Showing papers in "Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences in 1977"

Level Clustering in the Regular Spectrum

Abstract: In the regular spectrum of an f -dimensional system each energy level can be labelled with f quantum numbers originating in f constants of the classical motion. Levels with very different quantum numbers can have similar energies. We study the classical limit of the distribution P(S) of spacings between adjacent levels, using a scaling transformation to remove the irrelevant effects of the varying local mean level density. For generic regular systems P(S) = e -s , characteristic of a Poisson process with levels distributed at random. But for systems of harmonic oscillators, which possess the non-generic property that the ‘energy contours’ in action space are flat, P(S) does not exist if the oscillator frequencies are commensurable, and is peaked about a non-zero value of S if the frequencies are incommensurable, indicating some regularity in the level distribution; the precise form of P(S) depends on the arithmetic nature of the irrational frequency ratios. Numerical experiments on simple two-dimensional systems support these theoretical conclusions.

Thermal or electrical conduction through a granular material

Abstract: The material under investigation consists of particles of relatively large conductivity embedded or immersed in a matrix, the volume fraction of the particles being so high that they are in, or nearly in, contact. The particles are arranged randomly, and the material is statistically homogeneous. A general formula gives the effective conductivity of the material in terms of the average thermal (or electrical) dipole strength of a particle. The thermal flux across the surface of a particle is concentrated in areas near points of contact with another particle, and the dipole strength is approximately equal to a weighted sum of the fluxes across the areas near contact points. It is thus necessary to calculate the flux between two adjoining particles at different temperatures, and we do this by solving numerically an integral equation for the distribution of temperature over the (locally spherical) surface of one of the particles near the contact point. The flux between the two particles is found to be proportional to loge ah when a2 2h/a ≫ 1 and to log e a when a 2h/a ≪ 1, where h is the minimum gap between the particle surfaces, a~ 1 the mean of their local curvatures, and a the ratio of the conductivities of the particles and the matrix. In the case of two particles pressed together to form a circular flat spot of radius p , the flux occurs almost wholly in the particle material, and is proportional to p when ap/a ≫ 1. Explicit approximate results are obtained for the effective conductivity of the granular material in the case of uniform spherical particles. For a close-packed bed of particles making point contact the effective conductivity is found to be 4.0 k log e a where k is the matrix conductivity. This asymptotic relation (applicable when a ≫ 1) is seen to be consistent with the available measurements of the conductivity of packed beds of spheres. Values of the effective conductivity for packed beds of particles of different shape are not expected to be greatly different.

Exponentially Decreasing Distributions for the Logarithm of Particle Size

Abstract: The family of continuous type distributions such that the logarithm of the probability (density) function is a hyperbola (or, in several dimensions, a hyperboloid) is introduced and investigated. It is, among other things, shown that a distribution of this kind is a mixture of normal distributions. As to applications, the paper focuses on the mass-size distribution of aeolian sand deposits, with particular reference to the findings of R. A. Bagnold. The distribution family seems, however, to be of some potential usefulness in other concrete contexts too.

Shear behaviour of elastohydrodynamic oil films

Abstract: A simple constitutive equation is proposed for the isothermal shear of lubricant films in rolling/sliding contacts. The model may be described as nonlinear Maxwell, since it comprises nonlinear viscous flow superimposed on linear elastic strain. The nonlinear viscous function can take any convenient form. It has been found that an Eyring 'sinh law' fits the measurements on five different fluids, although the higher viscosity fluids at high pressure are well described by the elastic/perfectly plastic equations of Prandtl-Reuss. The proposed equation covers the complete range of isothermal behaviour: linear and nonlinear viscous, linear viscoelastic, nonlinear viscoelastic and elastic/plastic under any strain history. Experiments in support of the equations are described. The nonlinear Maxwell constitutive equation is expressed in terms of three independent fluid parameters: the shear modulus $G$, the zero-rate viscosity $\eta $ and a reference stress $\tau _{0}$. The variations of these parameters with pressure and temperature, deduced from the experiments, are found to be in broad agreement with the Eyring theory of fluid flow.

The thermodynamic theory of black holes

Abstract: The thermodynamic theory underlying black hole processes is developed in detail and applied to model systems. I t is found that Kerr-Newman black holes undergo a phase transition at a = 0.68 M or Q = 0.86 M , where the heat capacity has an infinite discontinuity. Above the transition values the specific heat is positive, permitting isothermal equilibrium with a surrounding heat bath. Simple processes and stability criteria for various black hole situations are investigated. The limits for entropieally favoured black hole formation are found. The Nernst conditions for the third law of thermodynamics are not satisfied fully for black holes. There is no obvious thermodynamic reason why a black hole may not be cooled down below absolute zero and converted into a naked singularity. Quantum energy-momentum tensor calculations for uncharged black holes are extended to the Reissner-Nordstrom case, and found to be fully consistent with the thermodynamic picture for Q M . For Q > M the model predicts that ‘naked’ collapse also produces radiation, with such intensity that the collapsing matter is entirely evaporated away before a naked singularity can form.

Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions

Abstract: In this paper it is shown that the analysis of Titchmarsh's book [32] for regular Sturm-Liouville problems on a finite closed interval carries over readily to regular problems involving the eigenvalue parameter in the boundary condition at one end-point. The manner in which this type of problem is associated with a self-adjoint operator in Hilbert space has recently been pointed out by Walter in [36], and his operator-theoretic formulation is adopted here. The use of the eigenfunction expansion is illustrated by applying it to solve a heat-conduction problem for a solid in contact with a fluid.

Radiation from Moving Mirrors and from Black Holes

Abstract: We extend our previous work on scalar quantum particle production by moving mirrors in two-dimensional flat space-time to models with asymptotically null trajectories. This proves to have considerable heuristic value in understanding the mechanism of quantum particle emission from black holes. We demonstrate that Hawking’s derivation of that phenomenon using ray-tracing is mathematically identical to the geometrical optics associated with a certain class of mirror trajectory. Investigation of the simpler system clarifies the relation between particles and energy in quantum field theory. A mirror trajectory is presented by which a flux of particles is created, but no energy at all is radiated. We also show that the stimulated emission that occurs when a single particle is incident on the mirror simply corresponds to the classical reflexion of the associated wave, and that the total energy may decrease in this process.

Initial Movement of Grains on a Stream Bed: The Effect of Relative Protrusion

Abstract: Shields (1936) found that the dimensionless shear stress necessary to move a cohesionless grain on a stream bed depended only on the grain Reynolds number. He ignored the degree of exposure of individual grains as a separate parameter. This report describes experiments to measure the dimensionless threshold stress and its dependence on grain protrusion, which was found to be very marked. The threshold stress for grains resting on the top of an otherwise flat bed in a turbulent stream was measured and found to be 0.01 –considerably less than previously-reported values of 0.03–0.06 for beds where all grains were at the same level. It is suggested that the new lower value be used in all turbulent flow situations where the bed is of natural sediments or unlevelled material. An hypothesis is proposed that the conventional Shields diagram implicitly contains variation with protrusion between the two extremes of (i) large grains and large Reynolds numbers, with small relative protrusion, and (ii) small grains, low Reynolds numbers, and protrusion of almost a complete grain diameter. In view of this, the extent of the dip in the Shields plot is explicable in that it represents a transition between two different standards of levelling as well as the transition between laminar and turbulent flow past the grains, the range of which it overlaps considerably.

On the Dirichlet problem for weakly non-linear elliptic partial differential equations

Abstract: We study the existence of solutions of the Dirichlet problem for weakly nonlinear elliptic partial differential equations. We only consider cases where the nonlinearities do not depend on any partial derivatives. For these cases, we prove the existence of solutions for a wide variety of nonlinearities.

The Effect of Tangential Force on the Contact of Elastic Solids in Adhesion

Abstract: This paper describes a study of adhesion between elastic solids and in particular the effect of a tangential force upon the size of the contact area In the first part of the paper, the relation between the stress intensity factor of the normally loaded contact and the overall energy balance approach is discussed In the second and main part of the paper, an analysis is given for the influence of a tangential force on the adhesive contact The equation derived to describe its effect on the contact size has been verified by experiments carried out on rubber hemispheres pressed against a glass flat The experimental results show qualitatively a clear reduction in contact area when a tangential force acts and quantitatively a reasonable agreement with theory within the limits of experimental error

The Dry Deposition of Sulphur Dioxide to Land and Water Surfaces

Abstract: The direct absorption of sulphur dioxide has been measured in the field by the concentration gradient method and by using $^{35}$S as a tracer. The mean deposition velocities (the ratios of deposition flux to concentration) for grass, soil and water were close to 1 cm s$^{-1}$. No concentration gradient could be detected over coniferous forest indicating an upper limit of 2 cm s$^{-1}$ on the deposition velocity. Radioactive measurements suggest that the deposition velocity for a dry forest canopy varies from 0.1 to 0.6 cm s$^{-1}$. Laboratory experiments showed that the rate of deposition to soil increased with soil pH. Deposition to calcareous soil, water and short grass was largely controlled by the rate of turbulent mixing, while diffusion into the leaf or the rate of sorption controlled deposition to taller vegetation. The results, with those of other workers, indicate that the deposition velocity is not very dependent on land use, and support a mean value of 0.85 cm s$^{-1}$ for Britain. This figure can be used to show that, of the 6 Mt of sulphur dioxide emitted to the atmosphere above Great Britain annually, 1.64 Mt is removed by dry deposition within the country, while rain removes 0.4 Mt. The same mean deposition velocity is consistent with the atmospheric sulphur balance for Western Europe, where dry deposition and precipitation together limit the mean residence time in the atmosphere to about two days and dry deposition removes twice as much sulphur as does precipitation. A similar relation is expected to apply world wide but concentration measurements are not adequate to estimate the magnitude of the sulphur cycle.

On thermodynamics and the nature of the second law

Abstract: The contents of this paper represent a new approach to continuum thermo­dynamics and are chiefly concerned with ( a ) a procedure for obtaining restrictions on constitutive equations, ( b ) an appropriate mathematical statement of the second law and ( c ) the nature of restrictions placed by the latter on thermo-mechanical behaviour of single phase continua. Our point of departure is the introduction of a balance of entropy and the use of the energy equation as an identity for all motions and all temperature distributions after the elimination of the external fields. This is in contrast to the approach adopted in most of the current literature on continuum ther­modynamics based on the use of the Clausius-Duhem inequality. In order to gain some insight into the nature of our procedure we first study the case of an elastic material, which includes that of an ideal fluid as a special case, before the consideration of the second law. We then go on to postu­late an inequality which reflects the fact that for every process associated with a dissipative material, a part of the mechanical work is always con­verted into heat and this cannot be withdrawn from the medium as mech­anical work. The restriction on the heat conduction vector is considered separately and is confined to equilibrium cases in which heat flow is steady. A restriction is also obtained for the internal energy when the body is in mechanical equilibrium subjected to spatially homogeneous temperature fields. Using the above approach, next we study the nature of thermodynamic restrictions on the thermo-mechanical response of a viscous fluid and simple materials with fading memory. A drawback to the Clausius-Duhem inequality is discussed by means of an example. For a class of rigid heat conductors in thermal equilibrium, the Clausius-Duhem inequality requires that if heat is added to the medium, the resulting spatially homogeneous temperature of the conductor decreases . Moreover, the in­-equality denies the possibility of propagation of heat in the conductor as a thermal wave with finite speed. The inequalities proposed in this paper do not suffer from these shortcomings.

Polynomial conserved densities for the sine-Gordon equations

Abstract: Like a number of other nonlinear dispersive wave equations the sine–Gordonequation z , xt = sin z has both multi-soliton solutions and an infinity of conserved densities which are polynomials in z , x , z , xx , etc. We prove that the generalized sine–Gordon equation z , xt = F ( z ) has an infinity of such polynomial conserved densities if, and only if, F ( z ) = A e αz + B e – αz for complex valued A, B and α ≠ 0. If F ( z ) does not take the form A e αz + B e βz there is no p. c. d. of rank greater than two. If α ≠ – β there is only a finite number of p. c. ds. If α = – β then if A and B are non-zero all p. c. ds are of even rank; if either A or B vanishes the p. c. ds are of both even and odd ranks. We exhibit the first eleven p. c. ds in each case when α = – β and the first eight when α ≠ – β . Neither the odd rank p. c. ds in the case α = – β , nor the particular limited set of p. c. ds in the case when α ≠ – β have been reported before. We connect the existence of an infinity of p. c. ds with solutions of the equations through an inverse scattering method, with Backlund transformations and, via Noether’s theorem, with infinitesimal Backlund transformations. All equations with Backlund transformations have an infinity of p. c. ds but not all such p. c. ds can be generated from the Backlund transformations. We deduce that multiple sine–Gordon equations like z , xt = sin z + ½ sin ½ z , which have applications in the theory of short optical pulse propagation, do not have an infinity of p. c. ds. For these equations we find essentially three conservation laws: one and only one of these is a p. c. d. and this is of rank two. We conclude that the multiple sine–Gordons will not be soluble by present formulations of the inverse scattering method despite numerical solutions which show soliton like behaviour. Results and conclusions are wholly consistent with the theorem that the generalized sine–Gordon equation has auto-Backlund transformations if, and only if Ḟ ( z ) – α 2 F ( z ) = 0.

The interaction of entropy fluctuations with turbine blade rows; a mechanism of turbojet engine noise

Abstract: The theory relating to the interaction of entropy fluctuations ('hot spots'), as well as vorticity and pressure, with blade rows is described. A basic feature of the model is that the blade rows have blades of sufficiently short chord that this is negligible in comparison with the wavelength of the disturbances. For the interaction of entropy with a blade row to be important, it is essential that the steady pressure change across the blade row should be large, although all unsteady perturbations are assumed small. A number of idealized examples have been calculated, beginning with isolated blade rows, progressing to single and then to several turbine stages. Finally, the model has been used to predict the low-frequency rearward-radiated acoustic power from a commercial turbojet engine. Following several assumptions, together with considerable empirical data, the correct trend and level are predicted, suggesting the mechanism to be important at low jet velocities.

Higher Order Laue Zone Effects in Electron Diffraction and their Use in Lattice Parameter Determination

Abstract: By fitting small probe-forming lenses into a conventional electron microscope, we have been able to observe higher order Lane zone (h.o.l.z.) diffraction effects from high symmetry zone axes of a wide variety of materials. Cooling the specimen with liquid nitrogen both greatly reduces the contamination rate and increases the visibility of the h.o.l.z. lines. An interpretation of these lines is given in terms of the dispersion surface construction and conditions for the visibility of h.o.l.z. effects are deduced. A theory from which numerical solutions have been obtained is outlined. Using h.o.l.z. lines, we can deduce the microscope operating voltage or the lattice parameter of the specimen to approximately one part in a thousand; relative changes can be measured about five times more precisely. The spatial resolution of the technique is approximately 10 nm. Strain gradients within the illuminated area can produce fringe patterns.

Periodic Grain Boundary Structures in Aluminium. I. A Combined Experimental and Theoretical Investigation of Coincidence Grain Boundary Structure in Aluminium

Abstract: The results of a computer simulation of the structure of periodic grain boundaries between twin related crystals of aluminium are described. An interatomic potential derived on the basis of pseudo-potential theory was used. The algorithm employed allows simultaneous local atomic relaxation and rigid body translation of the adjacent grains. It was found that rigid body translation is a dominant contribution to relaxation, and that the energy of a boundary, $\gamma $, is not simply related to boundary periodicity. In addition, annealing twins in aluminium were observed by using transmission electron microscopy and detailed correspondence with theoretical predictions was found in two areas; the calculated $\gamma $'s of experimentally observed boundary planes were lower than those of geometrically possible alternatives, and excellent agreement between predicted and experimentally measured rigid body translations was obtained for two types of tilt boundary.

On the modulation of water waves in the neighbourhood of kh ≈ 1.363

Abstract: In 1967, T. Brooke Benjamin showed that periodic wave-trains on the surface of water could be unstable. If the undisturbed depth is $h$, and $k$ is the wavenumber of the fundamental, then the Stokes wave is unstable if $kh$ $\geq $ $\sigma \_{0}$, where $\sigma $$\_{0}$ $\approx $ 1.363. The instability is provided by the growth of waves with a wavenumber close to $k$. This result is associated with an almost resonant quartet wave interaction and can be obtained by examining the cubic nonlinearity in the nonlinear Schrodinger equation for the modulation of harmonic water waves: this term vanishes at $kh$ = $\sigma $$\_{0}$. In this paper the multiple-scales technique is adapted in order to derive the appropriate modulation equation for the amplitude of the fundamental when $kh$ is near to $\sigma $$\_{0}$. The resulting equation takes the form i$A\_{T}$ - $a\_{1}A\_{\zeta \zeta}$ - $a\_{2}A|A|^{2}$ + $a\_{3}A|A|^{4}$ + i($a\_{4}|A|^{2}A\_{\zeta}$ - $a\_{5}A(|A|^{2})\_{\zeta})$ - $a\_{6}A\psi \_{T}$ = 0, where $\psi \_{\zeta}$ = $|A|^{2}$, and the $a$$\_{i}$ are real numbers. [Coefficients $a\_{3}$-$a\_{6}$ are given on $kh$ $\approx $ 1.363 only.] This equation is uniformly valid in that it reduces to the classical non-linear Schrodinger equation in the appropriate limit and is correct when $a$$\_{2}$ = 0, i.e. at $kh$ = $\sigma $$_{0}$. The equation is used to examine the stability of the Stokes wave and the new inequality for stability is derived: this now depends on the wave amplitude. If the wave is unstable then it is expected that solitons will be produced: the simplest form of soliton is therefore examined by constructing the corresponding ordinary differential equation. Some comments are made concerning the phase-plane of this equation, but more analytical details are extracted by treating the new terms as perturbations of the classical Schrodinger soliton. It is shown that the soliton is both flatter (symmetrically) and skewed forward, although the skewing eventually gives way to an oscillation above the mean level.

A reactance theorem

Abstract: A formula is derived for the frequency derivative of the input reactance of any linear, passive, time-invariant electromagnetic system. It consists of five terms, each of which is identifiable uniquely with one of the following five physical attributes of the system: time-average stored magnetic energy, time-average stored electric energy, radiation, dispersion by the medium, and dissipation by the medium.

On the vacuum stress induced by uniform acceleration or supporting the ether

Abstract: The stresses induced in the vacuum by the uniform acceleration of an infinite plane conductor are computed for the massless scalar and electromagnetic fields. Both Dirichlet and Neumann boundary conditions are considered for the scalar field; far from the conductor it is found, independently of the boundary condition, that the vacuum stress is ‘local’ and corresponds to the absence from the vacuum of black body radiation. Approaching the conductor, the energy density in the Dirichlet case is slightly lower than the ‘local’ term, and in the Neumann case slightly higher. At very small distances it again has the same asymptotic form for both scalar fields. For the electromagnetic field the results are similar to those for the scalar field with Dirichlet boundary conditions. Far from the conductor the spectrum is again black-body, though not Planckian. In all cases the acausal nature of ‘ perfect conductor ’ boundary conditions prevents the stress tensor from being finite on the conductor.

A refinement analysis of the crystallography of the phospholipid, 1,2-dilauroyl-DL-phosphatidylethanolamine, and some remarks on lipid—lipid and lipid-protein interactions

Abstract: A least-squares refinement analysis of atomic positional and thermal parameters in a single crystal of 1,2-dilauroyl-DL-phosphatidylethanolamine: acetic acid has been based on the X-ray diffraction intensities of 1132 independent reflexions, assessed by automatic microdensitometry. The final unweighted discrepancy index is 0.16 with e.s.ds of the bond lengths ranging from 0.02 to 0.12 A. The general features of our earlier, lessprecise analysis are confirmed. The close-packed arrangement of the phospholipid molecules is discussed in relation to electron microscopy and diffraction studies of the structures of membranes from Acholeplasma laidlawii and Halobacterium halobium.

Asymptotic stability and energy decay rates for solutions of hyperbolic equations with boundary damping

Abstract: : This report deals with the asymptotic behavior of solutions of the wave equation in a domain omega a sebset or equal to (R sup n). The boundary Gamma, of omega consists of two parts. One part reflects all energy while the other part absorbs energy to a degree. If the energy absorbing part is non-empty the authors show that the energy tends to zero as t nears infinity. With stronger assumptions one is able to obtain decay rates for the energy. Certain relationships with controllability are discussed and used to advantage.

The laminar separation of an incompressible fluid streaming past a smooth surface

Abstract: Sychev's (1972) proposal, that in general the laminar separation and breakaway of an incompressible fluid streaming past a smooth surface (e.g. on a bluff body) takes place through a triple-deck structure around the separation point, is examined numerically in this paper. The proposed pattern for large Reynolds number ($Re$) flows is based on a modification of the classical Kirchhoff (1869) free streamline theory, in which a slight adverse pressure gradient is provoked in the inviscid motion immediately ahead of the breakaway. This pressure gradient is just enough to generate a triple-deck development closer to the separation point. The major task then is to decide whether or not a solution of the basic triple-deck problem exists, and is regular at separation, and if it is unique. The numerical investigation, an iterative calculation of the relevant boundary layer problem, together with the potential flow relation between the unknown pressure and displacement, points fairly firmly to both the existence and uniqueness of a solution. Thus, for the bluff body problem when $Re$ $\gg $ 1, the triple-deck determines exactly how far the separation point lies from the position implied by inviscid (Kirchhoff) theory. Comparisons with separating incompressible fluid motions determined numerically from the Navier-Stokes equations and measured experimentally give some support overall to the triple-deck description. For the flow past a circular cylinder the agreement in the variation of pressure and skin friction near separation is in general very encouraging, for Reynolds numbers as low as 30.

Computer simulations of polyatomic molecules. II - Molecular dynamics studies of diatomic liquids with atom-atom and quadrupole-quadrupole potentials

Abstract: The molecular dynamics method has been used to study a model system of 256 homonuclear diatomic molecules governed by intermolecular potentials of the form U = UAA + Z7QQ, where UAA is an atom-atom potential and C/QQ is the quadrupole-quadrupole potential. For UAA both the full Lennard-Jones and the repulsive part only of the Lennard-Jones potential have been used, and C7QQ has been used with reduced quadrupole moments, Q* = $/(e L* = L/cr (bond length/atom diameter) between 0.33 and 0.63 have been studied, at reduced temperatures * =kT/e from 0.98 to 3.48 and reduced densities p* = pal from 0.522 to 1.043 (where ae is the diameter of a sphere having a volume equal to that of the diatomic). Detailed orientational structure in the liquid has been examined by calculating as many as 22 terms in the spherical harmonic expansion of the angle dependent pair distribution function. At short distances these systems exhibit a high degree of angular correlation, which increases with increasing elongation and density. Pair correlation functions calculated from the Lennard-Jones and Lennard-Jones repulsive models are virtually identical, other parameters being equal, and are similar to those for hard diatomics, indicating that both radial and orientational structure are determined mainly by short-range repulsive forces. The addition of a moderately strong quadrupole term to the potential produces dramatic changes in structure and significant changes in thermodynamic properties. A potential with a moderately strong quadrupole term is found to give the correct qualitative features for the structure factor of liquid bromine.

The Mean Forces Exerted by Waves on Floating or Submerged Bodies with Applications to Sand Bars and Wave Power Machines

Abstract: Water waves transport both energy and momentum, and any solid body which absorbs or reflects wave energy must absorb or reflect horizontal momentum also. Hence the body is subject to a mean horizontal force. In low waves, the force may be calculated immediately when the incident, reflected and transmitted wave amplitudes are known. For wave power devices the mean force can be large, so that anchoring presents practical problems. Experiments with models of the Cockerell wave-raft and the Salter ‘duck’ accurately confirm the predicted magnitude of the force at low wave amplitudes. For steeper waves, however, the magnitude of the force can be less than that given by linear theory. By experiments with submerged cylinders, it is shown that this is due partly to the presence of a free second harmonic on the down-wave side. In breaking waves, it is confirmed that the mean force on submerged bodies is sometimes reduced, and even reversed. An explanation is suggested in terms of the ‘wave set-up’ produced by breaking waves. Submerged cylinders act as a kind of double beach. A negative mean force arises from an asymmetry in the breaking waves, associated with a time-delay in the response to the change in depth. Similar arguments apply to submerged reefs and sand bars. Experiments with a model bar show that long low waves propel the bar towards the shore, whereas steep, breaking waves propel it seawards. This is similar to the observed behaviour of off-shore sand bars. The existence of a horizontal momentum flux (or radiation stress) in water waves is demonstrated by using it to propel a small craft.

Quantum Vacuum Energy in Two Dimensional Space-Times

Abstract: The paper presents in detail the renormalization theory of the energy-momentum tensor of a two dimensional massless scalar field which has been used elsewhere to study the local physics in a model of black hole evaporation. The treatment is generalized to include the Casimir effect occurring in spatially finite models. The essence of the method is evaluation of the field products in the tensor as functions of two points, followed by covariant subtraction of the discontinuous terms arising as the points coalesce. In two dimensional massless theories, conformal transformations permit exact calculations to be performed. The results are applied here to some special cases, primarily space-times of constant curvature, with emphasis on the existence of distinct 'vacuum' states associated naturally with different conformal coordinate systems. The relevance of the work to the general problems of defining observables and of classifying and interpreting states in curved-space quantum field theory is discussed.

On the Instability of an Internal Gravity Wave

Abstract: The linear stability of an internal gravity wave of arbitrary amplitude in an unbounded stratified inviscid Boussinesq fluid is considered mathematically. The instability is shown to be governed by a Floquet system and treated by a generalization of the method of normal modes. Some properties of the Floquet system, and in particular those of its parametric instability, are analysed. The parametric instability is related to the theory of resonant wave interactions; and the surface of marginal stability in the control space of the amplitude and wavenumbers is shown to be describable by the catastrophe theory of Thom. Finally some results of numerical calculations of the marginal surface are shown. The main physical conclusion is to confirm that the internal gravity wave is unstable always, even when its amplitude is small and so its local Richardson number is large everywhere for all time. It is suggested, by various illustrations and arguments, that the methods developed in this paper are applicable to the instability of many symmetric nonlinear waves.

Ignition and flame quenching in flowing gaseous mixtures

Abstract: The influence of pressure, velocity, turbulence intensity, turbulence scale and mixture composition on minimum ignition energy and quenching distance in flowing gaseous mixtures is examined experimentally for methane and propane fuels. In some experiments, the nitrogen in the air is replaced by various inert gases such as carbon dioxide, helium or argon, while in others the nitrogen is either partly or totally replaced by oxygen. The tests are conducted at room temperature in a 9 cm square working section through which the combustible mixture is arranged to flow at various levels of pressure, turbulence and velocity. At each test condition, the spark energy required to ignite the flowing mixture is measured for several gap widths in order to establish the optimum gap width corresponding to minimum ignition energy. From analysis of the relevant combustion and heat transfer processes involved, expressions for the prediction of quenching distance in flowing mixtures are derived. Support for the model employed in this analysis is demonstrated by a close level of agreement between theoretical predictions of quenching distance and corresponding values calculated from the experimental data on minimum ignition energy obtained over a wide range of mixture compositions and flow conditions.

A Generalized Field Theory. I. Field Equations

Abstract: A field theory representing a natural generalization of the theory of relativity is being constructed by using a tetrad-space. A unique set of field equations exactly equal in number (16) to the unknowns used, and having the same strength as those of general relativity, is obtained. All physical elements of interest are related directly to the members of the geometrical structure.

The Spectral Analysis of Randomly Sampled Records by a Direct Transform

Abstract: The spectral analysis of a signal from a randomly sampled time series is discussed. Spectral estimates derived from the direct transform of this series are compared with those obtained by the correlation method of analysis discussed earlier by the authors (Gaster & Roberts 1975). As found previously, additional variability arises from the random character of the sampling instants. An expression for this variability is derived, and predictions based on it are compared, over a wide range of sampling rates and bandwidths, with computed values obtained from simulated data. The relation between variability and sampling rate is used to find an optimum rate at which this variability is a minimum for a given amount of computation. By this means analysis of a simulated record is carried out over three and a half decades of frequency in one-third octave steps. The relative merits of forming spectral estimates by the direct transform of the data are compared with those of transforming the autocorrelation function. It turns out that although the computational effort is in general less with the technique investigated here, a greater quantity of data is needed to achieve a given level of variability.

Predicting the Strain Rate in the Zone of Intense Shear in which the Chip is Formed in Machining from the Dynamic Flow Stress Properties of the Work Material and the Cutting Conditions

Abstract: In previous applications of an approximate machining theory in which account is taken of the strain rate and temperature dependence of the work material flow stress properties it has been found necessary to use an empirical relation to determine the maximum value of the maximum shear strain rate in the chip formation zone. In this paper the machining theory is further developed so that this strain rate can be obtained as part of the solution. Predicted values found in this way are shown to be in excellent agreement with the rather limited number of experimental strain rate results which are available. The paper ends by showing that if the work material is allowed to approach the ideal constant flow stress material usually assumed in slip-line field theory then the predicted strain rates become extremely large. However, it is still found necessary in calculating the corresponding hydrostatic stresses to use the stress equilibrium equations for a variable flow stress material as the variable flow stress terms do not diminish as rapidly as might have been expected.