Capillary length
= Capillary length
{title2=$\ell_c=\sqrt{\gamma/(\rho g)}$}
{wiki}
The <capillary length> measures when <gravity> and <surface tension> are comparably important in shaping a <liquid> interface. For a <liquid> of uniform <mass density> $\rho$, beneath a <gas> whose <mass density> is negligible, with uniform <gravitational acceleration> $g>0$ and constant <surface tension> $\gamma>0$,
$$
\ell_c=\sqrt{\frac{\gamma}{\rho g}}.
$$
A deformation of height $a$ and horizontal length scale $L$ has <hydrostatic pressure> variation of order $\rho ga$ and small-slope <capillary pressure> of order $\gamma a/L^2$. Equating these gives $L\sim\ell_c$: <surface tension> smooths small features, whereas <gravity> matters for broad ones. For two resting <fluids>, replace $\rho$ by their positive <mass density> difference $\Delta\rho$ in this static balance. This change alone does not give the two-fluid <wave> dynamics, because both <fluids> can have inertia. https://ocw.mit.edu/courses/18-357-interfacial-phenomena-fall-2010/04759cfc15c79928ca9f72a18b3864dd_MIT18_357F10_lec_all.pdf[John W. M. Bush, Interfacial Phenomena, chapters 5–6: curvature and hydrostatic pressure]{ref}
With the stipulated illustrative parameters $\gamma=0.072\,\mathrm{N\,m^{-1}}$, $\rho=1000\,\mathrm{kg\,m^{-3}}$, and $g=9.81\,\mathrm{m\,s^{-2}}$, the <capillary length> is $\ell_c=2.71\,\mathrm{mm}$. These parameters define the ideal calculations below; they are not a measurement of a particular sample.
\Image[/capillary-waves-crossover.png]
{title=The <gravity-capillary wave> crossover occurs at <wavelength> $2\pi\ell_c$, where <phase speed> is smallest}
{description=Calculated <phase velocity> and <group velocity> magnitudes, and the ratio of the <surface tension> and <gravity> terms in the deep-water <dispersion relation>, for the stipulated $\gamma$, $\rho$ and $g$. The <capillary length> is $2.71\,\mathrm{mm}$, while the equality <wavelength> is $17.02\,\mathrm{mm}$. Both panels show ideal theory, without experimental data.}
For a <sinusoid> with <wavenumber> $k>0$, the relevant horizontal scale is $L=1/k$, while its <wavelength> is $\lambda=2\pi/k$. Thus the <Bond number> for this scale and the reciprocal restoring-term ratio are
$$
\mathrm{Bo}_{1/k}=\frac{\rho g}{\gamma k^2}
=\frac{1}{(k\ell_c)^2}
=\left(\frac{\lambda}{2\pi\ell_c}\right)^2,
\qquad
\frac{\text{capillary term}}{\text{gravity term}}
=(k\ell_c)^2.
$$
Equality is at $k_c=1/\ell_c$, hence $\lambda_c=2\pi\ell_c=17.02\,\mathrm{mm}$ for these parameters. Using $L=\lambda$ instead produces a different numerical <Bond number>; state the length convention. The <capillary length> itself is not the equality <wavelength>. This distinction helps choose whether a measured ripple should be modeled as a <surface gravity wave>, a nearly pure <capillary wave>, or a <gravity-capillary wave>. https://davidtong.org/pdfs/teaching/fluid-mechanics/fluids3.pdf[David Tong, Fluid Mechanics, §4.1.3, equations (4.23)–(4.24) and the wavelength factor $2\pi$]{ref}