); http://dx.doi.org/10.1063/1.1721273View Description Hide Description
The internal friction of quartz was studied by measuring the logarithmic decrement of a bar vibrating in resonance at its fundamental mode at 21 kilocycles per second and at its second harmonic mode at 42 kilo‐cycles per second as a function of temperature from 200°C to 440°C. The logarithmic decrement was found to rise rapidly to a maximum of 16×10−5 at about 240°C, then fall slowly to a minimum of about 8×10−5 at about 350°C, and then rise rapidly up to and beyond 460°C. The high temperature portion shows a temperature hysteresis in which the cooling curve is lower than the heating curve. Furthermore the magnitude and, to a certain extent, the position of the peak may depend upon previous history but heating in a vacuum can reduce the logarithmic decrement curve to a reproducible one, independent of room temperature handling and thermal gradients (up to 1.5°C/cm) or stress amplitudes (up to 10 g/mm2) applied at higher temperatures.
The measured curve can be considered as the sum of two relaxation peaks and an exponential curve. One peak has an activation energy of 27.2×103 calories per mole, and the other, 12.6×103 calories per mole. The exponential curve appears to be characterized by an activation energy of about 22×103 calories per mole but may vary with heat treatment.
24(1953); http://dx.doi.org/10.1063/1.1721274View Description Hide Description
The linear rf mass spectrometer previously described is analyzed for small signal operation. The results are obtained in a closed form for any entrance phase where previously only a graphical solution for an optimized entrance phase was obtained. Expressions for resolving power and current efficiency are derived for both sine wave and square wave operation. It is shown that although high resolving power may be more readily obtained with square wave operation, the product of resolving power and efficiency is approximately the same for sine or square waves. Loss of resolving power caused by the thermal energies of the ions is shown to be less troublesome with square wave operation.
24(1953); http://dx.doi.org/10.1063/1.1721275View Description Hide Description
Young's moduli (E) of the directions 〈100〉 and 〈111〉 and the shear modulus (G) for 〈100〉 were determined in germanium from −195 to 255, 275, and 140°C, respectively. From these moduli, the elastic parameters, the compressibility, and Poissons's ratio were calculated.
The thermal expansion was measured from −196 to 275°C.
24(1953); http://dx.doi.org/10.1063/1.1721276View Description Hide Description
The solution is given for the electric field of a small current‐carrying loop which is immersed in a dissipative medium and is energized by a step‐function current. Approximate expressions for the magnetic fields are also derived. The propagation of an electromagnetic pulse in sea water is then discussed.