Showing papers in "Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences in 1959"
TL;DR: In this article, an investigation of the structure of the electromagnetic field near the focus of an aplanatic system which images a point source is made, and the results are illustrated by diagrams and in a tabulated form based on data obtained by extensive calculations on an electronic computor.
Abstract: An investigation is made of the structure of the electromagnetic field near the focus of an aplanatic system which images a point source. First the case of a linearly polarized incident field is examined and expressions are derived for the electric and magnetic vectors in the image space. Some general consequences of the formulae are then discussed. In particular the symmetry properties of the field with respect to the focal plane are noted and the state of polarization of the image region is investigated. The distribution of the time-averaged electric and magnetic energy densities and of the energy flow (Poynting vector) in the focal plane is studied in detail, and the results are illustrated by diagrams and in a tabulated form based on data obtained by extensive calculations on an electronic computor. The case of an unpolarized field is also investigated. The solution is riot restricted to systems of low aperture, and the computational results cover, in fact, selected values of the angular semi-aperture a on the image side, in the whole range 0 ≤ α ≤ 90°. The limiting case α → 0 is examined in detail and it is shown that the field is then completely characterized by a single, generally complex, scalar function, which turns out to be identical with that of the classical scalar theory of Airy, Lommel and Struve. The results have an immediate bearing on the resolving power of image forming systems; they also help our understanding of the significance of the scalar diffraction theory, which is customarily employed, without a proper justification, in the analysis of images in lowaperture systems.
2,636 citations
TL;DR: In this paper, the elastic field outside an ellipsoidal inclusion or inhomogeneity may be expressed entirely in terms of the harmonic potential of a solid elliptipsoid.
Abstract: The results of an earlier paper are extended. The elastic field outside an inclusion or inhomogeneity is treated in greater detail. For a general inclusion the harmonic potential of a certain surface distribution may be used in place of the biharmonic potential used previously. The elastic field outside an ellipsoidal inclusion or inhomogeneity may be expressed entirely in terms of the harmonic potential of a solid ellipsoid. The solution gives incidentally the velocity field about an ellipsoid which is deforming homogeneously in a viscous fluid. An expression given previously for the strain energy of an ellipsoidal region which has undergone a shear transformation is generalized to the case where the region has elastic constants different from those of its surroundings. The Appendix outlines a general method of calculating biharmonic potentials.
1,323 citations
TL;DR: In this article, an integral representation for the electromagnetic field in the image space of an optical system is obtained in the form of an angular spectrum of plane waves, which is closely related to that introduced by Luneberg (1944) as a vector generalization of well-known formulae of Debye (1909) and Picht (1925).
Abstract: An integral representation is obtained for the electromagnetic field in the image space of an optical system . This representation, which is not restricted to systems of low angular aperture, is in the form of an angular spectrum of plane waves, and is closely related to that introduced by Luneberg (1944) as a vector generalization of well-known formulae of Debye (1909) and Picht (1925). It is shown that the representation has a simple physical interpretation in terms of a modified Huygens—Fresnel principle which operates with secondary plane waves rather than with secondary spherical waves.
761 citations
TL;DR: In this paper, an unusual rupture process is described, which consists of the sudden appearance of internal cracks in bonded rubber cylinders at a well-defined and comparatively small tensile load, and it is assumed that they form when the negative hydrostatic pressure component of the applied stress reaches a critical value.
Abstract: An unusual rupture process is described. It consists of the sudden appearance of internal cracks in bonded rubber cylinders at a well-defined and comparatively small tensile load. The cracks occur in all the vulcanizates examined, and in particularly weak rubbers are found to cause failure. Their appearance may also lead to marked changes in the load-deflexion relationship. A theoretical treatment is presented on the basis of a proposed criterion for crack formation. It is assumed that they form when the negative hydrostatic pressure component of the applied stress reaches a critical value. It is shown by means of the theory of large elastic deformations that a critical value of the negative pressure exists at which any cavity in the rubber will burst, and the calculated value is shown to be in satisfactory agreement with experiment. Moreover, the theory successfully predicts the observed dependence of the cracking stress on the Young’s modulus of the rubber, its virtual independence of the rubber strength and extensibility, and the general form of the observed variation of cracking stress and crack disposition with the thickness of the test-piece.
523 citations
TL;DR: In this paper, it was shown that the free edge of a sheet of uniform thickness moves into an expanding sheet at the same speed as antisymmetrical waves, sweeping the fluid into roughly cylindrical borders.
Abstract: The free edge of a sheet of uniform thickness moves into it at the same speed, (2 T /ρ t ) ½ , as antisymmetrical waves, sweeping the fluid into roughly cylindrical borders. Here T , ρ and t are surface tension, density and thickness of the sheet. In a radially expanding sheet t decreases with increasing radius and beyond a radius R where (2 T /ρ t ) ½ is greater than u the radial velocity of the sheet, the edge moves inwards faster than it is convected outwards. Photographs show that the edge of an expanding sheet establishes itself near but inside the radius R . The sheet produced by a swirl atomizer expands as a cone but photographs show that its thickness fluctuates very greatly at the point where it emerges from the orifice. The edge of a conical sheet of varying thickness establishes itself at a point well inside the radius at which (2 T /ρ t ) ½ = u , t being the mean thickness. A moving sheet of uniform thickness can be bounded by a stationary free edge at angle sin -1 ( W ½ ) to the direction of motion. Here W , the Weber number, is 2 T /ρ tu 2 . Photographs show free edges at this angle and therefore parallel to antisymmetrical waves. If this remained true in an expanding sheet the edges would coincide with the cardioids discussed in part II, but reasons are given to show that this is not the case. A small obstacle can divide an expanding sheet forming two edges which lie at the same angle to one another as the two cardioids, namely, 2 sin -1 ( W ½ ) but photographs show that these edges do not subsequently lie on cardioids.
473 citations
TL;DR: In this paper, a study of the free and forced oscillations of the earth is made, and the natural periods for radial, torsional and spheroidal types of oscillation are determined for all models except the homogeneous one, for which this period is only 44$\cdot $3 min.
Abstract: A study is made of the free and forced oscillations of the earth. The natural periods are determined for radial, torsional and spheroidal types of oscillation. Several models of the earth are used: a homogeneous model, such as was assumed originally by Love, a model consisting of a homogeneous solid mantle enclosing a homogeneous liquid core, Bullen's model B, and Bullard's models I and II. It is found that the spheroidal oscillation of order 2 has a period of about 53$\cdot $5 min in all models, except the homogeneous one, for which this period is only 44$\cdot $3 min. The common period of 53$\cdot $5 min agrees to within the observational error with the period of 57 min observed by Benioff on the seismograms of the Kamchatka earthquake of 1952. In contrast to all the other models, Bullen's model B possesses an additional spheroidal oscillation of order 2 of a period of 101 min. The latter oscillation is confined mainly to the core, its amplitude in the mantle being relatively very small. Benioff's observation of a second oscillation of a 100 min period in the Kamchatka earthquake record might be considered as evidence favouring Bullen's model B. The latter differs from Bullard's models mainly by having a central density of around 18 instead of about 12 g/cm$^{3}$. However, a theoretical investigation of the relative excitation of the various free modes by an impulsive compressional point-source situated at a shallow focal depth, shows that the amplitude of the 100 min oscillation should be more than 1000 times weaker than that of the 53$\cdot $5 min oscillation. It is thus not clear how a near-surface earthquake could have excited the core-oscillation. The spectrum of the free modes of oscillation has also been determined for n = 3 and 4, including the fundamental and the first two overtones for each case. The computed free periods of spheroidal oscillation range from 53$\cdot $5 min down to a period of 8 min for the fourth overtone in the case n = 2. We have also treated the bodily tides for Bullen's and Bullard's models. Love's numbers were determined in the case n = 2 for tidal periods of 6, 6 $\surd $2, 12 h and $\infty $. The dependence of the Love numbers on the period is small, a maximum range of variation of 13% occurring in the k-values between the periods of 6 h and $\infty $.
467 citations
TL;DR: In this paper, it was shown that, provided bond compression is taken into account, this result follows inevitably from even the simple l.c.a.o. theory, and is independent of the analytic form of either /?( r ), the resonance integral, or f(r ), the cr bond energy.
Abstract: Ooshika (1957) has recently found, using the self-consistent molecular orbital theory, that a cyclicpolyene C 2n H 2n exhibits marked bond alternation if n is very large. Here we show that, provided bond compression is taken into account, this result follows inevitably from even the simple l.c.a.o. theory, and is independent of the analytic form of either /?( r ), the resonance integral, or f ( r ), the cr bond energy. An investigation of the linear polyenes C 2n H 2n+2 and C 2n+1 H 2n+3 leads to the same conclusions, which contradict those of Lennard-Jones (1937) and Coulson (1938) but agree with those of Ooshika (1957) and Labhart (1957). A simple calculation, based on an exponential form for leads to a value of about 0.04 A for the difference in length between adjacent bonds in the infinite chain.
409 citations
TL;DR: In this paper, the authors derived a relation involving the fracture density per unit dose ( p 0 ) and the density of cross-linked units per unit dosage ( q 0 ), which holds strictly for exponential distributions, whether or not main-chain fracture occurs simultaneously with cross-linking, and also holds at high doses for the other distributions considered, providing that crosslinking is accompanied by fracture.
Abstract: The equation for the sol fraction ( s ) of a cross-linked polymer network becomes readily tractable when applied to special cases of the generalized distribution function n( u ) = C ( u/u 1 ) λ -1 exp ( — λu/u 1 ). For values of λ = ∞, 1 and 0 respectively, this function yields the uniform distribution, the exponential distribution and a hypothetical pseudo-random distribution. Assuming that cross-linking and fracture occur at random and in proportion to the radiation dose, simple expressions are derived relating sol fraction to radiation dose ( r ) for each of the three distributions. The most useful of these is the relation involving the fracture density per unit dose ( p 0 ) and the density of cross-linked units per unit dose ( q 0 ). s +√ s = p 0 / q 0 + 1/ q 0 u 1 r . This holds strictly for exponential distributions, whether or not main-chain fracture occurs simultaneously with cross-linking, and also holds at high doses for the other distributions considered, providing that cross-linking is accompanied by fracture. This treatment is applied to experimental results on low-density and high-density polyethylenes, polyvinyl acetates, polyvinyl chloride, polypropylene and polyalkyl acrylates. The relevant radiation parameters p 0 , q 0 and the corresponding G values are deduced. It is found, in the case of polyethylene, that q 0 is, within experimental error independent of the molecular weight, degree of branching or crystallinity, but is affected by the presence of air. Similar values of q 0 are also observed for polyvinyl acetate and polyvinyl chloride.
366 citations
TL;DR: In this paper, the authors defined plane gravitational waves as non-flat solutions of Einstein's empty spacetime field equations which admit as much symmetry as do plane electromagnetic waves, namely, a 5-parameter group of motions.
Abstract: Plane gravitational waves are here defined to be non-flat solutions of Einstein’s empty spacetime field equations which admit as much symmetry as do plane electromagnetic waves, namely, a 5-parameter group of motions. A general plane-wave metric is written down and the properties of plane wave space-times are studied in detail. In particular, their characterization as 4 plane ’ is justified further by the construction of 4 sandwich waves ’ bounded on both sides by (null) hyperplanes in flat space-time. It is shown that the passing of a sandwich wave produces a relative acceleration in free test particles, and inferred from this that such waves transport energy.
330 citations
TL;DR: In this article, the density matrix theory of N -particle systems was developed and the density matrices of the whole system may be expressed in terms of those of the component electron groups.
Abstract: Many-electron wave functions are usually constructed from antisymmetrized products of one-electron orbitals (determinants) and energy calculations are based on the matrix element expressions due to Slater (1931). In this paper, the orbitals in such a product are replaced by ‘group functions’, each describing any number of electrons, and the necessary generalization of Slater’s results is carried out. It is first necessary to develop the density matrix theory of N -particle systems and to show that for systems described by ‘generalized product functions’ the density matrices of the whole system may be expressed in terms of those of the component electron groups. The matrix elements of the Hamiltonian between generalized product functions are then given by expressions which resemble those of Slater, the ‘coulomb’ and ‘exchange’ integrals being replaced by integrals containing the one-electron density matrices of the various groups. By setting up an ‘effective’ Hamiltonian for each electron group in the presence of the others, the discussion of a many-particle system in which groups or ‘shells’ can be distinguished (e. g. atomic K, L, M , ..., shells) can rigorously be reduced to a discussion of smaller subsystems. A single generalized product (cf. the single determinant of Hartree—Fock theory) provides a convenient first approximation; and the effect of admitting ‘excited’ products (cf. configuration interaction) can be estimated by a perturbation method. The energy expression may then be discussed in terms of the electon density and ‘pair’ functions. The energy is a sum of group energies supplemented by interaction terms which represent (i) electrostatic repulsions between charge clouds, (ii) the polarization of each group in the field of the others, and (iii) ‘dispersion’ effects of the type defined by London. All these terms can be calculated, for group functions of any kind, in terms of the density matrices of the separate groups. Applications to the theory of intermolecular forces and to π -electron systems are also discussed.
323 citations
TL;DR: In this paper, the authors extended earlier work on the adhesion mechanism of friction and considered in particular the growth in area of contact as the tangential force is increased to the point at which gross sliding occurs.
Abstract: This paper extends earlier work on the adhesion mechanism of friction and considers in particular the growth in area of contact as the tangential force is increased to the point at which gross sliding occurs The earlier studies assumed that the area of true contact A is the same as that produced under static loading so that A = W / p 0 where W is the normal load and p 0 the plastic yield pressure of the metal If the junctions have a specific shear strength s , the friction F , that is the force to shear them, will be F = As and the coefficient of friction becomes μ = s / p 0 (Bowden & Tabor 1954) Recent studies, however, show that as the tangential stress is applied the area of true contact increases according to a relation of the type p 2 + αs 2 = p 2 0 where p is the normal and s the tangential stress in the contact region and α an appropriate constant With thoroughly outgassed metals, junction growth generally proceeds until practically the whole of the geometric area is in contact and coefficients of friction of the order of 50 or more are observed (Bowden & Young 1951) If the interface is contaminated, the stresses transmitted through it cannot exceed the critical shear stress of the interface The new point developed in this paper based on the work of Courtney-Pratt & Eisner (1957), is that until the shear stress reaches this value junction growth occurs as for clean metals Beyond this point, however, further junction growth is impossible and gross sliding occurs within the interfacial layer itself The analysis given here shows that if the interface is only 5% weaker than the bulk metal, junction growth ceases and gross sliding occurs when the coefficient of friction is of the order of unity This corresponds to the experimental observation that minute amounts of oxygen or air reduce the friction of thoroughly clean metals from extremely high values to values of about 1 In the presence of a lubricant film the transmissible stresses are so small that little junction growth can occur before sliding takes place The expression for the coefficient of friction now reduces to a form resembling that given by the earlier simpler theory, namely μ = s i / p 0 , where s i is the critical shear stress of the lubricant layer The present treatment thus incorporates the effect of combined stresses and surface contamination into a more general theory of metallic friction
TL;DR: In this article, it was shown that a finite change in direction of flow can occur at a cardioid which therefore assumes the form of a sharp edge, and the predicted wave patterns agree with those revealed by the schlieren photographs.
Abstract: It is shown that capillary waves are of two kinds, symmetrical waves in which the displacements of opposite surfaces are in opposite directions, and antisymmetrical waves in which the displacements are in the same direction. Any disturbance can be regarded as composed of these two types of wave. The antisymmetrical waves are non-dispersive. In a sheet of uniform thickness a moving point disturbance produces two narrow line-like waves. In a radially expanding sheet a fixed disturbance point produces two narrow disturbances in the form of cardioids. It is shown theoretically that a finite change in direction of flow can occur at a cardioid which therefore assumes the form of a sharp edge. A method was found for producing and photographing a sheet with a sharp edge in the form of a cardioid. The symmetric waves are very different, they are highly dispersive and are propagated much more slowly than the antisymmetrical waves. Experimentally a point disturbance produces both kinds of wave simultaneously. Reflexion photographs show the antisymmetrical waves, while the schlieren method is needed to reveal the symmetrical waves. The symmetrical waves produced in a moving sheet by a point disturbance are parabolas when the sheet is uniform in thickness, and of a more complicated form when the sheet is expanding. The predicted wave patterns agree with those revealed by the schlieren photographs.
TL;DR: The apparent diffusion coefficient is the sum of the molecular and Taylor diffusion coefficients in the two phases and a term due to the finite rate of partition between them as discussed by the authors, and it is shown how the Taylor diffusion coefficient depend on the ratio of amounts of solute held in two phases, and how this gives a connexion between the coefficient a 2 U 2 /48 D found by Taylor (1953) for viscous flow in a circular tube and the 11 a 2U 2/48 D in his analysis of the distillation column.
Abstract: The method used in an earlier paper (Aris 1956) to discuss the dispersion of a solute in a fluid flowing through a tube is applied here to the case in which the solute can also pass into another fluid phase flowing in an annular region around the first. The apparent diffusion coefficient is the sum of the molecular and Taylor diffusion coefficients in the two phases and a term due to the finite rate of partition between them. It is shown how the Taylor diffusion coefficients depend on the ratio of amounts of solute held in the two phases and how this gives a connexion between the coefficient a 2 U 2 /48 D found by Taylor (1953) for viscous flow in a circular tube and the 11 a 2 U 2 /48 D found by Westhaver (1942) in his analysis of the distillation column. The use of these apparent diffusion coefficients in distillation and partition chromatography is illustrated.
TL;DR: In this paper, a simple solution of the equations representing the shape of an axially symmetric sheet of fluid is given, and the shapes so calculated are compared with photographs of a "water bell" produced by placing a plane or conical obstruction in the centre of a jet of water.
Abstract: A simple solution of the equations representing the shape of an axially symmetric sheet of fluid is given. The shapes so calculated are compared with photographs of a 'water bell' produced by placing a plane or conical obstruction in the centre of a jet of water. The effect of air friction is evident in one of the photographs, and calculation, using a formula due to Howarth, shows that it should have been expected, though previous discussions of water bells have assumed it negligible.
TL;DR: In this paper, the effect of differential transport on the gradient of concentration across a boundary is calculated for a substance existing in solution as a series of aggregates in mutual equilibrium, and general equations are derived which are then used to construct schlieren patterns for a model system.
Abstract: The effect of differential transport on the gradient of concentration across a boundary is calculated for a substance existing in solution as a series of aggregates in mutual equilibrium. General equations are derived which are then used to construct schlieren patterns for a model system in order to illustrate the types of pattern to which aggregation can give rise. It is shown that in electrophoresis experiments the boundary in one limb is diffuse, and in the other hyper-sharp. In sedimentation the boundary is diffuse in general, but may have a hypersharp leading edge if there is a strong dependence o f the sedimentation velocity of the individual species on solute concentration. Although it is not possible for true resolution to occur at the boundary of a system in which equilibrium is maintained, the shapes of the schlieren patterns of the diffuse boundaries can be such as to give the impression that partial resolution is occurring. It means then that aggregates composed of more than two molecules are present and that at least one type of higher aggregate is strongly accentuated.
TL;DR: In this paper, it was shown that there exist systems of c-number Hamiltonian dynamics possessing neither canonically conjugate dynamical variables nor Lagrangian formulation, and the possibility of a non-c-number classical dynamics is then considered and realized.
Abstract: Various ways of generalizing classical canonical dynamics are considered. It is found that there exist systems of c-number Hamiltonian dynamics possessing neither canonically conjugate dynamical variables nor Lagrangian formulation. The possibility of a non-c-number classical dynamics is then considered and realized. Elementary examples of both types of dynamics are examined, and emerge as classical analogues of a Fermi oscillator. This is remarkable in view of the commonly held belief that a Fermi oscillator does not have a classical analogue. Not all possible classical analogues possess a Lagrangian formulation: those which do are of particular interest, since they provide the means of setting up a kind of Feynman principle.
TL;DR: In this article, it is shown that unless hydrogen atoms are being captured very rapidly from interplanetary space they must be very rare indeed at the base of the exosphere, and stress is laid on the importance of the steepness of the temperature gradient above the E layer in connexion with the thermal economy.
Abstract: Consideration is given to the significance of recent rocket and satellite studies relating to the structure of the thermosphere. It is shown that unless hydrogen atoms are being captured very rapidly from interplanetary space they must be very rare indeed at the base of the exosphere. Stress is laid on the importance of the steepness of the temperature gradient above the E layer in connexion with the thermal economy. Though no final conclusion is reached, it is thought that the view that the ionizing radiation from the sun is the main source of heat is more attractive than any alternative view yet put forward. An analytic model of the thermosphere is described in the appendix.
TL;DR: In this article, the authors extend the Born-von Karman theory of lattice dynamics to apply to a simple model of the germanium crystal, in which each atom is regarded as a charged core coupled to an oppositely charged shell.
Abstract: Previous attempts to explain the frequency-wave number relations for the normal modes of germanium, which have been determined experimentally by Brockhouse & Iyengar, have required the assumption of force constants between atoms which are relatively far apart. The theory then involves a large number of undetermined parameters which have no obvious physical justification, and the fact that the elastic constants of germanium satisfy an identity which might suggest that only interactions between adjacent atoms are important, has to be dismissed as a coincidence. In this paper we extend the Born-von Karman theory of lattice dynamics to apply to a simple model of the germanium crystal, in which each atom is regarded as a charged core coupled to an oppositely charged shell. This gives the atom the property of polarizability, not only in an electric field but also under the influence of bonding interactions between adjacent atoms. On the basis of this model, the frequency-wave number relations can be reasonably well accounted for with only two disposable parameters, and a simple explanation is provided of the fact that the elastic constants satisfy Born's identity. The value deduced for the polarizability of a germanium atom, using the neutron spectroscopy data of Brockhouse & Iyengar, is in good agreement with that determined directly from the dielectric constant. An extension of the theory of Mashkevich & Tolpygo provides some theoretical justification for the use of a shell model for germanium.
TL;DR: In this paper, the displacements due to a radiating point source in an infinite anisotropic elastic medium are found in terms of Fourier integrals, yielding explicit expressions for displacements at points far from the source.
Abstract: The displacements due to a radiating point source in an infinite anisotropic elastic medium are found in terms of Fourier integrals. The integrals are evaluated asymptotically, yielding explicit expressions for displacements at points far from the source. The relative amplitudes of waves from a point source are thus determined, and it is found that although in general the decay of wave amplitudes is proportional to the distance from the source, it is possible that in certain directions the decay is less than this. The method used in this paper is also shown to be an alternative way of deriving known results concerning the geometry of the propagation of disturbances. As an example, the radiation in a transversely isotropic medium from an isolated force varying harmonically with time is discussed.
TL;DR: In this paper, the authors show that dislocations occur in piled-up groups as a result of attractions between the piled-ups and isolated dislocation on other systems, leading to the formation of short segments of the resultant dislocation and of networks.
Abstract: Experiments by transmission electron microscopy on thin foils of stainless steel have enabled the interaction of dislocations on different slip systems to be observed in detail. Interactions are most frequently observed in piled-up groups as a result of attractions between the piled-up dislocations and isolated dislocations on other systems. The combination of dislocations takes place over short lengths only, leading to the formation of short segments of the resultant dislocation and of networks. The interactions are first discussed theoretically in terms of undissociated dislocations and then in terms of ribbon dislocations. Various configurations of the interacting ribbons are derived corresponding to the possible combinations of Burgers vectors which lead to attraction. Interesting features are found to occur at the nodes of the interacting and resultant dislocations; some nodes are contracted, others are extended into wide regions of fault. In some cases the structures of nodes have been resolved, so that direct comparison with theory can be made. The agreement between theory and observations verifies the dislocation theory in fine detail; the concepts of stair-rod dislocations and LomerCottrell interactions all appear to be well founded. Other deductions from the observations include an estimate of stacking-fault energy from an extended node, and the probable nonexistence of extrinsic faults. The factors determining the lengths over which the interactions take place are also briefly considered.
TL;DR: In this paper, the velocity of ultrasonic waves has been measured in a number of gases at 25 degrees C and for values of the ratio, ultrasonic frequency/pressure, ranging from 2
Abstract: The velocity of ultrasonic waves has been measured in a number of gases at 25 degrees C and for values of the ratio, ultrasonic frequency/pressure, ranging from 2 $\times $ 10$^{5}$ to 2 $\times $ 10$^{7}$ c s$^{-1}$ atm$^{-1}$. Dispersion, corresponding to a single vibrational relaxation process was shown by acetylene, CD$_{3}$Br and hexafluoro-ethane; and, to a double relaxation process, by ethane. Incipient dispersion was shown by propane, ethyl chloride, ethyl fluoride and dimethyl ether. No dispersion was shown by 1.1-difluoro-ethane, n-butane, iso-butane, neo-pentane and ammonia. Correlation of these with previous results leads to the conclusion that: (a) For molecules with a distribution of fundamental frequencies, such that there is only a small gap between the lowest and the remaining frequencies, vibrational activation enters via the lowest mode and spreads rapidly to the other modes, giving rise to a single relaxation process involving the whole of the vibrational energy. The chief factors determining the probability of excitation of the lowest mode are its frequency and the presence or absence of hydrogen atoms in the molecule. Molecules containing two or more hydrogen atoms suffer translational-vibrational energy transfer very much more easily than other molecules. Deuterium has almost the same effect as hydrogen. (b) For molecules, in which there is a large gap between the lowest and the remaining fundamental frequencies, a double relaxation process occurs. The complex energy transfer probabilities involved do not fit the same quantitative functional relation with vibrational frequency as in (a) above. (c) Torsional oscillations due to hindered internal rotation behave similarly to other fundamental modes. For molecules in which there is a large gap between the torsional frequency and the other modes (e.g. ethane) a double relaxation process occurs as in (b). Where there is no such gap, vibrational energy enters all modes via the torsional mode as in (a).
TL;DR: In this article, a potential flow model is constructed, in which the vortex layer is replaced by a vortex sheet of spiral form, and the problem is then reduced to a two-dimensional one by the use of slender-body theory and the assumption of a conical velocity field.
Abstract: The flow at incidence to a slender delta wing with a sharp leading edge usually separates along this edge, i.e. a vortex layer extends from the edge into the main flow. This layer rolls up above and inboard of the leading edge to form a region of high vorticity which strongly influences the flow pattern. This paper gives a theoretical treatment, more complete than those hitherto available, of this type of flow. A potential flow model is constructed, in which the vortex layer is replaced by a vortex sheet of spiral form, and the problem is then reduced to a two-dimensional one by the use of slender-body theory and the assumption of a conical velocity field. The boundary conditions expressing that the vortex sheet is a stream surface and sustains no pressure difference determine, in principle, its shape and strength. In practice, the inner part of the spiral and the finite, outer part of the spiral, which joins the inner part to the leading edge, are treated separately. The inner part is regarded as small and a solution is given for it which is asymptotically correct as the centre of the spiral is approached. The outer part is replaced by a sheet whose shape and strength depend on a finite number of parameters; these parameters are determined by applying the boundary conditions at isolated points. Results are given for the shape of the sheet and the pressures, loadings and forces on the wing, as functions of the ratio of the incidence to the aspect ratio.
TL;DR: In this paper, a criterion that must be satisfied by the positive-ion energy distribution at the edge of a sheath surrounding a negative probe is derived for the case when negative ions are present.
Abstract: A criterion that must be satisfied by the positive-ion energy distribution at the edge of a sheath surrounding a negative probe is derived for the case when negative ions are present. This criterion is then used to derive the potential outside the sheath region surrounding a spherical probe immersed in an electro-negative plasma. It is found that the potential falls to low values when the ratio of negative ions to electrons exceeds 2. Under these circumstances the positive-ion current collected is the random current across the sheath edge. If, however, the ratio is much less than 2 then the collection of positive ions proceeds as for an electro-positive gas.
TL;DR: In this paper, it was shown that the ratio κ / Dϵ = 0·633, where D is the molecular diffusivity of the solute when the porous medium is absent, is a property of the porous material alone and can be determined directly by diffusion measurements.
Abstract: If a long vertical tube filled with porous material contains a viscous solution, the density of which increases with height as a result of the presence of the dissolved substance, the equilibrium of the liquid is stable provided that the density gradient does not anywhere exceed the value d ρ /d Z = 3·390 μκ / gkb 2 Here κ , the diffusivity of the solute through the saturated porous medium, is defined to be the quantity of solute diffusing across unit area within the porous medium per unit time under unit density gradient. The above expression for the density gradient at neutral stability has been compared experimentally with Taylor’s value for the corresponding density gradient in a vertical capillary tube. For a porous medium consisting of randomly packed glass spheres of mean diameter about 0·2 mm and porosity ϵ = 0·365, it has been found that the two results are consistent provided that the ratio κ / Dϵ = 0·633, where D is the molecular diffusivity of the solute when the porous medium is absent. As this dimensionless ratio is a property of the porous material alone, it can be determined directly by diffusion measurements. An alternative method of measuring κ / Dϵ , based upon an electrical analogue, has led to a value of 0·641 for the same porous material, which is in good numerical agreement.
TL;DR: In this paper, the Hartree and Hartree-Fock formulations for perturbed atomic systems are discussed and a quantitative assessment of the accuracy of the two procedures is provided by the calculation of the dipole and quadrupole polarizabilities of helium, a variation-iteration method used to solve the relevant equations.
Abstract: The equations of the Hartree and Hartree-Fock formulations for a perturbed atomic system are discussed. It is pointed out that there are two alternative procedures, one of which is correct to first order in the error of the unperturbed wave function, but not the other, and explicit expressions are written down for the error in the derived perturbed energies in the two cases. A quantitative assessment of the accuracy of the two procedures is provided by the calculation of the dipole and quadrupole polarizabilities of helium, a variation-iteration method being used to solve the relevant equations.
TL;DR: In this paper, a modified spherical polar co-ordinates were formulated in modified spherical coordinates and the results of calculations made for R = 5, 10, 20, 40 were presented in the form of stream function and vorticity distributions; and further results of pressure distributions, velocity distributions, and drag coefficients, calculated from them.
Abstract: Relaxation methods are outlined, and the present problem formulated in modified spherical polar co-ordinates. The results of calculations made for R = 5, 10, 20, 40 are presented in the form of stream function and vorticity distributions; and further results of pressure distributions, velocity distributions, and drag coefficients, calculated from them . These results are shown to compare favourably with experimental work, showing a steady trend from symmetrical Stokes’s flow, towards boundary layer flow. The phenomenon of separation of the forward flow and development of a circulating wake, is explained and illustrated, the first formation of a wake being a R = 17.
TL;DR: In this article, a set of generators for the centre of the integral group ring of the symmetric group on n letters is presented. And a new proof is obtained of Nakayama's criterion for two representations of the group to belong to the same p -block.
Abstract: A set of generators is exhibited for the centre of the integral group ring of the symmetric group on n letters. As an application, a new proof is obtained of Nakayama’s criterion for two representations of the group to belong to the same p -block.
TL;DR: In this paper, a Druyvesteyn analysis was carried out on a spherical probe, and the second derivative of the current-voltage characteristic was found from the amplitude of the sine wave with the chopping frequency present in the probe current.
Abstract: The theory of the positive column is commonly based on the assumption of a Maxwellian energy distribution for the electrons, though in many cases it is far from clear how such a condition may be brought about. Moreover, past work has shown that the distribution is by no means always Maxwellian. Indeed, in any detailed consideration of the rates of inelastic collision processes occurring in the plasma it is desirable to be able to determine the distribution experimentally. This paper reports a method of carrying out a Druyvesteyn analysis electronically, and gives a critical account of its performance. A high-frequency voltage of small amplitude chopped with a certain lower frequency is applied to a spherical probe, and the second derivative of the current-voltage characteristic is found from the amplitude of the sine wave with the chopping frequency present in the probe current. An adequate signal-to-noise ratio is obtained by using narrow band amplification with phase-sensitive detection. A large amount of data has now been obtained with this method in a variety of striated discharges. It is found that in every case studied so far the energy distribution takes the form of two well-separated groups of electrons with sometimes a very small third group. The high-energy group is generated by the potential difference across the striation head and becomes progressively attenuated towards its tail. An effect of the varying distribution is a sudden increase in the net rate of loss of electrons from the low-energy group resulting from the fall in number of electrons capable of ionizing and the change in potential difference between wall and discharge at the end of the striation. This loss of electrons causes a fall in the local discharge conductivity and so gives rise to another potential step and striation head.
TL;DR: The asymptotic shape of the schlieren pattern, which would be approached with time, has been found and is illustrated in which boundaries are present which tend to become sharper through interaction, instead of spreading continually.
Abstract: When a solution is analyzed by means of electrophoresis, or sedimentation in the ultracentrifuge, the schlieren pattern, which is related in a simple way to the gradients of concentration at the junction of solution with solvent, generally gives a direct measure of the concentrations of the various species of macromolecule present. This is no longer true if reversible interaction occurs between the primary molecules in the solution, and the present paper endeavours to show how interaction modifies the schlieren pattern. A reversible reaction A + B = C is treated, and equations for the schlieren pattern are deduced which cover all possible values of the concentrations of A and B , for a given equilibrium constant, and all values of the velocity of C relative to that of A and B . It has not been possible to include the effects of diffusion, and finite rate of reaction, but, instead, the asymptotic shape of the schlieren pattern, which would be approached with time, has been found. Examples of typical patterns are given, and cases are also illustrated in which boundaries are present which tend to become sharper through interaction, instead of spreading continually. Published experiments on the electrophoresis of interacting systems in which two components appear to be present in one boundary system and three in the conjugate boundary system find an explanation in these patterns of the model system.
TL;DR: In this article, the non-relativistic problem of scattering of a particle by a target possessing discrete excited states can be expressed in terms of physical resonance states, i.e. solutions of the wave equation for complex energy in which in the asymptotic form of wave function in each channel one of the two possible exponential terms (which for real energy represent the incoming and outgoing wave) vanishes.
Abstract: The non-relativistic problem of scattering of a particle by a target possessing discrete excited states can be expressed in terms of ‘physical’ resonance states, i.e. solutions of the wave equation for complex energy in which in the asymptotic form of the wave function in each channel one of the two possible exponential terms (which for real energy represent the incoming and outgoing wave) vanishes. This representation is possible provided the interaction between the particles and the target vanishes exactly beyond a certain distance. If the interaction decreases exponentially a similar representation may in some cases still be obtained by analytic continuation; it contains also ‘redundant’ eigenstates in which the coefficient of one of the asymptotic waves tends to infinity. Possible generalizations of the method are discussed.