Showing papers in "Philosophical Transactions of the Royal Society A in 1968"

On the thermal instability of a rotating-fluid sphere containing heat sources

Abstract: The theory of marginal convection in a uniformly rotating, self-gravitating, fluid sphere, of uniform density and containing a uniform distribution of heat sources, is developed to embrace modes of convection which are asymmetric with respect to the axis of rotation It is shown that these modes are the most unstable, except for the smallest Taylor numbers, T (a measure of the rotation rate); ie for any T and o) (Prandtl number), the lowest Rayleigh number (a measure of the temperature gradients in the sphere) is associated with an asymmetric motion This is demonstrated both by an expansion method suitable for small T, and by asymptotic theory for T oo For large T, the eigenmode most easily excited is small in amplitude everywhere except near a cylindrical surface, of radius about half that of the sphere, and coaxial with the diameter parallel to the angular velocity vector

Kinematics of Geomagnetic Secular Variation in a Perfectly Conducting Core

Abstract: The motions in the Earth’s electrically conducting fluid core which are the probable cause of the geomagnetic secular variations have time scales of the order of a few centuries or less. Seismic bounds on the kinematic molecular viscosity of the core and order-of-magnitude arguments about the eddy viscosity make plausible the hypothesis that at such short periods the core motion consists of a boundary layer of Ekman-Hartmann type close to the core mantle boundary, and an interior free-stream motion where the viscosity and resistivity can be set equal to zero. This boundary-layer approximation requires that the unknown vertical length scale of the poloidal geomagnetic field deep in the core be at least as long as the 600 km horizontal length scales inferred at the surface of the core from observations above the mantle. For periods shorter than a century the Ekman and magnetic boundary layers are probably thinner than 120 km. If magnetic flux diffusion is neglected (i.e. if electrical conductivity is considered infinite) in the free stream in the core then the external geomagnetic field is completely determined by the fluid motion at the top of the free stream. Therefore the hypothesis of negligible flux diffusion in the free stream has implications for the geomagnetic secular variation, and these implications can be used as a test of whether there is any motion of a perfectly conducting core which will produce the observed secular variation. If the observed secular variation passes this test, we can write down explicitly all ‘ eligible’ velocity fields, i.e. all velocity fields at the top of the free stream in the core which are capable of producing exactly the observed secular variation. The different eligible velocity fields are obtained by different choices of an arbitrary stream function on the surface of the core. We describe a method of selecting from among all eligible velocity fields those which are of particular geophysical interest, such as the one which is most nearly a rigid rotation (westward drift) or the one which is most nearly a latitude dependent westward drift with m degrees of freedom.

On the shape of progressive internal waves

Abstract: An expansion technique, analogous to that of Stokes in the study of surface waves, is used to investigate the effects of finite amplitude on a progressive train of internal gravity waves. The paper is divided into two main parts, a study of interfacial waves in a two-fluid system and an examination of internal waves in a continuously stratified fluid. Experimental evidence is presented which confirms some of the theoretical predictions. The validity of the Boussinesq approximation is examined and particular examples are taken to illustrate the general results.

Contributions to the Theory of Diophantine Equations. I. On the Representation of Integers by Binary Forms

Abstract: An effective algorithm is established for solving in integers x, y any Diophantine equation of the type/( x, y ) = m , where/ denotes an irreducible binary form with integer coefficients and degree at least 3. The magnitude, relative to m, of the bound furnished by the algorithm for the size of all the solutions of the equation is investigated, and, in consequence, there is obtained the first generally effective improvement on the well known result of Liouville (1844) concerning the accuracy with which algebraic numbers can be approximated by rationals.

A Unified Analysis of Tides and Surges Round North and East Britain

Abstract: Tide gauge records from six ports round north and east Britain are analysed in terms of the response of sea level to gravitational and radiational tide potentials, and to a six-component function which describes the major features of the weather over the relevant sea area. Tidal interaction effects due to shallow water are allowed for up to third order. The linear tidal admittances in species 1, 2 and 3 are found to be smooth and consistent from one port to another. The radiational components are found to have amplitudes around 18% of the 12 h tide due to gravity at all ports, with consistent phase leads on the gravitational tide. The second-order interactions producing tides of species 4 are found to have strong dependence on frequency, and are surprisingly noisy at Southend. The third-order tides at species 6 have neither of these properties, and their amplitudes are shown to vary approximately with the cube of the main tidal amplitude. This last fact is reconciled with a quadratic frictional law. On the whole, the response analysis of the tides is found to be more accurate than a harmonic analysis which uses a greater number of arbitrary constants, with an increasing improvement in predictable variance at the shallower ports. At the two estuarine ports, effects coherent with interaction between tides and random variations in local sea level are found by a bispectral analysis. The interaction coefficients are reasonably consistent with those derived from the tides alone, and with computations by G. W. Lennon of wave propagation in the Thames Estuary. The interaction with local sea level is insufficient to account for the tidal modulations with frequencies less than 1 cycle/month. The spectra of these modulations at the M2 frequency ± f are strongly coherent, and show similar proportions of phase- and amplitudemodulations at all stations. They may be due to oceanic, local, or instrumental effects. The seasonal effects at f — + 1 cycle/year stand out clearly, and point to a third distinct agent of tidal modulation. The functions chosen to describe the weather are the first six coefficients of a special Taylor expansion of atmospheric pressure. As well as describing the field of normal stress, these functions implicitly embody a linearized wind stress through the pressure gradients. Inter-correlation between the six components are allowed for. They are all found to give significant contributions to sea level variance, and their responses extend to at least 24 h time delay

The Bakerian Lecture, 1967: Reversals of the Earth's Magnetic Field

Abstract: This paper is an account of the Bakerian Lecture given to the Royal Society on 15 June 1967. Reversals of the Earth’s magnetic field can be studied in the magnetization of lavas and sediments on land, in the magnetization of deep sea cores and in the magnetic pattern on the ocean floor. The lavas give radiometric dates but not a continuous sequence; the cores give continuity, great detail and a resolution as fine as 1000 years; the magnetic pattern gives information all through the Tertiary and connects the spreading of the ocean floor with the radiometric time scale. The dynamo theory of the Earth’s magnetic field may be able to account for reversals as an instability in the dynamo, but only models with a finite number of degrees of freedom have been investigated. Spreading of the ocean floor is believed to be associated with convective motions in the upper mantle, although there are difficulties connected with the equality of the oceanic and continental heat flows. There is some evidence for the extinction of radiolaria at times of reversal of the magnetic field; it has been suggested that this is due to the effect of the field on cosmic rays but this appears impossible. If the extinctions are due to the reversals, the mechanism is unknown. Reversely magnetized rocks are more highly oxidized than normally magnetized ones. The cause of this is unknown and is one of the outstanding problems of Earth science.

The stability of spiral flow between rotating cylinders

Abstract: The effect of an axial pressure gradient on the stability of viscous flow between rotating cylinders is discussed on the basis of the narrow gap approximation, the assumption of axisymmetric disturbances, and the assumption that the cylinders rotate in the same direction. The onset of instability then depends on both the Taylor number (T) and the axial Reynolds number (R). For large values of R, the dominant mechanism of instability is of the Tollmien-Schlichting type and the present theory is based therefore on a generalization of the asymptotic methods of analysis that have been developed for the Orr-Sommerfeld equation. The present results, when combined with previous results for small values of R, give the complete stability boundary in the (R, T)-plane. Only limited agreement is found with existing experimental data and it is suggested therefore that it may be necessary to consider either non-axisymmetric disturbances or nonlinear effects.

Noise of Gears

Abstract: In the past there were mainly two important factors in determining the performance of gear units: load capacity and fatigue life. In recent years, however, the noise radiation of running gears is of growing importance. It is now becoming impossible to design gears without considering the significance of this new factor. While there are many data available in the literature for calculating the permissible load, speed, and fatigue life, no methods have so far been developed for a true prediction of the radiated gear noise. A definite noise level, however, is often subject to guarantee between manufacturer and customer. To meet these requirements two fundamental factors must be resolved. First, both partners must set up the conditions for acceptance tests, and secondly, the gear manufacturer must have the know-how to meet these particular requirements. Investigations were therefore, made to find out the main factors influencing gear noise in order to be able to reduce it. In addition eighty installed gear-units were tested to set up classifications regarding generated noise and operating conditions.

Contributions to the Theory of Diophantine Equations. II. the Diophantine Equation y 2 = x 3 + k

Abstract: This paper is a sequel to Part I (Baker 1968) in which an effective algorithm was established for solving in integers x, y any Diophantine equation of the type y) = m, where ^denotes an irreducible binary form with integer coefficients and degree at least 3. Here the algorithm is utilized to obtain an explicit bound, free from unknown constants, for the size of all the solutions of the equation. As a consequence of the cubic case of the result, it is proved that, for any integer 4= 9, all integers x, y satisfying the equation of the title have absolute values at most exp { (10101 A:|)10 }.

Characteristics of the defocused spherical Fabry-Pérot interferometer as a quasi-linear dispersion instrument for high resolution spectroscopy of pulsed laser sources

Abstract: Defocused spherical mirror Fabry—Perot etalons, in which the mirror separation is slightly less than the common radius of curvature, produce a multiple-beam fringe pattern of concentric rings, with quasi-linear spectral dispersion over an appreciable annular region corresponding to two free spectral ranges. The characteristics of these interferograms are discussed in relation to their many advantages for pulsed laser spectroscopy. These advantages include: (i) accuracy of frequency difference measurement; (ii) high illumination of the detector with moderate energy density in the laser beam; (iii) ease of alinement and permanent adjustment of the mirrors leading to the attainment in practice of a very high instrumental finesse (N R values of up to 90 have been achieved); (iv) measurement of degree of spatial coherence of laser beam; (v) ease of matching the interferogram to the spatial resolution of the detector. A simple optical path relation determines the positions of the fringes and the location of the quasilinear dispersion region. The interfering wavefronts, formed by multiple reflexion, have been numerically computed and summed to provide information on the finesse, fringe profiles, contrast and optimum conditions of use of this new, very high resolving power (107 to 108) quasi-linear spectrographic disperser. Constructional details are described and optical design criteria are discussed, together with the various experimental arrangements for employing the instrument. Comparison is made with the equivalent confocal and plane Fabry—Perot etalons and methods of simultaneously measuring

High resolution spectroscopy of narrow band giant pulse lasers: time-dependent frequency shifts in ruby

Abstract: Both plane and defocused spherical Fabry-Perot interferometers, with a limiting resolution of less than 10 MHz, have been employed to study the output spectral brightness of narrow band giant pulse ruby lasers. Rotating prism and Pockels cell Q -switched systems have been compared and the mode selection properties of resonant reflectors demonstrated. The advantages of uniform pumping of the laser rod, when combined with a dye solution isolator, are illustrated by interferograms showing the direct spectroscopic detection of pure transverse mode structure in giant pulse lasers. To elucidate apparent discrepancies between time integrated interferograms and pulse envelope beat patterns, time resolved spectroscopy with a fast image tube streaking camera was carried out. Time and intensity dependent blue and red frequency shifts were detected and measured and rapid (

Noise and Engineering Design

Abstract: The paper presents a short account of various investigations carried out on the basic mechanism of noise produced by machines involving solid structures. The emitted noise of the machine is generally due to bending vibration of either the outer or inner surfaces or both. In vehicles, for example, consideration of both the internal and external surfaces is of importance: vibrations of the internal surfaces produce noise inside the cab or car saloon and this affects the driver and passengers, while the external surfaces produce the noise which affects the community

Free Tidal Oscillations in Rotating Flat Basins of the Form of Rectangles and of Sectors of Circles

Abstract: A study is made of the periods of free tidal oscillations and of the corresponding wave patterns m rotating flat basins which have the form of rectangles or of sectors of circles. The analysis is based on a variational principle for tidal oscillations. It is shown that, if denotes the tide height and its complex conjugate, the sign of the integral ij(d/ds) ds, which is real, taken around the periphery of the basin, determines whether the tidal wave propagates around the basin m the direction of rotation (positive wave), or opposite to it (negative wave). The sense of propagation can also be told from the sign of dk 2 (r),where k 2 = (or 2 -4w 2 ) r = 2w/o denoting the speed of rotation, and o the frequency. A discussion is given of the removal by rotation of the degeneracy that exists in some modes in the absence of rotation. The method (A) of expansion of in terms of the eigenfunctions for no rotation (T = 0) was found to converge well only for T n = 2nn (gh)/p, p denoting the perimeter of the basin. It is shown that near a blunt corner of the coast the tide has a singularity like that in potential flow.

Reduction of noise by the use of damping materials

Abstract: level being thus reduced. Free bending and strain waves can transport sound energy via extended sheet-metal structures to points far away from the sources, causing sound radiation into the air. In this case too, the application of vibration damping materials will be beneficial because in this way the structure-borne sound propagation will be partially suppressed by the attenuation of the free-bending waves. Furthermore, the damping of the bending vibrations can essentially improve the sound insulation of light-weight structures, so-called sandwich walls, consisting of outer metal plates and an air cushion between, filled with sound-absorbing material. The damping of the metal plates diminishes the influence of resonances and coincidences which otherwise would lead to a diminution of the sound reduction of the sandwich wall (Stiuber I956, I965). Finally, it has been shown that the fatigue life of metal panels, for example near the outer walls of jets which are excited to flexural vibrations by the random pressure of the air-borne sound field of high intensity, is increased as a result of the damping of the panels (Kurtze & Westphal I965). There has been a systematic development of vibration damping materials for nearly 20 years. Formerly, the production of these materials was a matter of trial-and-error methods (Beranek I960). It was a long road to the high-quality products which are in use today. The measuring technique had to be developed with which the viscoelastic properties, especially the damping of the damping materials and of the combined systems consisting of metal plates or sheet metal and damping viscoelastic layers, could be investigated. Furthermore, the theory of the bending vibrations and the viscoelastic properties, especially the damping of the combined systems, had to be worked out to enable us to understand the dependence of those properties on the viscoelastic properties of the layers and their thicknesses. In the first stage, two-layer systems consisting of metal panels with a damping layer on one side were considered (Lienard I951, 1957; Oberst & Frankenfeld 1952; Oberst, Becker & Frankenfeld I954). From the results of the theory the conditions which the damping materials had to fulfil to provide high damping of the combined systems could be established. It soon became obvious that the highest damping values could be obtained with amorphous thermoplastic polymers. Next, the possibilities were studied as to how vibration damping materials with optimum damping efficiency could be

Neville's and Romberg's processes: a fresh appraisal with extensions

Abstract: In this paper Neville’s process for the repetitive linear combination of numerical estimates is re-examined and exhibited as a process for term-by-term elimination of error, expressed as a power series; this point of view immediately suggests a wide range of applications—other than interpolation, for which the process was originally developed, and which is barely mentioned in this paper—for example, to the evaluation of finite or infinite integrals in one or more variables, to the evaluation of sums, etc. A matrix formulation is also developed, suggesting further extensions, for example, to the evaluation of limits, derivatives, sums of series with alternating signs, and so on. It is seen also that Neville’s process may be readily applied in Romberg Integration; each suggests extensions of the other. Several numerical examples exhibit various applications, and are accompanied by comments on the behaviour of truncation and rounding errors as exhibited in each Neville tableau, to show how these provide evidence of progress in the improvement of the approximation, and internal numerical evidence of the nature of the truncation error. A fuller and more connected account of the behaviour of truncation errors and rounding errors is given in a later section, and suggestions are also made for choosing suitable specific original estimates, i.e. for choosing suitable tabular arguments in the elimination variable, in order to produce results as precise and accurate as possible.

Magnetohydrodynamic wave propagation from a localized source including hall effect

Abstract: Lighthill's theorem on magnetohydrodynamnic wave propagation from a localized source gives an asymptotic expression for the wave amplitude at large distance from the source and this theorem is applied to a laboratory plasma. Attention is focused mainly on conditions where the Alfven speed is large compared with the sound speed and where the wave frequency is of the same order as the ion cyclotron frequency. These conditions are also typical of certain natural plasmas. The manner in which the wave amplitude depends on the direction of propagation and on the ratio of the wave frequency to the ion cyclotron frequency is examined and illustrated by graphs which also cover a range of source sizes. An extension is made into a two-fluid theory such that electron inertia and electron source terms can be included and the effect of finite electrical conductivity is considered.

Deafness due to Impulse Noise [and Discussion]

Abstract: The evaluation of the hazards to hearing experienced by exposure of the ear to noise has resulted in the specification of the physical characteristics of the noise in terms of damage risk criteria. In the case of steady state and relatively long on-period intermittent noises the researches carried out have resulted in the formulation of many such criteria (Glorig, Ward & Nixon 1962; Burns 1965; C. H. A. B. A. 1965? Kryter, Ward, Miller & Eldredge 1966), the interpretation of which in terms of an internationally agreed standard is at present being undertaken by the International Organization for Standardization (I. S. O.). For exposure to very short duration steady-state noise, for noise superimposed with impulsive components, and for high-intensity impulsive noise, there is little information regarding damage risk. What does exist and the problems associated with the assessment of deafness due to impulse noise will be discussed in this paper. It must be borne in mind, however, that no criterion will clearly demark safety from danger and, being based on the light of the best available knowledge, may be subject to change as new evidence becomes available.

Sandwich Damping Treatment Applied to Concrete Structures

Abstract: Concentration of sources of noise and vibration into densely populated regions and into buildings of slender construction according to modern practice has shown the need for the introduction into civil engineering of definite methods of vibration damping. This is necessary in particular with prestressed concrete construction in which there is normally hardly any internal damping and the stronger the concrete the less the damping will be. The slender forms of construction and greater speed ranges of many machines leads to more chances of striking a condition of resonance in a vibrating structure, wherein the exciting forces will be opposed only by the damping forces. Too little damping will mean large amplitudes and much noise.

Noise of Electrical Machines

Abstract: The control of noise in electrical machines may be effected by two separate and distinct methods. The first is the reduction of the noise at the source and the second the attenuation of this noise before it reaches the ears of the observer. Since noise reduction is costly, the problem is resolved into one of finding the most economic means of obtaining the required noise reduction, without interfering with the efficient functioning of the plant. It is generally considered that a good machine should run smoothly and quietly, so from the earliest days the designers and manufacturers have put a great deal of effort into producing quiet machines and more recently considerable research effort has been expended in explaining the basic phenomena.

Noise Control on Textile Machinery

Abstract: In the textile industry as a whole the pattern of machine layout and operation has been changed little over the years. Banks of machines, housed in large factory areas, are crowded together to preserve floor space and optimize production in both the natural and man-made fibres industries. Noise in the mills and factories is increasing, and one of the main reasons is that in the face of keen competition and already on a 24 h basis, machines are being operated at speeds higher than ever before to increase production rates. This is possible to a greater extent with the man-made fibres; continuous threads can be produced and wound more quickly because of their inherent strength and other advantageous properties compared with natural fibres. As a direct result of the speed increase, out of balance forces and vibration increase and more energy is released as sound.

Noise and the Householder

Abstract: To the householder, noise is important mainly in respect of its capacity to annoy. Rarely in residential noise problems arising from industry are we concerned with levels which are sufficiently high to interfere with communication or to cause deafness. It is this fact that makes assessment of the intrusive value of industrial noise extremely difficult. One can make objective measurements of hearing loss, or of interference with communication, but one cannot describe, in objective terms, the emotional reactions to noise of people enjoying the privacy of their homes. The problem, therefore, is to decide how we can express the intrusive propensities of a noise in terms of its physically describable characteristics. It might be argued that previous experience is the real guide that is, a survey of noises which people have found to be acceptable in the past, which can be used to predict reaction in new cases. This is, in fact, the basis of criteria of acceptability as assessed by such methods as those currently proposed by the Wilson Committee on the Problem of Noise (1963), and the International Standards Organization; and, indeed, one finds that for noise from traffic, aircraft, and (apart from some extreme examples) noise from

Frequency Weighting Contours for Predicting the Speech Interfering Aspects of Noise

Abstract: Physical and psychophysical schemes that purported to measure the speech interfering aspects of noise were examined in a series of papers by Klumpp & Webster (1962, 1963) and Webster & Klumpp (1963). Sixteen diverse-spectrum noises were adjusted in level so that listeners hearing monosyllabic (rhyme) words at a constant level of 78 dB from a loudspeaker obtained 50 % word intelligibility scores. Articulation index (a.i) calculations (see Kryter 1962 a ) predicted the speech-interfering properties of the noises very well. However, other, and simpler, schemes worked just as well; for example, speech interference level (s.i.l.) calculations (see Beranek 1954), based on octaves centred at 425, 850 and 1700 Hz, or 500, 1000 and 2000 Hz. The A-weighting and Din 3 networks (see Peterson & Bruel 1957), of a sound level meter (s.l.m.) were good, but the conventional use of noise criteria (n.c.), or alternate noise criteria (n.c.a.) (see Beranek 1957), curves did not work well unless (1) only that part of the curves centering on the octaves 500, 1000 and 2000 Hz was used, and (2) the noise spectra were allowed to ‘average through’ a contour and not just touch it at a peak value.

A Study of Problems of Noise in Offices

Abstract: The work now being carried out by the Building Research Station on the problems of work in offices has arisen from a series of more general studies of the office environment (Langdon 1959, 1963, 1965; Collins & Langdon 1960; Langdon & Keighley 1964) in the course of which the attitudes of office workers toward the social, functional and physical features of their surroundings were related to measurements of the main physical aspects, such as space, lighting, heating, noise and so on. The principal aim of these inquiries was to reveal particular problems associated with the type of office being studied, and to assess physical standards as related to comfort and satisfaction, particularly when these could not be defined operationally.

Noise Reduction on Road-Breaking Drills

Abstract: The noise generated by impulsive road-breakers is probably the most annoying part of the noise generated at construction and demolition sites and at roadworks. The problem was put into its proper perspective in the Wilson Committee’s report on the problem of noise (1963). It was shown by implication there that orthodox pneumatic road-breakers were unacceptable as far as noise is concerned, a reduction of some 7 to 1 5 dB(A) being desirable (i.e. from a present level of 82 to 85 dB(A) at 50 ft. radius to a level of 70 to 75 dB(A) outside the nearest window). Hydraulic road-breakers available at that time were as noisy as their pneumatic counterparts but were considerably more powerful. Electric and hydraulic breakers commercially available now (1967) are, as will be seen, considerably quieter than pneumatic drills. The use of an enclosure for the operator was encouraged in the Wilson report and the reductions in the noise radiation so obtained are quite high, but this device is in some ways difficult for the contractor. The breaker is usually working at an edge beyond which is broken rubble or a hole. It is difficult then to move the enclosure on its wheels over such terrain.