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The effective properties associated with the propagation of electromagnetic waves in photonic crystals (PCs) are analyzed for one dimensional photonic crystals to illustrate the difference in those properties between infinitely extended and finite size photonic crystals (along the wave propagation direction). It is shown that the multiple reflections at the two end surfaces of a PC result in oscillations in transmittance, phase velocity, group velocity, as well as effective refractive indexes neff, which are often determined experimentally from the transmitted or reflected wave. At the frequencies far away from the bandgap, the oscillation in both the transmittance and neff can be modeled by treating the PC as an effective medium with the effective properties derived from the dispersion curves of the corresponding infinitely extended PC. As the frequencies approaching the band edges, however, the oscillations in both transmittance and neff are markedly enhanced due to the reflections from the internal interfaces in the PC. The results clearly demonstrate the importance of the two end faces on the effective properties of a finite size PC, which can be quite different from those deduced for an infinitely extended PC.

The current collected by a finite‐length cylindrical probe is calculated numerically from a self‐consistent solution to the Poisson–Vlasov system of equations. A Boltzmann distribution is assumed for the electrons, and results are compared to the standard ‘‘orbital‐motion‐limited’’ theory. When the probe is very long compared to the sheath thickness and the ratio of probe radius to electron Debye length, Rp/λDe ≲ 1, the collected current agrees with the orbital‐motion theory for cylindrical probes; when the probe is very short compared to the sheath thickness and Rp/λDe ≲ 1, the collected current agrees with the orbital‐motion theory for spherical probes. When the probe length is comparable to the sheath thickness, there is a critical potential above which the probe behaves as a sphere and below which it behaves as a cylinder. Ion charge density, space potential, and ion particle orbits are displayed graphically in the sheath plasma surrounding the probe tip.

A system of one‐dimensional equations of motion for AT‐cut quartz strip resonators with narrow width and for frequencies up to and including the fundamental thickness shear is deduced from the two‐dimensional, first‐order equations for piezoelectric crystal plates by Lee, Syngellakis, and Hou [J. Appl. Phys. 61, 1249 (1987)] by expanding the mechanical displacements and electric potentials in a series of trigonometric functions of the width coordinate. By neglecting the piezoelectric coupling and the weak mechanical coupling through c56, four groups of coupled equations of motion are obtained. For the equations of each group, closed form solutions are obtained and the traction‐free conditions at four edges are accommodated. Dispersion curves and frequency spectrum are computed for quartz strips. The predicted frequency as a function of the length‐to‐thickness ratio and as a function of the width‐to‐thickness ratio of the quartz strips is compared with experimental data with good agreement.

The electronic structure of a finite superlattice exposed by an external electric field along the growth direction is studied using the Fourier expansion in electron wave functions. We have found surface localized [I. Tamm, Phys. Z. Sowjetunion 1, 733 (1933)] states and, in addition, two Stark ladders whose states are localized in the quantum wells and evenly spaced in energy. Strong interactions between Tamm states and Stark ladders are studied in detail.

In this paper the smallest or optimal dimensions of a Halbach cylinder of a finite length for a given sample volume and desired flux density are determined using numerical modeling and parameter variation. A sample volume that is centered in and shaped as the Halbach cylinder bore but with a possible shorter length is considered. The external radius and the length of the Halbach cylinder with the smallest possible dimensions are found as a function of a desired internal radius, length of the sample volume and mean flux density. It is shown that the optimal ratio between the outer and inner radius of the Halbach cylinder does not depend on the length of the sample volume. Finally, the efficiency of a finite length Halbach cylinder is considered and compared with the case of a cylinder of infinite length. The most efficient dimensions for a Halbach cylinder are found and it is shown that the efficiency increases slowly with the length of the cylinder.

The screening effects on the field enhancement factors of (3,3) and (5,0) finite length single-walled carbon nanotubes (SWCNTs) with respect to different kinds of tube terminations, either capped or H-terminated, are examined using first-principles calculations. Results show that for capped SWCNTs, the field enhancement factors of the (3,3) tube are slightly greater than those of (5,0) ones, while for H-terminated tubes exhibit the opposite effect. The effects of different endings on the SWCNT work functions are also investigated. Calculations show that for the capped SWCNTs, the work functions of (3,3) tubes are larger than those of (5,0) tubes, whereas for the H-terminated cases, the work functions of (3,3) tubes are smaller than those of the (5,0) tubes. Moreover, a saturated behavior of enhancement factors β versus the length of tubes is observed. Finally, the field enhancement factors of fixed length tubes can be enhanced to eventually achieve a convergent tendency as their tube-tube distance increases.

Dual-coated optical fibers are fabricated at elevated temperatures and operated at low temperature conditions. It is imperative that the fibers remain elastically stable, i.e., do not buckle as a result of the thermal contraction mismatch of the dissimilar materials of the fiber and the secondary coating or, in the case of an optical fiber interconnect, of the fiber and the material of an enclosure, if any. We develop a simple and easy-to-use predictive model that enables one to establish the condition of elastic stability for a dual-coated optical fiber of finite length, and/or to select the coating materials with the most favorable physical properties for such a fiber, and/or to establish the adequate thickness of the primary coating. The fiber is idealized as a free-free beam (i.e., a beam not supported at the ends), and the effect of the embedding medium is mimicked by a continuous elastic foundation. The obtained results can be applied also, with some minor modifications, for the evaluation of the el...

By inserting a grid in a beam‐generated plasma column, single waves can be produced as a result of the beam‐plasma instability exciting a finite‐length cavity mode. The oscillation frequency is found to increase with beam velocity, in contrast to the frequency‐velocity relationship of the wave with the maximum growth rate. The nonlinear evolution of the beam‐generated wave and satellites due to a test wave are observed.

The magnetic fields on the surface produced by a buried current‐carrying cable of finite length are considered. A homogeneous half‐space model of the earth and quasistatic conditions are assumed. The cable is grounded at its end points and oriented at an arbitrary angle to the interface. While, in general, numerical integration is required, the low‐frequency limit may be treated analytically. Small cable tilts are shown to modify the direction of the surface field but not the magnitude. The results have possible application to uplink communication and radio location of trapped miners.

We present simple analytic expressions for the classical energies of previously obtained nonlinear‐standing‐wave and single‐soliton solutions to the sine‐Gordon equation for a finite‐length system with fixed‐ and free‐boundary conditions. Plots of these energies are presented for several system lengths and the energy expressions are shown to reduce to those appropriate to familiar solutions in the infinite system as the length approaches infinity.

Theoretical analysis and numerical calculations are carried out of the skin current in a finite‐length cylindrical plasma when a step‐like electric field is applied externally, within the framework of classical theory. The spatial and temporal evolutions of the skin current are numerically calculated for two cases of plasmas with uniform and parabolic density distributions.

Green’s function for the solution of Poisson’s equation in cylindrical coordinates is calculated using the underlying Sturm–Liouville problem. The potential distribution produced by an arbitrarily oriented eccentric dipole located in a homogeneous conducting circular finite‐length cylinder is given; three alternative forms are presented, which enable us to compute the potential everywhere in the cylinder. Previous works devoted to this subject were incomplete in the sense that the formulas which were presented there do not converge everywhere. For our expressions of the Green function,we show that Gabor and Nelson’s theorem applies for the present geometry. We also consider the case where the current source and sink are remote from each other in order to check the consistency between Gabor and Nelson’s theorem and Helmholtz’s principle. Applications of our formulas are presented in a situation which is representative of the human trunk activated by the heart and discussion of the results is made in relation with electrocardiology. It is shown that the eccentricity of the dipole is responsible for curving the line of the greatest potentials and for wrapping the zero equipotential plane.

An analysis of the annular induction MHD (magnetohydrodynamic) generator is carried out for a model employing a finite‐length winding and a constant‐velocity fluid flowing in the annulus. An expression for the phase impedance is derived which accounts for entrance and exit of the fluid into the winding region. Impedance characteristics are presented which allow comparison with the infinite‐length case in terms of dimensionless parameters.

A successive approximation theory is developed for the thick film lubrication of complete journal bearings of finite length for arbitrary positions of the lubricant source. It has been carried through in detail up to the third approximation, which permits studies of systems having journal eccentricities up to the value of ½. The range of bearing length for which the theory is valid includes such as are of the order of, or greater than, half the bearing perimeter. Detailed calculations are reported on the inter‐relationships between the attitude of the line of centers of the journal and bearing, the journal eccentricity, the lubricant source strength, the ``Sommerfeld variable,'' (r2/c2)μN/P—where r=journal radius, c=radial clearance, μ=lubricant viscosity, N=journal speed and P=bearing load per unit projected bearing area—and the coefficient of friction, both for lubricant sources set at the crown and base of the bearing. Most of the numerical results refer to the case where the bearing length equals the bearing perimeter. In general it is found that when the lubricant source is at the crown or unloaded portion of the bearing, it tends to increase the journal eccentricities, oppose the journal supporting forces caused by the journal rotation, and increase the friction coefficients. When the source is set at the base, or loaded side, of the bearing, and has a strength less than the ``critical value,'' determined by the bearing length, it tends to decrease the journal eccentricities, to aid in supporting the load, and to decrease the friction coefficients on the journal. At the ``critical values'' of source strength the journal becomes concentric with the bearing and the lubricant source alone supports the load. For higher values of source strength the journal is raised above the center of the bearing so that the journal rotation detracts from the load‐carrying power of the film, and the friction coefficient on the journal begins to rise. Moreover, when the source is set at the base of the bearing, the friction coefficient curves for both the bearing and journal, when plotted against the Sommerfeld variable, all go through the origin in contrast to the infinitely long bearing theory where the coefficient of friction on the journal has a unit friction axis intercept. From the hydrodynamic point of view there are therefore no minima in the curves, or suggestions of thin film friction rises when the lubricant source is at the base of the bearing.

An electromagnetic plane wave in either E or H polarization is normally incident on a cylinder which is assumed to be an infinitely thin‐walled, perfectly conducting, circular tube of finite length with complete generality both in radius and length. For either polarization both axial and transverse currents are induced. The problem is transformed into determining an infinity of Fourier components by expanding each function of θ into a Fourier series. For each Fourier component a pair of integral equations for the total currents is obtained. These are decoupled in a special sense permitting the transverse total current to be obtained first, followed by the axial total current. The diflerence current is then defined and expressed in terms of the total current, making possible the determination of the inside and the outside total current. These are experimentally measurab e quantities. The theoretical formulas for the scattered fields, the far field patterns, and the scattering cross sections are also derived in terms of the total current. Finally, sample results, both theoretical and experimental, are included.

A method is presented for extending the theory of wall stabilized arcs to free burning arcs. It is based on accounting for the finite length of the arc column and utilizing the fact that such arcs are normally very thin compared to this length. Detailed calculations are performed for an arc column with constant properties, and an outline is given for incorporating more realistic properties. An interesting result of the analysis is that the axis temperature must increase greatly in front of the electrodes.

A successive approximation theory is developed for the hydrodynamics of journal bearing systems of finite length, the expansion parameter being the journal eccentricity. This is applied through the second approximation to full journal bearings of lengths exceeding the bearing perimeter. The existence of the lubricant source and associated end leakage is explicitly taken into account, it being assumed in the actual analysis that the source is set 90° in advance of the maximum film thickness. It is found that contrary to prevailing opinion, based on the infinitely long bearing theory, the friction coefficients and eccentricities for the systems are not uniquely determined by the ``Sommerfeld variable''—(r2/c2)(μN/P), where r=journal radius, c=radial clearance, μ=lubricant viscosity, N=journal speed in r.p.s., P=bearing load per unit projected bearing area. It depends also on the leakage flux or source strength. For the position of source assumed in the analysis the friction coefficients and eccentricities i...

Instabilities in one and two drifting‐electron streams neutralized by nondrifting ions and bounded transversely and axially are presented in mathematical and physical detail. Solutions for growth rates are obtained from the dispersion relation for a cylindrical stream of cold electrons and cold ions plus the boundary conditions at the gridded ends of the drift region. The new results in addition to the critical current (or critical perveance=190×10−6) given by Pierce in 1944 are: (i) less approximate inclusion of finite drift length, (ii) ac motion of ions, (iii) detailed physical description, (iv) growth rates over a range of parameters. The results show instability for immobile ions with the critical perveance value agreeing with that given by Pierce, but show marginal stability for mobile ions, contrary to Pierce and to the experiments of Atkinson. The difference between mobile and immobile ion solutions is ascribed to the ability of ions to neutralize (or not) excess electron charge in the vicinity of the output grid. The relation to stream‐ion wave amplification is discussed. The relation of the solutions presented here to those for a cylindrical stream which is not neutralized but is thin enough to have only small velocity variation radially is also discussed.

Two simple formulas are presented for the capacitance of a finite‐length, conducting right‐circular cylindrical tube which resides in a homogeneous medium of permittivity e. The formulas may be used to determine capacitance for all ratios (radius/length). The range of (radius/length) over which each formula applies is established by comparisons with capacitance calculated from numerical solutions of an integral equation.