Showing papers in "Philosophical Transactions of the Royal Society A in 1950"

The westward drift of the Earth's magnetic field

Abstract: The westward drift of the non-dipole part of the earth's magnetic field and of its secular variation is investigated for the period 1907-45 and the uncertainty of the results discussed. It is found that a real drift exists having an angular velocity which is independent of latitude. For the non-dipole field the rate of drift is 0.18$\pm $0.015 $ ^{\circ} $ /year, that for the secular variation is 0.32$\pm $0.067 $ ^{\circ} $ /year. The results are confirmed by a study of harmonic analyses made between 1829 and 1945. The drift is explained as a consequence of the dynamo theory of the origin of the earth's field. This theory required the outer part of the core to rotate less rapidly than the inner part. As a result of electromagnetic forces the solid mantle of the earth is coupled to the core as a whole, and the outer part of the core therefore travels westward relative to the mantle, carrying the minor features of the field with it.

Diffraction from an irregular screen with applications to ionospheric problems

Abstract: An analysis is made of the diffraction effects produced when a plane wave is incident upon an irregular diffracting screen, and the results are applied to the problem of the reflexion of radio waves from an ionosphere which is irregular in the horizontal plane. The nature of the irregular screen is assumed to be given in terms of the variation of electric wave-field in a plane just beyond the screen, and it is assumed that variations occur over the plane in one direction only. It is further assumed that the screen is 9random’ in the sense that it is one of an assembly all of which differ from each other, but have statistical properties in common, and deductions are made about the diffraction patterns averaged over the assembly. It is shown that many aspects of the problem can be investigated by use of the theory of ‘random’ electrical noise as developed by Rice and Uhlenbeck. The angular spectrum (Fraunhofer diffraction pattern) and the Fresnel diffraction pattern are described in terms of their spatial auto-correlation functions, and there is some discussion of a related method of dealing with Fresnel diffraction problems from completely determined screens. In part II of the paper the irregular ‘fading’ exhibited by a radio wave returned from the ionosphere is discussed in terms of two models in which the fading is assumed to be produced by movements of the diffracting centres in the ionosphere. The temporal auto-correlation function of the amplitude of the irregularly fading signal is related to the velocity of the ionospheric diffracting centres.

The theory of axisymmetric turbulence

Abstract: The present paper completes the theory of axisymmetric tensors and forms to the extent that is needed for the development of a theory of turbulence in which symmetry about a certain preferred direction is assumed to exist. Particular attention is given to the manner in which tensors, solenoidal in one or more indices, can be derived, uniquely, in a gauge-invariant way, as the curl of a suitably defined skew tensor. The explicit representation of the fundamental velocity correlation tensor ( ) in terms of two defining scalars is found; and the differential equations governing these scalars is also derived. In the theory of axisymmetric turbulence these latter equations replace the equation of von Karman & Howarth in the theory of isotropic turbulence.

Calcium Carbonate and Its Hydrates

Abstract: This paper is concerned with observations on the preparation of phases which may exist in the system CaCO 3 -H 2 0 in the presence o f ‘additives’, with a qualitative explanation of the appearance of unstable phases under such conditions. It has been found that the tendency for metastable phases of calcium carbonate to be precipitated from aqueous solutions is increased not only by raising the rate of precipitation but also by increasing the concentration of certain added substances. Thus in the presence of small amounts of Calgon* or magnesium-ion, and in saturated sodium chloride solution in the presence of magnesium-ion, a range of products is obtained according to the conditions of precipitation, particularly the stirring and the temperature. Calcite (the stable phase), aragonite, vaterite (/^-calcium carbonate) and calcium carbonate hexahydrate have been prepared In this way together with a new hydrate, CaCO 3 .H 2 O . By allowing reaction mixtures containing additive to stand until the first-formed gel has crystallized, metastable phases may be produced with considerable regularity. Metastable phase diagrams can be constructed which show the dependence of the phases on conditions of precipitation, e.g. sodium carbonate concentration and concentration of additive. By exploration using such diagrams, all the previously mentioned phases except vaterite have been prepared in the presence of magnesium-ion, and in the presence of Calgon all but aragonite have been identified. This suggests that to some extent the action of the additives is specific. In stirred conditions of reaction, calcium carbonate may be grown, without fresh nucleation, on seed crystals of calcite and vaterite at a much faster rate in the presence of limited amounts of Calgon than in its absence. At relatively high concentrations other phases may appear and growth of the calcite and vaterite ceases in spite of the presence of seed crystals. Compounds containing calcium carbonate formed in the presence of magnesium-ion or Calgon have been shown by analytical methods to contain magnesium or phosphorus in appreciable quantities. The presence of the additive leads to modification in the crystal habit of calcite and of the hexahydrate. It has also been noted that distortion of X-ray patterns of the crystalline species is common. This is consistent with absorption of the additive in the distorted phase. Absorption of additive seems, therefore, to be important in influencing nucleation and the formation of metastable phases. These observations are explained on the grounds that germs of the crystalline phases are subject to varying degrees of distortion depending on the constitution of the solution. The growth is therefore hindered to different degrees in different cases. In the limit the effect of distortion may be such that growth of nuclei of certain types may be inhibited, and there is a corresponding change in the conditions for the spontaneous crystallization from solution of the various phases.

Excitation and Ionization of Atoms by Electron Impact-The Born and Oppenheimer Approximations

Abstract: It is important, for many applications, to have reliable data on the magnitudes of the cross-sections for excitation and ionization of atoms and ions by electrons. In part I the usual approximations (those of Born and of Oppenheimer) which are made to obtain theoretical values are critically examined. It is pointed out that the assumption of separable bound wave functions may often lead to considerable errors. In the case of the Oppenheimer approximation the errors may even be such as to give results violating the principle of detailed balancing. Circumstances in which these errors are likely to be serious are analyzed, and precautions which may be taken to reduce them are proposed. The conditions under which the approximations are likely to fail, even when exact bound wave functions are used, appear to be related to the magnitude of certain coupling terms which are ignored in obtaining the approximations. The usefulness of certain conservation theorems which limit the possible size of collision cross-sections is also pointed out. A summary of those general properties of inelastic cross-sections which are reliably given by the theory is included. In part II the available experimental data are compared with the predictions of the Born and Oppenheimer approximations. The collision processes studied include the following: excitation of H, He, Na, Ne and Hg; ionization of H$\_{2}$, He, Ne, Hg, Ni (K) and Ag (K and L$\_{\text{III}}$). The investigation shows that the Born approximation is the one that should generally be used in the treatment of transitions which can take place without electron exchange having to be invoked. For these the approximation achieves a considerable degree of success. As far as can be judged from the comparison data available, the main defects are that the maxima of the predicted cross-section energy curves tend to be too pronounced, and to be located too close to the critical potentials. In the case of transitions involving a reversal of electron spin the Oppenheimer approximation must be used. Unfortunately, it proves to be very unsatisfactory. Thus for non-hydrogenic systems it may give very different results according to whether a prior or a post interaction is adopted. It leads to frequent violations of the conservation theorem and cannot be relied upon even to give the detailed shape of cross-section against energy curves. By generalizing from the evidence collected, an attempt is made to specify the conditions under which the Born and Oppenheimer approximations are most reliable; on this basis, proposals for systemization are made. Attention is drawn to the fact that some (but by no means all) of the observed excitation functions possess an extremely sharp peak just beyond the critical potential. The theory seems unable to reproduce this peculiar feature. It does not appear in the observed ionization functions.

Relaxation Methods Applied to Engineering Problems. XIV. Plastic Straining in Two-Dimensional Stress-Systems

Abstract: Only experiment can decide the criterion of elastic failure, and the relation of stress to strain during plastic distortion, in real materials such as steel; and (since stress is not directly measurable) it can do this only by verifying relations deduced from theory in regard to total displacements and resultant actions. Consequently practical value attaches to computational methods whereby, on the basis of some assumed criterion, relations of that kind can be formulated. This paper draws conclusions regarding two-dimensional systems (plane stress and plane strain) from the 'Mises-Hencky hypothesis', according to which failure occurs when (p$\_{2}$-p$\_{3}$)$^{2}$+(p$\_{3}$-p$\_{1}$)$^{2}$+(p$\_{1}$-p$\_{2}$)$^{2}$ = const. (p$\_{1}$,p$\_{2}$,p$\_{3}$ denoting the principal stresses), and from the relation $\Delta \gamma \_{1}$: $\Delta \gamma \_{2}$: $\Delta \gamma \_{3}$ = q$\_{1}$: q$\_{2}$: q$\_{3}$ assumed to hold during the subsequent plastic distortion ($\Delta \gamma \_{1}$, $\Delta \gamma \_{2}$, $\Delta \gamma \_{3}$ denoting the incremental plastic shear-strains and q$\_{1}$, q$\_{2}$, q$_{3}$ the principal shear stresses). Its methods could be applied to other hypotheses. In its worked examples some regions remain elastic while in other regions (here termed enclaves) the strain is partly plastic. Such cases present special difficulty in an orthodox treatment.

The Numerical Solution of Elliptic Differential Equations When the Boundary Conditions Involve a Derivative

Abstract: The numerical solution of differential equations involves the replacement of derivatives by finitedifference equivalents. The idea of using approximate equations and subsequently correcting for the higher differences, already applied to second-order equations with specified boundary values, is here extended to the case where conditions at the boundary involve a derivative. The method is applied with examples to second- and fourth-order equations. The more difficult problems associated with curved boundaries are discussed, with particular reference to problems of stretching of flat elastic plates. An alternative but more laborious method of obtaining accurate solutions, the method of ‘the deferred approach to the lim it’, is illustrated by examples.

The Levels in the North Sea Associated with the Storm Disturbance of 8 January 1949

Abstract: Observations, around the North Sea, have been collected for twenty stations in the British Isles and for fifteen stations on the Continent during the period 6 to 10 January 1949, and used in a discussion of a large storm disturbance on 8 January. The primary object of the investigation has been to get a picture of the water movements inside the North Sea, and of the way in which these movements are produced, in the course of a storm. Maps of co-disturbance lines in the North Sea have been drawn at frequent intervals and compared with the simultaneous meteorological conditions. The disturbance around the coast, and in the Thames and Humber estuaries, has been examined in detail, and the progression around the coast has been shown to be similar to that of the diurnal tide. Estimates have been made of the changes, during the storm, in the average level of the North Sea, and it has been shown that when the level was rising there was a large inflow of water down the western half; when the level was falling the outflow was up the eastern North Sea. New light has been thrown on several problems connected with storm surges. In particular, it would appear that storm surges of external origin, which hitherto could not be explained in terms of the winds, may be a direct result of an earlier outflow of water produced during a surge when the level has been lowered; also the excessive damping normally observed in the disturbed tide at Southend, which leads to an average value of eddy viscosity in the North Sea somewhat larger than that normally accepted, may be due to a reflexion from the German bight which arrives near the time when the lowest levels are expected. Two estimates have been made of the frictional constant, on the assumption that the tractive force of the wind varies as the square of the wind velocity, and are in agreement with accepted values. Prediction of the disturbance at Southend using a previously established formula has given good results. The disturbance transmitted through the Straits of Dover has been investigated.

The lateral instability of yielded mild steel beams of rectangular cross-section

Abstract: The critical load causing secondary failure of a deep beam by lateral buckling may be calculated by standard methods for those cases in which the beam behaves elastically under the applied load. When, however, the load is sufficiently great to cause partial yield of the beam, these methods give an estimate for the critical load which is too high. In the present paper the phenomenon of lateral buckling in deep mild steel beams of rectangular cross-section is studied from both a theoretical and an experimental standpoint. The paper is divided into three parts. In part I the critical lateral buckling load is shown to depend on the flexural rigidity of the beam about its weaker principal axis while the applied load, causing flexure about its stronger principal axis, is held constant. The dependence of this rigidity on the extent to which the beam has yielded is calculated, and the results are confirmed by tests on beams of rectangular and circular cross-section. It is also shown that the critical load depends on the initial torsional rigidity of the beam, defined as the initial slope of the torque against angle of twist per unit length relation for torsion about the longitudinal axis of the beam while the applied bending load is held constant. In part II it is first shown that in a beam which has partially yielded the shear force due to the variation of the applied bending moment along the length of the beam is carried entirely in the central elastic core of the beam. Using the theory of combined elastic and plastic deformation, it is then shown that the initial torsional rigidity remains constant at its value for elastic torsion, and experimental evidence in favour of this conclusion is presented. Using the results of parts I and II, the conditions causing lateral instability in deep mild steel beams of rectangular cross-section are determined in part III. For a beam bent by pure terminal couples these conditions may be deduced directly, but for the cases of beams subjected to central concentrated loads and of cantilevers a step by step solution of the governing differential equation is necessary. Experimental confirmation is given for the case of pure bending.

Thermodynamics and rheological behaviour of elasto-viscous systems under stress

Abstract: The systems discussed were obtained from dissolving aluminium and calcium soaps of fatty and naphthenic acids in hydrocarbons with the aid of peptizers like water, xylenols, alcohols, ‘Cello- solve’ or cresols. They behave similarly to solutions of rubber in benzene. Rheologically they exhibit certain peculiarities associated with secondary flow phenomena. It was observed that when such systems were subjected to stress, their free energies increased. The first series of experiments dealt with the behaviour of jets in the form of expanding conical sheets. Newtonian liquids broke up when the kinetic energy of the jet exceeded a certain function of the surface energy which tended to stabilize the sheets. With these elasto-viscous systems, the kinetic energy required to break up the expanding conical sheets was much greater than this function and thus it was concluded that the stressed sheets had a higher free energy than the un­ stressed relaxed body of the same liquid. The second series of experiments dealt with the quantitative evaluation of the free energy on subjecting the systems to stress at different temperatures. A rod was rotated inside a stationary sleeve, the space between being full of the liquid under consideration. From the drop in the level of the liquid, it was deduced that the internal energy and the entropy were both raised by shearing the system. The increase in internal energy was greater than the rise in entropy, and hence the free energy, as a whole, increased on shearing. These effects became less noticeable as the temperature was raised. Next a series of experiments was conducted on the behaviour of these systems when subjected to different states of shear between discs, cones or cylinders in which one member was rotated and the other held stationary. In all experiments secondary spontaneous flow from regions of high shear to regions of low shear confirmed the hypothesis that the increase in free energy on straining (or stressing) the system resulted in local instability. Surfaces of uniform shear acted as if they were semi-permeable membranes with a concentration gradient across them due to the structural units moving from regions of high to those of low shear values. Thus a species of osmotic pressure was generated which could be measured. These peculiar phenomena only occurred when there was a marked difference in the rate of shear between different points in the system; under uniform rates of shear they did not arise. Finally, experiments with pipes showed that these systems possessed two peculiarities: (1) the inlet end effects were far greater than those encountered with Newtonian systems under similar conditions; and (2) there appeared to be a retarding layer effect in the pipe. It is concluded that for these systems studies of the balance between input energy and the portions which are lost and stored would be better approaches to the study of the structure of the systems than a study of the viscosity of the liquids as such.

A Purification Method for Computing the Latent Columns of Numerical Matrices and Some Integrals of Differential Equations

Abstract: The emphasis is on practical arithmetical methods suitable for application to matrices of so many rows and columns that their determinants are intractable. The arithmetic begins from an arbitrary, or guessed, column of numbers. This may be regarded as a raw material composed of the desired constituent plus an unknown assortment of impurities. Although the impurities are not known, their behaviour when subjected to certain computing operations can be foreseen sufficiently to allow them to be cleaned away gradually. These operations depend on all the latent roots. There is an initial stage of groping-about to find roughly where the latent roots are located. This leads on to a final stage of rapid routine in which roughly known latent roots are used to purify the column associated with the desired root. It is shown that the cleaning operation spreads over a considerable region of the Argand diagram; so that very imperfect information is, nevertheless, often effective. The proof of the numerical result is by comparison of independent estimates, and not by theorems about infinitely many operations. A closely similar purification method was applied in 1910 to a wide class of problems (so-called jury problems) involving differential equations, such as Laplace’s, or the equation of plane stress. In the present paper the choice of cleaning agents for such purposes is made more systematic by employing Legendre’s principle of least squares.

The Determination of Gaseous Densities

Abstract: The microbalance technique for the determination of gaseous densities is examined and the require­ ments stated for its use as a method of high precision and reliability. A detailed description is given of the construction of a sensitive and stable quartz microbalance and of the precautions necessary to obtain reproducible gas-pressure measurements of high precision. The improved technique is used for the determination of the atomic weights of carbon and nitro­ gen by the comparison of the pressure ratios at * limiting density ’ of the pure gases nitrogen, methane and oxygen; the values obtained are: nitrogen 14.0078 (± 0*0003) carbon 12.0112 (± 0*0005) Nitrogen prepared by heating pure, dry sodium azide is shown to contain traces of hydrogen (either in the free state or combined with nitrogen as ammonia), and a new method is described of purifying nitrogen made by this method.

Diophantine Inequalities with an Infinity of Solutions

Abstract: Many results in the geometry of numbers assert, in effect, that inequalities of a certain type are soluble in integers, the constant on the right of the inequality being the best possible. Recent work of Mahler often enables one to prove that such an inequality has infinitely many solutions. In this paper we develop the theory of inequalities with infinitely many solutions, and investigate more deeply some of the questions which naturally arise.

The Propagation of a Sound Pulse in the Presence of a Semi-Infinite Open-Ended Channel. I

Abstract: The behaviour of a sound pulse approaching and progressing beyond the open end of a semi-infinite channel is discussed. A succession of diffracted waves is created at the open end for which a general formula is obtained, by operational methods, when the pulse originates inside the channel. With the aid of a simple reciprocity relation the asymptotic behaviour of these diffracted waves can be used to deduce the form of the wave returning along the channel when the original pulse approaches the open end from an arbitrary direction. Ultimately the returning wave becomes sensibly plane and separates into regions of length equal to the width of the channel, the form of the potential depending on the number of diffracted waves which contribute to each particular region. Explicit expressions are obtained for the potential in the first two regions at the head of the returning wave and for the third region when the pulse originates inside the channel. The case of an initial velocity distribution given by the Heaviside unit pulse is treated in detail.