Free-wave propagation in an irregularly stiffened,fluid-loaded plate

Techniques developed for analysis of the dynamic behaviour of random, composite media are applied to the study of free waves in irregularly stiffened plates, with or without fluid-loading. It is well known that the free wave in an exactly periodic structure is a Floquet wave which possesses a structure of “pass-”, and “stop-bands”. In this paper a method is presented for studying a structure in which the exact positions of the stiffening ribs are subject to some degree of randomness. In particular, the dispersion relation for free waves in the plate is derived, some solutions of which are presented and compared with the corresponding solutions for the exactly periodic structure.     

Solitary wave solutions for nonlinear Schrödinger equation with non-polynomial nonlinearity

A new kind of non-polynomial nonlinearity is introduced in the nonlinear Schrödinger equation (NLSE) and the conditions are determined for which it admits solitary wave solutions. The study is done for two cases: one in which the nonlinear interaction is of the non-polynomial form and second in which cubic nonlinearity is also included along with the radical nonlinearity. Dark and bright solitary waves solutions are obtained in the respective cases. Further, later case is extended to conditions for which corresponding equation reduces to driven quadratic-cubic NLSE possessing cnoidal solutions with plane wave phase, which reduces to bright soliton for a certain parameter.     

Optical devices based on materials with negative refraction

The Abbe invariant is obtained for media with negative refractive index, which allows calculations of optical devices with elements characterized by negative refraction. Formulas for calculating the source-image distances and the magnification provided by various lenses with negative refraction are derived. Block diagrams are given for the focusator, microscope, and spectrograph in which materials with negative refraction are used. The microscope and focusator magnifications are estimated. The linear dispersion is calculated for the spectrograph in which a plane-parallel plate with negative refraction is used as a spectral element.     

Radiation Reaction of a Nonrelativistic Quantum Charged Particle

An alternative approach to analyze the nonrelativistic quantum dynamics of a rigid and extended charged particle taking into account the radiation reaction is discussed with detail. Interpretation of the field operators as annihilation and creation ones, theory of perturbations and renormalization are not used. The analysis is carried out in the Heisenberg picture with the electromagnetic field expanded in a complete orthogonal basis set of functions which allows the electromagnetic field to satisfy arbitrary boundary conditions. The corresponding coefficients are the field operators which satisfy the usual commutation relations. A nonlinear equation of motion for the charged particle is obtained. A careful consideration of the quantum effects allows the derivation of a linear equation of motion which is free of both runaway solutions and preacceleration, even for a point charge. Also, the electromagnetic mass, which is defined as the coefficient of the acceleration operator, vanishes for a point particle. However, this does not mean that the results are free of ambiguities which are exhibited and discussed.     

Analytic supernova models and black holes

Presented are new analytic solutions to Einstein's field equations with properties normally associated with supernovas. These are the first analytic supernova models with pressure, temperature, and luminosity. These solutions are used to compare a radiative nonzero model (for which the pressure is continuous accross the outer boundary of the star) with a radiative zero model [(standard model) for which the pressure within the star is zero at the outer boundary].     

A new semiclassical elementary particle model

A semiclassical model in which elementary particles are represented as systems of charged shells with associated quark-like quantum numbers is presented. Specifically the baryons are considered. Formulas are obtained which express baryon masses and magnetic moments in terms of model parameters which relate baryon and quark properties. Basically, the mass and moment formulas are expressions for mass ratios and magnetic moment ratios. Simple identifications for the model parameters lead to a prediction for the proton-electron mass ratio and to fairly accurate predictions for the baryon magnetic moments in units of the proton moment. The mass and moment formulas, which relate corresponding properties of different particles, are generalised such as to express relationships between the members of a sequence of particles, where such a sequence is conceived of as containing only one (normal) baryon. A specific sequence, containing the proton and electron, is proposed; various physical properties of the particles in the sequence are determined. In particular, a second prediction for the proton-electron mass ratio is obtained; the two predictions differ numerically but both agree with the measured value of the mass ratio within experimental error.     

Strongly interacting particles with strongly singular potentials

A statistical system of particles is considered for which interaction potentials are strongly singular so that the standard perturbation theory cannot be used. A regular procedure for constructing a mass operator is suggested, having no ultraviolet divergences and giving the possibility of finding corrections for any approximation chosen. In this procedure, the divergences connected with the potential singularity are eliminated with the help of a smoothing function, for which a simple equation is given and whose properties are analyzed both analytically and numerically. Two effective regularization methods are formulated, eliminating divergences occurring while iterating propagator equations. A continuous iterative procedure is invented for calculating observable quantities and the fast convergence conditions for this procedure are shown to be equivalent to the fixed-point conditions.     

Analysis of the Element-Free Galerkin Method with Penalty for Stokes Problems

The element-free Galerkin (EFG) method with penalty for Stokes problems is proposed and analyzed in this work. A priori error estimates of the penalty method, which is used to deal with Dirichlet boundary conditions, are derived to illustrate its validity in a continuous sense. Based on a feasible assumption, it is proved that there is a unique weak solution in the modified weak form of penalized Stokes problems. Then, the error bounds with the penalty factor for the EFG discretization are derived, which provide a rationale for choosing an efficient penalty factor. Numerical examples are given to confirm the theoretical results.     

Robertson-Walker Fluid Sources Endowed with Rotation Characterised by Quadratic Terms in Angular Velocity Parameter

Einstein's equations for a Robertson-Walker fluid source endowed with rotation are presented up to and including quadratic terms in angular velocity parameter. A family of analytic solutions are obtained for the case in which the source angular velocity is purely time-dependent. A subclass of solutions is presented which merge smoothly to homogeneous rotating and non-rotating central sources. The particular solution for dust endowed with rotation is presented. In all cases explicit expressions, depending sinusoidally on polar angle, are given for the density and internal supporting pressure of the rotating source. In addition to the non-zero axial velocity of the fluid particles it is shown that there is also a radial component of velocity which vanishes only at the poles. The velocity four-vector has a zero component between poles.     

Free vibration analysis of the completely free rectangular plate by the method of superposition

While the subject of free vibration analysis of the completely free rectangular plate has a history which goes back nearly two centuries it remains a fact that most theoretical solutions to this classical problem are considered to be at best approximate in nature. This is because of the difficulties which have been encountered in trying to obtain solutions which satisfy the free edge conditions as well as the governing differential equation. In a new approach to this problem, by using the method of superposition, it is shown that solutions which satisfy identically the differential equation and which satisfy the boundary conditions with any desired degree of accuracy are obtained. Eigenvalues of four digit accuracy are provided for a wide range of plate aspect ratios and modal shapes. Exact delineation is made between the three families of modes which are characteristic of this plate vibration problem. Accurate modal shapes are provided for the response of completely free square plates.     

Coupled longitudinal and bending vibrations of a rotating shaft with an open crack

The coupling of longitudinal and bending vibrations of a rotating shaft, due to an open transverse surface crack is investigated. The assumption of the open crack leads to a system with behaviour similar to that of a rotor with dissimilar moments of inertia along two perpendicular directions. The local flexibility due to the presence of the crack can be represented by way of a 6×6 matrix for six degrees of freedom in a short shaft element which includes the crack. This matrix has off-diagonal terms which cause coupling along the directions which are indicated by these terms. Here shear is not considered and three degrees of freedom are used: bending in the two main directions and extension. This leads to a 3×3 stiffness matrix with coupling terms. The undamped free and forced coupled vibration are first considered. The coupling is investigated and the effects of unbalance and gravity are examined. Then damped coupled vibration is considered for free and forced vibration. The existence of coupling between longitudinal and bending vibration due to the crack is a very useful property which, together with the sub-critical resonance due to crack, can form a basis for crack identification in rotating shafts. New and interesting phenomena of coupled transverse and longitudinal motion are presented and discussed.     

Tunable metastability of surface nanostructure arrays

A Fokker-Planck equation is used to model the coarsening of surface nanostructure arrays. Metastable states are identified which are associated with a narrow size distribution and a coverage dependent mean island size. This is a general feature linked to nanostructures which, as a function of island size, are associated with a minimum in formation energy per atom and a positive chemical potential gradient. This has important implications for the self-organization of quantum dots.     

The isobar-doorway theory for pion-nucleus interactions

Pion-nucleus scattering and reactions are treated in a theory which explicitly introduces the pion-nucleon resonances. Using a separation in Hilbert Space, doorway states of isobar-nuclear systems are introduced and nonresonant processes are clearly separated from resonance interactions. With one choice of doorway states a multiple scattering series is derived which corresponds to the conventional theory with binding energy and other corrections included. When another choice the isobar-doorway model is derived, with parameterization explicitly related to specific dynamic effects, our framework provides a phenomenological model for treating meson-nucleus interactions to all orders. Moreover, the parameters of the model have clear theoretical significance which can extend our knowledge of strong interactions physics. A numerical study is given for elastic scattering.     

Generalized Canonical Noether Theorem and Poincare—Cartan Integral Invariant ofr a System with a Singular High—Order Lagrangian and an Application

Based on the canonical action,a generalized canonical first Noether theorem and Poicare-Cartan integralinvariant for a system with a singular high-order Lagrangian are derived.It is worth while to point out that the constraints are invariant under the total variation of canonical variables including time.We can also deduce the result,which differs from the previous work to reuire that the constraints are invariant under the simultaneous variations of canonical variables.A counter example to a conjecture of the Dirac for a system with a singular high-order Lagrangian is given,in which there is no linearization of constraint.     

A novel technique for imaging with inhomogeneous fields

We introduce a simple, efficient, low-SAR method for magnetic resonance imaging in the presence of a static field with a permanent, and possibly large gradient. The technique, which is called slant-slice imaging is essentially a spin-echo imaging sequence except that the imaging slice is oriented such that the static field gradient can be used in conjunction with applied gradients during readout. Data are collected for 2D slices. Unlike single point imaging techniques, entire lines of k-space are acquired with each readout. The slant-slice pulse sequence is used to obtain high quality images, using a clinical scanner to simulate a static field with a large permanent gradient. The effects of the inhomogeneity are quantified by two parameters nu and q, which are useful for assessing the utility of a magnet design for 3D-MR imaging.     

Piezoelectric transducer vibrations in a one-dimensional approximation

Summary The theory of piezoelectric transducer vibrations, which may be treated as onedimensional, is developed in detail for thin discs vibrating in a pure thickness extensional mode. An effort has been made to obtain relations of general validity, which include losses, and which are in a simple explicit form convenient for practical calculations. The behaviour of transducers is discussed with special attention to their characteristics at the two fundamental frequencies, the so-called parallel and series resonances. Several peculiarities occur when transducers are coupled to media with considerably different acoustic impedances. These peculiarities are discussed and illustrated by numerical results for quartz and PZT 4 piezoelectric discs radiating into water, air and liquid hydrogen. The application of the theory to different types of vibrations is briefly illustrated for thin bars vibrating longitudinally. Short discussions are included on compound transducer systems, and on the properties of thin discs as receivers.     

The transmission of radiation through a spatially stochastic medium

The transport of radiation through a medium which is spatially random is studied using diffusion theory and the method of smoothing. Equations are established for the average flux and current in the medium, together with the variance of these quantities. The theory is applied to a plane slab one side of which is irradiated by a uniform source of radiation. The reflection and transmission factors are calculated and a measure of their fluctuations is obtained. For more generality, the boundary conditions allow internal reflection of the radiation using the Fresnel coefficient, which is particularly useful for applications to optical tomography where we believe this problem to have some relevance. The results are illustrated numerically using stochastic models for weak and strong clumping and applied to transmission through adult brain tissue. Stochastic effects are seen to be significant.     

Finite Element Analysis of the Dominant Mode Variation in Groove Guide

In this paper, the dominant mode in groove guide is analyzed by finite element method. For the guide with a shallow groove, the electric field lines (abbreviated as E lines) are perpendicular to the parallel plate, which is different from the literatures, and the dominant mode distribution is irrelevant to the groove width; for the guide with a narrow but deeper groove, the E lines are parallel to the plate region, which represents the characteristics of lower attenuation, for the guide with a deeper and wider groove, a complicated E lines are obtained. These results reveal that the dominant mode distribution in groove guide is varied, which replenish our understanding of groove guide, and the results have important values in design, manufacture, and application of groove guide.