Showing papers on "Dirac delta function published in 1971"

The foundations of acoustics

Abstract: Preface The Symbols Historical Introduction I. Equations and Units II. Complex Notation and Symbolic Methods III. Analytic Functions: Their Integration and the Delta Function IV. Fourier Analysis V. Advanced Fourier Analysis VI. The Laplace Transform VII. Intergral Transforms and the Fourier Bessel Series VIII. Correlation Analysis IX. Wiener's Generalized Harmonic Analysis V. Transmission Factor, Filters, and Transients ("Kupfmuller’s Theory") XI. Probability Theory, Statistics, and Noise XII. Signals and Signal Processing XIII. Sound XIV. The One-Dimensional Wave Equation and Its Solutions XV. Reflection and Transmission of Plane Waves at Normal Incidence XVI. Plane Waves in Three Dimensional XVII. Sound Propagation in Ideal Channels and Tubes XVIII. Spherical Waves, Sources, and Multipoles XIX. Solution of the Wave Equation in General Spherical Coordinates XX. Problems of Practical Interest in General Spherical Coordinates XXI. The Wave Equation in Cylindrical Coordinates and Its Applications XXII. The Wave Equation in Spheroidal Coordinates and Its Solutions XXIII. The Helmholtz Huygens Integral XXIV. Huygens Principle and the Rubinowicz—Kirchhoff Theory of Diffraction XXV. The Sommerfeld Theory of Diffraction XXVI. Sound Radiation of Arrays and Membranes XXVII. The Green's Functions of the Helmholtz Equation and Their Applications XXVIII. Self and Mutual Radiation of Impedance Tables References Subject Index List of Symbols

On the Singularity Expansion Method for the Solution of Electromagnetic Interaction Problems

Abstract: : This note develops a new method for the solution of EMP interaction problems. Basically it involves expanding the solution in terms of its singularities in the Laplace transform or complex frequency (or s) plane. In the time domain each term comes from an inverse transform of the corresponding term in the singularity expansion. Finite size objects with well behaved media have only poles in the finite s plane for their delta function response. These factor into terms involving the classical natural frequencies and modes but in addition bring out factors which we call coupling coefficients as well as the possiblity of higher order poles besides simple poles, but still of finite order in the finite s plane. If the incident waveform has singularities in the finite s plane the response can be generally split into an object part (containing object poles) and a waveform part containing the waveform singularities. The object poles directly give amplitudes, frequencies, damping constants, and phases for the damped sinusoidal waveforms seen so commonly in EMP tests using pulsed waveforms. There is some latitude in the calculation of coupling coefficients and some difficulties are discussed. (Author)

One-Dimensional Electron Gas with Delta-Function Interaction at Finite Temperature

Abstract: In recent papers Gaudin) and Yang) gave the ground state energy of a onedimensional electron gas with a delta-function interaction)-S) as a solution of a set of coupled integral equations. We try to treat the thermodynamic properties of this system as a one-dimensional Bose gas and a one-dimensional Heisenberg model.> For this purpose it is necessary to obtain all of the energy eigenvalues of the Hamiltonian. In § 2 we review the work of Gaudin and Yang on the wave function. There appear two kinds of parameters k and A. In § 3 we make conjectures on the distributions of k's and A's in the complex plane. In § 4 the energy spectrum of the Hamiltonian for repulsive interaction is obtained and the integral equations which describe the thermodynamic properties are derived. In § 5 these integral equations are solved for some special cases. In §§ 6 and 7 we treat the electron gas with an attractive delta-function interaction.

Solution of the Delta Function Model for Heliumlike Ions

Abstract: A one‐dimensional Fredholm integral equation is derived for the ground state solution of the delta‐function model for two‐electron heliumlike ions. This equation is solved numerically; the perturbation series is developed through E(20) and compared with the solution of the integral equation. The series is further analyzed in terms of the singularity which determines its radius of convergence.

Kinetics of Charge Scavenging in γ-Irradiated Liquid Cyclohexane: A Comparison Between Different Ion–Electron Spacial Distribution Functions Used with Two Diffusion Approximations

Abstract: The form of the spacial distribution function of ions and electrons in an irradiated liquid is not accurately known and several types of function have been assumed by different workers. The various forms are compared and used in the calculation of electron–solute reaction yields in liquid cyclohexane. Our previously used semi-empirical distribution function (YSE) and a power function (YP) give satisfactory I results, but a three dimensional, single parameter Gaussian (YG1) and a delta function (YD) do not. The prescribed diffusion approximation has also been inserted into the model for comparison with earlier work. No improvement was obtained. Whereas our previous approximations may lead to an underestimate of the geminate neutralization times, the prescribed diffusion approximation may lead to an overestimate of them.

Extension of the Theory of the Boundary Diffraction Wave to Systems with Arbitrary Aperture-Transmittance Function

Abstract: In 1962, Miyamoto and Wolf succeeded in formulating the boundary-diffraction-wave theory for a general incident wave. In this paper, the theory is applied to the aperture having an arbitrary transmittance distribution, and it is found that every point where the gradient of the transmittance distribution is not zero is the origin of a secondary wave. The boundary diffraction wave can be expressed by the wave that originates at the points where the gradient of the transmittance distribution is the Dirac delta function. Hence, it seems reasonable that the diffraction wave, which is generated from the aperture point where the gradient of transmittance is not zero, is more general for discussion of diffraction problems.

Shadow channels in potential theory.

Abstract: The effect of coupling a shadow channel to a real channel is studied in the framework of nonrelativistic potential-scattering theory. For simplicity, $s$-wave scattering with energy-independent $\ensuremath{\delta}$-function potentials is considered. This model has the virtue of being exactly solvable in configuration space. The main features of the solution which are analogous to those expected in a relativistic field theory whose divergences have been removed using Sudarshan's shadow states, are as follows. The coupled problem reduces in the real channel to scattering by an effective energy-dependent pseudopotential. The physical scattering amplitude still satisfies elastic unitarity at all energies, but is piecewise analytic, with a point of nonanalyticity occurring at the threshold for the shadow channel.

Response of a Semi-Infinite Elastic Solid to an Arbitrary Line Load Along the Axis

Abstract: The axisymmetric problem of a line load acting along the axis of a semiinfinite elastic solid is solved using Hankel transforms. In this solution the line load is interpreted as a body force loading and by assuming the line load to be of the form of a Dirac delta function the solution of Mindlin's problem of a point load within the interior of the half space is obtained. Solutions of this problem presented in the literature have been obtained using semiinverse techniques whereas the solution given here is obtained in a systematic step-by-step manner.

Distribution theory and thin shells in general relativity

Abstract: Integral containing Heaviside and Dirac delta function in integrand with minus and plus infinity limits, computing value using thin spherical shells in general relativity

Could classical states also be distributions with extended supports.

Abstract: Unlike states in quantum mechanics or kinetic theory, it is generally believed that a classical state is a function with no spread; i.e., it does not involve any statistics. But a unified theory of dynamical processes suggests the plausibility of substituting for it the functional of error distribution.

Wave propagation in a rod with distributed-yielding hysteresis☆

Abstract: This study presents the first-order non-linear solutions for the propagation of stress and strain waves sent by a sinusoidal force from one end of a semi-infinite rod with distributed-yielding hysteresis to its other end at infinity. It is found that the width of the distribution spectrum of the yield limits has a strong influence on the amplitudes, the phase differences, and the velocity of propagation of the waves unless the level of external excitation is below a certain value. The results show that: (a) the asymptotic value of the decaying wave amplitude is highly dependent on the width of the distribution spectrum of the yield limits; (b) both the stress and the strain waves have a considerable phase lag behind the external excitation in the portion near the remote end of the rod, but the stress wave leads the strain wave by a small amount of the first order; (c) in general, waves move slower in the portion near the driven end and faster in the portion near the remote end. Solutions for the bilinearly hysteretic rod are obtained in a special subcase in which the distribution function for the yield limits takes the form of a Dirac delta function.