Exponential Index Research Articles

Propagation from a Point Source in a Randomly Refracting Medium

This paper considers the propagation of scalar (acoustic) waves from a single-frequency point source imbedded in a medium with random refractive index, in contrast with the usual plane-wave case in which the source is far removed from the medium. With the index being a statistically homogeneous and isotropic function of position, but not a function of time, the average complex field $u_{o}(r) = \langle u(r)\rangle$ and the spatial covariance $\langle u_{i}(r)u*_{i}(\rho;)\rangle$ of the fluctuation field u i (r) = u(r) - u o (r) are calculated. Beyond a few correlation lengths from the source, the average field can be approximated by a spherical wave with the same complex wavenumber found in the plane-wave case. A near-source wave number is also obtained. Under an improved far-field condition, the spatial covariance is reduced to spectral integration formulas for both transverse and longitudinal separation of the receiving points. These formulas reveal that correlation lengths are much longer in the point-source case than in the plane-wave case, even though the relative variances are the same. We illustrate this result with plots for an exponential index spectrum and for a constant spectrum.

Asymptotic Distributions of Zero Crossing Intervals

Suggestions are made on the form of the expression for the exponential index in the tail of the distribution of intervals between successive zeros for a stationary Gaussian process. These are generalized from proved results and are also based on dimensional grounds, and on the one completely solved (Markov) case; and they are confirmed by experimental figures. The low-frequency limit w(o) of the power spectrum determines the rough order of magnitude of the index; this is increased by the influence of high-frequency terms in a way which has not yet been ascertained.

Refraction of radio waves at low angles within various air masses

The refractive index structure and bending of radio rays within air masses of nonexponential refractive index height structure is treated in terms of the value expected in an average atmosphere of exponential form. It is demonstrated that refraction differences within air masses arise from departures of refractive index structure from the normal exponential decrease with height. The effect upon radio ray refraction of these departures from the normal exponential refractive index structure is most pronounced for small initial elevation angles of the radio ray.

A Course of Pure Mathematics.

1. Real variables 2. Functions of real variables 3. Complex numbers 4. Limits of functions of a positive integral variable 5. Limits of functions of a continuous variable: continuous and discontinuous functions 6. Derivatives and integrals 7. Additional theorems in the differential and integral calculus 8. The convergence of infinite series and infinite integrals 9. The logarithmic, exponential, and circular functions 10. The general theory of the logarithmic, exponential, and circular functions Appendices Index.

On the developement of exponential functions; together with several new theorems relating to finite differences

The subject here considered by Mr. Herschel relates to the celebrated theorems of Lagrange, expressing the connection between sim­ple exponential indices and those of differentiation and of integration. Since the theorems have been demonstrated by various subsequent analysts, as by Laplace, by Arbogast, and by Dr. Brinkley, the author takes them for granted; but observes that in their original form they are but abridged expressions of their meaning; and that in order to become practically useful, their exponential functions require further development.

III. On the developement of exponential functions; together with several new theorems relating to finite differences

In the year 1772, Lagrange, in a Memoir, published among those of the Berlin Academy, announced those celebrated theorems expressing the connection between simple exponential indices, and those of differentiation and integration. The demonstration of those theorems, although it escaped their illustrious discoverer, has been since accomplished by many analysts, and in a great variety of ways. Laplace set the first example in two Memoirs presented to the Academy of Sciences,* and may be supposed in the course of these researches, to have caught the first hint of the Calcul des Fonctions Generatrices with which they are so intimately connected ; as, after an interval of two years, another demonstration of them, drawn solely from the principles of that calculus appeared, together with the calculus itself, in the memoirs of the Academy. This demonstration, involving, however, the passage from finite to infinite, is therefore (although preferable perhaps in a systematic arrangement, where all is made to flow from one fundamental principle) less elegant ; not on account of any confusion of ideas, or want of evidence ; but, because the ideas of finite and infinite, as such, are extraneous to symbolic language, and, if we would avoid their use, much circumlocution, as well as very unwieldy formulæ must be introduced. Arbogast also, in his work on derivations, has given two most ingenious demonstrations of them, and added greatly to their generality ; and lastly, Dr. Brinkley has made them the subject of a paper in the Transactions of this Society,* to which I shall have occasion again to refer. Considered as insulated truths, unconnected with any other considerable branch of analysis, the method employed by the latter author seems the most simple and elegant which could have been devised. It has however the great inconvenience of not making us acquainted with the bearings and dependencies of these important theorems, which, in this instance, as in many others, are far more valuable than the mere formulæ.