Capillary Waves

In the deep water limit $k\,d\gg 1$ , the dispersion relation (11.121) simplifies to

$$\omega^{\,2} = g\,k + \frac{\gamma\,k^{\,3}}{\rho}.$$ (11.122)

It is helpful to introduce the capillary length,

$$l = \left(\frac{\gamma}{\rho\,g}\right)^{1/2}.$$ (11.123)

(See Section 3.4.) The capillary length of an air/water interface at s.t.p. is $2.7\times 10^{-3}\,{\rm m}$ (Batchelor 2000). 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

$$v_c = (2\,g\,l)^{1/2}.$$ (11.124)

This critical velocity takes the value $0.23\,{\rm m/s}$ for an air/water interface at s.t.p. (Batchelor 2000). It follows from Equation (11.122) that the phase velocity, $v_p=\omega/k$ , of a surface water wave can be written

$$\frac{v_p}{v_c} = \left[\frac{1}{2}\left(k\,l+\frac{1}{k\,l}\right)\right]^{1/2}.$$ (11.125)

Moreover, the ratio of the phase velocity to the group velocity, $v_g=d\omega/dk$ , becomes

$$\frac{v_g}{v_p} =\frac{1}{2}\left[\frac{1+3\,(k\,l)^2}{1+(k\,l)^2}\right].$$ (11.126)

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

$$\frac{v_p}{v_0} \simeq \frac{1}{(2\,k\,l)^{1/2}},$$ (11.127)

and

$$\frac{v_g}{v_p} \simeq \frac{1}{2}.$$ (11.128)

We can identify this type of wave as the deep water gravity wave discussed in Section 11.3.

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

$$\frac{v_p}{v_c}\simeq \left(\frac{k\,l}{2}\right)^{1/2},$$ (11.129)

and

$$\frac{v_g}{v_p}\simeq \frac{3}{2}.$$ (11.130)

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, Equation (11.129) can be rewritten

$$v_p \simeq \left(\frac{k\,\gamma}{\rho}\right)^{1/2},$$ (11.131)

which demonstrates that gravity plays no role in the propagation of a capillary wave. In fact, its place is taken by surface tension. Finally, it is easily seen that the phase velocity (11.125) attains the minimum value $v_p=v_c$ when $\lambda=\lambda_c$ (i.e., when $k\,l=1$ ). Moreover, from Equation (11.126), $v_g=v_p$ 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, $v_c$ .