Dispersion relation surface waves

Surface waves at an interface between two innniscible fluids involve effects due to gravity (g) and surface tension (a) forces. (In this section, o denotes surface tension and a denotes the stress tensor. The two should not be coiifiised with one another.) In a hydrodynamic approach, the interface is treated as a sharp boundary and the two bulk phases as incompressible. The Navier-Stokes equations for the two bulk phases (balance of macroscopic forces is the mgredient) along with the boundary condition at the interface (surface tension o enters here) are solved for possible hamionic oscillations of the interface of the fomi, exp [-(iu + s)t + i V-.r], where m is the frequency, is the damping coefficient, s tlie 2-d wavevector of the periodic oscillation and. ra 2-d vector parallel to the surface. For a liquid-vapour interface which we consider, away from the critical point, the vapour density is negligible compared to the liquid density and one obtains the hydrodynamic dispersion relation for surface waves + s>tf. The temi gq in the dispersion relation arises from... 

Solving Maxwell s equations at the metal/dielectric interface at the appropriate boundary conditions yields the surface plasmon dispersion relation, that is, the relation of the angular frequency co and the x-component of the surface plasmon wave vector kSP,... 

At the surface of metals, the surface plasmon-polaritons, also called "surface plasmons," are not the same as the "bulk" plasmons these surface plasmons are affected (i.e., shifted slightly in energy) by monolayer adsorbates thus Surface Plasmon Resonance (SPR) spectroscopy yields information about the nature of the binding of the adsorbates onto a metal surface. The surface plasmons are excited by a p-polarized electromagnetic wave (polarized in the plane of the film) that crosses a glass medium (1), such as a prism, and is partially reflected by a metallic film (2) and back into the glass medium the dispersion relation is... 

Surface plasmons (SPs) are surface electromagnetic waves that propagate parallel along a metal/dielectric interface. For this phenomenon to occur, the real part of the dielectric constant of the metal must be negative, and its magnitude must be greater than that of the dielectric. Thus, only certain metals such as gold, silver, and aluminum are usually used for SPR measurements. The dispersion relation for surface plasmons on a metal surface is ... 

Fig. 2.2. (a) Schematic illustration of the free-electron-like ID dispersion relation. f is the Fermi energy, fcp is the Fermi wave vector, and a is the lattice constant, (b) 3D schematic view of the resulting Fermi surface. Dashed line without interstack overlaps. Solid lines with small transfer integrals... 

Theoretically, SPW is described as a charge density oscillation that goierates highly confined electromagnetic fields on the surfoce of a metal film (24, 26, 31-35). The criterion for the excitation of SPW is that the incident laser beam must be matched in both frequency and momentum with that of SPW. This can only occur, for example, if P wave (TM wave) is incident from the glass side at a specific angle of which the projection of k vector of the incident photon matches SPW s k vector (26, 36, 37). The dispersion relation for a semi-infinite metal plane surface of... 

When the shear waves propagate through the elastic layer, or the elastic plate and reach the steady state, the type of the wave, SH wave for example, and it s dispersion relation are determined by the boundary conditions at the plate surfaces [7]. We have assumed that the sound waves modulate the stress fields at the tip of the crack, and then solved the wave equations with the boundary conditions at the surfaces of the crack and the plate. If the analysis is extended to derive the higher order fields and the dispersion relation of the wave is then obtained, such a wave do exist in the steady state. In this case we could confirm the existence of such "new wave" associated with the crack. Much algebra is required to obtain the higher order fields, however, it is not difficult to see the structure of the fields with the boundary condition at the plate surfaces. We find the boundary conditions at the plate surfaces for the second order stress fields are satisfied by the factor, cos /5 z, in the similar manner to Eq. 

The interface between a water body on the earth and the superposed air is able to perform gravity wave motions that are characterized by a dispersion relation that describes the relation between the angular frequency <7 and the wave number k of the wave. The wave components propagate with a certain phase velocity c on the sea surface. The dispersion velocity of surface gravity waves of a water layer of depth H is... 

When the shelf scale is comparable with the internal Rossby radius, wave motions normal to the shelf induce vertical motions due to the inclined bottom that generate internal pressure gradients. Therefore, a separation between barotropic and baroclinic modes is not possible anymore and these modes are denoted as mixed or hybrid modes that have been called coastally trapped waves. To evaluate their dispersion relations with respect to frequency and longshore wave number and their modal structure in the vertical plain normal to the coast, a two-dimensional eigenvalue problem must be solved numerically (Brink, 1991). The nodal lines of the velocity modes of these hybrid modes are inclined with respect to the sea surface in contrast to the baroclinic modes in case of a flat-bottomed ocean. 

Figure 9-19 Phonon dispersion relation (angular frequency vs. relative wave vector) for the three-stripe phase of CH4 on the external surface of a bundle. LI, L2, and L3 are longitudinal branches, i.e., molecular motion parallel to the groove. The dotted curve is the result for a ID adsorbate at the same density. The remaining curves correspond to the dispersion relation of transverse modes. (Adapted from Ref. [89].)...

Erik s research focused on the interfacial properties of the ocean surface, and, in particular, how the chemistry of the air-sea interface affects the dynamics of short waves, nearsurface flows and interfacial fluxes of heat, mass and momentum. During his short career, he contributed to over 30 scientific publications in this area. His doctoral research, carried out under the tutelage of well-known colloid and surface chemist, Sydney Ross, concerned the propagating characteristics of surface waves in the presence of adsorbed films. That work was eventually published as a series of seminal papers on capillary ripples, and his theoretical treatment of ripple propagation and a corrected dispersion relation for surface waves in the presence of a surface dilational modulus (with J. Adin Mann, Jr.) still stand as the definitive word on the subject. 

Bock EJ, Mann JA, Jr. (1989) On ripple dynamics. II. A corrected dispersion relation for surface waves in the presence of surface elasticity. J Colloid Interface Sci. 129 501-505... 

Bock EJ (1987) On ripple dynamics I. Microcomputer-aided measurements of ripple propagation. J Colloid Interface Sci 119 326-334 Bock EJ, Hara T (1995) Optical measurements of capillary-gravity wave spectra using a scanning laser slope gauge. J Atmos Oceanic Tech 12 395-403 Bock EJ, Mann JA (1989) On ripple dynamics II. A corrected dispersion relation for surface waves in the presence of surface elasticity. J Colloid Interface Sci 129 501-505... 

As stressed earlier, the transition from surface polaritons to Coulomb surface excitons corresponds to the limiting transition c —> oo. For p-polarized waves it yields the dispersion relation... 

Lucassen (1968) and Lucassen-Reynders (1969) worked out the theory for longitudinal surface waves, which appear at elasticity modules higher than 30 mN/m and behaves like a stretched membrane. The related dispersion equation has the form... 

Equation (10.4.18) is the sought after dispersion relation for surface waves on deep water. It may also be written as a dependence of wave speed on wavelength ... 

The determination of the equation for the free surface shape and the eigenvalue relation between the amplification factor (3, the wave number k, and the integer n is carried out just as in the evaluation of the dispersion relation for plane waves. In particular, from the kinematic boundary condition Eq. 

The linear stability characteristics of the jet are specified by Eq. (10.4.32), where we note that (3 alpa, which may be compared with the plane capillary wave result where crlpX. This behavior is not surprising and can be deduced from dimensional arguments. Indeed, for the jet when a 1, that is, when the wavelengths are small compared with the jet radius, we have from the properties of the Bessel function that /(,( )/I (a) = 1. With f3 = io), Eq. (10.4.32) reduces to the dispersion relation o) - k crlp for stable, sustained surface capillary waves on deep water (Eq. 10.4.19). 

Plasmon surface polaritons (PSPs) or surface plasmons are transverse magnetic waves that propagate along a metal-dielectric interface, their field amplitudes decaying exponentially perpendicular to the interface [29,30]. Their dispersion relation is given by... 

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