Waves deep water

Equation (10.125) is valid only for waves in shallow water, i.e., for waves of great length and moderate amplitude relative to their depth. For so-called deep-water waves, as might be encountered in the ocean, for example, but still presuming small amplitudes, a more accurate equation is... 

Yuen HC, Lake BM (1982) Nonlinear dynamics of deep-water gravity waves, In Advances in Applied Mechanics (ed. Chia-Shun Yih), vol 22. Academic Press, pp.67-229... 

Rapp RJ, Melville WK (1990) Laboratory measurements of deep-water breaking waves. Philos Trans Roy Soc London A331 735-800... 

To test these estimates of bottom disturbance by waves, a wave recorder was placed on Cable and Anchor Reef, where the water is 12.5 m deep, during the winter months of 1975. Water pressure was recorded continuously for 3 min each hour. Cable and Anchor Reef is surrounded by deep water and there is a long, open fetch to the east. The recorder is sensitive to pressure changes equivalent to 2 cm of water the period and amplitude of all recorded signals greater than this were read out and analyzed. The wavelength of the wave responsible for a pressure fluctuation 8p is found by graphical solution of the equation ... 

Waves cause significant water velocities over the bottom of Long Island Sound within a wave-affected zone that is confined to a shoreside area where the water depth is less than 18 m. Wind-driven currents occur only in the upper third of the water column throughout the deep water of the Sound. The tide is the dominant source of power for bottom pro-... 

The velocity potential satisfies the Laplace equation and the free surface boundary condition on cf> is obtained by differentiating Eq. (10.4.14) with respect to t and eliminating d ldt from Eq. (10.4.16). For waves on deep water the second boundary condition on the potential is supplied by the requirement that there are no disturbances deep in the water, or that = constant as... 

A sinusoidal wave solution of Laplace s equation satisfying both the free surface condition and the deep water condition is... 

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 ... 

Much of the procedure for the analysis of jet stability has already been set down in connection with the discussion of undamped surface waves on deep water. A fundamental difference in the jet problem from plane deep water waves is that it is axisymmetric with an imposed characteristic length scale equal to the jet radius a. Since the undisturbed jet is considered to be inviscid and in uniform flow, it can be reduced to a state of rest simply by a Galilean transformation. With gravity neglected and only surface tension forces acting, the pressure at any point within the jet is -I- ala. This then describes the basic flow needed for the first step of the stability analysis. 

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). 

Eq. (17.65) detemiines the stability conditions for the jet. It should be noted that fP Llpa, whereas in the plane case involving capillary waves at the surface of a deep reservoir we had aP according to (17.52). These perturbations also could be derived from (17.73). Indeed, consider the case of a = ak 1 or a 2, i.e. perturbations whose wavelength is small in comparison with the jet s radius. It then follows from the properties of the Bessel function that In ct) 1. Putting p = ico into (17.65), one finds that aP = k L/p L/pX, which corresponds to neutrally stable capillary waves on the surface of a deep water reservoir. 

First, we consider the influence of the liquid s viscosity on the damping of plane capillary waves on deep water. Suppose the liquid has low viscosity so that viscous effects only manifest themselves inside a thin boundary layer near the interface. Hence, outside the boundary layer, the liquid flow is potential, and the potential is described by the Laplace equation, while the liquid flow near the surface is described by the boundary layer equations with the accompanying condition that the tangent viscous stress at the free interface must be zero. The solution of this problem can be found in [2]. The main difference from the case of a non-viscous liquid is the appearance of a coefficient of the form exp(—jSjt) in the... 

Nafe, J.E., and Drake, C.L. 1957. Variation with depth in shallow and deep water marine sediments of porosity, density, and the velodties of compressional and shear waves. Geophysics, 22(3) 523-552. 

Anonymous (1965). Charles L. Bretsehneider. Civil Engineering 35(12) 73. P Anonymous (1975). Bretsehneider, Charles Leroy. Who s who in America 38 365. Anonymous (1994). Bretsehneider, Charles L. American men and women in science 1 872. Bretsehneider, C.L. (1952). The generation and decay of wind waves in deep water. Trans. AGU 381-389. 

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