Showing papers in "Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences in 1967"

The strength of highly elastic materials

Abstract: Under repeated stressing, cracks in a specimen of vulcanized rubber may propagate and lead to failure. It has been found, however, that below a critical severity of strain no propagation occurs in the absence of chemical corrosion. This severity defines a fatigue limit for repeated stressing below which the life can be virtually indefinite. It can be expressed as the energy per unit area required to produce new surface ( T 0 ), and is about 5 x 10 4 erg/cm 2 . In contrast with gross strength properties such as tear and tensile strength, T 0 does not correlate with the viscoelastic behaviour of the material and varies only relatively slightly with chemical structure. It is shown that T 0 can be calculated approximately by considering the energy required to rupture the polymer chains lying across the path of the crack. This energy is calculated from the strengths of the chemical bonds, secondary forces being ignored. Theory and experiment agree within a factor of 2. Reasons why T 0 and the gross strength properties are influenced by different aspects of the structure of the material are discussed.

Variational Methods and Applications to Water Waves

Abstract: This paper reviews various uses of variational methods in the theory of nonlinear dispersive waves, with details presented for water waves. The appropriate variational principle for water waves is discussed first, and used to derive the long-wave approximations of Boussinesq and Korteweg & de Vries. The resonant near-linear interaction theory is presented briefly in terms of the Lagrangian function of the variational principle. Then the author’s theory of slowly varying wavetrains and its application to Stokes’s waves are reviewed. Luke’s perturbation theory for slowly varying wavetrains is also given. Finally, it is shown how more general dispersive relations can be formulated by means of integro-differential equations; an important application of this, developed with some success, is towards resolving longstanding difficulties in understanding the breaking of water waves.

New Conduction and Reversible Memory Phenomena in Thin Insulating Films

Abstract: It has been observed that when thin (200 to 3000 A) film insulators have been formed by the electrolytic introduction of gold ions from one of the electrodes they can draw appreciable currents. They then show temperature-independent conductivity, voltage-controlled negative resistance and reversible voltage and thermal-voltage memory effects. It is postulated that the injected ions introduce a broad band of localized impurity levels within the normally forbidden band of the insulator. The electrons are assumed to move through the insulator by tunnelling between adjacent sites within the impurity band; it is also assumed that the electrons can, under certain conditions, be trapped within the impurity band. A model based on these ideas accounts in a self-consistent manner for all the experimental observations, and calculations of current-voltage characteristics based on the model are in fact in agree­ment with them.

On the Theory of Hertzian Fracture

Abstract: The fracture of a brittle solid under a spherical indenter is the best studied case of fracture in a strongly inhomogeneous, well defined, stress field. Two principal topics are discussed, the path of a crack in a field of non-uniformly directed stress, and the stability of cracks of various length when the prior stress on the crack path is non-uniform. For the first, it is shown that the crack growth should, to a first approximation , be orthogonal to the most tensile principal stress, and thus correspond, in a torsion-free stress field, to a surface delineated by the trajectories of the other two principal stresses: while, to a second approximation, the crack should deviate from this path by having a larger radius of curvature at every bend, thus exhibiting a pseudonertia even in slow growth. This is in accordance with the known experimental facts about the Hertzian crack, particularly the fact that the crack at the surface forms systematically outside the edge of the circle of contact, at which the maximum tensile stress occurs. On the second question, it is found that there are four crack lengths, c 0 , c 1 , c 2 , c 3 , corresponding to stationary values of energy. c 0 and c 2 represent unstable equilibria, and diminish with increasing load; cx and c3 represent stable equilibria and increase with increasing load. With small indenters, c 0 soon becomes less than the size of pre-present surface flaws, and an unobserved shallow ring crack of depth c 1 is produced: the critical condition for observed fracture is then the merging of c 1 with c 2 , allowing unstable growth to the cone crack of depth c 3 . This explains Auerbach’s law, that the critical load for production of a cone crack is proportional to the radius, r , of the indenter sphere. With larger indenters, of several centimetres radius for a typical case, c 1 , and c 2 merge and disappear before c 0 exceeds the size of pre-present flaws. The critical load for cone fracture then becomes nearly propor¬tional to r 2 , as observed. The previous calculations of Roesler (1956 a, b ) relate to the second stable crack dimension, c 3 , though his energy scaling principle is also applicable to the critical condition at which c 1 and c 2 merge. The Hertzian fracture test, within the validity range of Auerbach’s law, affords a means of measuring surface energy at the fracture surface independent of knowledge about the pre-present flaws.

Instability of Periodic Wavetrains in Nonlinear Dispersive Systems

Abstract: The defining property of the class of physical systems under consideration herein is that, by striking a balance between nonlinear and frequency-dispersive effects, they can transmit periodic waves of finite amplitude but constant form. For any such system, therefore, in respect of propagation in the x direction relative to a state of rest, the dynamical equations have exact periodic solutions of the form η ( x, t ) = H ( x - ct ), say, where c is a constant phase velocity depending on wave amplitude as well as on frequency or wavelength. This paper is concerned with the proposition that in many cases these uniform wavetrains are unstable to small disturbances of a certain kind, so that in practice they will disintegrate if the attempt is made to send them over great distances. The outstanding example only recently brought to light is that finite gravity waves on deep water are unstable: unmistakable experimental evidence of this property is now available, and it has also been demonstrated analytically. In §2 the essential factors leading to instability are explained in general terms. A disturbance capable of gaining energy from the primary wave motion consists of a pair of wave modes at side-band frequencies and wavenumbers fractionally different from the fundamental frequency and wavenumber. In consequence of a nonlinear effect on these modes counteracting the detuning effect of dispersion on them, they are forced into resonance with second-harmonic components of the primary motion and thereafter their amplitudes grow mutually at a rate that is exponential in time or distance. In §3 a detailed stability analysis is presented for wavetrains on water of arbitrary depth h , and it is shown that they are unstable if the fundamental wavenumber k satisfies kh > 1·363, but are otherwise stable. Finally, in §4, some experimental results regarding the instability of deep-water waves are discussed, and a few prospective applications to other specific systems are reviewed.

The occurrence of singularities in cosmology. III. Causality and singularities

Abstract: The questions about singularities that remain to be answered are discussed. It is shown, without any assumption about causality, that there are fully general solutions which evolve from a non-singular state to an inevitable singularity. An observationally testable condition is given which would imply the existence of a singularity if a reasonable assumption about causality were made. This condition would also be satisfied in an approximately spherical collapsing star and so would enable one to prove the occurrence of a singularity in such circumstances without assuming that space-time admits a Cauchy surface. If the assumption about causality held, the singularity could not be of the Misner type but would presumably involve infinite curvature. This would probably indicate that the Einstein theory broke down but only in very strong fields.

Surface-Wave Damping in Closed Basins

Abstract: The damping of surface waves in closed basins appears to be due to ( a ) viscous dissipation at the boundary of the surrounding basin, ( b ) viscous dissipation at the surface in consequence of surface contamination, and ( c ) capillary hysteresis associated with the meniscus surrounding the free surface. Boundary layer approximations are invoked in the treatment of ( a ) and ( b ) to reproduce and extend results that have been obtained previously by more cumbersome procedures. The surface film is assumed to act as a linear, viscoelastic surface and may be either insoluble or soluble; however, the relaxation time for the equilibrium of soluble films is neglected relative to the period of the free-surface oscillations. Capillary hysteresis is analysed on the hypothesis that both the advance and recession of a meniscus are opposed by constant forces that depend only on the material properties of the three-phase interface. The theoretical results for the logarithmic decrements of gravity waves in circular and rectangular cylinders are compared with the decay rates observed by Case & Parkinson and by Keulegan, which typically exceeded the theoretical value based on wall damping alone by factors of between two and three. It is concluded that both surface films and capillary hysteresis can account for, and are likely to have contributed to, these observed discrepancies. The theoretical effect of a surface film on wind-generated gravity waves is examined briefly and is found to be consistent with the observation that the addition of detergent to water can increase the minimum wind speed (required to produce waves) by one order of magnitude.

Transport of Lycopodium spores and other small particles to rough surfaces

Abstract: Measurements have been made in the field and in a wind tunnel of the transport of Lycopodium spores to grass an d other surfaces, and wind tunnel experiments also have been done with aerosols of various smaller particle sizes. The spores and other particles were made radioactive to enable the deposition of small numbers on rough surfaces to be detected. In principle the experiments in the wind tunnel were similar to those previously done with gases (Chamberlain 1966), but the mechanisms by which particles and gases are transported across the boundary layer are different. The velocity of deposition v g of the particle to the surface is equal to the terminal velocity v s if the wind speed is very small, but at higher speeds deposition by impaction on roughness elements becomes progressively more important. If the roughness elements are of a form which gives good impaction efficiency, and have a sticky surface, v g is determined by the rate of eddy diffusion in the turbulent boundary layer above the surface, and may equal or even exceed the analogous velocity of deposition of momentum. The effect of surface texture and stickiness was investigated by comparing the catch of particles on segments of real leaves with the catch on similarity shaped segments of PVG treated with adhesive. Stickiness is important in determining v g for particles of about 10 μ m diameter upwards, but not for smaller particles. In the field experiments, the use of radioactive tagging enabled the presence of a few Lycopodium spores in several grams of grass or soil to be detected, and the deposition could be measured at ranges up to 100 m from the source. At low wind speeds, v g was only a little greater than v s but at higher speeds the contribution of impaction became evident. A particularly high value was obtained when the grass was wet after recent rain. The field results with Lycopodium give a ratio of velocity of deposition to wind speed of 0·01, and this value is used to calculate the percentage of large spores or pollen grains which will travel various distances in normal meteorological conditions. It is found that the median range is about 1 km if the particles are liberated at a height of 50 cm, but 10 km if the height is 10 m. The relative importance of direct deposition to the ground and washout by rain of the air spora is considered, and is shown to depend on the effective height of the cloud of particles. For an effective height of 500 m, derived from vertical profiles of concentration observed from aircraft, it is calculated that about 25% of the total deposition of pollen grains may be in rain.

The strength of fused silica

Abstract: Silica fibres and flame polished rods having strengths around 600 Kg/mm2 were prepared and the preparation conditions studied. The effects on fibre strengths of atmospheric and temperature changes, and of variations in rate of stressing and previous heat treatment, were extensively investigated. From the results obtained a comprehensive qualitative picture of the strength of silica has been deduced; it is found to be strongly dependent on the testing conditions and to extend over the range 280 to 1500 Kg/mm2. INTRODUCTION With the increasing interest in composite materials, in which the load is borne by fibres of a brittle material, it becomes more and more important to understand the factors which govern the strength of brittle materials. This work on fused silica formed part of a more general study of composites and the context was briefly described by Morley (i964). The strength of silicate glasses has been studied for many years but until quite recently all work was carried out with 'weak' specimens which had suffered abrasive surface damage in their preparation or testing. Since the importance of surface damage was scarcely recognized and handling techniques varied widely, this led to very variable strengths, irreproducibility of results from one laboratory to another and a large apparent size dependence of strength. Further, true variations in the strength of the material (due to temperature, moisture, etc.) were often obscured or confused by the effects these variables had on the nature of the existing surface flaws. In this work high strength silica samples were prepared by adapting and extending the techniques of fibre production and handling described by Morley, Andrews & Whitney (i964). Stringent precautions were taken to avoid any contact whatso- ever with the surface of the silica under test and strengths up to 150OKg/mm2 were obtained under certain test conditions. It seems possible that the true strength of the material was being measured. These same high strength samples were studied under a wide range of conditions and the strength was seen to be a complex function of the atmosphere in which the silica was stressed, the temperature, and the rate of stressing. Specific damaging or weakening effects due to heat treatment or specimen preparation could be superimposed on the other factors. In attempting to present a clear account of the strength behaviour we describe each effect in a separate section of the paper. Some discussion of proposed mechan- isms has been introduced at each stage since this appeared necessary to explain the results described in following sections.

Quasi-static crack propagation

Abstract: When cracks of any size spread in brittle materials and when large cracks spread in large structures of ductile materials, irreversible deformation is confined to a small volume of material contiguous with the crack surfaces. A general theory of quasi-static crack propagation under such conditions is formulated. The stability of crack propagation is subject to chosen constraints, and general stability criteria under monotonically increasing load and displacement are presented. Experiments in which cracks are spread quasi-statically are described, and by recording both load and corresponding displacement of load, the local specific work of crack spreading, or fracture toughness ( R ) may be deduced without calculation of the elastic stress distribution or even measuring the shape of the test piece. By causing the crack to spread at a range of speeds, variation of R with speed of crack front may be found. When R is independent of scale, it is shown that the stress intensity to propagate geometrically similar cracks in geometrically similar structures varies inversely as the square root of the size. A principle controlling the path of a quasi-static crack is proposed and some experimental confirmation is offered. An interpretation of crack spreading under cyclic loading is given.

Partial Cone Crack Formation in a Brittle Material Loaded with a Sliding Spherical Indenter

Abstract: In the preceding paper Frank & Lawn (1967) investigated theoretically the development of a cone crack in the strongly inhomogeneous Hertzian stress field. The analysis outlined in that paper is now extended to incorporate a sliding motion of the spherical indenter across the specimen surface, assuming a uniform coefficient of friction over the contact area. Sliding is found to have a large influence on the quasistatic stress field in the loaded specimen, and this in turn affects the ultimate geometry of the cracks. The precise shape of the partially developed cones thus formed is a function only of the Poisson ratio of the specimen material and the coefficient of friction. Criteria determining when surface fracture will occur, expressed as relationships between the critical normal load P c acting on the specimen and the indenter radius r , are calculated as before. The Auerbach law found for purely normally loaded specimens, namely that P c is proportional to r , over a certain range of r , should cease to hold when the coefficient of friction exceeds about 0.02. P c then becomes very nearly proportional to r 2 , which corresponds to a critical stress criterion. The effect of sliding on the value of P c becomes large with larger values of the coefficient of friction; this is of particular relevance to studies of the surface damage of brittle materials.

The mechanics of wave formation in explosive welding

Abstract: Strong bonds can be produced by the explosive welding process and usually the weld interface has a characteristic wavy form. In this paper the mechanism of explosive welding is discussed, the present theories of wave formation are critically examined and a new mechanism of wave formation is proposed. According to this mechanism the materials of the impacting plates in the region of collision behave in a similar manner to liquids of low viscosity. As a result the impacting or flyer plate divides into a re-entrant jet and a salient jet. Very high pressure is produced at the stagnation point of the divided jet. The parent plate deforms under the stagnation point and consequently a hump is formed in the parent plate ahead of the point of collision. The hump builds up and eventually traps the re-entrant jet. The stagnation point then transfers to the top of the hump and then descends it and starts forming a new hump, and in this manner successive waves are formed. The proposed mechanism of wave formation seems to explain the experimentally observed behaviour reasonably well. Furthermore, experiments are reported in which one of the surfaces to be bonded was copper plated and the second surface was nickel plated and by this means the movements of the surfaces being bonded were traced. These experiments gave strong support to the proposed theory of wave formation.

Resonant Interactions between Planetary Waves

Abstract: In contrast to surface gravity waves, planetary waves can interact resonantly at the second order. Hence triplets of planetary waves may occur which are in resonance with each other. For simplicity the situation is studied first on a $\beta$ plane. The geometrical conditions for two waves to form a triplet with a given third wave are determined, and so also is the rate of energy transfer. Some conservation theorems are proved. The analysis allows for a non-zero horizontal divergence of the motion. For waves which cover a complete sphere it is shown that the resonant interactions between three different harmonic components take place, if at all, in the neighbourhoods of two latitude circles, situated symmetrically north and south of the equator.

On the solution of the Bénard problem with boundaries of finite conductivity

Abstract: The Benard problem in hydrodynamic stability is formulated under conditions where the media bounding the fluid have finite thermal diffusivity. It is shown that the principle of the exchange of stabilities remains valid in this case so that instability in the fluid first sets in as stationary convection. Solutions are obtained for various values of the ratio of the thermal diffusivity of the fluid to that of the bounding media; the critical Rayleigh number at which the instability occurs is markedly reduced when this ratio is large.

A Note on the Solution of Transient Dispersion Problems

Abstract: A solution is developed for the problem of transient mass dispersion in fully developed laminar flow in a tube. The method can be extended to more general geometries, and by superposition to include more general boundary conditions such as those which occur in forced convection heat transfer.

The elastic field of an inclusion in an anisotropic medium

Abstract: A new general method of solving problems of anisotropic elasticity, by successive approximations, is applied to find the elastic field of a basic physical problem. In this problem an arbitrary part of an unbounded homogeneous anisotropic medium is a misfitting inclusion that would undergo an arbitrary uniform strain if its surrounding material was absent. An inclusion of ellipsoidal shape is particularly considered; a convergent series of successive approximations is used to generate, and bound as restrictively as desired, the resulting uniform strain in this inclusion. Detailed bounds are given for a spherical inclusion in a medium of cubic anisotropy. Some applications are mentioned, especially the determination of the elastic field when the material in the ellipsoidal inclusion differs from that in its surroundings.

The recombination of nitrogen atoms and the nitrogen afterglow

Abstract: The recombination of nitrogen atoms has been studied photometrically in a fast flow system. The concentration of nitrogen atoms was determined by nitric oxide titration. The measured bimolecular rate constant has the form k$\_B$ = A + B[M] for M = N$\_2$, Ar and He with a constant value of A. The surface process was proved to be second order by inducing a small first order catalytic recombination involving CN and showing by computer analysis that a first order surface process would have been measured readily in our system. Third order rate constants (expressed as d[N$\_2$]/dt) had values: $$k\_{\mathrm{N}\_2} = k\_{\mathrm{Ar}} = (1\cdot38 \pm 0\cdot11) x 10^{15}, \quad k\_{He} = (1\cdot92 \pm 0\cdot18) x 10^{15}$$ in cm$^6$ mole$^{-2}$ s$^{-1}$ units at 298 $^\circ$K. The surface process in the 26 mm i.d. flow tube had a value of (4.4 $\pm$ 0.1) x 10$^8$ cm$^3$ mole$^{-1}$ s$^{-1}$ at 298 $^\circ$K. In the range 196 to 327 $^\circ$K, activation energies were - (975 $\pm$ 140) cal/mole for the homogeneous process and + (620 $\pm$ 50) cal/mole for the surface reaction. The intensity of the nitrogen afterglow was shown to be proportional to [N]$^2$ and independent of total pressure for nitrogen carrier in the range 2 to 10 mmHg. Partial replacement of the nitrogen carrier by helium or argon enhanced the nitrogen afterglow on a mole fraction basis. This effect was shown to be associated with an efficient quenching of the emitting B$^3$II$\_g$ state by molecular nitrogen. This view is supported by Jeunehomme & Duncan's work on the pressure dependence of the lifetime of this state. Measurement of the absolute intensity of the afterglow when combined with their data show that about 50% of the recombination passes through the B$^3\Pi\_g$ state. On this basis it is concluded that the A$^3\Sigma^+\_u$ state and not the shallow $^5\Sigma^+\_g$ state is the precursor of the afterglow. Levels of the B$^3\Pi\_g$ state around v' = 12, 6 and 2 are populated by collision induced transition from the A state.

Quantum yields of triplet formation in solutions of chlorophyll

Abstract: The method of flash photolysis has been used for direct measurement of the quantum yields of triplet formation in dilute chlorophyll solutions at 23 $^\circ$C. In ether solution, the triplet yields were found to be 0.64 and 0.88 for chlorophyll a and b, respectively. In very dry hydrocarbon solution, the yields decrease by at least a factor of 5. It is suggested that the low yields of fluorescence and intersystem crossing in dry solvents are due to dissociation of excited singlet dimer chlorophyll into ground state monomer species.

Gas Hydrates Containing Argon, Krypton and Xenon: Kinetics and Energetics of Formation and Equilibria

Abstract: Inert gases have been found to react fully and rapidly with ice crystals, with a mixture of crystals of chloroform and of ice, or with pre-formed chloroform hydrate, provided the crystals were rapidly agitated with small steel ball-bearings. Reaction still occurred readily at temperatures as low as-183°C (argon and ice). The above finding made it possible to measure sorption isotherms in hydrates of types I and II over a range of pressures and at various temperatures. For the type II chloroform hydrate heats of intercalation, ∆ H , of rare gases were evaluated from the temperature coefficients of the isotherms, which closely followed Langmuir’s isotherm equation. For Ar, Kr, and Xe, ∆ H was respectively -6.1, -6.7 and -7.9 kcal/mole respectively. In the type I hydrates the sorption isotherms necessarily terminated at the dissociation pressures. These pressures were measured over a range of temperatures. From them overall heats of reaction, ∆ H 1 , between ice and inert gas were derived for degrees of occupancy of the intracrystalline cavities of over 90%. For Ar, Kr and Xe ∆ H 1 was -2.94, -3.98 and - 5.77 kcal/mole respectively. Corresponding heats of intercalation estimated indirectly were -5.5, -6.0 and -7.4 kcal/mole, and the averaged heat of formation of the empty host lattice of type I from ice was estimated as 0.35 ± 0.10 kcal/mole H 2 O. The results have been discussed theoretically from viewpoints of energetics and equilibria, and of the significance of d P /d T in Clapeyron’s equation. Equilibria in the chloroform hydrates show deviations from the ideal statistical thermodynamic treatment of clathrate solid solutions, the best result being obtained for Xe and the worst for Ar.

The Density of Electrons in a Perfect or Imperfect Lattice

Abstract: The spatial distribution of electronic charge is calculated exactly for three simple cases. These are the perfect lattice, perfect lattice with a single vacancy and perfect lattice with a single substitutional impurity. The potential is assumed to be of muffin tin form. The results are similar in appearance to the equations for the Kohn-Rostoker band structure calculations and appear ideally suited to numerical computation. Examples are given to illustrate the results. These are rather simple, but two interesting conclusions can be drawn: the charge at a vacancy is not directly proportional to the density of states of the host lattice and an impurity can cause two bound states to appear from a narrow band.

On the Radiation Field of Pulse Solutions of the Wave Equation

Abstract: It was shown in two earlier papers that, if u(x, t) = u(x$\_1$, x$\_2$, x$\_3$, t) satisfies the wave equation in the exterior of some fixed sphere r = a and vanishes for t $\leqslant$ r, then ru(r$\xi$, r + $\tau$) $\sim$ f($\xi$, $\tau$) as r $\rightarrow \infty$, provided that $\xi$ is a fixed unit vector and t - r = $\tau$ remains bounded. The function f was called the radiation field of the pulse solution u. The problem of determining u when f is given was then considered: this is called the radiation field problem in the present paper. It was proved that (i) the radiation field determines the pulse solution uniquely, and (ii) that if two radiation fields f$\_1$ and f$\_2$ coincide, for all $\tau \geqslant$ 0, in any open set of the unit sphere, then f$\_1$ = f$_2$. In the present paper new proofs of these results are given, by means of a simple transformation of variables and an appeal to Holmgren's uniqueness theorem. The same method is then applied to self-adjoint linear second order equations of normal hyperbolic type, which can be considered as wave equation in a curved spacetime. Radiation fields are defined by means of a family of characteristics that represent outgoing wave fronts. It is shown that if the metric, expressed in coordinates adapted to these characteristics, satisfies certain conditions at infinity (similar to those that have been used in the theory of gravitational waves by Bondi, van den Berg & Metzner, and Sachs), then radiation fields exist. It is also shown that the radiation field determines the associated pulse solution uniquely, and that radiation fields form a coherent family of functions with unique continuation properties.

Triplet state quantum yields for some aromatic hydrocarbons and xanthene dyes in dilute solution

Abstract: Quantum yields of triplet state formation and extinction coefficients of the triplet states have been determined by direct depletion methods for solutions of anthracene, phenanthrene, 1,2,5,6-dibenzanthracene, fluorescein, dibromofluorescein, eosin and erythrosin. The values obtained for the hydrocarbons are in reasonable agreement with those obtained by other workers using energy transfer and heavy atom perturbation techniques. In all cases which we have studied, the sum of the quantum yields of fluorescence and triplet state formation is equal to unity within the limits of experimental error, showing that radiationless transfer from the excited singlet to the ground state is negligible.

Electron correlations in narrow energy bands V. A perturbation expansion about the atomic limit

Abstract: A perturbation expansion about the atomic limit in powers of the interaction between atoms is developed and a graphical analysis of the terms of the series is described. The manner in which the series degenerates in the band theory limit is discussed and a rule for generating approximations valid in both the atomic and band theory limits is given and applied to discuss the electron propagator.

The Association of Oxygen Atoms and their Combination with Nitrogen Atoms

Abstract: The rate constants of the reactions \begin{equation*}\tag{2}N &+ O + M = NO + M \\ \tag{4}O &+ O + M = O\_2 + M\end{equation*} have been determined in active nitrogen systems, nitric oxide being added to result in the partial production of oxygen atoms. The concentrations of these atoms were monitored by measurements of the intensity of the N$\_2$ First Positive emission and NO $\beta$ emission. The following rate constants (in cm$^6$ mole$^{-2}$ s$^{-1}$) were obtained at room temperature (298 $^\circ$K) $$\quad \mathrm{N}\_2$ \quad \mathrm{Ar} \quad \mathrm{He} \\ 10^{-15}k\_2 3\cdot88 \pm 0\cdot30 \quad 2\cdot98 \pm 0\cdot35 \quad 1\cdot36 \pm 0\cdot17 \\ 10^{-14}k_4 \quad 11\cdot3 \pm 1\cdot1 \quad 6\cdot0 \pm 0\cdot6 \quad 4\cdot6 \pm 0\quad4$$ In the range 196 to 327 $^\circ$K, the temperature coefficient of reaction (2) corresponds to a T$^-\frac{1}{2}$ dependence or an activation energy of -270 $\pm$ 120 cal/mole. This is unusually small for a three body recombination and contrasts with more 'normal' activation energy of -1420 $\pm$ 350 cal/mole found for reaction (4). The NO $\beta$ emission associated with reaction (2) has a similar temperature coefficient to the overall reaction, but is slightly enhanced by replacing the nitrogen carrier by argon. Our kinetic studies of this emission generally confirm the mechanism of Young & Sharpless (1962).

Vibrations of random dilute alloys

Abstract: The vibrational properties of a crystal with a randomly arranged small concentration c of impurities are considered by a thermal Green function technique. Detailed calculations are made for the case when the impurities have changed mass. The Green functions lead directly to average and weighted displacement and momentum correlations which may in turn be related to observable properties such as density of states, thermal energies, optical absorption, neutron scattering, etc. Detailed calculations are presented in a number of cases based on realistic models and compared with experiment. The method treats correctly multiple interactions of a vibrational quantum with a single defect but treats interference between defects only approximately. It is therefore correct to order c but only approximately correct to higher orders. Resonances of the impurities which occur within the phonon spectra are satisfactorily described. The impurity bands which occur around the frequencies of the localized modes of the isolated defects are also described but the approximations are less satisfactory in this case.

Viscoelastic relaxation of supercooled liquids. II

Abstract: In addition to the six liquids for which the results of viscoelastic measurements are reported in part I, studies have also been made on a further eight liquids which differ widely in molecular composition, namely: squalane, tri($\beta$-chloroethyl) phosphate, tri(o-tolyl) phosphate, tri(m-tolyl) phosphate, tris(2-ethyl hexyl) phosphate, tetra(2-ethyl hexyl) silicate, m-bis(m-phenoxy phenoxy) benzene and bis(m-(m-phenoxy phenoxy) phenyl) ether. It has been confirmed for each liquid that the limiting shear modulus, G$\_\infty$, varies with temperature according to the relation previously found, $\frac{1}{G\_\infty}=\frac{1}{G\_0}+C(T-T\_0).$ Plotting the quantities $R\_L/(\rho G\_\infty)^\frac{1}{2}$ and $X\_L/(\rho G\_\infty)^\frac{1}{2}$ against log$\_{10}$($\omega\eta$/G$\_\infty$), where R$\_L$ and X$\_L$ are the resistive and reactive components of the shear mechanical impedance, shows that the results for six of these liquids are indistinguishable within experimental error. Moreover, the curves drawn through the experimental points represented in this manner are identical with the corresponding curves found for five of the liquids for which results are described in part I. The results for these eleven liquids show a striking agreement. A new theoretical model is put forward from which the viscoelastic properties of pure liquids in the supercooled region can be predicted within experimental error. This is regarded as a limiting case for liquids which follow the free volume description of steady flow viscosity. In the light of available evidence, it is envisaged that the viscoelastic relaxational behaviour approaches that represented by a single (Maxwell) relaxation process as the degree of co-operative motion of the molecules decreases. Only three of the fourteen liquids investigated show a behaviour which does not follow the theoretical prediction. The observed deviations are not great but are outside the limit of experimental error. Measurements on mixtures of liquids which individually exhibit behaviour in agreement with theory show that an impurity content of less than 2% can result in similar departures from theory. The lack of agreement for the three liquids in question is therefore attributed to impurities in the samples measured.

Some Special Cases Treated by the Whitham Theory

Abstract: The theory of the preceding paper (by G. B. Whitham) may be valuable in problems of nonlinear dispersive wave systems governed by equations too complicated for analytical or numerical treatment. It suggests, in fact, that the development of wave groups, whose parameters vary gradually enough, can be represented by much less complicated approximate equations, derived from the assumption that in each separate small region the waves are closely like plane periodic ones. Nevertheless, the effort demanded to solve even those approximate equations would normally be very substantial, and therefore it is important (see §1) to get a preliminary idea of whether the solutions would correspond well with reality or not, by calculating the implications of the theory in detail for some rather simple system or systems and comparing them with experiment.

Nonlinear Interactions Treated by the Methods of Theoretical Physics (with Application to the Generation of Waves by Wind)

Abstract: The ‘Feynman’ diagram method for analysing wave-wave interactions in random wave fields is generalized to include non-conservative interactions between wave fields and external fields The particle interpretation is no longer applicable, but the transfer expressions can still be conveniently summarized in terms of ‘transfer’ diagrams, which correspond to collision diagrams in the particle picture The method is applied to interactions between gravity waves and the turbulent atmospheric boundary layer The complete set of lowest order transfer diagrams contains the Phillips and Miles mechanisms of wave generation and an additional set of wave-turbulence interactions, which have not been considered previously The closure hypothesis invoked in the derivation of the transfer expressions is discussed briefly in appendix A It is pointed out that Benney & Saffman’s recent derivation of the transfer expressions without the usual closure hypothesis contradicts the irreversibility of the transfer expressions and is valid only initially The relevant statistical properties depend on the distinction between coarse grained and fine grained distributions This is illustrated in appendix B by a discussion of the Gaussian property of linear, random wave fields

The Symmetry of Nonrigid Molecules: The Schrodinger Supergroup

Abstract: The symmetry groups of nonrigid molecules possess two subgroups: one is that of all the symmetry operations of the Schrodinger group of the molecule and the other is a group of a new type of operation called an isodynamic operation. The total group, which is called the Schrodinger supergroup, is a semidirect product of the Schrodinger group and the isodynamic group. Examples are given for borondifluoride methyl, ethane, dimethylacetylene and borontrimethyl. It is shown that in certain cases the double groups of the isodynamic group have to be used. The results do not agree always with those recently given by Longuet-Higgins: a careful analysis of the nature of symmetry and isodynamic operations indicates the origin of the discrepancy.

Numerical Evaluation of Kramers-Kronig Relations

Abstract: Optical rotatory dispersion and circular dichroism are closely connected phenomena. Their interdependency is contained in the Kramers-Kronig relations which are, just as thermodynamic relations, of a very fundamental nature. As a matter of fact they apply to every physical system, where the response to an action from outside is linear and where the principle of causality holds. This means that the system does not squeal before it is hurt. For our purpose it is of interest to realize that the relations imply that a complete knowledge of circular dichroism as a function of frequency contains the same in­formation as a complete knowledge of rotary dispersion. However, our knowledge about the two effects is always limited to a small region of the spectrum only. A region that extends from 200 to 800 nm, say. Therefore it is of great importance to use the Kramers-Kronig relations in order to derive as much information as possible from the two types of measurements together at the available wavelengths.