Volume 14, Issue 11, 01 November 1943
Index of content:
14(1943); http://dx.doi.org/10.1063/1.1714935View Description Hide Description
The law of Arrhenius which correlates the viscosity of a colloidalsolution logarithmically to the volume percent concentration of solid matter has previously been tested only for dilute suspensions which appeared to the investigators to behave like true Newtonians. Similar suspensions have been tested at higher concentrations of solid matter and the authors found that although Arrhenius' law could not be checked, a similar exponential law evolved, which, however, is correct only within the tested region of viscosity. The authors realized that with the increase of solid matter the suspension can become plastic or pseudo‐plastic and may also show thixotropic behavior. Therefore it was considered as a major part of this investigation to determine the validity of Arrhenius' law in that region of concentration, where plasticity occurs. Although Arrhenius' law was not found to be applicable for plastic materials, two exponential laws, similar to Arrhenius' law, have been established between the plastic viscosity and the volume percent of pigment content of a plastic suspension on the one hand, and between the yield value and the volume percent of pigment content of a plastic suspension on the other hand. The two constants in the exponents of the two equations have been shown to be logarithmically related to the average diameter, d 3, of the pigment particles contained in the suspension. Oil mixtures with the same and with different type constituents have been tested below a certain rate of shear, called ``the limiting rate of shear,'' where they are Newtonian liquids, and also above this limiting rate of shear where they behave like thixotropic plastics. Their Newtonian viscosities, obtained at rates of shear below the limiting rate of shear, have been found to increase logarithmically with the volume percent of one of the oils contained in the mixture, which is in agreement with Arrhenius' exponential law. However, above the limiting rate of shear, oil mixtures deviate from the exponential law, which is logically expected, since their limiting rate of shear depends on the Newtonian viscosity and therefore is different for each oil mixture.
14(1943); http://dx.doi.org/10.1063/1.1714936View Description Hide Description
In Part I several forms of the solution of the transmission‐line equations are discussed including, in particular, a completely hyperbolic one in which the over‐all attenuation and phase shift of the terminal impedances are expressed in a form involving terminal functions which simplify the analysis of many complex problems. Formulas for the input resistance, reactance, conductance, and susceptance of a section of line of any length and terminated in an arbitrary impedance are given in completely general terms. Curves computed from these formulas for four typical terminations are shown. A general circle diagram for determining the input impedance and admittance as well as the terminal functions is described. It consists of circles of constant over‐all attenuation and circles of constant over‐all phase shift. The distribution of current is represented in general terms suitable for a line driven at one end or by a generator coupled loosely at any point along the line. In Part II the general formulas described in Part I are applied to practical problems. A simple and completely general formula is derived for determining the transfer of power and the efficiency of transmission for a line of any length terminated in an arbitrary load. An equally simple and general formula for the standing wave ratio is given. Application of the formulas for input impedance and admittance to the problem of bead spacing on resonant and non‐resonant lines and to series and shunt impedance transforming or matching sections is outlined. The experimental determination of attenuation and phase constants and of ``Q'' for the line and for the terminations is discussed. The application of formulas and experimental methods to include hollow pipe transmission lines or wave guides is considered briefly in Part III. A method for defining and measuring terminating impedances for hollow pipes is described.
14(1943); http://dx.doi.org/10.1063/1.1714937View Description Hide Description
The Resolution of Boundary Value Problems by Means of the Finite Fourier Transformation: General Vibration of a String14(1943); http://dx.doi.org/10.1063/1.1714938View Description Hide Description
The finite sine transformation and the finite cosine transformation are defined as the linear functional operations and , respectively. The inversion of the product of two transforms can be made by means of four Faltung theorems. The finite sine transformation was applied to a boundary value problem of a general vibrating string, in which the partial differential equation has coefficients which may be functions of the time. A resolution was made of the transformedboundary value problem by the introduction of a fundamental set of solutions of the homogeneous transformed problem; an inversion in closed form was accomplished by the use of the Faltung theorems. A formal verification of this solution was made as well as a short survey of the applicability of the finite Fourier transformation to problems in engineering and physics.