Dispersion Relation Of Surface Waves Research Articles

Electromagnetic Surface Waves in a Relativistic Magnetized Plasma Propagating in a Cylindrical Column

Dispersion relations of electromagnetic surface waves in a magnetized relativistic plasma propagating in a cylindrical column surrounded by vacuum are derived by means of the relativistic fluid equations. The zero order relativistic beam is assumed to stream along the direction of the axial magnetic field. The coupled equations of the E- and B-wave modes are solved analytically in terms of the cylindrical coordinates in weak or intermediate values of the magnetic field in which the cyclotron frequency is much smaller than the relativistic Doppler-shifted frequency. The entangled field components of the E- and B-waves are decoupled by assuming that the flow is incompressible. The surface wave dispersion relations of the E- and B-wave modes show a strong similarity and symmetry. The static magnetic field splits the E- and B-waves, thus doubling the modes in unmagnetized plasma. By taking the radius of the cylinder infinite, the dispersion relations in a semi-infinite plasma are derived.

On effective surface elastic moduli for microstructured strongly anisotropic coatings

The determination of surface elastic moduli is discussed in the context of a recently proposed strongly anisotropic surface elasticity model. The aim of the model was to describe deformations of solids with thin elastic coatings associated with so-called hyperbolic metasurfaces. These metasurfaces can exhibit a quite unusual behaviour and concurrently a very promising wave propagation behaviour. In the model of strongly anisotropic surface elasticity, strain energy as a function of the first and second deformation gradients has been introduced in addition to the constitutive relations in the bulk. In order to obtain values of surface elastic moduli, we compare dispersion relations for anti-plane surface waves obtained using the two-dimensional (2D) model and three-dimensional (3D) straightforward calculations for microstructured coatings of finite thickness. We show that with derived effective surface moduli, the 2D model can correctly describe the wave propagation.

Effects of nonlinearity on Anderson localization of surface gravity waves

Anderson localization is a multiple-scattering phenomenon of linear waves propagating within a disordered medium. Discovered in the late 50s for electrons, it has since been observed experimentally with cold atoms and with classical waves (optics, microwaves, and acoustics), but whether wave localization is enhanced or weakened for nonlinear waves is a long-standing debate. Here, we show that the nonlinearity strengthens the localization of surface-gravity waves propagating in a canal with a random bottom. We also show experimentally how the localization length depends on the nonlinearity, which has never been reported previously with any type of wave. To do so, we use a full space-and-time-resolved wavefield measurement as well as numerical simulations. The effects of the disorder level and the system’s finite size on localization are also reported. We also highlight the first experimental evidence of the macroscopic analog of Bloch’s dispersion relation of linear hydrodynamic surface waves over periodic bathymetry.

Instability of viscous annular liquid film jet under high air-to-fluid velocity rate

Based on the surface wave dispersion relation equation of viscous annular liquid film jet established by the research group in the early stage, a dispersion relation equation calculation program is written according to Mathematica. A linear stability analysis is carried out for the high air-fluid velocity ratio viscous annular liquid film jet entering a compressible gas medium environment. The relationship between the surface wave growth rate of an annular liquid film jet and the surface wave number under different characteristic parameters is obtained. The connection based on the dominant wave’s growth rate or the dominant wave’s number with dimensionless parameters such as air-fluid velocity rate, air-fluid density rate, Weber number of liquid, liquid Reynolds number, and gas Mach number was analyzed. The consequences reveal that under the high air-fluid velocity ratio, the increase of the above dimensionless parameters promotes the fragmentation course of the annular liquid film jet. Moreover, the augment of the air-fluid velocity rate has the greatest influence on the stability of the viscous annular liquid film jet. In contrast, the Reynolds number of the liquid has less influence than the reliability of the annular liquid film jet. Linear stability analysis under this condition lays the foundation for the prediction model of the most unstable parameters of annular liquid film jet in practical engineering applications.

Theoretical analysis of surface waves in piezoelectric medium with periodic shunting circuits

The investigations of surface waves in the piezoelectric medium bring out great possibility in designing smart surface acoustic wave (SAW) devices. It is important to study the dispersion properties and manipulation mechanism of surface waves in the semi-infinite piezoelectric medium connected with periodic arrangement of shunting circuits. In this study, the extended Stroh formalism is developed to theoretically analyze the dispersion relations of surface waves under different external circuits. The band structures of both the Rayleigh wave and the Bleustein-Gulyaev (BG) wave can be determined and manipulated with proper electrical boundary conditions. Furthermore, the electromechanical coupling effects on the band structures of surface waves are discussed to figure out the manipulation mechanism of adjusting electric circuit. The results indicate that the proposed method can explain the propagation behaviors of surface waves under the periodic electrical boundary conditions, and can provide an important theoretical guidance for designing novel SAW devices and exploring extensive applications in practice.

Comment on: “Tunable surface waves supported by graphene-covered left-handed material structures” [Opt. Commun. 489 (2021) 126874

In a recent article (Yaqoob et al., 2021), Yaqoob et al. studied the tunable electromagnetic surface wave propagation on the graphene-covered left-handed material (LHM) slab backed by the perfect electric conductor (PEC). They derived the dispersion relations for both transverse electric (TE) and transverse magnetic (TM)-polarized surface waves, and then used the relations for two types of LHMs according to the spectral range, i.e., GHz- and THz-LHM. The present Comment points out although the system under consideration (i.e., graphene-covered LHM) is itself physically challenging, the results for THz-LHM are also rather unreal because the frequency regime to accomplish the calculations falls within the high THz frequency band. In such a frequency band, the conductor substrate no longer behaves like a PEC and thus the results are rather unreal when considering THz-LHM. We note that the metal, for example, gold or silver may be modeled as a PEC, but only for a few THz. Moreover, when considering GHz-LHM, we obtain new forms of dispersion relations for both TE and TM surface waves. We find that our dispersion relations do not agree with the results obtained by Yaqoob et al.

Single-point dispersion measurement of surface waves combining translation, rotation and strain in weakly anisotropic media: theory

SUMMARY A new approach is proposed for measuring the dispersion relation of surface waves in weakly anisotropic media using a single station, which consists of translational displacement and rotation or strain. The azimuth-dependent surface wave dispersion curve can be directly obtained by using the amplitude ratio of translational displacement to rotation or strain. Using observations from earthquakes from a variety of azimuths allows us to characterize the anisotropy of subsurface media. The amplitude ratio gives the localized azimuth-dependent dispersion relations of the formation beneath a receiver without requiring knowledge of the source or structure along the path. The theory shows that in anisotropic media the coupled quasi-Rayleigh wave and quasi-Love wave will result in elliptically polarized rotational motions. In addition, rotational motion information allows the determination of backazimuth even in general anisotropic media. We carry out numerical experiments, investigate the effects of noise and degree of anisotropy and demonstrate the potential for field studies.

Excitation of surface waves in 3D ion beam neutralization

Neutralization of beams with 2D and 3D geometries by the electrons emitted from an external source is studied using particle-in-cell simulations. Our work reveals that the high-energy electrons excite Trivelpiece–Gould (TG) surface waves in the beams with 3D axisymmetric geometries. These high-energy electrons are generated because of a large amplitude electrostatic solitary wave (ESW) that forms near the electron source and has an electric potential amplitude more than three times the electron thermal energy. We also find that surface wave excitation only happens when the beam radius is large enough at the ion source to attract enough electrons that could form the large amplitude ESW. A comparison of the 3D TG surface wave dispersion relation with an expression for 2D surface waves reveals that they become excited in 3D axisymmetric but not in 2D planar beam because of a higher phase speed requirement in the latter case.

An accurate method for dispersion characteristics of surface waves in layered anisotropic semi-infinite spaces

In this paper, we develop a novel method for the dispersion characteristics of surface waves in layered anisotropic semi-infinite spaces. Compared to solving the algebraic transcendental eigenequations directly, this method can find all surface waves accurately. For solving accurately, the first-order state equation is formed by introducing the dual variable. Some properties of the complex Hamiltonian matrix are presented and proved, then based on these properties, the radiation condition for the homogeneous anisotropic half-space is given. And, based on the scaling and squaring algorithm and mixed energy matrix, the precise integration method (PIM) is presented for solving accurately the eigenvalue problem of ordinary differential equations (ODEs) corresponding to surface waves. All eigenfrequencies can be found with certainty by using the concept of the eigenvalue count in the Wittrick-Williams (W-W) algorithm. Numerical examples show that the proposed method is highly accurate and efficient for solving dispersion relations of surface waves in layered anisotropic semi-infinite spaces, which involves the comparison of other previously published methods.

Love-type waves in thermoelastic solid with double porosity structure

In the present work, we consider a model consisting of an elastic layer of finite thickness resting over an elastic half-space in perfect contact. The layer and the half-space both are assumed to be composed of different thermoelastic materials possessing different double porosity structures. The dispersion relation for Love-type surface waves has two factors, one of which matches exactly with the dispersion relation of Love waves of classical elasticity, while the second factor arises due to the presence of double porosity and thermal field in the model. Each of these factors (dispersion relation) provides two types of Love waves called ‘first front’ and ‘second front’ of Love waves. For a specific model, these two dispersion relations are computed numerically to obtain the phase speed of corresponding front of propagating Love waves. It is found that the speed of second front is much faster than the speed of first front. The phase speed of fundamental mode of the second front increases with the inclusion of thermal property, while it decreases with the inclusion of either type of voids/porosities in the model. Some earlier known results are also recovered from the present formulation as special cases.

Surface Waves on Non-conducting Structured Interfaces

Surface waves are demonstrated theoretically to propagate along periodically corrugated surfaces made of non-conductive lossless materials with positive permittivity. An analytic derivation of the surface wave dispersion relation is presented in the case of the deeply subwavelength period of a corrugated structure using impedance boundary conditions (IBCs) at the interfaces. Thus obtained dispersion relation is verified numerically and limitations as well as the physicality of IBCs when modeling surface waves are discussed. Finally, suitable materials, potential experimental realization, and sensing applications of surface waves in the terahertz spectral range are discussed.

Functionally Graded Materials Pile Structure for Seismic Noise Cancellation

This study explores surface-wave propagation through periodic arrangements of multilayered piles in a single-layer soil medium from the viewpoint of bandgaps. The influence of the geometrical parameters of the model on the bandgaps is discussed. The periodic theory of solid-state physics is imposed to derive the dispersion equation. The finite-element method is implemented to calculate the dispersion relation and attenuation zones for surface waves in a periodic pile and soil system. The results show that the bandgaps are sensitive to the size and arrangement of the piles. The practicability of ground vibration-isolation by periodic pile barriers are verified in the time domain. The location and width of bandgaps are further authenticated through finite unit cell–based frequency response analyses. It is found that the frequency range of the Rayleigh wave reduction agrees well with the theoretical attenuation zone. This work provides new insights into the design of periodic piles as wave barriers. This approach has scope for protecting infrastructures, such as nuclear power plants, which pose high environmental risks. It is found that the dispersion analysis gives essential information for Rayleigh wave isolation using piles, including the frequency range and the form of decline.

Tunable terahertz Bloch surface waves in one-dimensional photonic crystals with a Dirac semimetal cap layer

The dispersion relation and localization of surface waves in semi-infinite one-dimensional photonic crystals in the presence of Dirac semimetal as a cap layer are theoretically and numerically investigated, using the transfer matrix method at terahertz band. It is demonstrated that the frequency and localization of surface waves can be tuned by changing the cap layer's Fermi energy for both transverse electric and transverse magnetic polarizations. The numerical results show that the dispersion relation of surface waves and consequently their localization are more sensitive to changes in Fermi energy. Therefore, the surface states with the desired localization can be achieved. In addition, the effect of the cap layer's thickness and the arrangement of the photonic crystal layers on surface waves are investigated. The results show that these parameters have considerable effect on modification of surface waves.

Rayleigh-like waves in multilayered elastic media containing voids: Use of the Haskell matrix method

Using the Haskell matrix method, the dispersion relation of Rayleigh-like surface waves propagating through a multilayered elastic solid half-space is derived. Each layer as well as the half-space is assumed to have voids (pores) distributed evenly throughout. This dispersion relation is then reduced for a 2-layered model (single layer over a half-space) to study the characteristics of phase speed of Rayleigh-like wave. For a particular model, the numerical computations are performed to observe the effect of voids on the fundamental mode of Rayleigh-like waves for n = 2 and 3. For the 2-layered model, it is also shown that the particle motion remains elliptical but influenced by the presence of voids. In the absence of voids from the model, the dispersion relation earlier obtained by Haskell (1953) for the case n = 2 is recovered successfully.

Nonlinear surface waves on overdense plasma

Nonlinear perturbations have significant consequences on excitation of surface waves on overdense plasma. Linear surface modes on overdense plasma appear at specific frequencies of the incident wave. By nonlinear considerations, however, this phenomenon takes place for frequencies different from frequencies of the linear regime. Here, to study the nonlinear effects on the modifications of the proper frequencies of exciting the surface modes, we consider the framework of the warm electron-fluid model and Poisson equation. We apply a perturbation procedure to linearize the nonlinear set of magnetohydrodynamic equations. Obtaining the analytical solutions at the first- and second-order corrections provides a powerful tool to perform our study. By employing proper boundary conditions, we derive dispersion relations for both corrections. The range of the allowed frequency of the incident wave that leads to the stable surface plasma waves is determined at the linear approximation. Then, at the second-order corrections, we investigate the nonlinear modifications of the dispersion relation of the plasma surface waves and find a new proper frequency region. We also discuss the influences of the presence of the nonlinearities and the thermal effects on the involved physical quantities.

Experimental Dispersion Relation of Surface Waves along a Torus of Fluid.

We report the observation of gravity-capillary waves on a torus of fluid. By means of an original technique, a stable torus is achieved by depositing water on a superhydrophobic groove with a shallow wedge-shaped channel running along its perimeter. Using a spatiotemporal optical measurement, we report the full dispersion relation of azimuthal waves propagating along the inner and outer torus borders, highlighting several branches modeled as varicose, sinuous, and sloshing modes. Standing azimuthal waves are also studied leading to polygonlike patterns arising on the two torus borders with a number of sides different when a tunable decoupling of the two interfaces occurs. The quantized nature of the dispersion relation is also evidenced.

Characteristics of hybrid surface plasmon polaritons at a chiral graphene metal interface in cylindrical waveguides

A detailed theoretical analysis of the hybrid surface plasmon mode generation and propagation at a chiral graphene metal (CGM) interface for a cylindrical structure is presented. The conductivity of graphene is modeled using the Kubo formalism and the dispersion relation for the hybrid surface waves is computed on the basis of the Kubo formalism and impedance matching boundary conditions. The hybrid mode consisting of lower and upper plasmon modes is witnessed due to the presence of the chiral medium. The frequency band gap between the lower and upper plasmon modes is found to be sensitive and tunable with respect to the chiral strength, radial distance of the waveguide, and chemical potential of graphene. The propagation length and effective refractive index can also be modulated by varying the chiral strength and chemical potential. For very high values of chirality and biasing voltage, the Goos–Hänchen effect is observed at the CGM interface. The cutoff values of chiral strength as a function of normalized frequency make the proposed structure applicable for chiroptical and chemical sensing and enantiomeric detection in the THz frequency regime.

Slab-geometry surface waves on steep gradients and the origin of related numerical issues in a variety of ICRF codes

In the ion cyclotron range of frequencies, electromagnetic surface waves are physically relevant for wave–filament interactions, parasitic edge losses and sheath–plasma waves. They are also important numerically, where non-physical surface waves may occur as side effects of slab-geometry approximations. We give new, completely general, mathematical techniques to construct dispersion relations for electromagnetic surface waves between any two media, isotropic or anisotropic, and first-order corrections for when the material interface is steep but continuous. We discuss numerical issues (localized non-convergence, undesired power generation) that arise in numerical calculations due to the presence of surface waves.

Slanted-eye featured forbidden bands in a water channel with undulated sidewalls

Resonances induced by structure periodicity can always lead to frequency gaps, which are forbidden bands of propagating waves. Besides the traditional Bragg gaps, non-Bragg gaps have been found in periodic waveguides for years, which have been proved to be caused by the different transverse mode interference. Here, we demonstrate another observable slanted-eye feature of non-Bragg gaps by introducing the sidewall corrugations into a water channel. Based on the multiple scale expansion and group theory, we have derived the dispersion relations for water surface waves propagating in a channel with periodically corrugated sidewalls and introduced the resonance selection rule to identify the characteristics of forbidden bands. It has been experimentally found that the creation and enhancement of Bragg and non-Bragg gaps highly depend on the phase difference between the two sidewall corrugations. Furthermore, the non-Bragg nature forbidden bands are quite different from those of Bragg ones when the duty ratio of corrugations varies from 0 to 1. The non-Bragg gaps have unexpectedly formed a slanted-eye structure rather than the Bragg’s normal eye in spectra, though the both forbidden bands experience a process of producing, enhancement, decaying, and disappearing. The slanted-eye feature confirms the different nature of non-Bragg gaps over Bragg ones and could benefit applications in wave control engineering.

An estimation of the axial structure of surface-wave produced plasma column

An analytical expression to predict the axial distribution of electron density along the surface-wave sustained plasma column has been derived. The model is based on the square root approximation of the dispersion relation that gives a linear roll-off of electron density distribution in the plasma column. The point is to use a more tractable analytical approximation of dispersion relation of surface waves. An expression of attenuation coefficient, which includes the collisional loss, has been applied in studying the structure of the surface-wave produced plasma column. The derived attenuation model predicts a linear axial attenuation of electron density, as well as a nonlinear relation between the glass permittivity dependent frequency sensitivity and the wavenumber. The nonlinearity arises from non-negligible dielectric thickness. This result is consistent with early Aliev's results of the electron density profile, which correspond to the volume collisional absorption regime of surface wave sustained plasmas. The combined effects of surface-wave amplitude attenuation due to collisions and frequency dependence of group velocity cause an axial decrease in the surface wave axial power flux as the wave moves away from the launcher. A short analysis of the sensitivity of the normalized frequency to the permittivity of the glass based on the quasistatic approximation of the dispersion relation of surface waves is given.