Dispersion Equation For Waves Research Articles

On the possibility of the full stabilization of the capillary instability of a dielectric liquid jet by a longitudinal static field

The dispersion equation for axisymmetric capillary waves at the surface of a cylindrical jet of an ideal incompressible dielectric liquid in a symmetry axis electrostatic field collinear to the jet has been analyzed. It was shown that, in contradiction with the idea put forward previously, no full stabilization of the capillary instability occurs at any arbitrarily large values of the field intensity.

Transverse surface waves in a piezoelectric material carrying a gradient metal layer of finite thickness

The existence and propagation behavior of transverse surface waves in a layered structure concerning a piezoelectric substrate and a gradient metal layer are theoretically investigated in this paper. The Wentzel–Kramers–Brillouin (WKB) method is applied to obtain the analytical solutions in the gradient metal layer. The dispersion equation for transverse surface waves in such structure is obtained in a quite simple mathematical form, where the material gradient of the metal layer assumes arbitrary functions. Effects of material gradient on three types of dispersion behavior are discussed in detail based on a proper classification. Numerical results show that the material gradient in the metal layer evidently affects the fundamental mode shape of the transverse surface waves but has negligible effects on the higher order modes.

On the electron whistler dispersion law in a cold plasma with light ions and heavy charged particulates

The dispersion equation of electron whistler waves in a cold plasma with two light ions of comparable gyrofrequencies and heavy charged particulates is derived. It is valid in a very wide frequency range above the highest ion cutoff frequency when the wave frequency is essentially less than the electron plasma frequency. The derived electron whistler dispersion law is expressed through the relative contents of the two light ions and the electrons, as well as the characteristic frequencies of the magnetized plasma, as the lower hybrid resonance frequency, the two highest ion cutoff frequencies, the gyrofrequencies of the light ions, and the electron gyro- and plasma frequencies. The approximation of vanishingly small gyrofrequencies of the heavy ions permits to determine with a relevant accuracy the electron whistler dispersion law using the features of electron whistler spectrograms only. Estimates of the relative charge density of the light ions are obtained and the dispersion laws of the adjacent branches, i.e., the electron whistler waves and the so-called ion cyclotron whistlers are calculated. For the electron whistler waves, the presence of negative ions can be the origin of a manyfold increase in the lower cutoff frequency; a merging effect of the cutoff frequencies of the adjacent branches can also appear.

Effect of tensor g-factor on the spectrum of eigen modes in magnetoactive plasma

The influence of the tensorial nature of the gyromagnetic coupling on a spectrum of eigen modes in magnetoactive plasma is studied. A dispersion equation for waves propagating in such a medium is obtained in the approximation of frigid hydrodynamics.

DISPERSION OF ELECTROMAGNETIC WAVES GUIDED BY AN OPEN TAPE HELIX I

The dispersion equation for electromagnetic waves guided by an open tape helix for the standard model of an inflnitesimally thin and perfectly conducting tape is derived from an exact solution of a homogeneous boundary value problem for Maxwell's equations. A numerical analysis of the dispersion equation reveals that the tape current density component perpendicular to the winding direction does not afiect the dispersion characteristic to any signiflcant extent. In fact, there is a signiflcant deviation from the dominant-mode sheath- helix dispersion curve only in the third allowed region and towards the end of the second allowed region. It may be concluded that the anisotropically conducting model of the tape helix that neglects the above transverse-current contribution is a good approximation to the isotropically conducting model that takes into account this contribution except at high frequencies even for wide tapes.

Dispersion relation of degenerated electron–positron plasma in an ultra-relativistic regime

AbstractThe dispersion relation for superluminal waves in degenerated and isotropic electron–positron plasmas is investigated. The dispersion equation of linear waves is derived from the relativistically correct form of the dielectric function and the Fermi distribution function. Analytical dispersion laws for the real part of the wave frequency are derived by applying the long-wavelength approximation and the short-wavelength approximation. Using the numerical simulation method, we obtain the full dispersion curve which cannot be given by an analytic method.

Waves on the surface of a vapor film under conditions of intensive heat fluxes

We investigate surface waves on the interface between a thin vapor film and a layer of liquid in the presence of a high steady heat flux. This problem arises when a metal surface heated to a high temperature is immersed into a cold liquid. The general boundary conditions, which take into account the temperature dependence of saturation pressure on the vapor-liquid interface, are derived. These boundary conditions generalize the traditional conditions on the free surface of liquid in the gravity field. The stability of the planar vapor-liquid interface is investigated analytically with linear approximations. The dispersion equation for surface waves on the vapor-liquid interface in the presence of strong heat flux is derived. A number of different, distinct from the classical surface wave problem, effects arise in the problem under consideration. The thermal processes, which occur on the phase boundary and are possible in the absence of gravity force, lead to the generation of weakly decaying periodic waves of low amplitude, whose velocities may exceed significantly those of gravity waves. The heat flux through the interface may cause periodic surface waves of small length (ripple), which are not capillary. The processes of phase transition on the interface are capable of providing the stability of vapor film under a layer of liquid in the gravity field.

Dispersion relation and growth rate in electromagnetically pumped free-electron lasers with ion-channel guiding

A theory of electromagnetic-wave wiggler with ion-channel guiding is presented. Steady-state electron trajectories are obtained by solving the equations of motion for a single electron in a free-electron laser with electromagnetic-wave wiggler and ion-channel guiding field. The eighth-degree polynomial dispersion equation for electromagnetic and space charge waves in this configuration is derived. The characteristics of dispersion relation are analyzed in detail by numerical solutions. It is found that the growth rate for group I and II orbits has been affected by the presence of the ion-channel guiding and the growth rate for group II orbits is more affected.

Probing Arrays of Circular Magnetic Microdots by Ferromagnetic Resonance

X-band ferromagnetic resonance (FMR) was used to characterize in-plane magnetic anisotropies in rectangular and square arrays of circular nickel and Permalloy microdots. In the case of a rectangular lattice, as interdot distances in one direction decrease, the in-plane uniaxial anisotropy field increases, in good agreement with a simple theory of magnetostatically interacting uniformly magnetized dots. In the case of a square lattice a four-fold anisotropy of the in-plane FMR field H(r) was found when the interdot distance a gets comparable to the dot diameter D. This anisotropy, not expected in the case of uniformly magnetized dots, was explained by a non-uniform magnetization m(r) in a dot in response to dipolar forces in the patterned magnetic structure. It is well described by an iterative solution of a continuous variation procedure. In the case of perpendicular magnetization multiple sharp resonance peaks were observed below the main FMR peak in all the samples, and the relative positions of these peaks were independent of the interdot separations. Quantitative description of the observed multiresonance FMR spectra was given using the dipole-exchange spin wave dispersion equation for a perpendicularly magnetized film where in-plane wave vector is quantized due to the finite dot radius, and the inhomogenetiy of the intradot static demagnetization field in the nonellipsoidal dot is taken into account. It was demonstrated that ferromagnetic resonance force microscopy (FMRFM) can be used to determine both local and global properties of patterned submicron ferromagnetic samples. Local spectroscopy together with the possibility to vary the tip-sample spacing enables the separation of those two contributions to a FMRFM spectrum. The global FMR properties of circular submicron dots determined using magnetic resonance force microscopy are in a good agreement with results obtained using conventional FMR and with theoretical descriptions.

On new electromagnetic waves in a multicomponent insulator

The dispersion equation for additional transverse electromagnetic waves in a multicomponent amorphous insulator is analyzed in the vicinity of a narrow absorption line. Such waves can be excited due to spatial dispersion associated with fluctuation of the polarizability of insulator molecules. The conditions under which the dispersion relation of an additional (new) wave is mainly determined by the mean square of the polarizability fluctuations are determined.

Linear wave reflection by trench with various shapes

Two types of analytical solutions for waves propagating over an asymmetric trench are derived. One is a long-wave solution and the other is a mild-slope solution, which is applicable to deeper water. The water depth inside the trench varies in proportion to a power of the distance from the center of the trench (which is the deepest water depth point and the origin of x-coordinate in this study). The mild-slope equation is transformed into a second-order ordinary differential equation with variable coefficients based on the longwave assumption [Hunt's, 1979. Direct solution of wave dispersion equation. Journal of Waterway, Port, Coast. and Ocean Engineering 105, 457–459] as approximate solution for wave dispersion. The analytical solutions are then obtained by using the power series technique. The analytical solutions are compared with the numerical solution of the hyperbolic mild-slope equations. After obtaining the analytical solutions under various conditions, the results are analyzed.

Propagation of SH waves in an irregular monoclinic crustal layer

The present paper discusses the dispersion equation for SH waves in a monoclinic layer over a semi-infinite elastic medium with an irregularity. In the absence of the irregularity, the dispersion equation reduces to standard dispersion equation for SH waves in a monoclinic layer over an isotropic semi-infinite medium. The dispersion curves for different size of the irregularity are computed and compared for the half-space without any irregularity. It can be seen that the phase velocity is strongly influenced by the wave number and the depth of the irregularity.

Dispersion equations for steady waves in an anisotropic elastic plate or a layered plate

Steady waves propagating in a plate that consists of one or more layers of general anisotropic elastic material are studied. The surface of the plate can be a traction-free (F), rigid (R) or slippery surface (S). The interface between any two layers in the plate can be perfectly bonded (b) or in sliding contact (s). The thickness of the layers need not be the same. The purpose of this paper is to present dispersion equations for all possible combinations of the boundary and interface conditions. If the thickness h of one of the layers is very small, the dispersion equation allows us to expand the solution in an infinite series in the power of h from which an approximate solution can be obtained by keeping the terms up to O ( h n ) for any n . The special case of a sandwich plate that consists of a centre layer and two identical outside layers is studied. In the literature, the dispersion equations for a sandwich plate were studied for special elastic materials. The results presented here are for elastic materials of general anisotropy.

An Analytic Solution for Wave Scattering by a Circular Cylinder Mounted on a Conical Shoal

In this paper, the combined refraction-diffraction of plane monochromatic waves by a circular cylinder mounted on a conical shoal in an otherwise open sea of constant depth is solved analytically based on the mild-slope equation. At first, Hunt's approximate direct solution for the wave dispersion equation [Hunt, 1979] is used to transform the mild-slope equation with implicit coefficients into an equation with all coefficients being explicitly expressed. This equation is then solved analytically in terms of combined Fourier series and Taylor series. Comparisons are made between the present method and the analytical solutions based on the linear shallow-water equation [Zhu and Zhang, 1996] and the Helmholtz equation [MacCamy and Fuchs, 1954] for long waves and short waves, respectively, and excellent agreements are obtained. For waves in intermediate water depth, comparisons are made with numerical results based on the mild-slope equation and an equally good quality of agreement is achieved. By varying the shoal size, it is found that the wave amplification normally enhances with the increase of shoal, due to the increased contribution from wave refraction. For relatively short wave, the trapped waves by refraction in front of the island interact with the reflected waves and form the partial-standing waves there.

Modified Newton–Raphson Solution for Dispersion Equation of Transition Water Waves

Modified Newton–Raphson solution for dispersion equation of transition water waves is proposed for practical applications. The wave dispersion equation is a nonlinear equation. Therefore, one has to apply a time-consuming trial-and-error method. However, it may be solved by utilizing an iterative technique commonly referred to as Newton–Raphson (NR) iteration technique and Chebyshev approximation, which are used to solve the system of nonlinear equations. Chebyshev approximation has the advantage of requiring less iteration. In this study, a numerical solution model based on utilizing modified NR technique with Chebyshev approximation to determine value of the wave number (k) is developed. It is shown how iteration problems can be solved by modified NR technique with Chebyshev approximation. This computational model is applied by computer programs that have visual basic (VBA) code prepared under the Microsoft Excel Macro. The wave dispersion equation for transition water waves were solved by modified NR technique with Chebyshev approximation implemented in the computer programs prepared with VBA computer language. An example for the given numerical solution model is presented.

The group velocity variation of Lamb wave in fiber reinforced composite plate

Experimentally measured Lamb wave group velocities in composite materials with anisotropic characteristics are not the same as the theoretical group velocities which is calculated with the Lamb wave dispersion equation. This discrepancy arises from the fact that the angle between the group velocity direction and the phase velocity direction in anisotropic materials exists. Wave propagation in a composite material with anisotropic characteristics should be considered with respect to magnitude correction in addition to direction correction. In this study, S0 mode phase velocity dispersion curves are depicted with the variation of degree with respect to the fiber direction using a Lamb wave dispersion relation in the unidirectional, bidirectional, and quasi-isotropic composite plates. Slowness surface is sketched by the reciprocal value of the phase velocity curves. The magnitude and direction of the group velocity could be calculated from the slowness surface. The recalculated group velocities with consideration of the magnitude and direction from the slowness surface are compared with experimentally measured group velocities. The proposed method shows good agreements with theoretical and experimental results.

An analytic solution to the mild slope equation for waves propagating over an axi-symmetric pit

An analytic solution to the mild slope equation is derived for waves propagating over an axi-symmetric pit located in an otherwise constant depth region. The water depth inside the pit decreases in proportion to an integer power of radial distance from the pit center. The mild slope equation in cylindrical coordinates is transformed into ordinary differential equations by using the method of separation of variables, and the coefficients of the equation in radial direction are transformed into explicit forms by using the direct solution for the wave dispersion equation by Hunt (Hunt, J.N., 1979. Direct solution of wave dispersion equation. J. Waterw., Port, Coast., Ocean Div., Proc. ASCE, 105, 457–459). Finally, the Frobenius series is used to obtain the analytic solution. Due to the feature of the Hunt's solution, the present analytic solution is accurate in shallow and deep waters, while it is less accurate in intermediate depth waters. The validity of the analytic solution is demonstrated by comparison with numerical solutions of the hyperbolic mild slope equations. The analytic solution is also used to examine the effects of the pit geometry and relative depth on wave transformation. Finally, wave attenuation in the region over the pit is discussed.

Elastic waves along a cylindrical borehole in a poroelastic medium saturated by two immiscible fluids

The propagation of elastic waves along a cylindrical borehole filled with/without liquid and embedded in an infinite porous medium saturated by two immiscible fluids has been studied. The theory of porous media saturated by two immiscible fluids developed by Tuncay and Corapcioglu (1997) is employed. Frequency equations determining the phase velocity of axial symmetric waves are obtained. It is found that the surface waves along cylindrical borehole are dispersive. The dispersion equation of Rayleigh-type surface waves along the boundary of a poroelastic solid half-space saturated by two immiscible fluids is also obtained. Some special cases have been deduced and the dispersion curves are obtained numerically for a peculiar model. It is found that the density of fluids affects the Rayleigh mode.

Multiple scattering by a random configuration of cylinders in a poro‐elastic medium: Properties of the coherent waves

An extension of Twersky’s formalism is developed in order to obtain the dispersion equation of the coherent waves in a poro‐elastic medium in which cylindrical scatterers are randomly placed. The high frequency regime only, where no dispersion effects occur in the absence of scatterers, is investigated. The scatterers lay within a slab‐like region of the host medium, and an incident harmonic plane wave gives rise to coherent plane waves in the slab. At normal incidence, the coherent waves are damped, while they are inhomogeneous at other incidence angles. The dispersion equation obtained shows that the longitudinal (fast and slow) coherent waves are coupled, while the shear one obeys the Waterman and Truell’s well‐known dispersion equation.

Scattering and Trapping of Wave Energy by a Submerged Truncated Paraboloidal Shoal

In this paper, we study the scattering and trapping of wave energy by a submerged truncated paraboloidal shoal. The analytical approach of Liu et al. in 2004 for solving wave amplification around a circular island mounted on a shoal is extended in this study to investigate waves above a submerged shoal. In this approach, the mild-slope equation is first transformed into an explicit equation by using Hunt's 1979 Pade approximation to the wave dispersion equation. It is then solved analytically in terms of combined Fourier series and Taylor series. For incident waves varying from short to long waves, wave amplifications in the vicinity of the shoal are calculated using the newly derived analytical model and they agree excellently with the finite difference solutions of Lin in 2004. It is found that, for the particular shoal studied in this paper, there exists an energy focal point where the wave amplitude is the maximum. This point is the result of combined wave refraction, diffraction, and reflection. The point is always located along the x-axis and it is behind the shoal center for very short waves. As the increase of wave period, it moves upstream and it can come in front of the shoal center for very long waves when wave reflection is significant. For a particular incident wave, wave amplification increases and the focal point moves toward the shoal center with the reduction of the water depth above the shoal center. The maximum wave amplitude increases exponentially as the decrease of local water depth. It is also found that there exists the so-called calm areas for shallowly submerged shoal, which are located behind the focal point and the local amplitude can be as small as nearly zero.