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A new type of plasma instability is described in an electron‐beam ionized discharge. This instability occurs in the form of current oscillations in gas mixtures in which the dissociative attachment rate increases strongly with electric field. It has been observed experimentally in He : H2O 74 : 2 and in He : CO2 1 : 1 and 9 : 1 mixtures. A theoretical analysis which describes the onset of instability is presented, and a physical explanation is given. The region of instability in parameter space of ionization source function S and electric field E is presented for certain gas mixtures. The transient phenomena in this type of discharge appear to be controlled by electron attachment, and a theoretical explanation of the observed current overshoot has been provided.

We consider the effect of velocity fields upon the transfer of line radiation by two-level atoms. We show that a simultaneous solution of the radiation transfer equation and the time-dependent rate equations leads to an equation for the source function which contains the Lagrangian derivative. We discuss a physical interpretation of the derivative term and present a method for solving this type of problem. We exhibit calculations which show that, for quite reasonable velocity fields, large errors can be produced if the derivative terms in the rate equations are neglected.

Recently, determination of the resolvent kernel of Milne's integral equation for a spherical, isotropically-scattering medium with internal sources has been made by several authors (cf. Heaslet and Warming ; (1) Nagirner ; (2) Wilson and Sen (3) ). In this paper, it is shown how to compute the resolvent kernel of the above Milne's equation in terms of the modified Sobolev's Φ-function, which is reduced to the angular integration of the source function in the diffuse radiation field by a finite slab. In other words, once the X - and Y -functions of a slab with twice the optical radius of the sphere have been computed and a Cauchy system for the source function has been solved, the resolvent kernel under consideration can be determined by integration of the modified Sobolev Φ-function.

The fully renormalized kinetic theory (FRKT) previously discussed by the author in the case of self-diffusion is extended to the case of density fluctuations. In the theory techniques are developed for calculating classical phase-space time-dependent correlation functions. The method centers around the development of an exact expression for the memory function associated with the phase-space-correlation function. This exact expression is written in a compact and symmetric form that is convenient for making approximations. It is shown explicitly how one can make contact with the Boltzmann-Enskog approximation for the memory function which is valid for moderate densities, as well as the terms which lead to the much-discussed ${t}^{\ensuremath{-}\frac{2}{3}}$ long-time behavior. The most striking difference between self-diffusion and the case of density fluctuation arises in the region of the critical point. It is shown in the theory how to make contact with the mode-mode coupling results of Kawasaki. From a formal point of view the development of the FRKT is carried forward three important steps in this paper. It is shown how to relate the memory function to the two-particle source function in a more powerful and direct method than the projection-operator approach developed in FRKT I. After a simple rearrangement, the memory function is written in a far more symmetric form than in FRKT I, and finally it is shown how the ideas of connectedness and cumulants can be successfully introduced into the theory.

An important problem in radiative transfer is finding the radiative fields produced by various illuminations (both external and internal) of a plane-parallel, inhomogeneous, absorbing, emitting, and anisotropically-scattering finite medium. One approach to a solution is to find the source function, which represents the rate of production of scattered radiation per unit volume and solid angle, at each point in the medium. The present study develops the existence of a Green's function, called the fundamental source function, which separates the optical properties of the medium from the driving illumination. Radiative linearity then allows the representation of all possible source functions as convolutions of the illumination with the fundamental source function. Parametric differentiation (invariant imbedding) is used to replace the governing linear integral equation for the fundamental source function with a set of differential equations appropriate for numerical integration. This approach for finding the fundamental source function leads naturally to the introduction of fundamental scattering and transmission functions. Our inclusion of anisotropic internal illumination (sources) allows us to develop four new reciprocity relations involving these functions. The reciprocity relations state general equivalences between an internally and an externally driven medium and thus greatly reduce the complexity of radiative transfer.

Considering a homogeneous, isotropically scattering target slab of optical thickness x, containing an internal plane emitting source, the aim of the present paper is to determine the distribution of the internal emitting source by measuring the angular distribution of the intensity of radiation emergent from the slab. First, a system of differential recurrence relations for the intensity of finite order scattered radiation emergent from the top is deduced. In other words, a Cauchy system for determining the finite order emergent intensity is expressed in terms of a source function in the diffuse radiation field and the Fredholm resolvent that were computed in a preceding paper [cf. Bellman, Ueno, and Vasudevan, Math. Biosci. 15, 195–203 (1972)]. Then, the quasilinearization technique is used to solve the inverse problem of determining characteristics of the internal source by measuring the radiation field emergent from the top. This method will be employed in future computations relating to realistic situations in the radiation diagnosis and therapy.

Summary A two-dimensional brittle unilateral shear fracture was numerically simulated to form and stop in an elastic isotropic continuum. The final displacements and stresses for unilateral fracture were found to be similar to those for bilateral fracture. However, during and shortly after fracture, dynamic differences in the two types of fracture exist. Displacements normal to the fracture are more complex in the unilateral case because the fracture rotates about its instantaneous centre which moves as the fracture grows larger. For displacements normal and parallel to the fracture, the rise times are substantially shorter at the terminal end than at the starting end of the fracture. Overshoot is observed on all displacements near the unilateral fracture model which incorporated a dynamic coefficient of friction equal to 0·05 along the fracture surface.

We discuss a model for the formation of the chromospheric Ca ii K line which does not make the usual assumption of complete redistribution. Using a physically reasonable scattering model, we find significant departures due to the frequency dependence of the line source function, particularly in the relative intensity and centre-to-limb behaviour of the K1 parts of the line and in the asymmetry produced by differential velocity fields. We conclude that the frequency dependence of the K line source function must be considered in quantitative models for the formation of the K line.

This is a study of source characteristics of 13 earthquakes with magnitudes between 2.4 and 5.1 located near the San Andreas fault in central California. On the basis of hypocentral locations and fault-plane solutions the earthquakes separate into two source groups, one group clearly related to the throughgoing northwest-trending San Andreas fault zone and the other apparently associated with generally north-trending bifurcations such as the Calaveras fault. The basic data consist of broad-band recordings (0.03 to 10 Hz) of these earthquakes at two sites of the San Andreas Geophysical Observatory (SAGO). Epicentral distances range between 2 and 40 km, and maximum ground displacements from 4 to 4000 microns were recorded. The whole-record spectra computed from the seismograms lend themselves to source parameter studies in that they can be interpreted in terms of low-frequency level, corner frequency, and high-frequency slope. Synthetic seismograms have also been used to estimate source parameters in both the time domain and frequency domain, and the results compare favorably with those estimated directly from the spectra. The influences of tilts and nonlinear response of the seismometer were considered in the interpretation of the low frequencies. Seismic source moments estimated from the low-frequency levels of the spectra show a linear dependence on magnitude with a slope slightly greater than 1. The geology at the recording site can contribute an uncertainty factor of at least 3 to the estimated moments. Observed corner frequencies are only weakly dependent on magnitude. Interpreted in terms of source dimension, these corner frequencies imply values of 1 to 2 km for the earthquakes of this study. The corner frequencies may also be interpreted in terms of the rise time source function, yielding values in the range 0.5 to 1.0 sec. The data indicate that the earthquakes of this study are all surprisingly similar in their fundamental source parameters, with only the seismic moment showing a strong dependence on magnitude.

Exact solutions are given for some problems of radiative transfer through media of circular cylindrical symmetry surrounding a line source. Absorption, emission, and scattering cross sections are taken to decrease inversely with distance from the cylinder axis. The source function in the media is assumed to be proportional to the mean intensity. It is shown that the surface intensity can be obtained without explicit knowledge of the source function and that the radiation field for these problems remains anisotropic at arbitrarily large optical depths. In spite of infinitely large optical depth between surface and axis, physical solutions exist for the case of admixture of a stimulated-emission component. The maximum allowed contribution from this component is calculated. The method of variable-moment quotients for computing the mean intensity iteratively in cylindrical geometries is discussed.

The classical ray-series method for electromagnetic wave propagation in inhomogeneous media is applied to the problem of wave propagation in isotropic, homogeneous, linear viscoelastic media characterized by virtually arbitrary time-dependent relaxation or creep functions. The full three-dimensional treatment is presented, followed by the specialization to the one-dimensional propagating pulse problem. In this last case, the ray-series is evaluated numerically for the creep function $$\psi (t) = \frac{1}{\mu }\left\{ {1 + \frac{q}{\alpha }\left. {\left[ {\left. {\left( {1 + \frac{1}{\tau }} \right)^\alpha - 1} \right]} \right.} \right\}} \right.H(t)$$ for various model parameter ranges and for various initial source functions.

A non-LTE formulation is given for the transfer of radiation in the autoionizing lines of neutral aluminum at λ1932 and λ1936 through both the Bilderberg and Harvard-Smithsonian model atmospheres. Numerical solutions for the common source function of these lines and their theoretical line profiles are calculated and compared with the corresponding LTE profiles. Our results show that the non-LTE profiles provide a better match with the observations. They also indicate that the continuous opacity of the standard solar models should be increased in this wavelength region if the center-limb variations of observed and theoretical profiles of these lines are to be in reasonable agreement.

This paper presents various methods of synthesizing sources placed on a conical surface to produce a prescribed radiation pattern. The sources considered are a series of electric dipoles placed in free space on the surface of the cone around a set of circumferences. These circumferences are equidistant from each other, and the dipoles are oriented in the circumferential direction with no circumferential variation in intensity and phase. A successive approximation method is employed to obtain the source function for those sources which are placed in a region where the circumferences measure approximately less than two wavelengths. For sources placed in the region where the circumference is less than approximately (4/3)\pi wavelengths, an expansion formula of the product of the Bessel and exponential functions is employed. When the sources are located far from the cone tip, a method utilizing the maximum points of the Bessel function is used to compute the source function. In considering the beamwidth and the sidelobe level, the Tchebycheff pattern with the tapering effect is employed. Numerical examples are given to illustrate the effectiveness of the methods.

The examination of array records from underground nuclear explosions has produced the qualitative picture of the generation of simple signals which are transmitted to teleseismic distances along paths which exert comparatively little effect. At the recording sites, complications in the recorded signal are caused by crustal inhomogeneities local to the site. By using records from exceptionally good sites, or those where processing techniques eliminate the complications in the records, the combined effects of source signal and transmission path can be studied. To a first approximation, explosion source functions have been quantitatively defined and therefore quantitative data about the transmission path can be deduced. Specifically, data on the non-elastic properties of the mantle have been obtained. With the source and transmission paths defined, the character of the record complication at the poorer sites can be examined with the hope of producing deconvolution operators largely to eliminate the effects of crustal inhomogeneities. Conversely where the recording site characteristics are known, then records from earthquakes at teleseismic distances can be examined with a view to determining the source function. Preliminary work on the quantitative evaluation of earthquake source functions will be described.

A singular integral equation is derived for the three-dimensional Fourier transform of the source function, when the scattering atmosphere is contained in a finite convex volume.This equation is shown to reduce to the usual equation in the case of an isotropic point source in a finite spherical atmosphere of radius R0, and is used to solve the same problem when the source is anisotropic.It is shown that in the latter case an expansion in Legendre polynomials results, in which the coefficients are obtained from the integral equations of a similar construction to those for an isotropic source. The error in taking the dominant part is now however of order 1/R0 and not e–R0.

By using an especially simple form of the Sommerfeld-Watson transform, the equivalence is established of quite different appearing expressions for the electrostatic potential obtained by expanding the source function in orthogonal functions of two different coordinates.

Equations are constructed to represent quasi-stationary mean flow of momentum and heat on a spherical earth, averaged over a long period of time such as a year and over latitude circles. The crucial shearing Reynolds stress associated with meridional transfer of zonal velocity is assumed to depend linearly on a product of the earth's angular velocity, Ω, and the meridional gradient of mean temperature; the shearing stresses associated with vertical transfer of zonal velocity and of meridional velocity are assumed to depend linearly on the vertical gradients of zonal and of meridional mean velocities respectively, and the mean eddy transfer of heat along a meridian is assumed to depend linearly on the mean meridional temperature gradient. All proportionality coefficients are taken to be independent of latitude. Two forms are assumed for the non-adiabatic atmospheric heat source function, Q, used in the thermodynamic equation. In the first case Q is assumed known (from analyses of observations) as a function of height and latitude. In the second case, Q incorporates a heating term which is partly controlled by the model itself and represents some of the characteristics of sensible and latent heat transfer. A solution of the basic equations is obtained in both cases in the form of double expansions in powers of two parameters, one depending on Ω and the other on ΔT, the mean annual temperature difference between equator and pole. The solution is evaluated using Fourier techniques. The series expansions are found to be reasonably convergent for realistic values of the various parameters involved, three terms only being required in the ΔT expansion and five terms at most in the Ω expansion, but extensive numerical evaluation by digital computer is involved: the region considered is bounded by the tropopause and lies between the equator and 70° latitude. The computed zonal velocity has the characteristic east-west variation with latitude and a broad band maximum of 19 m sec−1 and the meridional velocity the characteristic tricellular structure. A poleward eddy angular momentum flux and polar inversion are predicted. The results, through verification of the postulates, add support to the Rossby view of the general circulation in which the cyclonic-scale eddies act to release potential energy of the atmosphere to supply their own kinetic energy and form the mean zonal kinetic energy. They further indicate the value of the reconstructed ‘austausch’ approach for this problem.

Summary The source and character of signal generated noise at Eskdalemuir array (EM) is demonstrated. Analysis of records elsewhere demonstrates the the site dependence of the phenomenon. Criteria for selecting sites for standard three component instrumentation are formulated on the basis of these observations. A plane P-wave front impinging on a plane free surface will be totally reflected as body waves, but if the surface is irregular part of the energy is converted to surface waves (Hudson, in press). We describe these surface waves as signal generated noise. They are impossible to identify on standard seismographs, and the magnitude of the effect escaped notice until large explosions were studied by means of large arrays of seismometers. Thirlaway (1966) discussed the experimental evidence for the hypothesis that between distances of 30-90" (the ' source window ') the P-wave initiated by an explosive source was recorded as a pulse which was undistorted during transmission. Carpenter (1966) showed that the pulse observed at these distances could be quantitatively derived from the known source function of explosions by assuming an Earth model in which the elastic parameters are a function only of the Earth's radius. This hypothesis is, apparently, not supported by the records at some standard stations. h these, the P-wave coda is unacceptably extended, and the coherence from station to station is low. We suggest that the presence of signal generated noise may be one reason for the inconsistency. Evidence for this came from a detailed analysis of explosion records from four U.K.A.E.A. type arrays. They are located at Eskdalemuir, Scotland; Yellowknife, Canada; Gauribidanur, India and at Tennant Creek, Australia. Each array consists two lines of vertical component seismometers at right angles, each line having instruments, with the exception of Gauribidanur, which at the time of this study, bad only half that number. The seismometers are spaced at intervals of 24 km except Eskdalemuir where the spacing is 896 metres. used to identify signal generated noise by measurements its origin, and by comparing the total delayed and summed ements in the array. In some cases particle orbits have been ee components. e reality of these P-generated surface waves, we searched for e examples in records of stations located in topographically rugged areas. (Records of explosion and deep focus earthquake signals at stations like La Paz, for example, display P-codas that are nearly always extended). Detailed analysis of

The recent work of Sen and White [1] on the excitation of large-scale oscillations in an atmosphere on a rotating globe by gravitational and thermally induced forces is extended to include heating in any portion of the atmosphere. The region where the heating occurs is unrestricted regarding temperature profile. The solution χ(z) of the radial differential equation of motion consists of (1) a complementary function yex/2, and (2) a particular solution q(z)/γH(z) + ex/2 I(x) where I is obtained by integrating the heat source function (with appropriate weight factor) from the top of the atmosphere to the level in question, γ is the ratio of specific heats, H(z) is the scale height, x = ∫dz/H where z is the altitude, q is the heat source function, and χ(z) is the divergence of the velocities. With this representation of the wave function, one recovers the expressions for the velocity components and pressure variations of Pekeris [2] and of Weekes and Wilkes [3] for the purely gravitational case, provided the wave function y is replaced by [y + I]. General expressions are also obtained for the vertical and horizontal displacements and for the amplification over equilibrium tide at any level. The latter expression is useful for establishing the so-called solar control of the dynamics of a given layer by radiation absorbed in any other layer. The boundary conditions of Weekes and Wilkes [3] are shown valid for a "top" with constant positive temperature gradient, believed to be the case above 100 km. Finally, the solar and lunar semi-diurnal winds as collected by Briggs and Spencer [4] for the E region when tested by present theory are found to have the same azimuthal dependence as exists at the ground. This is consistent with the original assumption of “variables separable.”

It is shown that light and dark contrasts observed in the radiation of strong lines may arise from lateral differences in the local value of the scattering term alone, independently of the local values of temperature and pressure. Mechanisms for the production of such differences are suggested, and regions are pointed out in which they occur as 'features.' From numerical examples presented, it is seen that the effect of such features on the emergent radiation field is determined by their position with respect to the shape of the source function.

The thermal maps of Murray and his collaborators have been analyzed statistically into solar effects, latitude variations, and limb darkening. Seeking a pole position for minimum residual variance leads to α = 7.4 ± 2.0°, δ = −68.1 ± 0.9°. The solar effect is opposite to the simple expectation of an increasing temperature from antisolar to subsolar point; this suggests that the main deposition of solar energy takes place somewhat below the levels relevant to the measured emission. Inversion of the limb-darkening curves yields a source function which, if interpreted as a thermal source function, indicates an increasing temperature with depth and cold polar regions, in agreement with microwave data. Finally, analysis of the terminator position indicates a very deep and variable scattering layer and gives some support to a predominantly scattering source function in the 10-μ region.

In the theory of radiative transfer the equation of transfer formulated from the local viewpoint is expressed in terms of two optical parameters, i.e. an albedo for single scattering and an indicatrix of scattering. These optical parameters are assumed classically to be constant throughout the layer under consideration. In the present paper we restrict our discussion to the exact solutions of stationary transfer problems of radiation. The Laplace transform method due to Wiener and Hopf [l] and the Chandrasekhar limiting process [2] are considered to be the local approach for solving exactly the equation of transfer. The above two methods have further been extended by Miss Busbridge [3, 41, Huang [5], and Kourganoff [3], respectively. Reducing the Milne first integral equation to the auxiliary equation by the idea of linear aggregation, Ambarzumian [l] has elegantly developed a new method giving the exact solution of the transfer equation. Then the extension of Ambarzumian’s first method has been done in several transfer problems of current interest in astrophysics: line formation in semi-infinite, coherently and non-coherently scattering atmosphere (Busbridge [4] ; Ueno [6]), and line formation in a coherently scattering atmosphere of finite thickness (Busbridge [7, 41). Furthermore, in 1943 Ambarzumian [8] initiated a new method based on the principle of invariance. The method is considered to be the global approach, because a nonlinear integral equation in +-function can directly be obtained by physical analysis of the condition of transfer without actually solving the Milne first integral equation for the source function on the way. The extension of Ambarzumian’s physical method has been made by Miss

An approximate solution for Milne’s equation for the source function in a homogeneous plasma sphere of large optical thickness, is derived for a class of line profiles with different importance of the wings. The dependence of the strength of the stationary radiation field in the sphere on the relative weight of the wings is shown. In particular, it is demonstrated that when the line wings tend to be so strong that the profile is no longer normalizable, the system behaves for any dimension as if it were optically thin. The mean number of scatterings a photon experiences in the plasma sphere, which is closely connected with the source function, is also considered.

While wave propagation in gyromagnetic media has been investigated extensively in the technical literature, little attention has been given to the excitation of such waves by a prescribed source. This problem is treated in the present paper. To render the analysis tractable, the source function is taken as a time‐harmonic line source of electric currents with a rapidly varying phase, embedded in an infinite homogeneous gyromagnetic medium whose axis of magnetization is perpendicular to the source direction. For further simplification, the operating frequency is taken to be near gyromagnetic resonance. The resulting model incorporates relevant anomalies of the radiation process, and also serves as a prototype for more practical thin slab configurations. The analysis is performed for permeability tensors in which spin exchange effects are ignored and included. In the former case, the refractive index surface may have open branches and gives rise to an infinity in the total radiated power (infinity catastrophe); this anomaly is removed when spin exchange is accounted for. Detailed study of the radiation fields and radiated power densities shows that except for certain initial directions, the outward power flow is almost entirely in the electromagnetic waves, and the total radiated power in the electromagnetic waves far exceeds that in the spin exchange waves, even when the source dimensions tend to zero. These results imply that a localized electromagnetic current source does not strongly excite the spin exchange waves.

In this paper an explicit approximate form for the generalized Biberman-Holstcin factor r~m is derived. The generalized form of the coefficient ~,~,~ for the spectral-radiation distribution "~t (source function) was introduced in (1), in the frame of a pseudoOnm thermodynamic model. The deviation from thermal equilibrium of a bmmdcd plasma is caused by leakage of the radiation energy. This effect perturbs the populations of the excited levels of the atoms, but the velocity distributions of the free particles are assumed to be Maxwcllian. In order to maintain the steady statc thc radiation energy loss is assumed to be compensated by an external source (thermal source, electric current, etc.). The densities AT~ of clectrons and hr~ of atoms in the ground state are assumed to be space independent. The generalized coefficient Y .... is to be determined from the radiation transport equation. For this, in the first step, the isotropic transport equation is solved

Theoretical line contours calculated for fixed values of the line constants and a given model atmosphere show an increase of the stokes parameters Q, U, and V but a decrease of I if the portion of noncoherent scattering increases. These effects increase from the center of the solar disk to the limb. The action of scattering may be approximately simulated in LTE contours by increasing the gradient of the source function and fitting in this way theoretical contours to observed ones. There remains, however, the effect of V- reversal near the line core, which is caused by anomalous dispersion and is abnormally increased by scattering. (auth)

Conventional plane wave array theory does not apply to arrays in an inhomogeneous medium. In a stratified waveguide such as the ocean there are many modes of propagation, and each of them is dispersive. As expressed in normal mode formalism, the transmission between vertical arrays can be considered as a filter problem. In this paper we consider the response of the array filters to an ambient noise field, and maximize the signal‐to‐noise ratio for transmission in a noisy waveguide. The resulting optimum, or matched array, filter is given by the conjugate of the product of the source function and the waveguide transmission function. The response of the matched array filter is weighted with the reciprocal of the noise power in each mode and the attenuation.

A probabilistic model for solving transfer problems in non-homogeneous, isotropic, and non-coherent scattering cylindrical shell media has been proposed. The source function is considered to be frequency independent. The scattering and transmission functions have been defined for the case of complete redistribution in frequency. A tractable integrodifferential equation for the scattering function has been derived.

General solutions of the heat conduction equation are presented for axially symmetric pulse heating of a cylindrical sample with constant surface conditions. The solutions, established by means of Green's functions, are applicable to cases in which the radial and axial components may be treated separately and the heat source function is of the form Q f(x) g(r) ψ(t). Particular results already presented in the literature are derived, as well as more general formulae. Reference is made to a novel form of digital computing technique for finding the values of parameters which give the best correlation of theoretical and experimental temperature-time characteristics.