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Capillary Waves

Water in contact with air actually possesses a finite surface tension, $ T\simeq 7\times 10^{-2}\,{\rm N\,m}^{-1}$ (Haynes and Lide 2011b), which allows there to be a small pressure discontinuity across a free surface that is curved. In fact,

$\displaystyle [p]_{z=0_-}^{z=0_+} = T\,\frac{\partial^2 \zeta}{\partial x^2}$ (987)

(Batchelor 2000). Here, $ (\partial^2\zeta/\partial x^2)^{-1}$ is the radius of curvature of the surface. Thus, in the presence of surface tension, the boundary condition (938) takes the modified form

$\displaystyle T\,\frac{\partial^2\zeta}{\partial x^2} = \rho\,g\,\zeta- \left. p_1\right\vert _{z=0},$ (988)

which reduces to

$\displaystyle \left.\frac{\partial \phi}{\partial z}\right\vert _{z=0} = \frac{...
..._{z=0} -\frac{1}{g}\left.\frac{\partial^2\phi}{\partial t^2}\right\vert _{z=0}.$ (989)

This boundary condition can be combined with the solution (943), in the deep water limit $ k\,d\gg 1$ , to give the modified deep water dispersion relation (see Exercise 21)

$\displaystyle \omega = \sqrt{g k + \frac{T}{\rho} k^3}.$ (990)

Hence, the phase velocity of the waves takes the form

$\displaystyle v_p = \frac{\omega}{k} = \sqrt{\frac{g}{k} + \frac{T}{\rho} k},$ (991)

and the ratio of the group velocity to the phase velocity can be shown to be

$\displaystyle \frac{v_g}{v_p} = \frac{k}{\omega}\,\frac{d\omega}{dk}=\frac{1}{2}\left[\frac{1+3\,T\,k^2/(\rho\,g)}{1+T\,k^2/(\rho\,g)}\right].$ (992)

We conclude that the phase velocity of surface water waves attains a minimum value of $ \sqrt{2} (g T/\rho)^{1/4}\sim 0.2 {\rm m s}^{-1}$ when $ k=k_0\equiv (\rho g/T)^{1/2}$ , which corresponds to $ \lambda\sim 2\,{\rm cm}$ . The group velocity equals the phase velocity at this wavelength. For long wavelength waves (i.e., $ k\ll k_0$ ), gravity dominates surface tension, the phase velocity scales as $ k^{-1/2}$ , and the group velocity is half the phase velocity. As we have already mentioned, this type of wave is known as a gravity wave. On the other hand, for short wavelength waves (i.e., $ k\gg k_0$ ), surface tension dominates gravity, the phase velocity scales as $ k^{1/2}$ , and the group velocity is $ 3/2$ times the phase velocity. This type of wave is known as a capillary wave. The fact that the phase velocity and the group velocity both attain minimum values when $ \lambda\sim 2\,{\rm cm}$ means that when a wave disturbance containing a wide spectrum of wavelengths, such as might be generated by throwing a rock into the water, travels across the surface of a lake, and reaches the shore, the short and long wavelength components of the disturbance generally arrive before the components of intermediate wavelength.


next up previous
Next: Exercises Up: Dispersive Waves Previous: Ship Wakes
Richard Fitzpatrick 2013-04-08