Showing papers in "Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences in 1983"

Atmospheric angular momentum fluctuations, length-of-day changes and polar motion

Abstract: Variations in the distribution of mass within the atmosphere and changes in the pattern of winds, particularly the strength and location of the major mid-latitude jet-streams, produce fluctuations in all three components of the angular momentum of the atmosphere on timescales upwards of a few days. In a previous study (Hide et al . 1980) it has been shown that variations in the axial component of atmospheric angular momentum during the Special Observing Periods in 1979 of the First GARP Global Experiment (FGGE, where GARP is the Global Atmospheric Research Program) are well correlated with changes in length-of-day. This would be expected if the total angular momentum of the atmosphere and ‘solid’ Earth were conserved on short timescales (allowing for lunar and solar effects) but not if angular momentum transfer between the Earth’s liquid core and solid mantle, which is accepted to be substantial and even dominant on timescales upwards of several years, were significant on timescales of weeks or months. Fluctuations in the equatorial components of atmospheric angular momentum should contribute to the observed wobble of the instantaneous pole of the Earth’s rotation with respect to the Earth’s crust, but this has not been shown conclusively by previous studies. In this paper we re-examine some aspects of the underlying theory of non-rigid body rotational dynamics and angular momentum exchange between the atmosphere and solid Earth. Since only viscous or topographic coupling between the atmosphere and solid Earth can transfer angular momentum, no atmospheric flow that everywhere satisfied inviscid equations (including, but not solely, geostrophic flow) could affect the rotation of a spherical solid Earth. Currently available meteorological data are not adequate for evaluating the usual wobble excitation functions accurately, but we show that partial integration leads to an expression involving simpler functions ─ here called ‘equatorial angular momentum functions’ ─ which can be reliably evaluated from available meteorological data. The length-of-day problem is treated in terms of a similar ‘axial angular momentum function’ ; and ‘effective angular momentum functions’ are defined in order to allow for rotational and surface loading deformation of the Earth. Daily values of these atmospheric angular momentum functions have been calculated from the ‘initialized analysis global database’ of the European Centre for Medium-Range Weather Forecasts (ECMWF). They are presented for the period 1 January 1981─30 April 1982, along with the corresponding astronomically observed changes in length-of-day and polar motion, published by the Bureau International de l’Heure (BIH). Changes in length-of-day during this period can be accounted for almost entirely by angular momentum exchange between the atmosphere and solid Earth, and the existence of a persistent fluctuation in this exchange, with a timescale of about 7 weeks, is confirmed. We also demonstrate that meteorological phenomena provide an important contribution to the excitation of polar motion. Our work offers a theoretical basis for future routine determinations of atmospheric angular momentum fluctuations for the purposes of meteorological and geophysical research, including the assessment of the extent to which movements in the solid Earth associated with very large earthquakes contribute to the excitation of the Chandlerian wobble.

On the Joint Distribution of Wave Periods and Amplitudes in a Random Wave Field

Abstract: A theoretical probability density is derived for the joint distribution of wave periods and amplitudes which has the following properties: (1) the distribution is asymmetric, in accordance with observation; (2) it depends only on three lowest moments m 0 , m 1 , m 2 of the spectral density function. It is therefore independent of the fourth moment m 4 , which previously was used to define the spectral width (Cavanie et al . 1976). In the present model the width is defined by the lower-order parameter v = ( m 0 m 2 / m 2 1 - 1) ½ . The distribution agrees quite well with wave data taken in the North Atlantic (Chakrabarti & Cooley 1977) and with other data from the Sea of Japan (Goda 1978). Among the features predicted is that the total distri­bution of wave heights is slightly non-Rayleigh, and that the interquartile range of the conditional wave period distribution tends to zero as the wave amplitude diminishes. The analytic expressions are simpler than those derived previously, and may be useful in handling real statistical data.

A cluster approach to the structure of imperfect materials and their relaxation spectroscopy

Abstract: A new theory is proposed for the explanation of observed relaxation phenomena, which differs significantly from theories suggested by the authors before. The theory is based on a model of structural organization of macroscopically sized samples of imperfectly structured materials, both solids and liquids, and is intermediate in character. In terms of the model, a microscopic structure is maintained over a cluster containing a number of microscopic units, with an array of clusters described by a steady-state distribution completing the macroscopic picture. The structural regularity of each level of morphological organization is precisely defined by a coarse-grained index, which is given a thermodynamic interpretation in terms of binding energies and configurational entropy. The limiting cases of an ideal liquid and a perfect crystal are recovered as asymptotic extremes in terms of this definition. The consequences of this model for the relaxation dynamics of the structure are examined and it is shown that prepared fluctuations decay in a time-power law manner as coupled zero-point motions evolve either within clusters or between clusters, with a power determined by the relevant regularity index. As a result, the origin of power law noise in materials is explained in terms of configurational entropy, and its relation with gaussian and white noise, which appear as asymptotic limits, outlined. The shape of the steady-state distribution of the array of clusters is also determined without any a priori assumptions, and it is shown to range from an unbounded form to a δ function as the regularity of the array superstructure increases. Experimental examples of dielectric relaxation spectroscopy have been used to illustrate these structural concepts and outline the way in which this technique can be used to deduce the structural organization of the sample. Finally, a short description is given of some commonly observed forms of response and their structural interpretation.

Lines of circular polarization in electromagnetic wave fields

Abstract: A monochromatic fully polarized electromagnetic wave propagating in free space possesses, in general, two families of singular lines ( C lines) on which the transverse field is circularly polarized. The distribution of polarization ellipses around a C line shows that it obeys the same classifi­cation scheme as the isotropic points of a two-dimensional symmetric tensor: that is, a given section of a C line may belong to one of three different line patterns and it may be elliptic or hyperbolic. In addition it may be left-or right-handed. However, the way in which the polarization ellipses are executed in time shows that C lines may also be regarded as singularities of phase, analogous to line dislocations or interference fringes in scalar waves. From this point of view, a given line is of edge or screw type, according to its orientation, or, more generally, is a curved line of mixed edge-screw type. The whole field is divided into regions of opposite hand by surfaces S of linear polarization, and each family of C lines is confined to just one of these regions.

The effective conductivity of a periodic array of spheres

Abstract: We consider the problem of determining the effective conductivity k$^*$ of a composite material consisting of equal-sized spheres of conductivity $\alpha$ arranged in a cubic array within a homogeneous matrix of unit conductivity. We modify Zuzovski & Brenner9s (1977) method and thereby obtain a set of infinite linear equations for the coefficients of the formal solution equivalent to that derived by McKenzie et al. (1978) using a method originally devised by Rayleigh (1892). On solving these equations we derive expressions for $k^{\ast}$ to $O(c^9) - c$ being the volume fraction of the spheres - for simple, body-centred and face-centred cubic arrays, and also obtain numerical values for k$^*$ over the whole range of $\alpha$ and $c$. We show that these results for cubic arrays can be used to estimate $^{\ast}$ for random arrays of identical spheres. For arrays of highly conducting and nearly touching spheres, Batchelor & O9Brien (1977) showed that \begin{equation*} k^{\ast} \sim \begin{cases} -K_1\ln (1-\chi)-K_2 (\alpha = \infty, \chi=(c/c_{\max})^{\frac{1}{3}}\rightarrow 1), \\ 2K_1\ln \alpha-K9_2\quad (\chi = 1, \alpha \rightarrow \infty),\end{cases} \end{equation*} where $c_{\max}$ corresponds to the volume fraction when the spheres are actually touching each other, and determined $K_1$ for the three cubic arrays. Our numerical results are consistent with the above asymptotic expressions except for the fact that the numerical values for the constants $K_2$ and $K9_2$ thereby obtained do not quite satisfy the relation $K9_2 = K_1(3.9 - \ln 2) + K_2$ given by Batchelor & O9Brien. We have been unable to find the reason for this slight discrepancy.

The Nature of Machining Damage in Brittle Materials

Abstract: The micromechanics of failure emanating from machining-induced cracks in brittle materials is investigated. In situ monitoring of crack response during breaking tests (with use of acoustic wave scattering), strength measurements and post-failure fractography all indicate that the crack response is dominated by residual stresses. Two components of residual stress have been identified: a crack-wedging force due to the plastic zone beneath the strength-controlling machining groove, and a compressive surface layer due to adjacent grooves. The wedging force dominates and causes stable equilibrium crack extension during a breaking test. The implications of the results for non-destructive evaluation of surface damage by acoustic wave scattering is discussed.

Polarization Effects in the Diffraction of Electromagnetic Waves: The Role of Disclinations

Abstract: Three-dimensional diffraction patterns of monochromatic electromagnetic waves contain moving line singularities where the magnitude of the transverse field is zero, and its direction is therefore indeterminate. They are called disclinations, by analogy with the corresponding linear features in liquid crystals. A disclination in a vector wave is a natural generalization of a dislocation in a scalar wave. Where the scalar wave approximation in optics predicts a dislocation, or interference null, the full vector theory reveals a double helix structure: a disclination line in the electric field and another in the magnetic field winding around each other with a spacing of order (wavelength/2$\pi$). A perturbing plane wave causes this composite structure itself to become coiled. When there is a 'polarization effect' in the diffraction pattern a disclination in the electric field becomes a moving helix or, more generally, a coiled coil. As it moves it sweeps out a surface on which the polarization is everywhere linear. In optical experiments this observable surface is the most significant effect of disclinations. In general, however, the disclinations constitute elements of structure of the electromagnetic field, and their arrangement thus provides an effective way of describing the three-dimensional geometry of even very complicated diffraction fields, for example of microwaves.

Wave set-up, percolation and undertow in the surf zone

Abstract: Waves approaching a sloping beach induce a tilt in the mean water level within the surf zone. The existence of this ‘set-up’ is here demonstrated by observing the mean flow in a straight tube laid parallel to the incoming waves; also by showing that the waves induce a siphon in a U-tube laid on the sloping bottom. It is argued theoretically, and confirmed by experiment, that the set-up should help to drive an offshore bottom current (the undertow) between the shoreline and the breaker line. Seawards from the breaker line the bottom current is reversed. The consequent convergence of the bottom currents may contribute to building up the ‘breaker bar’. Further experiments show that the mean onshore pressure gradient drives a circulation of water within a porous beach. The associated pattern of streamlines also extends into the land, inshore from the run-up line. Theoretically, the injection of dye at the sediment-water interface might be used to probe the porosity of the beach material.

Diffusion-limited reaction rate theory for two-dimensional systems

Abstract: The Noyes and Smoluchowski diffusion-limited reaction rate theories are proved to be equivalent on a lattice. The Noyes theory is analysed and used to predict the kinetics of the two-dimensional irreversible reaction A + B $\rightarrow$ P. Only condensed phase reactions with molecules of A and B undergoing Brownian motion (diffusion) are discussed. For comparison, all calculations are done in both two and three dimensions. The two-dimensional rate function k$\_N$(t) in the equation $d\[A]/dt = d[B]/dt = -k\_N(t)[A\]\[B\]$ asymptotically goes to zero as ($\ln$ t)$^{-1}$ as t increases; the asymptotic expansion of k$\_N$(t) is derived from the expansion for the first-return probability in a random walk on a square lattice. The theoretical rate function is determined as a function of the probability $\alpha$ of reaction given an encounter. Although k$\_N$(t) is not significantly different from 'empirical' rate functions in a Monte Carlo simulation of a two-dimensional chemical reaction, it does differ from the rate function in a two-dimensional fluorescence quenching experiment.

Simulation of Electrostatic Systems in Periodic Boundary Conditions. III. Further Theory and Applications

Abstract: This paper makes some developments and clarifications of the theory for the application of periodic boundary conditions to the numerical simulation of the statistical mechanics of a cubic sample of dipolar particles. The reaction-field effect is treated rigorously. The anisotropies inherent in the periodic boundary condition Hamiltonian are allowed for in the derivation of a new fluctuation formula. A perturbation theory to account for anisotropic long-ranged terms is described, giving two di­-electric constant estimates from one simulation. These new results are illustrated with Monte Carlo simulations of the Stockmayer system at reduced density 0.8, reduced square dipole moment 2.0 and scaled temperature 1.35, giving a dielectric constant estimate of 25 ± 2 from all the data, and showing that the perturbation theories are very accurate. It appears possible to claim that periodic boundary conditions should be used with infinite external dielectric constant in almost all circumstances, because they then give a chain of configurations that provide compar­atively very stable estimates of dielectric constant.

Solution to the equations of parallel transport in Kerr geometry; tidal tensor

Abstract: It is shown how the special separability properties of the Kerr solution can be used to obtain an explicit analytic solution to the problem of constructing an orthonormal tetrad that is parallel-propagated along an arbitrary time-like geodesic. The components of the tidal tensor are calculated explicitly in terms of this tetrad. The special case of geodesics lying in the equatorial plane is examined.

The structure of an aqueous solution of nickel chloride

Abstract: The method of multipattern analysis based on neutron diffraction from isotopically different samples has been applied to a 4.35 molal solution of NiCl 2 in D 2 O. Seven distinct scattering patterns were obtained at a temperature of 21 °C and the particle-particle pair functions for correlations between Cl - and D 2 O, Ni 2+ and D 2 O, Cl - and Cl - , Ni 2+ and Ni 2+ and Ni 2+ and Cl - were extracted from the data. The results are used to obtain a real-space description of the ions and their immediate environment.

Soliton solutions of the Korteweg de Vries and the Kadomtsev-Petviashvili equations: the Wronskian technique

Abstract: The well known soliton solutions of the Kadomtsev-Petviashvili equations are written in terms of determinants of Wronskian form. By using this compact representation together with the Hirota bilinear form of the equations, it is demonstrated by elementary algebraic methods that the N -soliton solution satisfies the evolution equation and the N and N + 1-soliton solutions satisfy the associated Backlund transformation. The relation of these results to the eigensolutions of the inverse scattering method and to the more usual representation of the N -soliton solution is also given.

Vibration control of multi-mode rotor-bearing systems

Abstract: This paper presents a computationally fast and efficient least-squares method to minimize the vibration of any general rotor-bearing system by the application of external control forces. The D-optimality concept is used to optimize the force locations. The proposed method provides a wide range of statistical information, and the sensitivity of the optimum response to changes in the control forces. Magnetic bearings can be applied to implement the open-loop adaptive vibration control strategies outlined in the paper. These components can also be used to inject a multi-frequency test signal as required for identi­fication studies.

Flame Propagation Through Dust Clouds of Carbon, Coal, Aluminium and Magnesium in an Environment of Zero Gravity

Abstract: The mechanism and the rate of flame propagation through dust clouds of carbon, coal, aluminium and magnesium have been investigated. Any errors due to the upward buoyant motion of burnt gases and the downward settling velocity of dust particles were eliminated by conducting these experiments in a zero-gravity environment. A technique of flat-flame propagation was developed to measure the burning velocity accurately. The results show that the burning velocity is influenced by particle size, fuel transfer number, dust concentration, volatile matter (for coal), oxygen enrichment and heat loss by radiation from the burning fuel particles. A simple model to elucidate the structure and the mechanism of flame propagation is developed. Then expressions to predict the flame thickness and the burning velocity are derived. Attention is drawn to the similarity that exists between the mechanisms of flame propagation through dust clouds and through fuel mists. The importance of radiation heat loss is emphasized. It is shown that for a dust cloud of particle size 10 μ m of graphite or aluminium in oxygen, radiation loss from particles can reduce the burning velocity by as much as 40% or 25% respectively.

Complex Dynamics of Compliant Off-Shore Structures

Abstract: The dynamics of advanced compliant off-shore structures can be extremely complex owing to the inherent nonlinearities. Subharmonic resonances can coexist with stable small-amplitude solutions, the response observed depending solely on the starting conditions of the motion. So care must be taken in digital, analogue and model studies to explore a comprehensive set of initial conditions. 'Efficient' automated digital computations could miss an entire subharmonic peak by locking onto a coexisting small-deflexion fundamental solution. Chaotic, non-periodic motions of strange attractors can also arise in well defined deterministic resonance problems. The waveforms of these look like the result of a stochastic process, and because of their extreme sensitivity to initial conditions a statistical description must be sought. Genuinely chaotic solutions can be identified by looking for nearby period-doubling bifurcations, and for the exponential divergence of adjacent starts leading to a loss of correlation. The period-doubling cascades give rise to subharmonics of arbitrarily high order close to the chaotic regimes, so the duration of digital, analogue and laboratory experiments must be long and chosen with care. The resonance of simple bilinear and impact oscillators is used as a vehicle to illustrate these general ideas.

The elastohydrodynamic lubrication of piston rings

Abstract: The piston seal that separates the hostile environment of the combustion chamber from the crankcase that contains the lubricant is an essential machine element in reciprocating engines. The sealing force pressing the piston rings against the cylinder liner varies with the combustion chamber pressure to form an effective self-adjusting mechanism. The conjunctions between piston rings and cylinder liners are thus subjected to cyclic variations of load, entraining velocity and effective lubricant temperature as the piston reciprocates within the cylinder. Recent theoretical and experimental studies have confirmed that piston rings enjoy hydrodynamic lubrication throughout most of the engine cycle, but that a transition to mixed or boundary lubrication can be expected near top dead centre. The purpose of the present paper is to examine the suggestion that elastohydrodynamic lubrication might contribute to the tribological performance of the piston seal, particularly near top dead centre. The mode of lubrication in eight four-stroke and six two-stroke diesel engines is assessed in terms of the dimensionless viscosity and elasticity parameters proposed by Johnson (1970), and the associated map of lubrication regimes. The survey indicates unequivocally that elastohydrodynamic action can be expected during part of the stroke in all the engines considered. In the second part of the paper a detailed examination of the influence of elastohydrodynamic action in one particular engine is presented to confirm the general findings recorded in the study of lubrication regimes. Current analysis of the lubrication of rigid piston rings already takes account of the variation of surface temperature along the cylinder liner and its influence upon lubricant viscosity. It is shown that, when the enhancing influence of pressure upon viscosity is added to the analysis of rigid piston rings, the predicted cyclic minimum film thickness is more than doubled. Full elastohydrodynamic action, involving both local distortion of the elastic solids and the influence of pressure upon viscosity, results in a fourfold increase in film thickness. It is further shown that it is necessary to take account of the variation of squeeze-film velocity throughout the lubricated conjunction at each crank angle if reliable predictions of film shape and thickness are to be achieved. It is thus concluded that the wave of elastic deformation, which ripples up and down the cylinder liners many times each second in diesel engines, together with the associated local elastic deformations on the piston rings themselves, combine with the influence of pressure upon lubricant viscosity to enhance the minimum oil film thickness in the piston seal by elastohydrodynamic action.

A model of adsorption-desorption hysteresis in which hysteresis is primarily developed by the interconnections in a network of pores

Abstract: A porous material is described in terms of a model of its structure. The pores have a single internal cavity and four constrictions connecting this cavity to adjacent ones. Full and empty pores have various configurations of full and empty neighbours and these different configurations are used in the analysis of the behaviour of the material when it is being filled by adsorption and emptied by desorption. Their role is similar to that of intermediates in a chain of chemical reactions. The model for adsorption predicts a simple relation between adsorption scanning curves inside the hysteresis envelope irrespective of the number of interconnections of the pores. The desorption model gives results similar to published experiments.

Acoustic wave radiated by head-on collision of two vortex rings

Abstract: An attempt has been made to detect the acoustic wave radiated by the collision of two vortex rings and to compare it with predictions made by the theory of aerodynamic sound with and without fluid viscosity. The vortex rings generated in the experiment had translational speed 40 ≾ U ≾ 80 m/s and radius 5 ≾ R ≾ 9 mm, and their motion was studied optically by means of a photo-sensor system. The pressure signals of the emitted wave were measured by a microphone and analysed by digital methods. It has been found that the detected acoustic pressure p obeys the scaling law p ∝ ( ρ 0 / c 2 ) U 4 R / r predicted by the theory ( ρ 0 and c are the density and sound speed respectively in the undisturbed state, r is the distance of the observation point from the collision centre), and p possesses a typical peak value p * ≈ 5 Pa for U = 76 m/s and r/R = 93, and that the total acoustic energy of radiation is about 0.4 ( U/c ) 5 times the initial kinetic energy of the vortex system. Analysis of the observed signals shows that the acoustic pressure is com­posed of quadrupole and monopole radiation. Although the inviscid theory can account for the initial part of the detected signal quantita­tively, it fails to predict the appearance of the monopole, and it is also not possible to give even a qualitative description of the quadrupole in the later stages. A viscous theory is presented to obtain an improved fit to the observed profile. In the latter theory the monopole radiation is related to the viscous decay of total kinetic energy, and the main quadruple is affected by the decay of the vortex strength as a result of pair annihilation under the action of viscosity. Thus resonable agree­ment has been obtained between the detected and predicted waves. This has made it clear that the initial inviscid phase is followed by phases dominated by the action of viscosity.

Lower-tail approximations for the probability of failure of three-dimensional fibrous composites with hexagonal geometry

Abstract: The chain-of-bundles model for the strength of unidirectional fibrous materials is extended to cover 3D situations wherein the parallel filaments are arranged laterally in a hexagonal array. Within each bundle, the strengths of the fibres vary statistically and share load according to a local load-sharing rule, a rule which describes how the loads of failed fibres are redistributed on to nearby surviving fibres. We consider two idealized versions of this rule, one geometrically motivated and the other more mechanically motivated. We extend earlier asymptotic techniques for the 2D planar problem to the present 3D case, and obtain various approximations for the probability distribution for material strength. The Weibull distribution again emerges as central to the results, but the calculation of its shape and scale parameters is greatly complicated by the large number of new failure configurations that may arise in the hexagonal array of fibres. Earlier 2D results connecting the Weibull shape parameter to the critical failure sequence size do not in general hold in 3D settings. The general character of the results, however, is the same as in the 2D setting, with 3D materials being stronger because of the reduced severity of the fibre overloads in the hexagonal array. Also, the two local load-sharing rules though quite different in character yield surprisingly similar numerical results.

On a conjecture related to the number of solutions of a nonlinear Dirichlet problem

Abstract: In an earlier paper (1981), the present authors made a conjecture about the number of solutions of a semilinear elliptic boundary value problem which has been investigated extensively in the past decade. The conjecture is proved in the one-dimensional case.

Derivation of the Power-Zienau-Woolley Hamiltonian in Quantum Electrodynamics by Gauge Transformation

Abstract: The different forms of Hamiltonian for the coupled system consisting of the electromagnetic field and a non-relativistic charged particle are considered in the context of gauge-transformation theory. The conventional Lagrangian of the system in an arbitrary gauge is converted to a new form by transformation to another arbitrary gauge, and a new formulation of the theory is obtained by expressing the new Lagrangian in terms of the initial potentials. Thus different gauge transformations produce different momenta $\pi$ conjugate to the initial vector potential $\mathbf{A}$, and hence different forms of Hamiltonian. The transformations that produce the Coulomb-gauge and Power-Zienau-Woolley (P.Z.W.) Hamiltonians are considered in detail. It is shown that $\Pi$ is transverse in both cases and only the transverse part of $\mathbf{A}$ is accordingly involved in the field quantization; neither the longitudinal part of A nor the scalar potential appears explicitly, the instantaneous Coulomb energies being included via an electronic polarization determined by the gauge generator. The transformations between gauges are illustrated by simple diagrammatic representations of $\mathbf{A}$ and $\Pi$. Compararison with the commonly used unitary transformation derivation of the P.Z.W. Hamiltonian emphasizes the need for a careful reinterpretation of the physical significance of $\Pi$ after the unitary transformation has been made.

Diffraction of Elastic Waves by a Penny-Shaped Crack: Analytical and Numerical Results

Abstract: The diffraction of time-harmonic stress waves by a penny-shaped crack in an infinite elastic solid is an important problem in fracture mechanics and in the theory of the ultrasonic inspection of materials Martin (Proc R Soc Lond A 378, 263 (1981)) has proved that the corresponding linear boundary-value problem has precisely one solution, and that this solution can be constructed by solving a two-dimensional Fredholm integral equation of the second kind However, this integral equation has a complicated matrix kernel and the components of its vector solution are coupled The main purpose of the present paper is to show how Martin's integral equation can be explicitly solved in terms of a sequence of functions, each of which satisfies a very simple scalar integral equation of the second kind; this simplification may be made for any incident wave For an incident plane wave, further simplifications are possible We show that the solution at an arbitrary angle of incidence can be derived from the solution at a particular angle of incidence, namely grazing incidence The resulting computational procedure is especially attractive if only the stressintensity factors or the far-field displacements are required Finally, we present some numerical results for the scattering of a P-wave at normal incidence and an SV-wave at oblique incidence, and compare these with those of other authors

Characteristic Initial Data and Wavefront Singularities in General Relativity

Abstract: The structure of singularities (caustics), self-intersections of wavefronts (null hypersurfaces) and wavefront families (null coordinates) in arbitrary space-times is discussed in detail and illustrated by explicit examples of stable wavefront singularities in Minkowski space. It is shown how characteristic initial data determine the caustics and the self-intersections of the characteristics of Einstein’s field equations.

The prolongation structures of quasi-polynomial flows

Abstract: We look closely at the process of finding a Wahlquist-Estabrook prolongation structure for a given (system of) nonlinear evolution equation(s). There are two main steps in this calculation: the first, to reduce the problem to the investigation of a finitely generated, free Lie algebra with constraints; the second, to find a finite-dimensional linear representation of these generators. We discuss some of the difficulties that arise in this calculation. For quasi-polynomial flows (defined later) we give an algorithm for the first step. We do not totally solve the problems of the second step, but do give an algebraic framework and a number of techniques that are quite generally applicable. We illustrate these methods with many examples, several of which are new.

Magnetic hysteresis and minor loops: models and experiments

Abstract: The features of minor loop behaviour in magnetic hysteresis are examined by comparing some experimental data on a Mn-Al-Ge film with two phenomenological models. The 'independent particle' model of Preisach and Everett proves to describe qualitatively all the observed behaviour. This model is generalized slightly to allow for the interaction of the 'particle' through an average demagnetizing field, and a convenient method for calculating the consequences of the model is described.

The generation of possible crystal structures of primary amides

Abstract: Procedures are outlined for generation of crystal structures of primary amide molecules by constructing the possible ways in which the molecules may pack. For each given one- or two-dimensional hydrogen-bonded array, ensembles of three-dimensional crystal structures are generated by considering the possible ways in which the arrays may be juxtaposed. Observed and generated hypothetical molecular arrangements are analysed to highlight both favourable and unfavourable features, par­ticularly in terms of close packing principles, the size and shape of the molecule, van der Waals and Coulomb interactions and N-H ∙ ∙ ∙ O bonding geometry.

On the Jones birefringence

Abstract: The possibility that a physical system may possess simultaneously two independent linear birefringences is a consequence of the Jones calculus formulated in 1948. Before this only one linear birefringence was known. The new one is characterized by a pair of orthogonal fast and slow axes that bisect those of the traditional linear birefringence. Apparently over­looked since it was initially postulated, the Jones birefringence is the subject of the present investigation. Pictorial symmetry arguments and a Mueller matrix method are used to identify systems in which both linear birefringences may exist. They are found to occur naturally in certain magnetic and non-magnetic crystals, for which the relevant uniaxial classes are listed. In addition, it is shown that the Jones birefringence may be induced in a fluid by the application of uniform electric and magnetic fields E and B , parallel to each other and transverse to the light path, the dependence being linear in EB . A quantitative theory relates both birefringences to physical property tensors and confirms the symmetry predictions. From the theory an assessment is made of the magnitude of the effect in a crystal and a gas, and although very much smaller than known linear birefringences, it should be capable of measurement.

Optical Microscopy with Extended Depth of Field

Abstract: A method is described that allows the depth of field in optical microscopy to be extended, in principle without limit, while high-resolution, diffraction-limited imaging is retained. Experimentally, an extension of more than two orders of magnitude has been achieved. The images are found to be of high optical quality, and this is borne out by theoretical predictions. The resolution is shown theoretically to be similar to that in a conventional microscope of the same numerical aperture, and for some criteria an improvement is obtained.

Monocular depth perception from optical flow by space time signal processing

Abstract: A theory of monocular depth determination is presented. The effect of finite temporal resolution is incorporated by generalizing the MarrHildreth edge detected operator - V2G(r) where V2 is the Laplacian and G(r) is a two-dimensional Gaussian. The constraint that the edge detection operator in space-time should produce zero-crossings at the same place in different channels, i.e. at different resolutions of the Gaussian, led to the conclusion that the Marr-Hildreth operator should be replaced by - LI2G(r, t) where E2 is the d'Alembertian V2- (1/u2)(a2/at2) and G(r, t) is a Gaussian in space-time. To ensure that the locations of the zerocrossings are independent of the channel width, G(r, t) has to be isotropic in the sense that the velocity u appearing in the defintion of the d'Alembertian must also be used to relate the scales of length and time in G. However, the new operatior - LI2G(r, t) produces two types of zero-crossing for each isolated edge feature in the image I(r, t). One of these, termed the 'static edge', corresponds to the position of the image edge at time t as defined by V2I(r, t) = 0; the other, called a 'depth zero', depends only on the relative motion of the observer and object and is usually found only in the periphery of the field of view. When an edge feature is itself in the periphery of the visual field and these zeros coincide, there is an additional cross-over effect. It is shown how these zero-crossings may be used to infer the depth of an object when the observer and object are in relative motion. If an edge feature is near the centre of the image (i.e. near the focus of expansion), the spatial and temporal slopes of the zeros crossing at the static edge may be used to infer the depth, but, if the edge feature is in the periphery of the image, the cross-over effect enables the depth to be obtained immediately. While the former utilizes sharp spatial and temporal resolution to give detailed three-dimensional information, the cross-over effect relies on longer integration times to give a direct measure of the time-to-contact. We propose that both mechanisms could be used to extract depth information in computer vision systems and speculate on how our theory could be used to model depth perception in early visual processing in humans where there is evidence of both monocular perception of the environment in depth and of looming detection in the periphery of the field of view. In addition it is shown how a number of previous models are included in our theory, in particular the directional sensor proposed by Marr & Ullman and a method of depth determination proposed by Prazdny.