Showing papers in "Philosophical Transactions of the Royal Society A in 1955"

On Certain Integrals of Lipschitz-Hankel Type Involving Products of Bessel Functions

Abstract: This paper is concerned with the evaluation and tabulation of certain integrals of the type (* 00 I(p, v; A) = J J fa t) ) e~cttxdt. In part I of this paper, a formula is derived for the integrals in terms of an integral of a hypergeometric function. This new integral is evaluated in the particular cases which are of most frequent use in mathematical physics. By means of these results, approximate expansions are obtained for cases in which the ratio b/a is small or in which b~a and is small. In part II, recurrence relations are developed between integrals with integral values of the parameters pt, v and A. Tables are given by means of which 7(0, 0; 1), 7(0, 1; 1), 7(1, 0; 1), 7(1,1; 1), 7(0, 0 ;0), 7(1, 0;90), 7(0, 1; 0), 7(1, 1; 0), 7(0,1; - 1 ), 7(1,0; - 1 ) and 7(1,1; - 1 ) may be evaluated for 0

On the stability of three-dimensional boundary layers with application to the flow due to a rotating disk

Abstract: A phenomenon of boundary-layer instability is discussed from the theoretical and experimental points of view. The china-clay evaporation technique shows streaks on the surface, denoting a vortex system generated in the region of flow upstream of transition. Experiments on a swept wing are described briefly, while experiments on the flow due to a rotating disk receive much greater attention. In the latter case, the axes of the disturbance vortices take the form of equi-angular spirals, bounded by radii of instability and of transition. A frequency analysis of the disturbances shows that there is a narrow band of disturbance components of high amplitude, some frequencies within this band corresponding to disturbances fixed relative to the surface and others corresponding to moving waves. Furthermore, the determination of velocity profiles for the rotating-disk flow is described, the agreement with the theoretical solution for laminar flow being quite satisfactory; for turbulent flow, however, the empirical theories are not very satisfactory. In order to explain the vortex phenomenon just discussed, the general equations of motion in orthogonal curvilinear co-ordinates are examined by superimposing an infinitesimal disturbance periodic in space and time on the main flow, and linearizing for small disturbances. An important result is that, within the range of certain approximations, the velocity component in the direction of propagation of the disturbance may be regarded as a two-dimensional flow for stability purposes; then the problem of stability formally resembles the well-known two dimensional problem. However, it is important to emphasize that this result—namely, that the flow curvature has little influence on stability—is applicable only to the possible modes of instability in a local region. The nature of three-dimensional flows is discussed, and the importance of co-ordinates along and normal to the stream-lines outside the boundary layer is examined. In accord with the formal two-dimensional nature of the instability, there is a whole class of velocity distributions, corresponding to different directions, which may exhibit instability. The question of stability at infinite Reynolds number is examined in detail for these profiles. As for ordinary two-dimensional flows, the wave velocity of the disturbance must lie somewhere between the maximum and minimum of the velocity profile considered. The points where the wave velocity equals the fluid velocity are called critical points, of which most of the profiles considered have two. Then Tollmien’s criterion that velocity profiles with a point of inflexion are unstable at infinite Reynolds number is extended to the case of profiles with two critical points. One particular profile—namely, that for which the point of inflexion lies at the point of zero velocity—may generate neutral disturbances of zero phase velocity, corresponding to the disturbances visualized by the china-clay technique. A variational method for the solution of certain of the eigenvalue problems associated with stability at infinite Reynolds number is derived, found by comparison with an exact solution to be very accurate, and applied to the rotating disk. The fixed vortices predicted by the theory have as their axes equi-angular spirals of angle 103°, in good agreement with experiment, but the agreement between theoretical and experimental wave number is not good, the discrepancy being attributed to viscosity. Finally, the correlation between the experimentally observed and theoretically possible disturbances is discussed and certain conclusions drawn therefrom. The streamlines of the disturbed boundary layer show the existence of a double row of vortices, one row of which produces the streaks in the china clay. Application of the theory to other physical phenomena is described.

On the Assessment of Optical Images

Abstract: In the formation of an optical image, each surface element of the object gives rise to a more or less blurred distribution in the image surface, of total brightness proportional to that of the object element. The image is the sum of these distributions in the appropriate sense: when the object is coherently lit, the image is built up by adding their complex amplitudes; when the object elements are regarded as incoherent it is the intensities which are added. In both cases the image can be expressed as the convolution of the object with a spread function which characterizes the optical system. In systems for which the spread function does not change appreciably from one part of the field to another, the Fourier transform of the image is obtained to a sufficient approximation on multiplying the Fourier transform of the object with that of the spread function. More generally, this holds for any part of the field of a non-isoplanatic system over which the changes in the form of the spread function are small enough to be disregarded; we call such an area an 'isoplanatism-patch'. Working over such an area, an optical system can be regarded as a linear filter in which the Fourier components of the object reappear in the image multiplied by 'transmission factors'. These factors, first considered by Duffieux, depend on the aperture and aberrations of the system, and in Section 2 they are evaluated in terms of an ikonal function. The qualities required of an optical image are so varied that an assessment valid over the whole range of practical applications seems out of the question. Two extreme cases are considered in the present paper. In the first of these it is assumed that the aim of an optical design is to produce an image which is directly similar to the object. This is appropriate when no process of image interpretation or reconstruction is envisaged. In the second case, the aim is to produce an image containing the greatest possible amount of information about the object, without regard to the complexity of the interpretation processes which may be needed to extract it. For the first case, a criterion of image fidelity is proposed in Section 2.4 which gives a numerical measure of the resemblance of image to object in terms of the transmission factors of the optical system. In the second case, assessment is based on the information content of the image in Shannon's sense. This depends not only on the transmission factors of the system but also on the statistical properties of the presumed object set and of the unpredictable fluctuations which necessarily disturb observation; the analysis is carried through in Section 3. In Section 4 the assessment of optical images is discussed in terms of these two criteria.

On the product of three homogeneous linear forms and indefinite ternary quadratic forms

Abstract: Isolation theorems for the minima of factorizable homogeneous ternary cubic forms and of indefinite ternary quadratic forms of a new strong type are proved. The problems whether there exist such forms with positive minima other than multiples of forms with integer coefficients are shown to be equivalent to problems in the geometry of numbers of a superficially different type. A contribution is made to the study of the problem whether there exist real $\phi $, $\psi $ such that x|$\phi $x-y||$\psi $x-z| has a positive lower bound for all integers x > O, y, z. The methods used have wide validity.

Large elastic deformations of isotropic materials X. Reinforcement by inextensible cords

Abstract: The theory of large elastic deformations of incompressible, isotropic materials developed in previous papers of this series is employed to examine some simple deformations of elastic bodies reinforced with cords. The cords are assumed to be thin, flexible and inextensible, and to lie parallel and close together in smooth surfaces in the undeformed body, which is thus divided into sections by boundary surfaces which are inextensible in certain directions. In the simple problems considered, the cords impose relationships upon the parameters which specify the deformation. The following examples are examined from this point of view: (i) the pure homogeneous strain of a thin uniform sheet containing a double layer of cords lying in a plane midway between its major surfaces; (ii) the combined pure homogeneous strain and flexure of a cuboid containing a double layer of cords lying in a plane parallel to a pair of opposite faces, the two sets of cords being unsymmetrically disposed in this plane with respect to the remaining faces of the cuboid, and the symmetrical case being obtained from this by a suitable choice of constants; (iii) the combined extension and flexure of a thin rectangular sheet with two sets of cords placed symmetrically in a plane parallel to its major surfaces, the problem being considered as a limiting case of (ii); (iv) the simultaneous extension, inflation and torsion of a cylindrical tube containing one or two sets of cords lying in helical paths concentric with the axis of the cylinder. In all cases, relations are obtained for the determination of the tensions in the cords in terms of the applied forces and the parameters which define the deformation.

The thermal conductivity of metals at low temperatures

Abstract: The thermal conductivity of high-purity samples of thirty-two metals has been measured. These were Ag, Al, Au, Be, Cb, Cd, Ce, Co, Cu, Fe, Ga, In, Ir, La, Mg, Mn, Mo, Ni, Pb, Pd, Pt, Rh, Sb, Sn, Ta, Ti, Tl, U, V, W, Zn and Zr. For most metals measurements were taken from 2 to 40°K, but where necessary they were extended to 90°K. For superconductors they were taken in both the superconducting and normal states. The conductivity was found to be entirely electronic except for Sb and U. Most of the specimens were polycrystalline, but single crystals of Zn, Cd, Sn, Pb, Ga and Ti were measured. For Zn and Ga, specimens of different orientations with respect to the rod axis were obtained, and in both these metals the thermal conductivity was found to be anisotropic. The thermal resistance, W, at low temperatures of nearly all the metals is of the form W — a T 2 + gjT, and the constants a and /? have been calculated. If is the limiting thermal conductivity at high temperatures and 6 is the Debye temperature, then the value of aK^d2 is the same for the metals in any one chemical group. For some metals the electrical resistance was measured at the same time as the thermal conductivity over the full temperature range and hence the Lorenz number, L, was calculated. The limiting value of L at low temperatures for several metals was found to be considerably higher than the theoretical value, in particular for Ti and Zr. A corresponding effect to the minimum in the electrical resistance of Mg has been found in the thermal resistance. A large increase in the thermal conductivity of Fe after a period of time has been ascribed to the precipitation of impurities in the metal. A method is given for estimating the thermal conductivity of a metal at low temperatures.

An Interpretation of Gravity Anomalies in the Eastern Mediterranean

Abstract: This paper gives an interpretation of the results of the 1950 British Submarine Gravity Survey The area covered by the survey is divided into four separate problems: the environs of Malta, the Crete island arc, Cyprus, and the Nile delta A 2000 ft submarine scarp south of Malta is associated with a change in the Bouguer anomaly of about 34 mgal This scarp is interpreted as a fault scarp, and geological sections consistent with the gravity and seismic data are drawn The positive anomalies in the southern Aegean Sea and the negative anomalies outside the Crete island arc lead to the hypothesis that, in this area, the granitic and intermediate layers are of constant thickness, the Mohorovicic discontinuity remaining parallel to the surface topography Large positive anomalies on Cyprus indicate the presence of a large buried basic or ultrabasic mass; this mass is discussed with relation to the geological structure of the island The thickness of sediments on the Nile delta and the crustal sag caused by them are estimated from gravity anomalies and topographic considerations

The scattering of a scalar wave by a semi-infinite rod of circular cross section

Abstract: The form of the exact solution for the scattering of a plane harmonic scalar wave by a semi-infinite circular cylindrical rod of diameter 2a is found when the boundary condition is u=0 or $\partial $u/$\partial u $ = 0, where u represents the scalar field and $ u $ is the normal to the rod. When the angle of incidence is $\pi $, i.e. the angle between the direction of propagation of the incident wave and the normal (out of the rod) to the end is $\pi $, the average pressure amplitude on the end of the rod and the scattering coefficient are found for the boundary condition $\partial $u/$\partial u $ = 0. Graphs are given showing the behaviour of these quantities for the range 0 $\leq $ ka $\leq $ 10, where k is the wave-number. When ka reaches 10, the quantities have almost become constant. For small values of ka the scattering coefficient is shown to be $\frac{1}{4}$(ka)$^{2}$; it appears from the numerical results that this is, in fact, a fairly close approximation for ka<2. It is further shown that the average pressure amplitude on the end for other angles of incidence is approximately the product of the average pressure amplitude for an angle of incidence of $\pi $ and the amplitude of the symmetric mode (ka < 3$\cdot $83) which the incident field would produce inside a hollow semi-infinite cylinder occupying the same position as the rod. When the boundary condition is u=0 and ka is small it is proved that the scattered field is the same as that due to a semi-infinite hollow cylinder longer by an amount 0$\cdot $1a approximately. A similar result does not hold for the boundary condition $\partial $u/$\partial u $ = 0. The theory is extended to the case when a pressure pulse falls on a circular rod. It is found that the pressure on the end drops almost to its final value in the time taken for a wave to travel the diameter of the rod, and that the average pressure during this process is given, at time t, by {0$\cdot $915 + 0$\cdot $745(2 - a$\_{0}$t/a)$^{2}$}$^{\frac{1}{2}}$ approximately, where a$\_{0}$ is the speed of sound. Tables, in the range 0(0$\cdot $25)10 of ka, of the 'split' functions which arise in connexion with a semi-infinite cylinder are given.

The de Haas-Van Alphen Effect. III. Experiments at Fields up to 32 kG

Abstract: The periodic field dependence of magnetic anisotropy (de Haas-van Alphen effect) has been studied for bismuth and zinc crystals by the torque method between about 1.5 and 32 kG at 4.19 degrees K and about 1.5 degrees K; in each case the orientation was chosen so that only a single fundamental periodicity was present. Particular attention was paid to the phase and harmonic content of the oscillations and to the form of the field dependence of amplitude. For bismuth good agreement was found with the theoretical formula except that the signs of the fundamental and the odd harmonics had to be reversed. For zinc the field dependence of amplitude at high fields was quite at variance with the theoretical formula and (probably in consequence) the harmonics were much weaker than in bismuth; the phases of the fundamental and the first harmonic were quite different from those of the theoretical formula. A slight field dependence was found for bismuth even at 85 degrees K, similar to the rather more marked one previously found in zinc. Oscillations of very much shorter period than the main ones were observed in zinc at the highest fields and their characteristics were briefly studied. Qualitative observations were made of the very short-period oscillations in a lead crystal, previously only observed by an impulsive method at much higher fields. No oscillations could be detected in copper, silver or gold crystals.

The Numerical Solution of the Differential Equations Governing the Reflexion of Long Radio Waves from the Ionosphere. II

Abstract: This paper presents a series of curves which are the results of some calculations of the reflecting properties of various models of the ionosphere for radio waves of frequency 16 kc/s. The method of calculation was described in a previous paper (Budden 1955). No attempt is made to deduce a model of the ionosphere capable of explaining all the observations, but the aim has been rather to establish some general principles which may indicate how future theoretical and experimental work should be planned. In most of the calculations it was assumed that the earth’s magnetic field is vertical and that the electron collision frequency in the ionosphere is constant. The limitations imposed by these restrictions are discussed. The first half of the paper describes some calculations for a model of the ionosphere in which the electron density increases exponentially with height, and the second half deals with a model having both D - and E -layers. The results in both cases are compared with observations.

Trapped Flux in Superconductors

Abstract: When a magnetic field is applied to a superconductor the normal state may be restored, and on removing the field the superconducting state is re-established, usually with a proportion of the field trapped in normal channels. The amount of flux trapped has been studied systematically as a function of temperature in rods of pure tin and of tin alloyed with indium up to 3%. In order to obtain significant results the specimens must be single crystals, homogenized by prolonged annealing, and having well-polished surfaces. The proportion of flux trapped is very small ($\sim $0$\cdot $1%) in pure tin, increasing steadily as the indium concentration is increased. For indium concentrations less than about 2$\cdot $3% the proportion trapped tends to zero as the temperature tends to the transition temperature. For greater indium concentrations there is a sharp rise in trapping to a very high value ($\sim $50%) at the transition temperature. The trapped flux is rather firmly bound. In order to account for these results a model of the superconducting state is developed, based on the theories of London & London and of Gorter & Casimir, and incorporating the idea of coherence. Typical processes such as spontaneous nucleation of the superconducting phase are analyzed and used to discuss the factors influencing the coalescence of adjacent superconducting domains, which is an essential part of the trapping mechanism. It is concluded that for not too great indium concentrations coalescence is achieved only through the presence of flaws, and that the sudden change in behaviour at 2$\cdot $3% indium marks the beginning of spontaneous coalescence. The model appears to be capable of accounting qualitatively for most of the details of the observed behaviour.

The Moments of the Number of Points of a Lattice in a Bounded Set

Abstract: A certain average value of the k th power of the number of points of a lattice, with determinant 1, in a bounded n -dimensional set S is evaluated in terms of the volumes of certain sets of dimensions n, 2n, ..., kn associated with S , provided n is sufficiently large in comparison to k . The result is used to obtain a slight improvement of the Minkowski—Hlawka theorem.

An electrically calibrated bomb calorimeter

Abstract: After a review of the physical foundations of bomb calorimetry, a bomb calorimeter is described which can be heated electrically or by combustion. The evaporation of water is prevented by sealing the vessel completely. Other modifications in the normal technique are introduced after consideration of (a) the difference in temperature between the outer surface of the calorimeter and the thermometer immersed in it, (b) the effective 'boundary' of the calorimeter in electrical and combustion experiments, and (c) variation with temperature of the constants of the apparatus. The heat of combustion of samples from a batch of thermochemical quality benzoic acid is found to be 26436 J/g under 'standard bomb conditions'. The standard error of the determination, a combination of errors in reproducibility of electrical and combustion experiments and in measurement of the corrections, is estimated to be 2$\cdot $2 J/g.

A New High Precision Calorimeter for the Measurement of Heats of Combustion: And the Heat of Combustion of Succinic Acid

Abstract: The construction, calibration and use of a high-precision bomb calorimeter of the aneroid type are described. It consists of a stainless-steel combustion bomb as designed by Carlton-Sutton (1933) in a mantle of aluminium alloy, the temperature of which is measured by a platinum-resistance thermometer. A simple means of measuring the electrical ignition energy has been devised. Because the time required for the attainment of a thermal steady state is greater for this type than for the conventional water calorimeter, the theory underlying methods of allowing for heat loss from the calorimeter to its surroundings has been critically re-examined, and the limitations of the commonly used Dickinson method (1914) have been clearly stated. For high precision to be attained, it proves to be particularly necessary that the conditions under which calibrations and measurements are performed shall be as similar as possible. Working under optimum conditions, chosen after systematic trials, it is possible to reduce to 0.012% the standard deviation of the values for the calibration constant, with benzoic acid as the standard substance. The heat of combustion of succinic acid has been redetermined as -$\Delta $U$_{R}$ = 3020$\cdot $57 $\pm $ 0$\cdot $43 cal/g (vac.). This agrees very well with the high-precision determination by Huffmann (1938) 3020$\cdot $47 $\pm $ 0$\cdot $43. Succinic acid is suitable as a secondary standard for combustion calorimetry; and the best value is taken to be 3020$\cdot $5 $\pm $ 0$\cdot $5 cal/g (vac.).

An Assessment of the System of Optimum Coding Used on the Pilot Automatic Computing Engine at the National Physical Laboratory

Abstract: The paper describes the system of optimum coding which is used on the Pilot ACE, the electronic computer at the National Physical Laboratory. It includes a number of simple examples of programs prepared for the machine and gives an assessment of the gain in speed which results from the use of optimum coding in general. It concludes with a description of the design of the full-scale ACE which takes full advantage of the general principles embodied in the design of the Pilot ACE.