Finite Space Interval Research Articles

On Tamm's problem in the Vavilov-Cherenkov radiation theory

We analyse the well known Tamm problem, examining the motion of a charge in a finite space interval, with a velocity exceeding the velocity of light in the medium. By comparing Tamm's formulae with the exact ones we prove that the former do not properly describe the conditions for Cherenkov radiation. We also investigate Tamm's formula cos = 1/n, defining the position of the maximum of the field strengths' Fourier components for the infinite uniform motion of a charge. Numerical analysis of the Fourier components of the field strengths shows that they have a pronounced maximum at cos = 1/n only for the charge motion in the infinitely small interval. As the interval is increased, many maxima appear. For the charge motion in an infinite interval there is an infinite number of maxima of the same amplitude. Quantum analysis of Tamm's formula leads to the same results.

Stability and Instability of the Wave Equation Solutions in a Pulsating Domain

The behavior of energy is studied for the real scalar field satisfying d'Alembert equation in a finite space interval 0<x<a(t); the endpoint a(t) is assumed to move slower than the light and periodically in most parts of the paper. The boundary conditions are of Dirichlet and Neumann type. We give sufficient conditions for the unlimited growth, the boundedness and the periodicity of the energy E. The case of unbounded energy without infinite limit (0< lim inf t→+∞E(t) < lim sup t→+∞E(t)=+∞) is also possible. For the Neumann boundary condition, E may decay to zero as the time tends to infinity. If a is periodic, the solution is determined by a homeomorphism [Formula: see text] of the circle related to a. The behavior of E depends essentially on the number theoretical characteristics of the rotation number of [Formula: see text].

Rindler solutions and their physical interpretation

We show that the singular behavior of Rindler solutions near horizon testifies to the currents of particles from a region arbitrarily close to the horizon. Besides, the Rindler solutions in right Rindler sector of Minkowski space can be represented as a superposition of only positive-or only negative-frequency plane waves; these states require infinite energy for their creation and possess infinite charge in a finite space interval, containing the horizon. The positive-or negative-frequency representations of Rindler solutions analytically continued to the whole Minkowski space make up a complete set of states in this space, which have, however, the aforementioned singularities. These positive (negative)-frequency states are characterized by positive (negative) total charge, the charge of the same sign in right (left) Rindler sector and by quantum number κ. But in other Lorentz invariant sectors they do not possess positive (negative)-definite charge density and have negative (positive) charge in left (right) Rindler sector. Therefore these states describe both the particle (antiparticle) and pairs, the mean number of which is given by Planck function of κ. These peculiarities make the Rindler set of solutions nonequivalent to the plane wave set and the inference on the existence of thermal currents for a Rindler observer moving in empty Minkowski space is unfounded.

Subfringe integration profilometry of three-dimensional diffuse objects

A fringe analysis technique is proposed to automatically mea- sure 3-D object shape. A grating pattern is projected onto an object, and the grating pattern deforms in accordance with the 3-D object. This de- formed grating image is analyzed by means of the subfringe integration algorithm (SIA), in which the phase is retrieved using N bucket integra- tion. Only one frame of the image is sufficient to obtain the phase distri- bution and the object shape at every pixel on measured object. Theoret- ical analysis, experimental results, and simulation calculations are presented. © 1997 Society of Photo-Optical Instrumentation Engineers. (S0091-3286(97)03009-2) Optical measurements of 3-D object shapes has found wide application in various fields because of its potential advan- tages: high accuracy and noncontact and fully automated measurement. Most of the current research is aimed at de- veloping the phase measurement technique, and several methods of phase shifting to measure the 3-D object shape have been developed. The phase shifting moiretopography technique 1 can achieve much higher accuracy, but it re- quires a complicated phase shifting arrangement. In con- trast to moiretopography, space-phase demodulation ana- lyzes the deformed grating itself instead of the moirefringe pattern, and requires only one image pattern to obtain the phases of the deformed grating. Some impressive research has been done. Fourier transform profilometry 2 ~FTP! is free from the shortcomings associated with the moireto- pography, but the FTP method is computationally intensive in transformation and inverse transformation. Since the choice of the filtering window is related to the practical problem, it is not easy to automatically process the fringe signals. Spatial phase detection 3 ~SPD! and finite-duration impulse response low-pass digital filter 4 ~FIR DF! are two excellent methods in which the phase is demodulated by sinusoidally fitting and numerical integration similar to that used in communication techniques. But the phase demodu- lation of the SPD method is at the middle point of every carrier interval, not at every pixel. Only an average phase value is obtained in every sampling space period, which limits the resolution. The phase calculation of the FIR DR method, similar to FTP, is very complex. It seems that one- step phase-shift grating projection 5 ~OPGP! is simpler than other phase demodulation algorithms, but it requires the fringe period to occupy the special carrier interval ~e.g., either 4-pixel width or 8-pixel width!. The aim of this paper is to present an alternative space- phase extraction method, the subfringe integration algo- rithm ~SIA!, for measuring the 3-D objects shape. The SIA technique is similar to the form of projection optics geom- etry, but the phase demodulation algorithm is quite differ- ent. The SIA technique extends the phase extraction algo- rithm proposed by Wyant, 6 which integrated the intensity over a finite time interval. In our work, an SIA that inte- grates the intensity over a finite space interval is discussed. The phase is demodulated using an N integrating bucket algorithm similar to correlation detection, which is used in communication techniques. Since SIA evaluates the fringe pattern in the space domain instead of the temporal domain, it is easier to program than other phase extraction methods. Because it demodulates the space phase at every pixel on points, it has high measurement accuracy and spatial reso- lution. Another advantage of the SIA technique is that the processing speed of this technique is faster than that of FTP, which is very useful in on-line inspection.

Determination of the source term in the heat conduction equation

The inverse problem involving determination of the forcing function, depending not only on the unknown variable but also the space variable, is considered. However, the form of the function is assumed to be of the form f(u)+xg(u) assuming only an affine dependence on the space variable. The problem considered is defined on a finite space interval and non-negative time axis. The auxiliary conditions are prescribed at the boundaries of the space interval.

Stochastic processes in a finite space interval

Stochastic processes in a finite space interval

Nonlinear dynamic analysis of reinforced concrete structures

Dynamic ultimate load calculations mainly for reinforced concrete beams and plates, are discussed. Starting from the corresponding differential equations, the calculations also include the rotational inertia of single beam or plate elements as well as the shear deformations. With actual structural dynamic problems in nuclear power plants, the shear behaviour of reinforced concrete beams and plates is more important than it is usually, as is shown by examples. The finite propagation velocity of bending and shear waves are taken into account. Solution of the equations of motion is obtained by numerical intergration using finite time and space intervals. The calculations are performed using time dependent bending and shear laws for reinforced concrete up to the point of failure with realistic deformations. These latest scientific developments are of great significance for dynamic ultimate load analysis in practice. Elastic-plastic examples of application are compared with corresponding linear-elastic solutions. It is shown that the design of construction members based on elastic-plastic dynamic stress calculations in general is economically advantageous. This important conclusion is proven by numerical results. Also the relation to the approximation of a one-degree-of-freedom dynamic system, including or excluding the plastic ductility of the structural member, is demonstrated. Finally, lumped-mass multi-degree systems calculated by integrating numerically the corresponding equations of motion, are dealt with briefly. A nonlinear dynamic calculation of a foundation of a recently built reactor building is presented as an example for blast resistant analysis.

Extremes of realizations of continuous time stationary stochastic processes on closed intervals

In many engineering problems it is of great interest to know the distribution of the extremes in finite realizations of continuous time, stationary processes. From a mathematical point of view, it is extraordinarily difficult to give a rigorous and systematic theory for this subject. The asymptotic behavior of the extremes for some special processes is known, but only few results exist concerning the extremes of the sample function within a closed interval on the time axis (Darling and Siegert [4], Slepian [6]), except for the rather special and strange process where the values of the sample function at different times are complete independent. As all engineering problems are restricted to finite time or space intervals, it would be worth-while to try to get even rather crude results. As an example we may consider a test bar of a ductile material. When an increasing tension force is applied to such a bar, it will start to yield in the weakest cross-section, when the force reaches some level. This level defines the yield strength of the bar. Now it is obvious that the yield force in the weakest cross-section is a minimum value of a finite sample function of a stochastic process, which in many cases may be assumed to be stationary. Thus, we just are concerned with the problem of finding the distribution of an extreme of a sample function defined on a closed interval of length as the length of the test bar.