bluuebunny/crossref_metadata_2025_split huggingface dataset:
On the motion of a Gaussian wave-packet in a parabolic potential field
Authors: Coulson C. A. , Rushbrooke G. S. 
Published: November 2008

The paper deals with the motion of a Gaussian wave-packet in a parabolic potential field, contrasting the quantum-mechanical behaviour of the packet with that which follows from pure classical mechanics.The behaviour is entirely classical only if the initial phase of the space-wave-function is linear in the space coordinate; in which case the wave-packet is initially a ‘best-possible’ packet according to Heisenberg's uncertainty relationship.The paper concludes with a brief reference to the similar problem for a free particle.Note added in Proof, 16 February 1946. Through the courtesy of Mr M. S. Bartlett, we have been privileged to see the manuscripts of some work by Mr J. E. Moyal and himself, in which the motion of a probability wave-packet is discussed in more general terms than our own: and in which it appears that the motion of our wave-packet, which we have described as non-classical, is indeed actually classical provided that a certain correlation exists between momentum and position in the p, q-distribution function. With this knowledge we now find that classical and quantal predictions agree precisely provided that the initial classical probability function is given bythis is a particular case of their more general results.
Simple progressive solutions of the wave equation
Authors: Friedlander F. G. 
Published: November 2008

The object of this paper is to determine all the solutions of the wave equationwhich are of the simple formwhere F denotes an arbitrary function. It will be shown that, in addition to the obvious cases of plane or spherical progressive waves, such solutions exist only when the wave frontsare certain algebraic surfaces of the fourth order, the cyclides of Dupin. These include, as degenerate cases, the sphere, the plane, the cylinder, the cone, and the torus.
The effect of a fixed vertical barrier on surface waves in deep water
Authors: Ursell F. 
Published: November 2008

In this paper the two-dimensional reflection of surface waves from a vertical barrier in deep water is studied theoretically.It can be shown that when the normal velocity is prescribed at each point of an infinite vertical plane extending from the surface, the motion on each side of the plane is completely determined, apart from a motion consisting of simple standing waves. In the cases considered here the normal velocity is prescribed on a part of the vertical plane and is taken to be unknown elsewhere. From the condition of continuity of the motion above and below the barrier an integral equation for the normal velocity can be derived, which is of a simple type, in the case of deep water. We begin by considering in detail the reflection from a fixed vertical barrier extending from depth a to some point above the mean surface.
Two-dimensional plastic flow of a Bingham solid
Authors: Oldroyd J. G. 
Published: November 2008

The rheological equations of state for an isotropic Bingham solid may be written in tensor form:whereThe notation is explained fully in a previous paper (1). The frame of reference is essentially Eulerian. Primes denote deviatoric components of the stress, strain and rate of strain tensors, pik, εik and eik (for example, ). The dilatation is denoted by Δ. The rigidity and bulk moduli μ and κ are, if not constants, essentially scalar functions of the stress tensor. The variable viscosity η involves two constants, the yield value ϑ and the reciprocal mobility η1.
Kolmogoroff's theory of locally isotropic turbulence
Authors: Batchelor G. K. 
Published: November 2008

A new and fruitful theory of turbulent motion was published in 1941 by A. N. Kolmogoroff. It does not seem to be as widely known outside the U.S.S.R. as its importance warrants, and the present paper therefore describes the theory in some detail before presenting a number of extensions and making a comparison of experimental results with some of the theoretical predictions.Kolmogoroff's basic notion is that at high Reynolds number all kinds of turbulent motion, of arbitrary mean-flow characteristics, show a similar structure if attention is confined to the smallest eddies. The motion due to these eddies of limited size is conceived to be isotropic and statistically steady. Within this range of eddies we recognize two limiting processes. The influence of viscosity on the larger eddies of the range is negligible if the Reynolds number is large enough, so that their motion is determined entirely by the amount of energy which they are continually passing on to smaller eddies. This quantity of energy is the local mean energy dissipation due to turbulence. On the other hand, the smaller eddies of the range dissipate through the action of viscosity a considerable proportion of the energy which they receive, and the motion of the very smallest eddies is entirely laminar. The analytical expression of this physical picture is that the motion due to eddies less than a certain limiting size in an arbitrary field of turbulence is determined uniquely by two quantities, the viscosity and the local mean energy dissipation per unit mass of the fluid.The mathematical method of describing the motion due to eddies of a particular size is to construct correlations between the differences of parallel-velocity components at two points at an appropriate distance apart. Kinematical results analogous to those for turbulence which is isotropic in the ordinary sense are obtained, and then the scalar functions occurring in the expressions for the correlations are determined by dimensional analysis. The consequences of the theory in the case of turbulence which possesses ordinary isotropy are analysed and various predictions are made. One of these, namely that dimensionless ratios of moments of the probability distribution of the rate of extension of the fluid in any direction are universal constants, is confirmed by recent experiments, so far as the second and third moments are concerned. In several other cases it can be said that relations predicted by the theory have the correct form, but further experiments at Reynolds numbers higher than those hitherto used will be required before the theory can be regarded as fully confirmed. If valid, Kolmogoroff's theory of locally isotropic turbulence will provide a powerful tool for the analysis of problems of non-uniform turbulent flow, and for the determination of statistical characteristics of space and time derivatives of quantities influenced by the turbulence.
The lattice properties of asymmetric hyperbolic regions
Authors: Cassels J. W. S. 
Published: November 2008

In this paper, all numbers are real and all radicals are positive.Let f(x, y) = ax2 + bxy + cy2 be an indefinite quadratic form, and let d = Δ2 = b2 − 4ac, where Δ > 0. A well-known theorem of Minkowski states that, if (x0, y0) is any pair of numbers, then there exists a pair (x, y), x ≡ x0 (mod 1), y ≡ y0 (mod 1), say (x, y) ≡ (x0, y0) (mod 1), such thatfor k ≥ ¼, and Davenport has shown how this result may be improved if we know a value assumed by f(x, y) for coprime integral values, (x, y) = (m, n) ≠ (0, 0). In this paper, we discuss the more general inequalitywhere R and S are constants, and use the method developed in the first paper of this series to obtain a sharper and more general theorem than Davenport's. We give an application to the theory of real Euclidean quadratic fields and to a problem in Diophantine approximation discussed by Khintchine.
Non-convex regions in three and more dimensions
Authors: Macbeath A. M. 
Published: November 2008

Blichfeldt (2) gave a method of proving the existence of lattice points in certain regions. His method depends on the following lemma (which he stated in a somewhat different form):Lemma 1. Let K′ be a closed region of volume V. Let K be another region so related to K′ that the difference of any two points in K′ is a point in K. Let Λ be any lattice whose determinant is positive and does not exceed V.
Notes on iterative processes
Authors: Hartree D. R. 
Published: November 2008

If ηn is the error in the result of repeating an iterative process n times, and ηn+1 is the iterative process is called Kth order. It is shown that if for a given equation there is an iterative process of the (K + 1)th order, the iterative process of the Kth order is not unique, and conversely if an iterative process of the Kth order is not unique, it is generally possible to construct from two of the Kth order processes a process of the (K + 1)th order. As an example, three second-order processes for a square root are exhibited, and a third-order process is derived from two of them.Iterative processes for positive and negative integer roots are given, of kinds suitable for use on machines in which division is a relatively slow process and one to be used sparingly. It is shown how a second-order process can be derived from the results of two repetitions of a first-order process. The extension of this iterative process for the solution of differential and integral equations is a development which is urgently required.
Note on the geometrical optics of diffracted wave fronts
Authors: Friedlander F. G. 
Published: November 2008

It is well known that a solution of the wave equationfor which u = ∂u/∂t = 0 initially outside a surface S0, vanishes at time t in the exterior of a surface St parallel to, and at normal distance ct from S0, so that the wave fronts of disturbances represented by the solutions of the wave equation obey the laws of geometrical optics. Analogous results hold for the solutions of any linear hyperbolic second-order partial differential equation with boundary value conditions of the ‘Cauchy’ type. But the wave fronts of solutions of problems in which some of the boundary conditions are of the type representing reflexion do not seem to have been treated, and in particular the case of diffraction, when there is a ‘shadow’, does not seem to have been considered at all.
Surface waves on deep water in the presence of a submerged circular cylinder. I
Authors: Ursell F. 
Published: November 2008

ABSTRACTA method is given for the calculation of the surface waves of small amplitude generated on deep water by a normal velocity distribution of period 2π/σ prescribed over a submerged circular cylinder. The method of solution involves a system of linear equations in an infinite number of unknowns; this system always possesses a solution. The unknowns may be obtained as power series in a parameter Ka, convergent for sufficiently small values of the parameter. When the parameter is not small, the equations can be solved by infinite determinants. It is shown that the reflexion coefficient of waves incident on a fixed circular cylinder vanishes, as was first shown by Dean. The pulsations of a submerged cylinder are discussed when the normal velocity is the same at all points of the cylinder at any given time.
The distribution of the integers N for which {θN} < φ
Authors: Slater N. B. 
Published: November 2008

Let ν(M) denote the number of integers N in the range (0 ≤ N ≤ M) for whichwhere {y} denotes the fractional part of y. It is a convenient but trivial restriction to assume always that θ and φ lie between 0 and 1. There is a well-known result, which will be called ‘Weyl's Theorem’, that if θ is irrational,
Random division of an interval
Authors: Mauldon J. G. 
Published: November 2008

We consider n − 1 points xj (j = 1, …, n − 1) selected independently at random from the interval (0, 1), the distribution of any xj being the rectangular distributionThe points xj divide the interval (0,1) into n intervals. Let λ be the sum of the lengths of the k largest of these n intervals. It is the purpose of this paper to give the exact sampling distribution of λ for any given n and k (see equation (5)). Fisher (1) in 1929 solved the problem for k = 1. We first investigate the distribution of μ = 1 − λ.
Trapping modes in the theory of surface waves
Authors: Ursell F. 
Published: November 2008

ABSTRACTIt is shown that a mass of fluid bounded by fixed surfaces and by a free surface of infinite extent may be capable of vibrating under gravity in a mode (called a trapping mode) containing finite total energy. Trapping modes appear to be peculiar to the theory of surface waves; it is known that there are no trapping modes in the theory of sound. Two trapping modes are constructed: (1) a mode on a sloping beach in a semi-infinite canal of finite width, (2) a mode near a submerged circular cylinder in an infinite canal of finite width. The existence of trapping modes shows that in general a radiation condition for the waves at infinity is insufficient for uniqueness.
The spectra of turbulence in a compressible fluid; eddy turbulence and random noise
Authors: Moyal J. E. 
Published: November 2008

AbstractThe state of a real fluid is completely specified by its velocity, density, pressure and temperature fields. When the fluid is in turbulent flow, all these quantities fluctuate in a disordered manner. The method of space Fourier spectra is used to show that these field variables separate into two physically distinct groups, one corresponding to fluctuating acoustical waves, orrandom noise, and the other to fluctuating vorticity, oreddy turbulence. The corresponding decomposition of the spectral and correlation tensors in a homogeneous field of turbulence is given. The noise Fourier components are shown to be coupled to the eddy Fourier components only through the non-linear inertia terms in the dynamical equations of the fluid; whereas the former propagate as acoustical waves, the wave character of the latter is due entirely to the mean motion of the fluid. The measurement of the noise component, its attenuation through absorption by walls and its effects on the eddy component are discussed. Finally, the dynamical equations for the eddy component of the velocity spectral tensor in a homogeneous field of turbulence are compared with the corresponding equations for an incompressible fluid.
On an extension of Riemann's method of integration, with applications to one-dimensional gas dynamics
Authors: Ludford Geoffrey S. S. 
Published: November 2008

AbstractThe object of this paper is to investigate the influence of the initial values on the behaviour of the one-dimensional unsteady isentropic motion of a perfect gas, and in particular the occurrence of singularities. For this purpose it is expedient to develop an unfolding procedure in the theory of partial differential equations of hyperbolic type. This not only resolves a difficulty in the classical treatment of the initial value problem associated with this type of equation, but also simplifies the presentation.The geometrical properties of the singularities (branch and limit lines, edges of regression, etc.) occurring in two-dimensional steady flow have been well discussed already, and their counterparts in one-dimensional unsteady motion have been indicated by Stocker and Meyer(5). However, here a method for finding the explicit analytic connexion between the occurrence of such singularities and the prescribed initial conditions is developed.Assuming quite general initial conditions, two examples are considered. In the first, a conjecture of Riemann is verified; in the second a special type of periodic solution is discussed, and it is shown that this solution cannot be continued to all values of t > 0. Finally, this latter result is shown to be true for arbitrary periodic initial conditions.
Mass transport in gravity waves
Authors: Ursell F. 
Published: November 2008

ABSTRACTA proof is given of the existence of a slow drift associated with the passage of gravity waves over the surface of a frictionless fluid when the wave motion is irrotational. The drift is in the direction of propagation and is greatest near the surface, decreasing steadily towards the bottom, where it may be negative if the fluid is of finite depth. The particle displacement due to a solitary wave has similar properties. The proof confirms the approximate calculations of Stokes (6) and extends the proof of Rayleigh applicable only to deep water.
The serial test for sampling numbers and other tests for randomness
Authors: Good I. J. 
Published: November 2008

AbstractIn the serial test for sampling numbers (3, 4) an expression is used. It is in the form of a sum of squares and has previously been supposed to have asymptotically a χ2 (γ-variate) distribution. By evaluating it is shown that this supposition cannot be correct, since is not equal to the number of degrees of freedom. The evaluation is based on some results which can be used for testing a wide class of properties of random sequences. The definition of is generalized to deal with subsequences of length ν (instead of length 2) and the generalized , or is then decomposed into a linear form of simple variates defined in terms of discrete Fourier transforms. When the number, t, of digits in the sample space is prime these simple variates are shown to have asymptotically independent χ2 distributions with one degree of freedom each. The coefficients of the linear form are not 1 but they become 1 when the first and second differences with respect to ν are taken. Thus and have asymptotically χ2 distributions. The work is closely related to that of Bartlett (1) and Vajda (6).
Note on hydrodynamics
Authors: Darwin Charles 
Published: November 2008

AbstractA study is made of the actual trajectories of fluid particles in certain motions of classical hydrodynamics. When a solid body moves through an incompressible fluid, it induces a drift in the fluid, such that the final positions of the particles are further on than those from which they started. The drift-volume enclosed between the initial and final positions is equal to the volume corresponding to hydrodynamic mass, that is, the mass of fluid to be added to that of the solid in calculating its kinetic energy. This result is proved quite generally. The work involves integrals which are not absolutely convergent, and these are discussed in relation to the general mechanics of fluids. When the trajectories are considered of the fluid surrounding a rotating body, it is shown that the fluid particles slowly drift round the body, even though the motion is irrotational and without circulation. There seems to be in some respects a closer resemblance between the behaviour of the idealized hydrodynamic fluid and a real fluid than might be expected from the well-known discrepancies between them.
A theorem of Minkowski on the product of two linear forms
Authors: Chalk J. H. H. 
Published: November 2008

1. Let Λ denote any plane lattice of determinant Δ ╪ 0. Then it is well known that the regioncontains a pair of generating points of Λ; and that, unless the lattice Λ is of the special formwhere both α│β, γ│δ are rational, there are an infinity of such pairs in the interior of the region. Minkowski (4) obtained this result by considering the ‘tangent’ parallelogram Пλ,inscribed in the region (1). With simple geometrical arguments, he showed that(i) Пλ always contains a point of Λ, other than 0,(ii) if Пλ contains two primitive points of Λ then these generate Λ, except possibly when Λ has points, other than 0, on both coordinate axes.
On the oscillations of a bowed string
Authors: Friedlander F. G. 
Published: November 2008

ABSTRACTThe small transverse oscillations of a string excited by bowing are treated by neglecting all external forces except that due to the bow, taking the ends of the string as fixed, and replacing the bow by a concentrated force acting transversely at the ‘bowing point’, whose magnitude depends on the velocity, v, of the string at this point. Non-linear recurrence equations for v are obtained, from which some of the principal features of the motion can be inferred, and some qualitative conditions about the dependence of the transverse force of the bow on v can be drawn. A detailed examination of the case when the string is bowed at the mid-point shows that, in order to account for the stability of the periodic motion excited, it appears to be necessary to allow for the effect of dissipative forces, such as air friction; to a first approximation this can be done by postulating a small concentrated dissipative force at the bowing point hi addition to the frictional force exerted by the bow. The theory predicts that for small bowing speeds a ‘noise’ is produced instead of a note; this phenomenon does not appear to have been noted in the literature, but can easily be verified in practice.
Further solutions of the Falkner-Skan equation
Authors: Stewartson K. 
Published: November 2008

ABSTRACTIn an attempt to understand the mechanism governing separation the possibility of there being further solutions to the Falkner-Skan equation f′′′ + ff″ + β(1 − f′2) = 0, in addition to those found by Hartree, is investigated.It is found that when β < − 0·1988 there are no solutions satisfying |f′| ⩽ 1 for all η, and that if 0>β> − 0·1988 there are two acceptable solutions, one with f″(0) < 0. The new ones are computed to three places of decimals for various values of β and tabulated. In addition, it is shown that if − 0·5 < β < 0 there is a family of solutions corresponding to boundary layers bounded on one side by free streamlines. These are also computed and graphs of f′(0) and δ1 are displayed. The impact of these solutions on the theory of separation is discussed.
Bounds for the least solutions of homogeneous quadratic equations
Authors: Cassels J. W. S. 
Published: November 2008

In this note I obtain bounds for the least integral solutions of the equationin terms ofFor ternary diagonal forms, such bounds have been given by Axel Thue (4) and, more recently, by Holzer (1), Mordell (2) and Skolem (3), but these lead only to bad estimates for general ternaries. So far as I know there have not been given estimates for n≽4. Here I generalize Thue's method to prove:Theorem. Suppose that n ≥ 2 and that f(ɛ) represents zero. Then there is an integral solution of f(a) = 0 with
A use of complex probabilities in the theory of stochastic processes
Authors: Cox D. R. 
Published: November 2008

ABSTRACTThe exponential distribution is very important in the theory of stochastic processes with discrete states in continuous time. A. K. Erlang suggested a method of extending to other distributions methods that apply in the first instance only to exponential distributions. His idea is generalized to cover all distributions with rational Laplace transforms; this involves the formal use of complex transition probabilities. Properties of the method are considered.
The derivation of the equations of motion of an ideal fluid by Hamilton's principle
Authors: Herivel J. W. 
Published: November 2008

ABSTRACTA new formulation of Hamilton's principle for the case of an ideal fluid is proposed which is claimed to be the uniquely proper form for such a system. The resulting derivation of the equations of motion on varying with respect to the position of the fluid particles is free from the difficulties encountered in previous treatments based on incorrect forms of Hamilton's principle. The treatment also differs from previous treatments in transforming to the fixed (a, b, c) space before carrying out the variation, and in allowing for the equations of mass and entropy conservation by means of undetermined multipliers. The necessity for conservation of entropy, if Hamilton's principle is to be applied, is emphasized. Finally, the method, first used by Eckart, of deriving the equations of motion for an ideal fluid by means of a variational principle of the same form as Hamilton's, but varying with respect to the velocities of the fluid particles, is extended to the general case of rotational motion.