Kinematically restricted phonon transmission in partly-unzipped tubes of square and triangular lattices

An analysis of the transmission of `scalar' phonons across partially unzipped square and triangular lattice tubes, assuming nearest neighbor interactions between particles, is presented. The phonon transport is assumed to involve the out-of-plane phonons in the unzipped portion and the radial phonons in the tubular portion. An exact expression of reflectance and transmittance for the waves incident from either portions of the waveguide are provided explicitly, in terms of the Chebyshev polynomials, which leads to the provision of a simple expression for the ballistic conductance.Comment: 24 pages, 16 figure

Synchronization of globally coupled oscillators without symmetry in the distribution of natural frequencies

The collective behavior in a population of globally coupled oscillators with randomly distributed frequencies is studied when the natural frequency distribution does not possess an even symmetry with respect to the average natural frequency of oscillators. We study the special case of absence of symmetry induced by a group of scaling transformations of the continuous distribution of frequencies. When coupling between oscillators is increased beyond a critical threshold favoring spontaneous synchronization, we found that the variation in the velocity of the traveling wave depends on the extent of asymmetry in the natural frequency distribution. In particular for large coupling this velocity is the average natural frequency whereas at the onset of synchronization it corresponds to the frequency where the Hilbert Transform of the frequency distribution vanishes

Transmission of waves across atomic step discontinuities in discrete nanoribbon structures

Scalar wave propagation across a semi-infinite step or step-like discontinuity on any one boundary of the square lattice waveguides is considered within nearest-neighbour interaction approximation. An application of the Wiener-Hopf method does yield an exact solution of the discrete scattering problem, using which, as the main result of the paper, the transmission coefficients for energy flux are obtained. It is assumed that a wave mode is incident from either side of the step and the question addressed is what fraction of incident energy is transmitted across the atomic step discontinuity. A total of ten configurations are presented that arise due to various placements of discrete Dirichlet and Neumann boundary conditions on the waveguide. Numerical illustrations of a measure of conductance are provided

Conductance of discrete bifurcated waveguides as three terminal junctions

An expression for the transmission matrix based conductance is provided for the propagation of scalar waves in certain bifurcated discrete waveguides using the paradigm of a three-terminal Landauer-Buttiker junction. It is found that the conductance across the terminals of bifurcated branches forming a sharp corner, interpreted as a controller of `leakage' flux, can be tuned by manipulating the number of channels and the type of lateral confinement. Natural applications in engineering and science arise in the context of nanoscale transport involving elastic, phononic, or electronic waves. In particular, the paper includes a discussion of temperature dependent thermal conductance, assuming only the contribution of out-of-plane phonons, along with some graphical illustrations.Comment: 25 pages, 9 Figure

Energy expense via lattice wave emission for mode III brittle fracture in square, triangular, and hexagonal lattices

The mode III fracture problem for a hexagonal lattice is discussed and compared with square and triangular lattices

A dislocation-dipole in one dimensional lattice model

A family of equilibria corresponding to dislocation-dipole, with variable separation between the two dislocations of opposite sign, is constructed in a one dimensional lattice model. A suitable path connecting certain members of this family is found which exhibits the familiar Peierls relief. A landscape for the variation of energy has been presented to highlight certain sequential transition between these equilibria that allows an interpretation in terms of quasi-statically separating pair of dislocations of opposite sign from the viewpoint of closely related Frenkel-Kontorova model. Closed form expressions are provided for the case of a piecewise-quadratic potential wherein an analysis of the effect of an intermediate spinodal region is included

Continuum limit of discrete Sommerfeld problems on square lattice

A low frequency approximation of the discrete Sommerfeld diffraction problems, involving the scattering of a time harmonic lattice wave incident on square lattice by a discrete Dirichlet or a discrete Neumann half-plane, is investigated. It is established that the exact solution of the discrete model converges to the solution of the continuum model, i.e. the continuous Sommerfeld problem, in certain discrete Sobolev space defined by W. Hackbusch. The proof of convergence has been provided for both types of boundary conditions when the imaginary part of incident wavenumber is positive

A Family of Solitary Waves in Frenkel-Kontorova Lattice

A family of solitary waves is constructed in Frenkel-Kontorova model and its continuum and quasi-continuum approximations. Each solitary waves is characterised by the number of local maxima in its profile and a relation between external force and the velocity of wave. Such waves may be interpreted as a coherent motion, with constant velocity, of two dislocations of opposite sign or of kink-antikink pair

Wiener-Hopf factorisation on unit circle: some examples from discrete scattering

I discuss some problems featuring scattering due to discrete edges on certain structures. These problems stem from linear difference equations and the underlying basic issue can be mapped to Wiener-Hopf factorization on an annulus in the complex plane. In most of these problems, the relevant factorization involves a scalar function, while in some cases a nxn matrix kernel, with n>=2, appears. For the latter, I give examples of two non-trivial cases where it can be further reduced to a scalar problem but in general this is not the case. Some of the problems that I have presented in this paper can be also interpreted as discrete analogues of well-known scattering problems, notably a few of which are still open, in Wiener-Hopf factorization on an infinite strip in complex plane

Discrete scattering by two staggered semi-infinite defects: reduction of matrix Wiener-Hopf problem

As an extension of the discrete Sommerfeld problems on lattices, the scattering of a time harmonic wave is considered on an infinite square lattice when there exists a pair of semi-infinite cracks or rigid constraints. Due to the presence of stagger, also called offset, in the alignment of the defect edges the asymmetry in the problem leads to a matrix Wiener-Hopf kernel that cannot be reduced to scalar Wiener-Hopf in any known way. In the corresponding continuum model the same problem is a well known formidable one which possesses certain special structure with exponentially growing elements on the diagonal of kernel. From this viewpoint the present paper tackles a discrete analogue of the same by reformulating the Wiener-Hopf problem and reducing it to a finite set of linear algebraic equations; the coefficients of which can be found by an application of the scalar Wiener-Hopf factorization. The considered discrete paradigm involving lattice waves is relevant for modern applications of mechanics and physics at small length scales