New Approaches for Computing Wave Growth Rate due to Wind Induced Shear Instabilities

摘要

In studying the instability of sensitive jets Rayleigh suggested an improvement in the theory by supposing a gradual change in velocity and thus proceeded to derive an equation which is known today as the Rayleigh equation. Miles (1957) proposed a model for the growth of wind waves on the basis of Rayleigh's equation. Later, Conte and Miles (1959) gave accurate computations of wave growth rates by numerical solution of Rayleigh's equation for a logarithmic wind profile. This work first considers a different approach of solving the Rayleigh equation for arbitrary mean wind profiles by implementing the ideas of Rayleigh, which render the equation analytically solvable in the immediate vicinity of the singular point hence providing the initial values for the numerical integration. Furthermore, the wave growth rate is obtained from the dispersion relation of the air-water interface, which involves the vertical integration of the disturbed vertical velocity. The growth rates obtained are then compared with those of Conte and Miles (1959) and found to be in excellent agreement. In closing, approximations to the vertical profile of the disturbed vertical velocity are suggested as a basis for the development of a vertically integrated model of coupled air-water system.