Water in contact with air actually possesses a finite surface tension,
(Haynes and Lide 2011b), which allows there to be a small pressure discontinuity
across a free surface that is curved. In fact,
![$\displaystyle [p]_{z=0_-}^{z=0_+} = T\,\frac{\partial^{\,2} \zeta}{\partial x^{\,2}}$](img3160.png) |
(9.325) |
(Batchelor 2000).
Here,
is the radius of curvature of the surface.
Thus, in the presence of surface tension, the boundary condition (9.276)
takes the modified form
![$\displaystyle T\,\frac{\partial^{\,2}\zeta}{\partial x^{\,2}} = \rho\,g\,\zeta- \left. p_1\right\vert _{z=0},$](img3162.png) |
(9.326) |
which reduces to
![$\displaystyle \left.\frac{\partial \phi}{\partial z}\right\vert _{z=0} = \frac{...
...\frac{1}{g}\left.\frac{\partial^{\,2}\phi}{\partial t^{\,2}}\right\vert _{z=0}.$](img3163.png) |
(9.327) |
This boundary condition can be combined with the solution (9.281), in the
deep water limit
, to give the modified deep water dispersion
relation (see Exercise 26)
![$\displaystyle \omega = \sqrt{g\,k + \frac{T}{\rho}\,k^{\,3}}.$](img3164.png) |
(9.328) |
Hence, the phase velocity of the waves takes the form
![$\displaystyle v_p = \frac{\omega}{k} = \sqrt{\frac{g}{k} + \frac{T}{\rho}\,k},$](img3165.png) |
(9.329) |
and the ratio of the group velocity to the phase velocity can be shown to
be
![$\displaystyle \frac{v_g}{v_p} = \frac{k}{\omega}\,\frac{d\omega}{dk}=\frac{1}{2}\left[\frac{1+3\,T\,k^{\,2}/(\rho\,g)}{1+T\,k^{\,2}/(\rho\,g)}\right].$](img3166.png) |
(9.330) |
We conclude that the phase velocity of surface water waves attains a minimum value of
when
, which corresponds to
. The group velocity equals the phase velocity at this wavelength. For long wavelength waves (i.e.,
), gravity dominates surface tension, the phase velocity scales as
,
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.,
), surface tension dominates gravity, the
phase velocity scales as
, and the group velocity is
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
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.