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,

$$[p]_{z=0_-}^{z=0_+} = T\,\frac{\partial^{\,2} \zeta}{\partial x^{\,2}}$$ (9.325)

(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 (9.276) takes the modified form

$$T\,\frac{\partial^{\,2}\zeta}{\partial x^{\,2}} = \rho\,g\,\zeta- \left. p_1\right\vert _{z=0},$$ (9.326)

which reduces to

$$\left.\frac{\partial \phi}{\partial z}\right\vert _{z=0} = \frac{... ...\frac{1}{g}\left.\frac{\partial^{\,2}\phi}{\partial t^{\,2}}\right\vert _{z=0}.$$ (9.327)

This boundary condition can be combined with the solution (9.281), in the deep water limit $k\,d\gg 1$, to give the modified deep water dispersion relation (see Exercise 26)

$$\omega = \sqrt{g\,k + \frac{T}{\rho}\,k^{\,3}}.$$ (9.328)

Hence, the phase velocity of the waves takes the form

$$v_p = \frac{\omega}{k} = \sqrt{\frac{g}{k} + \frac{T}{\rho}\,k},$$ (9.329)

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

$$\frac{v_g}{v_p} = \frac{k}{\omega}\,\frac{d\omega}{dk}=\frac{1}{2}\left[\frac{1+3\,T\,k^{\,2}/(\rho\,g)}{1+T\,k^{\,2}/(\rho\,g)}\right].$$ (9.330)

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.