CaltechAUTHORS: Combined
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A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenWed, 18 Sep 2024 18:50:40 -0700Döppler's principle for a windy atmosphere
https://resolver.caltech.edu/CaltechAUTHORS:BATmwr17
Year: 1917
DOI: 10.1175/1520-0493(1917)45<441:DPFAWA>2.0.CO;2
In view of the interest which is being taken in the effect of wind on the propagation of sound it may be worth while to recall the form which Döppler's principle assumes when a wind is blowing.https://resolver.caltech.edu/CaltechAUTHORS:BATmwr17Mathematical theory of sound ranging
https://resolver.caltech.edu/CaltechAUTHORS:BATmwr18
Year: 1918
DOI: 10.1175/1520-0493(1918)46<4:MTOSR>2.0.CO;2
The development of the art of concealing large guns so that they can not be easily seen by hostile airmen or observers in kite balloons has brought into prominence the study of methods of locating powerful guns by means of observation of the time of arrival of their gunfire at one or more observing stations. There are really two distinct problems to be discussed.
(1) The simple case when the flash is seen and the distance of the gun is to be determined from the observed interval of time between the instants when the flash is seen and the report is heard at a single station.
(2) The more complex case when the flash is not seen. The sound of the report must now be timed at three or four observing stations and the position of the gun estimated from the observed differences in time. A small error in the timing of the sound is more disastrous in the second case than in the first, consequently an accurate method of timing the arrival of the sound is very necessary for the successful application of the second method.https://resolver.caltech.edu/CaltechAUTHORS:BATmwr18The Structure of an Electromagnetic Field
https://resolver.caltech.edu/CaltechAUTHORS:BATpnas18
Year: 1918
This elementary field corresponds to a state of affairs in which electric charges of a concentrated form are created and travel along straight lines with the velocity of light, the directions of these lines being specified by the different values of the unit vector s. Whenever a concentrated electric charge is created an amount of electricity which will just compensate it is fired out in all directions and provides an elementary 'aether' which is the seat of the electromagnetic field of the concentrated charge. A concentrated electric charge and its elementary aether lie at any instant on a sphere whose centre is at the point where these charges originated [1]; if now this point moves with a velocity less than the velocity of light the different spheres bearing electricity that exist at time t do not intersect and if the arbitrary function f(a) is never zero there will be a sphere through each point of space so that our elementary aethers will fill the whole of space; if however the function f(a) is sometimes zero, for example if it is zero when a is less than ao, then the elementary aethers will not fill the whole of space.https://resolver.caltech.edu/CaltechAUTHORS:BATpnas18The Electromagnetic Vectors
https://resolver.caltech.edu/CaltechAUTHORS:BATpr18
Year: 1918
DOI: 10.1103/PhysRev.12.459
An electromagnetic fiels in the ether is usually specified by the values at each point and at each instant of two vectors E and H, but a more general specification is obtained by using the two vectors
F=E + 1/c x (v x H),
Q=H - 1/c x (v x E),
where v is an arbitrary velocity representing at each point and at each instant the velocity of an imaginary recording instrument and c is the velocity of light.
These vectors are of fundamental importance in electromagnetic theory for F is usually supposed to represent the force which the field would exert on a minute unit electric charge moving with velocity v and Q may be supposed to represent the force which the field would exert on a minute unit magnetic charge if such a thing could exist and move with velocity v.
On account of the importance of these vectors F and Q it will be worth while to get a clear conception of the way in which they vary when the field remains constant and v varies.https://resolver.caltech.edu/CaltechAUTHORS:BATpr18Radiation from a Moving Magneton
https://resolver.caltech.edu/CaltechAUTHORS:BATpnas19
Year: 1919
The rate of radiation of energy from a ring of electrons revolving in a circular orbit and from various other distributions of moving electric charges and magnetic poles has been calculated by G. A. Schott, [1] who finds that the rate of radiation of energy is almost invariably positive. This is certainly true in the case of a single electric pole describing a circular orbit as is indicated by the well known formulae of Larmor and Liénard for the rate of radiation. Thus electromagnetic theory in its present form lends no suport to Bohr's idea of non-radiating orbits.
A steady distribution, such as a Parson magneton which consists of a complete ring of electric charges following one another round the ring at a constant speed will evidently give no radiation when the ring is stationary as a whole, but as Schott remarks the ring may be expected to radiate energy when its centre has an acceleration.
Schott's results are so important that it is desirable that they should be confirmed by an independent method and an attempt has been made to devise a method by which the rate of radiation from a moving electric pole and magnetic doublet may be readily calculated. In two important cases we have confirmed Schott's surmise that the rate of radiation is positive.https://resolver.caltech.edu/CaltechAUTHORS:BATpnas19On a differential equation occurring in Page's theory of electromagnetism
https://resolver.caltech.edu/CaltechAUTHORS:BATpnas20
Year: 1920
In a recent article Mr. Leigh Page(1) has generalized the electromagnetic equations by introducing the idea of a rotation of the field round a moving electric pole.https://resolver.caltech.edu/CaltechAUTHORS:BATpnas20The stress-energy tensor in electromagnetic theory and a new law of force
https://resolver.caltech.edu/CaltechAUTHORS:BATpr22
Year: 1922
DOI: 10.1103/PhysRev.20.243
Modified Electromagnetic Theory.—By suitably modifying the equations for the components of the stress-energy tensor it is possible to reconcile electromagnetic theory with the idea of non-radiating electronic orbits. The change, however, is equivalent to assuming that an element of electricity is acted upon by a new force which depends on the gradient of the density of electricity and balances the usual electromagnetic force. By assuming a certain distribution of density within an electron it is also possible to account for the existence of discrete electronic charges.https://resolver.caltech.edu/CaltechAUTHORS:BATpr22The decay of a simple eddy
https://resolver.caltech.edu/CaltechAUTHORS:20140804-112342128
Year: 1923
The principal result obtained in this report is a generalization of Taylor's formula for a simple eddy. The discussion of the properties of the eddy indicates that there is a slight analogy between the theory of eddies in a viscous fluid and the quantum theory of radiation. Another exact solution of the equations of motion of viscous fluid yields a result which reminds one of the well-known condition for instability in the case of a horizontally stratified atmosphere.https://resolver.caltech.edu/CaltechAUTHORS:20140804-112342128The Location of Energy
https://resolver.caltech.edu/CaltechAUTHORS:20141215-092411903
Year: 1923
DOI: 10.1126/science.57.1469.238
A calculation of the mass of an electron,
based upon the modification of electromagnetic
theory proposed in a recent paper, has led to
the surprising result that the mass inside an
electron in uniform motion, when calculated
from the momentum, is equal but opposite in
sign to the mass outside.https://resolver.caltech.edu/CaltechAUTHORS:20141215-092411903The Inertial Coefficients of an Airship in a Frictionless Fluid
https://resolver.caltech.edu/CaltechAUTHORS:20140804-113134879
Year: 1924
This report deals with the investigation of the apparent inertia of an airship hull. The exact solution of the aerodynamical problem has been studied for hulls of various shapes and special attention has been given to the case of an ellipsoidal hull. In order that the results for this last case may be readily adapted to other cases, they are expressed in terms of the area and perimeter of the largest cross section perpendicular to the direction motion by means of a formula involving a coefficient K which varies only slowly when the shape of the hull is changed, being 0.637 for a circular or elliptic disk, 0.5 for a sphere, and about 0.25 for a spheroid of fineness ratio 7. For rough purposes it is sufficient to employ the coefficients, originally found for ellipsoids, for hulls otherwise shaped. When more exact values of the inertia are needed, estimates may be based on a study of the way in which K varies with different characteristics and for such a study the new coefficient possesses some advantage over one which is defined with reference to the volume of fluid displaced. The case of rotation of an airship hull has been investigated also and a coefficient has been defined with the same advantages as the corresponding coefficient for rectilinear motion.https://resolver.caltech.edu/CaltechAUTHORS:20140804-113134879The derivation of electromagnetic fields from a basic wave-function
https://resolver.caltech.edu/CaltechAUTHORS:BATpnas24
Year: 1924
1. Derivation of a Logarithmic Wave Function. - Electromagnetic fields may be derived from wave-functions in at least two ways that are analytically distinct. In the first place four wave-functions satisfying a divergence relation may be chosen as the components of a 4-vector and field-vectors derived from these four electromagnetic potentials in the usual way. The four potentials may in their turn be derived by differential operations from the components of a 6-vector whose components may be taken to be any six wave-functions. This method is a generalization of the well-known methods of Fitzgerald and Hertz;(1) it has the disadvantage that the wave-functions cannot be chosen arbitrarily if magnetic poles are to be excluded.https://resolver.caltech.edu/CaltechAUTHORS:BATpnas24The radiation of energy and angular momentum
https://resolver.caltech.edu/CaltechAUTHORS:BATpr26
Year: 1926
DOI: 10.1103/PhysRev.27.606
When an electromagnetic field, derived from retarded potentials, is given it is generally possible to add to this a scalar field, derived from a retarded potential ψ, in such a way that the total radiation of energy to infinity is zero for each direction, the contribution of the scalar field being calculated with the aid of a tensor used in a former paper. The radiation of angular momentum to infinity is not zero for each direction unless the electromagnetic field is of a certain type. The approximate values of the field vectors at a great distance from the origin can be expressed in terms of a quantity α and when the field is of the type just mentioned α satisfies a certain partial differential equation. This result may be regarded as typical for attempts to solve the radiation problem in which the electromagnetic radiation in one of Bohr's stationary states is supposed to be balanced by radiation of a new type. Some remarks are made on the attempts which have been made to solve the problem with the aid of electromagnetic fields alone and a brief discussion is given of the radiation of angular momentum according to the classical theory.https://resolver.caltech.edu/CaltechAUTHORS:BATpr26Lagrangian functions and Schrödinger's rule
https://resolver.caltech.edu/CaltechAUTHORS:BATpnas27a
Year: 1927
In a recent paper Schrödinger(1) has extended a rule, used by writers in the theory of gravitation, for deriving a stress energy tensor from a Lagrangian function and has illustrated its application in the case of the tensor which he has associated with the system of equations proposed by Gordon.(2)https://resolver.caltech.edu/CaltechAUTHORS:BATpnas27aThe symmetry of the stress-tensor obtained by Schroedinger's rule
https://resolver.caltech.edu/CaltechAUTHORS:BATpnas27b
Year: 1927
In the recent developments of the calculus of variations used in the new quantum theory the problem arises of finding a general expression for a world-function such that a symmetrical stress-energy tensor may be derived from it by means of Schroedinger's rule.(1)https://resolver.caltech.edu/CaltechAUTHORS:BATpnas27bVariable flow in pipes
https://resolver.caltech.edu/CaltechAUTHORS:BATpr30
Year: 1930
DOI: 10.1103/PhysRev.35.177
The problem considered is that of finding a variable pressure gradient or forced motion of the pipe in a longitudinal direction which, at the time when it ceases to act, will have produced a prescribed distribution of velocity over the cross-section of the pipe. The subsequent changes in the distribution as the motion decays are also investigated and the cases examined point to the conclusion that an initial velocity profile with curvature of one sign will retain this property as it changes into the profiles for the different stages of the decaying motion.https://resolver.caltech.edu/CaltechAUTHORS:BATpr30Physical problems with discontinuous initial conditions
https://resolver.caltech.edu/CaltechAUTHORS:BATpnas30a
Year: 1930
These problems are usually treated by the methods developed by the great. French mathematician Jean Baptist Joseph Fourier who died 100 years ago.https://resolver.caltech.edu/CaltechAUTHORS:BATpnas30aSome properties of spherical harmonics
https://resolver.caltech.edu/CaltechAUTHORS:20200410-102857453
Year: 1930
DOI: 10.1090/s0002-9904-1930-04941-1
A Newtonian potential V(x, y, z) can often be derived from a four-dimensional potential W(x, y, z, w) by forming the definite integral V = 1/π ∫^∞_(-∞)Wdw.https://resolver.caltech.edu/CaltechAUTHORS:20200410-102857453Irrotational motion of a compressible inviscid fluid
https://resolver.caltech.edu/CaltechAUTHORS:BATpnas30b
Year: 1930
Variational Principles.--Let us assume that in the free two-dimensional irrotational motion of a compressible inviscid fluid the "density of mechanical energy" p + 1/2 ρ q^2 is an assigned differentiable function f(ρ) of the density ρ of the fluid.https://resolver.caltech.edu/CaltechAUTHORS:BATpnas30bSolutions of a certain partial differential equation
https://resolver.caltech.edu/CaltechAUTHORS:BATpnas31a
Year: 1931
The partial differential equation ∂u/∂t = x(∂^2u/∂x^2 – u) is readily seen to possess the two particular solutions U1 = xe^(-x tanh t) sech^2t, U2 = e^(-x coth t).https://resolver.caltech.edu/CaltechAUTHORS:BATpnas31aRelations between confluent hypergeometric functions
https://resolver.caltech.edu/CaltechAUTHORS:BATpnas31b
Year: 1931
Some of the functions mentioned in a recent paper may be expressed in terms of known functions.https://resolver.caltech.edu/CaltechAUTHORS:BATpnas31bLogarithmic solutions of Bianchi's equation
https://resolver.caltech.edu/CaltechAUTHORS:BATpnas33
Year: 1933
The partial differential equation ∂^nV/∂x1∂x2…∂xn = MV was discussed by Bianchi(1) with the aid of the methods of Riemann and Picard. The results were extended to a more general equation which was also studied by Niccoletti.(2) The original equation, for a constant value of M, was studied later by Sibirani(3) in connection with a generalization of the Bessel function and some partial differential equations were listed which could be solved with the aid of this function. The case in which M is constant has also been studied by Chaundy(4) who gives some solutions in the form of definite integrals which we wish to obtain here with the aid of Murphy's theorem.https://resolver.caltech.edu/CaltechAUTHORS:BATpnas33Functions orthogonal in the Hermitian sense. A new application of basic numbers
https://resolver.caltech.edu/CaltechAUTHORS:BATpnas34
Year: 1934
To find a particular set of functions Hn(u) satisfying the Hermitian relation Im,n ≡ ∫∞ -∞ e^-1/2x^2 Hm(ix)Hn(-ix)dx = 0 in which the exponential factor is exp (-x2/2) as also in (14) we may put z = e^iax, where a is an arbitrary positive constant and assume that Hn(ix) is a polynomial of the nth degree in z with real coefficients.https://resolver.caltech.edu/CaltechAUTHORS:BATpnas34Some expansions associated with Bessel functions
https://resolver.caltech.edu/CaltechAUTHORS:BATpnas35
Year: 1935
An Expansion for the Product of Two Bessel Functions.-1.1. An expansion for the product of two Bessel functions obtained by one of us(1) led to the discovery of a different expansion for the said product multiplied by the leading terms in the power series for the Bessel functions. Two proofs of this second expansion are given here.https://resolver.caltech.edu/CaltechAUTHORS:BATpnas35Functional differential equations and inequalities
https://resolver.caltech.edu/CaltechAUTHORS:BATpnas36a
Year: 1936
Let us first try to find the minimum value of the integral ∫02π[f'(x)+mf(x + π)+e(x)]^2dx where f(x) is a uniform function of period 2π which is integrable and such that ∫02π[f(x)]^2dx=1.https://resolver.caltech.edu/CaltechAUTHORS:BATpnas36aProgressive waves of finite amplitude and some steady motions of an elastic fluid
https://resolver.caltech.edu/CaltechAUTHORS:BATpnas36b
Year: 1936
1. Progressive Plane Waves.--When a gas is initially stationary and at uniform temperature and pressure the density p may be regarded as also uniform initially with a value p0 which for convenience may be taken as unity. The velocity of sound at this time will also be independent of position and equal, say, to c0.https://resolver.caltech.edu/CaltechAUTHORS:BATpnas36bThe Lift and Drag Functions for an Elastic Fluid in Two Dimensional Irrotational Flow
https://resolver.caltech.edu/CaltechAUTHORS:BATpnas38a
Year: 1938
The lift function Y and the drag function X are defined by the equation....https://resolver.caltech.edu/CaltechAUTHORS:BATpnas38aRayleigh Waves
https://resolver.caltech.edu/CaltechAUTHORS:BATpnas38b
Year: 1938
The transmission of plane longitudinal waves of unlimited extent from the ground to the air was investigated by C. G. Knott (1) many years ago. He found that the resulting air waves, which are propagated in almost a vertical direction, generally have only a small amount of energy. Earthquake sounds have been studied by many writers. C. Davison (2) put forward the theory that they originate from the margin of the region disturbed by the earthquake and travel some distance through the earth before being transmitted to the air. A summary of results relating to earthquake noises has been given by Landsberg (3). The type of air motion considered here is not the simple progressive wave in an unlimited atmosphere but is a type of free vibration of the air and ground having the characteristics of a Rayleigh wave except that its velocity of propagation is less than the velocity of sound in air instead of being slightly less than the velocity of a shear wave in the ground. The mathematical analysis is very similar to that used by Stoneley (4) in his study of Rayleigh waves in a plane homogeneous elastic earth below a compressible sheet of water of unlimited extent. It is assumed here, however, that the vertical velocity of the air is negligible at a height H above the ground while in Stoneley's work the boundary condition at the free surface of the water is one of constant pressure. His remarks on nodal planes indicate that his analysis may be applicable in our case but it has been thought worth while to give the analysis again in a form in which the velocity of the wind is taken into consideration and some of Stoneley's approximations are omitted. It is thought that the analysis may be of some interest in connection with the interpretation of the ground roll observed in geophysical field work. For information relating to the ground roll I am indebted to Dr. Gutenberg, Mr. Martin Gould and other members of the group connected with the Pasadena Seismological Laboratory. It has generally been assumed, of course, that the influence of the air on the propagation of seismic waves is slight but such an assumption ought to be justified by numerical work in the base of waves produced by an artificial explosion for there are some features of the phenomena that are not fully elucidated. The problem resembles that of the loud speaker with infinite baffle, the disturbed area of the earth corresponding to the membrane that is set in vibration. Now in the theory of the loud speaker the short circuiting of energy is a familiar phenomenon, there is not simply a radiation of energy outwards. If, then, there is a similar short circuiting of energy in the air after an explosion, an interaction of air and ground is to be expected. If the air and ground are treated as a coupled system, an explosion may be expected to give rise to a subsequent motion that is composed of free vibrations of the system and the particular type of motion to be studied is, indeed, a free vibration. The whole problem is, then, one of the partition of energy among a number of free vibrations including in particular the ordinary Rayleigh wave and the new type of Rayleigh wave. This second type of Rayleigh wave is called "new" merely to distinguish it from the old type but it has been known for a long time that there is more than one type of Rayleigh wave for a stratified medium. With regard to the likelihood of the existence of a marked interaction between the earth and the air it should be mentioned that many years ago the late Lord Rayleigh (5) concluded that in the vicinity of a vibrating body of linear dimensions small in comparison with the wave-length, the air acts as if it were almost incompressible while the great mass of air at some distance from the body is slightly compressed periodically. A similar conclusion has been reached more recently by Lennard Jones (6) after some elaborate calculations. In trying to apply this result to our problem we are led to surmise that when the ground rises initially after an explosion the air immediately above it will either move away laterally and produce a reaction on the ground somewhere else or will try to lift or compress the great body of air above it.https://resolver.caltech.edu/CaltechAUTHORS:BATpnas38bCoulomb's Function
https://resolver.caltech.edu/CaltechAUTHORS:BATpnas38c
Year: 1938
In his work on Rayleigh waves Coulomb (1) has studied the function ...[Eq. (1.1)].
The function ψ0 with a complex value of h occurs in the work of Buchholz (2) on the propagation of alternating currents in the earth between two electrodes connected above ground by a rectangular loop of wire whose vertical ends support the horizontal piece.https://resolver.caltech.edu/CaltechAUTHORS:BATpnas38cThe Transformation of a Lagrangian Series into a Newtonian Series
https://resolver.caltech.edu/CaltechAUTHORS:BATpnas39a
Year: 1939https://resolver.caltech.edu/CaltechAUTHORS:BATpnas39aThe Aerodynamics of Reacting Substances
https://resolver.caltech.edu/CaltechAUTHORS:BATpnas39b
Year: 1939
Speaking broadly, the subject of aerodynamics is concerned not only with the forces exerted by a single fluid on solid bodies but also with the behavior of mixtures such as those which occur in the gasoline engine and in the atmosphere. In a complete study of the motion of a fluid attention must be paid to reactions between its constituents, evaporation, conduction and radiation of heat, diffusion, viscosity and other phenomena.
It is uncertain to what extent equations governing the various processes can be derived from a single variational principle as some of the phenomena mentioned are known to present difficulties, but the situation may be clarified a little by an examination of the equations derived from a variational principle of a very general type.https://resolver.caltech.edu/CaltechAUTHORS:BATpnas39bThe Polynomial of Mittag-Leffler
https://resolver.caltech.edu/CaltechAUTHORS:BATpnas40
Year: 1940https://resolver.caltech.edu/CaltechAUTHORS:BATpnas40Some Asymptotic Relations
https://resolver.caltech.edu/CaltechAUTHORS:BATpnas42a
Year: 1942https://resolver.caltech.edu/CaltechAUTHORS:BATpnas42aAn Orthogonal Property of the Hypergeometric Polynomial
https://resolver.caltech.edu/CaltechAUTHORS:BATpnas42b
Year: 1942https://resolver.caltech.edu/CaltechAUTHORS:BATpnas42bNote on the Function F (a, b; c - n; z)
https://resolver.caltech.edu/CaltechAUTHORS:BATpnas44
Year: 1944
The generating function [equation] may be used to find an estimate of F(a, b; b - n; z) for large positive values of n. When the point t = 1 - 1/z lies outside the circle t = 1 the singularity t = 1 may be used to find an estimate by the method of Darboux [1] and the result is F(a, b; b - n; z) ~ 1.https://resolver.caltech.edu/CaltechAUTHORS:BATpnas44Two Integral Equations
https://resolver.caltech.edu/CaltechAUTHORS:BATpnas45
Year: 1945
[no abstract]https://resolver.caltech.edu/CaltechAUTHORS:BATpnas45An Extension of Schuster's Integral
https://resolver.caltech.edu/CaltechAUTHORS:BATpnas46
Year: 1946
Schuster's integral occurs in the theory of total reflection
of light.https://resolver.caltech.edu/CaltechAUTHORS:BATpnas46Higher Transcendental Functions [Volumes I-III]
https://resolver.caltech.edu/CaltechAUTHORS:20140123-104529738
Year: 1953
The work of which this book is the first volume might be described as an up-to-date version of Part II. The Transcendental Functions of Whittaker and Watson's celebrated "Modern Analysis". Bateman (who was a pupil of E. T. Whittaker) planned his "Guide to the Functions" on a gigantic scale. In addition to a detailed account of the properties of the most important functions, the work was to include the historic origin and definition of, the basic formulas relating to, and a bibliography for all special functions ever invented or investigated. These functions were to be catalogued and classified under twelve different
headings according to their definition by power series, generating functions, infinite products, repeated differentiations, indefinite integrals, definite integrals, differential equations, difference equations, functional
equations, trigonometric series, series of orthogonal functions, or integral equations. Tables of definite integrals representing each function and numerical tables of a few new functions were to form part of the "Guide". An extensive table of definite integrals and a Guide to numerical tables of special functions were planned as companion works.https://resolver.caltech.edu/CaltechAUTHORS:20140123-104529738Tables of Integral Transforms
https://resolver.caltech.edu/CaltechAUTHORS:20140123-101456353
Year: 1954
A considerable proportion of the tremendous amount of material collected by the late Professor Harry Bateman concerns definite integrals. The organization and presentation of this material is a very difficult task to which Bateman devoted considerable attention. It is fairly clear that the arrangement used in shorter tables of integrals is not very suitable for a collection about three times the size of Bierens de Haan, and the circumstance that a considerable proportion of these integrals involves higher transcendental functions with their manifold and not always highly standardized notations, does not make this task easier. Eventually, Bateman decided to break up his integral tables into several more or less self-contained parts, classifying integrals according to their fields of application. A collection of integrals occurring in the theory of axially symmetric potentials was prepared, and other similar collections were to follow. Clearly such a plan involves a generous amount of duplication if the resulting tables are to be self-contained, but it also has great advantages from the user's point of view.https://resolver.caltech.edu/CaltechAUTHORS:20140123-101456353