Surface Gravity Waves

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 

Deepwater waves are dispersive since long waves move faster than short waves. This results in the dispersion of an initial elevation of the sea surface composed of an ensemble of waves with different wave numbers while propagating on the sea surface. The flow field associated with a deepwater wave of amplitude a consists of circular motion whose amplitude decays exponentially with depth z 0 as 7 a exp(fe). A characteristic particle velocity of a wave with amplimde a 0(1 m) and a period of 6 s is near the sea surface m = 0(1 m/s). The amplitude of the particle velocity decreases at a depth z = L/2 to 4% of the velocity near the sea surface. 

Shallow water waves have large wavelengths compared to the water depth such that kH 1. Then the dispersion relation (2.26) is approximated by 

That means that shallow water waves are nondispersive, that is, an initial disturbance of the sea surface consisting of an ensemble of shallow water waves keeps its form along its path of propagation. 

The particle velocity associated with the shallow water wave is elliptic ally with the major axis aligned parallel to the bottom and the minor axis perpendicular to it. The amplimde of the particle velocity decays with depth but remains finitely by close to the bottom where the minor axis of the particle motion approaches to zero. This implies that particle motions of shallow water waves are quite capable of resuspending sediments as well as to develop a turbulent bottom boundary layer. 

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Precht, E., and Huettel, M. (2003) Advective porewater exchange driven by surface gravity waves and its ecological implications. Lirnnol. Oceanogr. 48, 1674—1684. 

Wind generated water waves (surface gravity waves) should be taken into consideration in the flood analysis for coastal sites. The calculations of extreme events (such as surges, seiches or tsunamis) and the associated wind waves should be performed together since the results are non-hnear and it is not appropriate to evaluate the partial effects separately and then add them to obtain the maximum flood level. 

Havelock applied Fourier integrals to develop a theory for surface gravity waves forced by circular wavemakers in water of both infinite and finite depth. The fluid motion may be obtained from the negative gradient of a scalar velocity potential [Pg.34]

Conformal and domain mappings are applications of complex variables to solve 2D boundary value problems. Conformal mapping is an angle preserving transformation that will compute exact nonlinear solutions for surface gravity waves of constant... 

H. Socquet-Juglard, K. Dysthe, K. Trulsen, H. Krogstad and J. Lin, Probability distribution of surface gravity waves during spectral changes, J. Fluid Mechanics 542, 195-216 (2005). 

Fig. 9.5. Spectrum of surface gravity waves in the ocean (modified from Ref. 68). Periods (upper scale) are in hours (h), minutes (min), and seconds (s).

Here, v denotes an eigenfunction, co means the eigenfrequency, / is the degree of the spherical harmonic function which describes the pattern of the mode on the stellar surface, and E,(r) is the gravity wave potential which consists of the inverse square of the Brunt-V isfiia frequency, N(r), multiplied by /(7+1) and an /-independent part ... 

However, the flow regime of a film cannot be defined uniquely as laminar or turbulent, as in the case of pipe flow, due to the presence of the free surface. Depending on the values of AFr and JVw , the free surface may be smooth, or covered with gravity waves or capillary or mixed capillary-gravity waves of various types. Thus, under suitable conditions, it is possible to have smooth laminar flow, wavy laminar or turbulent flow, where the wavy flows may be subdivided into gravity or capillary... 

It is clear that the flow regime is a complicated but predictable function of the physical properties of the liquid, the flow rate, and the slope of the channel. It has been shown that, for water films, gravity waves first appear in the region NrT = 1-2, capillary surface effects become important in the neighborhood of JVw = I, and the laminar-turbulent transition occurs in the zone ArRe = 250-500 (F7). 

In addition to the theories reviewed above, there are many treatments in the literature which deal with the hydraulics of wavy flow in open channels. Most of these refer to very small channel slopes (less than 5°) and relatively large water depths. Under these conditions, surface tension plays a relatively minor part and is customarily neglected, so that only gravity waves are considered. For thin film flows, however, capillary forces play an important part (K7, H2). In addition, most of these treatments consider a turbulent main flow, while in thin films the wavy flow is often... 

Hiihnerfuss H, Alpers W, Garrett WD, Lange PA, Stolte S (1983) Attenuation of capillary and gravity waves at sea by monomolecular organic surface films. J Geophys Res 88 9809-9816... 

By the way of contrast, a water surface covered with a freshly spilled crude oil exhibits quite different physicochemical characteristics. No surface tension gradients will be generated by the undulating water surface and thus the Marangoni effect plays no role. On the other hand, the crude oil layer may also dampen short gravity waves due to its relatively high viscosity as compared with a pure water surface. 

Lombardini PP, Fiscella B, Trivero P, Cappa C, Garrett WD (1989) Modulation of the spectra of short gravity waves by sea surface films slick detection and characterization with a microwave probe. J Atmos Oceanic Technol 6 882-890... 

Fig. 4. Theoretical surface divergence for a 15 Hz capillary-gravity wave. Curves are shown for an ideal inviscid flow, a real flow with zero viscoelasticity, and a surfactant flow with a finite viscoelasticity. Each curve is along the phase of each wave, with the horizontal position at zero corresponding to a wave crest. The wave amplitude wavelength ratio is 1 20...

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... 

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