Explicit Dispersion Relation Research Articles

On the existence of equatorial wind waves

In this paper we apply local bifurcation theory to prove the existence of steady, periodic, two-dimensional surface water waves in the equatorial region which have a general underlying vorticity distribution. Furthermore, we derive explicit dispersion relations for the flow in the case where the vorticity is constant.

Localization for a line defect in an infinite square lattice

Localized defect modes generated by a finite line defect composed of several masses, embedded in an infinite square cell lattice, are analysed using the linear superposition of Green's function for a single mass defect. Several representations of the lattice Green's function are presented and discussed. The problem is reduced to an eigenvalue system and the properties of the corresponding matrix are examined in detail to yield information regarding the number of symmetric and skew-symmetric modes. Asymptotic expansions in the far field, associated with long wavelength homogenization, are presented. Asymptotic expressions for Green's function in the vicinity of the band edge are also discussed. Several examples are presented where eigenfrequencies linked to this system and the corresponding eigenmodes are computed for various defects and compared with the asymptotic expansions. The case of an infinite defect is also considered and an explicit dispersion relation is obtained. For the case when the number of masses within the line defect is large, it is shown that the range of the eigenfrequencies can be predicted using the dispersion diagram for the infinite chain.

An approximate secular equation of generalized Rayleigh waves in pre-stressed compressible elastic solids

The present paper is concerned with the propagation of Rayleigh waves in a pre-stressed elastic half-space coated with a thin pre-stressed elastic layer. The half-space and the layer are assumed to be compressible and in welded contact with each other. By using the effective boundary condition method, an explicit third-order approximate secular equation of the wave has been derived that is valid for any pre-strains and for a general strain-energy function. When the pre-strains are absent, the secular equation obtained coincides with the one for the isotropic case. Numerical investigation shows that the approximate secular equation obtained is a good approximation. Since explicit dispersion relations are employed as theoretical bases for extracting pre-stresses from experimental data, the secular equation obtained will be useful in practical applications.

An approximate secular equation of Rayleigh waves propagating in an orthotropic elastic half-space coated by a thin orthotropic elastic layer

The present paper is concerned with the propagation of Rayleigh waves in an orthotropic elastic half-space coated with a thin orthotropic elastic layer and the main purpose of the paper is to establish an approximate secular equation of the wave. By using the effective boundary condition method an approximate secular equation of third-order in terms of the dimensionless thickness of the layer has been derived. From this equation two different third-order approximate secular equations are obtained for the case when the half-space and the layer are both isotropic, one of which recovers the secular equation of second-order derived by Bovik [P. Bovik, A comparison between the Tiersten model and O(H) boundary conditions for elastic surface waves guided by thin layers, J. Appl. Mech. 63 (1996) 162–167]. An explicit second-order approximate formula for the Rayleigh wave velocity has been created based on the obtained approximate secular equation. Since explicit dispersion relations are employed as theoretical bases for extracting the mechanical properties of the thin films from experimental data, the obtained secular equation and formula for the velocity may be useful in practical applications.

On the modelling of equatorial waves

The present theory of geophysical waves that either raise or lower the equatorial thermocline, based on the reduced‐gravity shallow‐water equations on theβ‐plane, ignores vertical variations of the flow. In particular, the vertical structure of the Equatorial Undercurrent is absent. As a remedy we propose a simple approach by modeling this geophysical process as a wave‐current interaction in thef‐plane approximation, the underlying current being of positive constant vorticity. The explicit dispersion relation allows us to conclude that, despite its simplicity, the proposed model captures to a reasonable extent essential features of equatorial waves.

Dyadic Green's function study of band structures of dispersive photonic crystals

We present here in terms of a dyadic Green's function (DGF) a general description of optical phenomena in photonic crystal (PC) structures, described particularly by frequency-dependent components, assuming that PC structures are decomposed into their relatively simple constituent parts via conductivity tensors. We demonstrate this approach by explicitly calculating the DGFs for electromagnetic waves propagating in the one- and two-dimensional dispersive PCs consisting of a periodic array of identical metallic quantum wells and a periodic square array of identical metallic quantum wires, each embedded in a three-dimensional dispersive medium. By means of the explicit analytic dispersion relations, which result from the frequency poles of the corresponding DGFs, we also calculate the band structures of these dispersive PCs by simple numerical means. Our analysis shows that the band structures calculated from our DGF approach conform well with those calculated from the traditional computational methods.

Dispersion of spoof surface plasmons in open-ended metallic hole arrays

Using phase sensitive microwave frequency measurements, we obtain the dispersion of spoof surface plasmon waves on a highly conducting sheet perforated with a two-dimensional array of subwavelength holes, and compare our results to an explicit analytical dispersion relation obtained by the modal matching method. We observe splitting into symmetric and antisymmetric surface modes, a behavior analogous to that of surface plasmons in thin metallic films at optical frequencies. We show that spoof surface modes play an important role in both near and far field transmissions. Specifically, we show that superfocussing effects which are present for surface plasmons in metallic films are absent for hole arrays (i.e., no amplification of near fields is possible for spoof surface plasmons). While many of the apparent properties of spoof surface plasmons resemble those of surface plasmons in thin metallic films, the analogy is therefore incomplete in the high frequency limit.

An Efficient Approach to Include Full-Band Effects in Deterministic Boltzmann Equation Solver Based on High-Order Spherical Harmonics Expansion

We present an efficient method to include full-band-structure effects for the case of a silicon conduction band in a deterministic Boltzmann equation solver based on the high-order spherical harmonics expansion method. This method employs the exact density of states and the group velocity obtained from band structure calculations, and it eliminates the modulus of the wave vector in the formulation such that an explicit invertible dispersion relation is not required. While the present method does not require additional central-processing-unit time and memory, compared with the analytic band model, the simulation results are significantly improved and in excellent agreement with those from the full-band Monte Carlo simulations and from an approach based on an invertible anisotropic band that matches several moments of the group velocity of the full band structure.

On linear internal waves on the sea, strongly vertically trapped

We study some explicit cases of marine thermocline. We focus our attention on the strongly vertically trapped internal waves, which in our cases allow an explicit dispersion relation and a simple behaviour in terms of elementary functions. The explicit form of the Vaisala-Brunt frequency N2{z) is proportional to 1 / \z—20| in one case and to A2—B2(z—zD)2 in the other. A comparison with some experimental data concerning the Ligurian Sea is actually in course.

Propagation behaviors of shear horizontal waves in piezoelectric-piezomagnetic periodically layered structures

This paper investigates shear horizontal (SH) waves propagating in a periodically layered structure that consists of piezoelectric (PE) layers perfectly bonded with piezomagnetic (PM) layers alternately. The explicit dispersion relations are derived for the two cases when the propagation directions of SH waves are normal to the interface and parallel to the interface, respectively. The asymptotic expressions for dispersion relations are also given when the wave number is extremely small. Numerical results for stop band effect and phase velocity are presented for a periodic system of alternating BaTiO 3 and Terfenol-D layers. The influence of volume fraction on stop band effect and dispersion behaviors is discussed and revealed.

Wave propagation in prestretched polymer nanofibers

Wave propagation in prestretched ultrathin polymer fibers (e.g., those as-electrospun polymer nanofibers) are under the influence of prestretch, surface energy, and nonlinear elasticity. A one-dimensional nonlinear elastic model is proposed to take into account such combined influence in the wave propagation phenomenon. In the model, the polymer nanofibers are considered to behave as hyperelastic Mooney–Rivlin solid. For small dynamic disturbance, linearized wave equation is established by superimposing the dynamic displacement as linear disturbance on the prestretched equilibrium state. Explicit wave dispersion relations are obtained and relevant numerical examples are demonstrated in examining the dependency of wave phase speed upon the wave number at varying surface properties, fiber radius, and prestretch. In the limiting case of neglecting the dynamic effect, the present wave equation can yield the governing equation of surface rippling in compliant nanofibers. This governing equation is capable of predicting the initiation condition of surface rippling and the critical fiber radius, below which compliant nanofibers cannot be produced due to surface instability. Results obtained in this study are applicable as the theoretical basis of dynamic characterization of compliant nanowires/nanofibers, nanofiber device design, and nanostructural analysis.

Propagation of Bleustein–Gulyaev wave in 6 mm piezoelectric materials loaded with viscous liquid

The influence of a viscous liquid on acoustic waves propagating in elastic or piezoelectric materials is of particular significance for development of liquid sensors. Bleustein–Gulyaev wave is a shear-type surface acoustic wave and has the advantage of not radiating energy into the adjacent liquid. These features make the B–G wave sensitive to changes in both mechanical and electrical properties of the surrounding environment. The Bleustein–Gulyaev wave has been reported to be a good candidate for liquid sensing application. In this paper, we investigate the potential application of B–G wave in 6 mm crystals for liquid sensing. The explicit dispersion relations for both open circuit and metalized surface boundary conditions are given. A numerical example of PZT-5H piezoelectric ceramic in contact with viscous liquid is calculated and discussed. Numerical results of attenuation and phase velocity versus viscosity, density of the liquid and wave frequency are presented. The paper is intended to provide essential data for liquid senor design and development.

Omnidirectional gaps of one-dimensional photonic crystals containing indefinite metamaterials

The explicit dispersion relation for one-dimensional photonic crystals consisting of alternative layers of indefinite metamaterial and ordinary positive-index material is obtained and analyzed in detail. It is shown that such a photonic crystal possesses an omnidirectional zero-averaged refractive-index gap when the indefinite metamaterial is a cutoff or anticutoff medium, yet it possesses an omnidirectional zero-effective phase gap when the photonic crystal contains two different always-cutoff media. The appearance of these two omnidirectional gaps does not require that all tensor elements of permittivity and permeability have negative values, quite different from that for the photonic crystal containing isotropic negative-index or single-negative material. Moreover, the new zero-averaged refractive-index gap and zero-effective phase gap are also insensitive to the incident angle and polarization of light and are invariant upon the change of scale length. These properties provide more degrees of freedom to design new types of optical devices.

Electroviscous potential flow in nonlinear analysis of capillary instability

We study the nonlinear stability of electrohydrodynamic of a cylindrical interface separating two conducting fluids of circular cross section in the absence of gravity using electroviscous potential flow analysis. The analysis leads to an explicit nonlinear dispersion relation in which the effects of surface tension, viscosity and electricity on the normal stress are not neglected, but the effect of shear stresses is neglected. Formulas for the growth rates and neutral stability curve are given in general. In the nonlinear theory, it is shown that the evolution of the amplitude is governed by a Ginzburg–Landau equation. When the viscosities are neglected, the cubic nonlinear Schrödinger equation is obtained. Further, it is shown that, near the marginal state, a nonlinear diffusion equation is obtained in the presence of viscosities. The various stability criteria are discussed both analytically and numerically and stability diagrams are obtained. It is also shown that, the viscosity has effect on the nonlinear stability criterion of the system, contrary to previous belief.

Quantization of the Low-Frequency Variability of the Double-Gyre Circulation

AbstractThe low-frequency dynamics of the double-gyre wind-driven circulation in large midlatitude oceanic basins is investigated. It is shown that for quasigeostrophic models linear (Rayleigh) friction is necessary to obtain realistic recirculation gyres and elongated jet streams with small meridional-to-zonal aspect ratio. It is also found that the use of either no-slip or free-slip boundary conditions does not change the drastic effects of bottom drag on the large scales. These long oceanic jets are alternatively destabilized and restabilized through successive (subcritical) supercritical symmetry-breaking bifurcations that are linked to the (non) existence of stationary Rossby waves. These waves are strongly localized along the oceanic front and are thus hardly affected by the basin geometry. Numerical and analytical results show that these waves are “quantized” with respect to the length of the jet, and an explicit dispersion relation is given. Numerical computations of branches of steady states, together with linear and nonlinear analysis, indicate that two classes of regimes characterize the low-frequency dynamics of the flow. The first class corresponds to supercritical regimes, which are associated with oceanic jets that have been destabilized by undamped stationary Rossby waves. In particular, these regimes allow the formation of gyre modes responsible for low-frequency relaxation oscillations of the jet. The second class corresponds to subcritical regimes, which are either quiescent or dominated by high-frequency instabilities and are characterized by jets that do not allow the formation of both stationary Rossby waves and gyre modes. Each of these regimes is characterized by typical spatial and time scales that are both quantized. The number n of “bumps” of the jet, which is related to the zonal wavenumber of the stationary Rossby waves, is used to distinguish between these regimes either in their supercritical or subcritical phase. For instance, the supercritical n = 2 regime is associated with a class of interdecadal gyre modes that extend up to 3000 km in the zonal direction. The quantization of the low-frequency dynamics and the existence of these regimes are also found to survive severe modifications of the basin geometry. These quantized regimes suggest that the low-frequency dynamics in turbulent regimes is likely to be autosimilar to the low-frequency dynamics found in a weakly nonlinear “ground” regime corresponding to n = 0.

Stability of a liquid jet into incompressible gases and liquids

We carry out an analysis of the stability of a liquid jet into a gas or another liquid using viscous potential flow. The instability may be driven by Kelvin–Helmholtz KH instability due to a velocity difference and a neckdown due to capillary instability. Viscous potential flow is the potential flow solution of Navier–Stokes equations; the viscosity enters at the interface. KH instability is induced by a discontinuity of velocity at a gas–liquid interface. Such discontinuities cannot occur in the flow of viscous fluids. However, the effects of viscous extensional stresses can be obtained from a mathematically consistent analysis of the irrotational motion of a viscous fluid carried out here. An explicit dispersion relation is derived and analyzed for temporal and convective/absolute (C/A) instability. We find that for all values of the relevant parameters, there are wavenumbers for which the liquid jet is temporally unstable. The cut-off wavenumber and wavenumber of maximum growth are most important; the variation of these quantities with the density and viscosity ratios, the Weber number and Reynolds is computed and displayed as graphs and asymptotic formulas. The instabilities of a liquid jet are due to capillary and KH instabilities. We show that KH instability cannot occur in a vacuum but capillary instability can occur in vacuum. We present comprehensive results, based on viscous potential flow, of the effects of the ambient. Temporally unstable liquid jet flows can be analyzed for spatial instabilities by C/A theory; they are either convectively unstable or absolutely unstable depending on the sign of the temporal growth rate at a singularity of the dispersion relation. The study of such singularities is greatly simplified by the analysis here which leads to an explicit dispersion relation; an algebraic function of a complex frequency and complex wavenumber. Analysis of this function gives rise to an accurate Weber–Reynolds criterion for the border between absolute and convective instabilities. Some problems of the applicability to physics of C/A analysis of stability of spatially uniform and nearly uniform flows are discussed.

Relativistic kinetic theory of electromagnetic waves in equilibrium magnetized plasma. General dispersion equations

The relativistic kinetic theory of parallel propagating electromagnetic waves in a magnetized equilibrium plasma is presented. On the basis of relativistic Vlasov–Maxwell equations, a general explicit dispersion relation is derived by a correct analytical continuation for all complex frequencies of electromagnetic waves.PACS Nos.: 52.25.Dg, 52.25.Xz, 52.27.Ep, 52.27.Ny, 52.35.Hr, 52.35.Mw, 52.35.Py

Instability of liquid film flowing down a linearly heated plate *

Abstract The full-scale linear stability equations for a liquid film flowing down a linearly heated inclined plate are derived from the N-S equations, and the stability characters of both temporal and spatial modes are computed by using Chebyshev spectral collocation method. The effects of Weber numbers and Marangoni numbers on growth rate, marginal curve, critical Reynolds number, etc. are investigated. An explicit dispersion relation Under long-wave approximation, which is in exact agreement with Miladinova's one, is obtained and the limits of long-wave approximation are discussed.

Traveling waves in a one-dimensional integrate-and-fire neural network with finite support connectivity

We study the existence of traveling waves in a one-dimensional network of integrate-and-fire neurons with finite support coupling. We show that when the reset after spiking is sufficiently low, the interspike intervals (ISIs) for a traveling wave with an explicit first wave front can be computed. Further, we analytically derive a self-consistency equation that generates an explicit dispersion relation between the velocity of periodic traveling waves and their corresponding ISIs. Finally, we prove that the ISIs of non-periodic traveling waves converge to the ISI of the periodic waves with the same speed.

A nonlinear RDF model for waves propagating in shallow water

In this paper, a composite explicit nonlinear dispersion relation is presented with reference to Stokes 2nd order dispersion relation and the empirical relation of Hedges. The explicit dispersion relation has such advantages that it can smoothly match the Stokes relation in deep and intermediate water and Hedgs’s relation in shallow water. As an explicit formula, it separates the nonlinear term from the linear dispersion relation. Therefore it is convenient to obtain the numerical solution of nonlinear dispersion relation. The present formula is combined with the modified mild-slope equation including nonlinear effect to make a Refraction-Diffraction (RDF) model for wave propagating in shallow water. This nonlinear model is verified over a complicated topography with two submerged elliptical shoals resting on a slope beach. The computation results compared with those obtained from linear model show that at present the nonlinear RDF model can predict the nonlinear characteristics and the combined refraction and diffraction of shallow-water waves.