Dispersion relation capillary waves

It can be shown that this fonn leads to an unphysical dispersion relation for capillary waves t/,7 -- f/", rather... 

We now turn to the exposition of the dispersion relation of Lucassen-Reynders-Lucassen for capillary waves at a fluid-fluid interface, given as... 

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 (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... 

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). 

Derive the dispersion relation for capillary waves on the surface of a fluid with nonzero viscosity. Since this is a dissipative system, the energy approach used in the text is not applicable and one must start from the hydrodynamic equations for an incompressible fluid. Assume that that amplitude of the wave and its velocity are small, linearize the hydrodynamic equations and calculate both the real and imaginary parts of the dispersion relation for these (possibly overdamped) waves. 

Time-resolved quasi-elastic laser scattering (QELS) experiments [306, 307] allow the measurement of the dispersion law (the relation between the frequency of capillary waves, co, and their wavelength, In/k). To derive the dispersion law theoretically, one must consider the dynamics of the two degrees of freedom - the position of the surface, z = (x, y, t), and the fluid velocity. In the limit of small viscosity, such a theory [291] gives... 

Both the frequency and capillary wave-vector are related to the material properties of the fluid via the dispersion relation. For pure liquid surfaces the dispersion relation has been derived independently by Lamb (1945) and Levich (1962), the frequency and damping of the capillary waves being given by... 

Lucassen-Reynders and Lucassen (Lucassen 1968, Lucassen-Reynders and Lucassen 1969) have derived the dispersion relation for a liquid surface in the presence of a surface film. They showed that periodic disturbance of such a film-covered surface results in a surface tension that varies from point to point on the surface because of the fluctuations in surface concentration. Consequently, in addition to a transverse stress being developed, a finite tangential surface stress is also present. The solution to this dispersion equation has two roots, one of which corresponds to the capillary waves (transverse motion) and one of which corresponds to longitudinal or dilational waves derived from the transverse stress. The dispersion relation (D( o)) obtained for a film at the interface between two media is... 

The opposite limit is that of a very thick annular region, i.e. a >> 1. Physically this is the problem of the capillary instability of a fluid thread immersed in an infinite fluid as first studied by Tomotika [77] Numerical solutions of the dispersion relation indicate that the squeezing mode disappears as a increases (this is expected since the squeezing mode has a band of unstable waves 0 < A < 1/a), and the stretching mode reaches a limit which is independent of 7 - this is because the outer interface is at infinity and has no effect on the stretching mode, to leading order. 

For high enough values of Ga the oscillatory mode is the capillary-gravity wave. The time scales tgr, and cap associated with this twofold wave are much smaller than the viscous and thermal timescales (at least for Pr l, Bo l). Then dissipative effects are relatively weak and the dispersion relation is... 

The latter relation is known as Kelvin s equation. Methods for creating propagating capillary ripples typically involve either a mechanical or electrocapillary disturbance of the fluid interface [189-191]. The laser is more appropriate because it does not necessitate physical contact with the fluid surface [497]. The wave characteristics, which are necessary for the evaluation of the interfacial properties through the dispersion relation, are often determined by the reflection of a laser beam from the fluid surface to a position-sensitive photodiode. 

Longitudinal waves are, to a major extent, related to the tangential stress with a frequency that depends on the viscosity and surface elasticity. They do not exist in non-viscous liquids. Capillary-gravity waves, however, have a frequency that depends on gravity and on surface tension(Laplace-Kelvin overpressure) but not on the viscosity and are admissible in the absence of dissipation. The latter only appears in the damping factor and frequency-deviation in the dispersion relation. 

Once we know threshold values, dispersion relations and necessary conditions to be fulfilled in order to excite and eventually sustain oscillations we must show that the nonlinear terms saturate the exponential growth thus stabilizing the interfacial waves. This is a formidable task and,for simplicity,we now restrict consideration to capillary-gravity (Kelvin-Laplace) transverse waves at an a/r-liquid interface.In this case we know that in order to have sustained oscillations the elasticity Maiangoni number must be negative When due consideration is taken of the nonlinear terms -that we have omitted in Section 2-we have shown that, indeed, past the instability threshold the evolution corresponds to a nonlinear wave propagation. 

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