Wind Driven Waves in Deep Water

Consider the scenario described in the previous section. Suppose that the lower fluid is a body of deep water at rest, and the upper fluid is the atmosphere. Let the air above the surface of the water move horizontally at the constant velocity $V'$ . Suppose that $\rho$ is the density of water, $s=\rho'/\rho$ the specific gravity of air with respect to water, and $\gamma$ the surface tension at an air/water interface. With $V=0$ , $k\,d\rightarrow \infty$ , $k\,d'\rightarrow \infty$ , the dispersion relation (11.132) reduces to

$$\rho\,(1-s)\,g + \gamma\,k^{\,2} = \rho\,k\,c^{\,2}+s\,\rho\,k\,(V'-c)^2.$$ (11.139)

This expression can be rearranged to give

$$c^{\,2} - \left(\frac{2\,V'\,s}{1+s}\right)c + \frac{V'^{\,2}\,s}{1+s} =c_1^{\,2},$$ (11.140)

which is a quadratic equation for the phase velocity, $c$ , of the wave. Here,

$$c_1^{\,2}= \frac{g}{k}\left(\frac{1-s}{1+s}\right) + \frac{\gamma\,k}{\rho\,(1+s)},$$ (11.141)

where $c_1$ is the phase velocity that the wave would have in the absence of the wind. In fact, we can write

$$c_1^{\,2} = \frac{1}{2}\left(\frac{\lambda}{\lambda_c}+\frac{\lambda_c}{\lambda}\right)c_0^{\,2}$$ (11.142)

where $\lambda_c=2\pi\,l$ is the capillary wavelength, and $l$ and $c_0$ are defined in Equations (11.134) and (11.135), respectively.

For a given wavelength, $\lambda$ , the wave velocity, $c$ , attains its maximum value, $c_m$ , when $dc/dV'=0$ . According to the dispersion relation (11.140), this occurs when

$$V' = c_m = (1+s)^{1/2}\,c_1.$$ (11.143)

If the wind has any other velocity, greater or less than $c_m$ , then the wave velocity is less than $c_m$ .

According to Equation (11.140), the wave velocity, $c$ , becomes complex, indicating an instability, when

$$V'^{\,2} > \frac{(1+s)^2}{s}\,c_1^{\,2}= \frac{(1+s)^2}{2\,s}\left(\frac{\lambda}{\lambda_c}+\frac{\lambda_c}{\lambda}\right)c_0^{\,2}.$$ (11.144)

We conclude that if the wind speed exceeds the critical value

$$V'_c = \frac{(1+s)}{s^{1/2}}\,c_0=6.6\,{\rm m/s} = 12.8\,{\rm kts}$$ (11.145)

then waves whose wavelengths fall within a certain range, centered around $\lambda_c$ , are unstable and grow to large amplitude.

The two roots of Equation (11.140) are

$$c = \frac{V'\,s}{1+s} \pm\left[c_1^{\,2}- \frac{s\,V'^{\,2}}{(1+s)^2}\right]^{1/2}.$$ (11.146)

Moreover, if

$$V' < (1+s^{\,-1})^{1/2}\,c_1$$ (11.147)

then these roots have opposite signs. Hence, the waves can either travel with the wind, or against it, but travel faster when they are moving with the wind. If $V'$ exceeds the value given previously then the waves cannot travel against the wind. Because $c_1$ has the minimum value $c_0$ , it follows that waves traveling against the wind are completely ruled out when

$$V' > (1+s^{\,-1})^{1/2}\,c_0 = 6.6\,{\rm m/s} = 12.8\,{\rm kts}.$$ (11.148)