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Capillary Waves

In the deep water limit $k\,d\gg 1$, the dispersion relation (1227) simplifies to
\begin{displaymath}
\omega^2 = g\,k + \frac{\gamma\,k^3}{\rho}.
\end{displaymath} (1228)

It is helpful to introduce the capillary length,
\begin{displaymath}
l = \left(\frac{\gamma}{\rho\,g}\right)^{1/2}.
\end{displaymath} (1229)

(See Section 4.4.) The capillary length of an air/water interface at s.t.p. is $2.7\times 10^{-3}\,{\rm m}$. The associated capillary wavelength is $\lambda_c=2\pi\,l=1.7\times 10^{-2}\,{\rm m}$. Roughly speaking, surface tension is negligible for waves whose wavelengths are much larger than the capillary wavelength, and vice versa. It is also helpful to introduce the critical phase velocity
\begin{displaymath}
v_c = (2\,g\,l)^{1/2}.
\end{displaymath} (1230)

This critical velocity takes the value $0.23\,{\rm m/s}$ for an air/water interface at s.t.p. It follows from (1228) that the phase velocity, $v_p=\omega/k$, of a surface water wave can be written
\begin{displaymath}
\frac{v_p}{v_c} = \left[\frac{1}{2}\left(k\,l+\frac{1}{k\,l}\right)\right]^{1/2}.
\end{displaymath} (1231)

Moreover, the ratio of the phase velocity to the group velocity, $v_g=d\omega/dk$, becomes
\begin{displaymath}
\frac{v_g}{v_p} =\frac{1}{2}\left[\frac{1+3\,(k\,l)^2}{1+(k\,l)^2}\right].
\end{displaymath} (1232)

In the long wavelength limit $\lambda \gg \lambda_c$ (i.e., $k\,l\ll 1$), we obtain

\begin{displaymath}
\frac{v_p}{v_0} \simeq \frac{1}{(2\,k\,l)^{1/2}},
\end{displaymath} (1233)

and
\begin{displaymath}
\frac{v_g}{v_p} \simeq \frac{1}{2}.
\end{displaymath} (1234)

We can identify this type of wave as the deep water gravity wave discussed in Section 10.3.

In the short wavelength limit $\lambda\ll \lambda_c$ (i.e., $k\,l\gg 1$), we get

\begin{displaymath}
\frac{v_p}{v_c}\simeq \left(\frac{k\,l}{2}\right)^{1/2},
\end{displaymath} (1235)

and
\begin{displaymath}
\frac{v_g}{v_p}\simeq \frac{3}{2}.
\end{displaymath} (1236)

This corresponds to a completely new type of wave known as a capillary wave. Such waves have wavelengths that are much less than the capillary wavelength. Moreover, (1235) can be rewritten
\begin{displaymath}
v_p \simeq \left(\frac{k\,\gamma}{\rho}\right)^{1/2},
\end{displaymath} (1237)

which demonstrates that gravity plays no role in the propagation of a capillary wave--its place is taken by surface tension. Finally, it is easily seen that the phase velocity (1231) attains the minimum value $v_p=v_c$ when $\lambda=\lambda_c$ (i.e., when $k\,l=1$). Moreover, from (1232), $v_g=v_p$ at this wavelength. It follows that the phase velocity of a surface wave propagating over a body of water can never be less than the critical value, $v_c$.


next up previous
Next: Capillary Waves at an Up: Waves in Incompressible Fluids Previous: Surface Tension
Richard Fitzpatrick 2012-04-27