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The effect of randomness on the stability of capillary gravity waves in the presence of air flowing over water

Majumder D. P., Dhar A. K.

International Journal of Applied Mechanics and Engineering

2015

Vol. 20, no. 4

835--855

A nonlinear spectral transport equation for the narrow band Gaussian random surface wave trains is derived from a fourth order nonlinear evolution equation, which is a good starting point for the study of nonlinear water waves. The effect of randomness on the stability of deep water capillary gravity waves in the presence of air flowing over water is investigated. The stability is then considered for an initial homogenous wave spectrum having a simple normal form to small oblique long wave length perturbations for a range of spectral widths. An expression for the growth rate of instability is obtained; in which a higher order contribution comes from the fourth order term in the evolution equation, which is responsible for wave induced mean flow. This higher order contribution produces a decrease in the growth rate. The growth rate of instability is found to decrease with the increase of spectral width and the instability disappears if the spectral width increases beyond a certain critical value, which is not influenced by the fourth order term in the evolution equation.

Fourth order nonlinear evolution equation for capillary gravity waves in deep water in the presence of a thin thermocline including the effect of wind

Debsarma S., Das K. P.

International Journal of Applied Mechanics and Engineering

2003

Vol. 8, no 2

177-193

A fourth order non-linear evolution equation is derived for a capillary-gravity wave packet in deep water in the presence of a thin thermocline including the effect of wind and viscous dissipation in water. In deriving this equation it has been assumed that the wind induced basic current in water is exponential and the effect of shear in air flow and viscous dissipation in water is accounted for by including a term in the evolution equation. The nonlinear evolution equation is used to study the stability of a uniform capillary-gravity wave train. Expressions for the maximum growth rate of instability and wave number at marginal stability are obtained. From results shown graphically it is found that the inclusion of wind effect increases the growth rate of instability irrespective of the presence of a thin thermocline. For waves with a small wave number, a thin thermocline has a stabilizing influence both in the presence and in the absence of wind input and the maximum growth rate of instability decreases with the increase of thermocline depth. But for waves with a large wave number a thin thermocline has no influence.