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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 $\gamma >0$,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. ^[ref]

With the stipulated illustrative parameters $\gamma =0.072\mathrm{N}{\mathrm{m}}^{-1}$, $\rho =1000\mathrm{kg}{\mathrm{m}}^{-3}$, and $g=9.81\mathrm{m}{\mathrm{s}}^{-2}$, 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.

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 areEquality 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. ^[ref]

A gravity-capillary wave has restoring contributions from both gravity and surface tension. Consider an initially resting, horizontally unbounded liquid layer of constant depth $h$, with uniform mass density $\rho$, incompressible flow, inviscid flow and irrotational flow. The upper gas exerts constant pressure and has negligible inertia. Assume a clean interface with constant surface tension $\gamma$: no surfactant stresses, imposed current or wind. The wave amplitude $a$ must satisfy $ka\ll 1$ and $a\ll h$, so linearized boundary conditions apply. Deep water means $kh\gg 1$, rather than a particular depth in metres. ^[ref]

Let $z=0$ be the undisturbed free surface, with liquid below it. Potential flow gives a velocity potential $\varphi$ with $\mathbf{u}=\nabla \varphi$, and incompressible flow gives the Laplace equation. The linearized kinematic boundary condition for a free-surface graph and the Unsteady Bernoulli equation, including gravity and the Young–Laplace equation, giveHere $\eta (x,t)$ is the interface displacement; the signed small-slope pressure jump is $p_{\mathrm{liquid}}-p_{\mathrm{gas}}=-\gamma {\eta }_{xx}$. A wave crest has ${\eta }_{xx}<0$, so its capillary pressure is positive. Choose a right-travelling Fourier mode of positive wavenumber $k$ and angular frequency $\omega$:The factor $e^{kz}$ decays into the deep liquid, satisfies the Laplace equation, and makes ${\varphi }_z={\eta }_t$ at $z=0$. Substituting in the dynamic boundary condition leavesThis derivation displays the two restoring contributions with the same sign. Dimensional analysis also checks the result: $\gamma /\rho$ has units ${\mathrm{m}}^3{\mathrm{s}}^{-2}$, so both $gk$ and $(\gamma /\rho )k^3$ have units ${\mathrm{s}}^{-2}$. ^[ref]

At finite depth, a flat impermeable bottom at $z=-h$ requires ${\varphi }_z=0$. Replacing $e^{kz}$ by the normalized vertical factor $\cosh (k(z+h))/\cosh (kh)$ makes the surface ratio ${\varphi }_z/\varphi =k\tanh (kh)$. Eliminating the mode amplitudes then givesThus the same model supplies its own depth check: use the deep-water formula only when $\tanh (kh)$ is sufficiently close to one for the required accuracy. For example, $kh=3$ reduces $\omega$ by about $0.248\%$ relative to deep water, while $kh=1$ reduces it by about $12.7\%$. These percentages are calculations from this factor. ^[ref]

In deep water the phase velocity magnitude follows a wave crest, while the group velocity magnitude follows the wave envelope of a narrow wave packet. Differentiating the positive-frequency dispersion relation givesFor $k{\ell }_c\ll 1$, the surface gravity wave limit has $c_g\simeq c_p/2$. For $k{\ell }_c\gg 1$, the deep-water capillary-wave dispersion relation has $c_g\simeq 3c_p/2$: a wave crest slips backward relative to the faster wave envelope, even while both advance. Neither speed is the oscillating fluid element's own velocity. ^[ref]

There is a particularly neat coincidence at the crossover. Differentiate $c_p^2=g/k+\gamma k/\rho$: its unique minimum is at $k_c=\sqrt{\rho g/\gamma }=1/{\ell }_c$, exactly where the restoring terms are equal. ThereforeFor the stipulated parameters in capillary length, the minimum is $0.231\mathrm{m}{\mathrm{s}}^{-1}$. At $\lambda =3\mathrm{mm}$, direct substitution gives $f=\omega /(2\pi )\simeq 131\mathrm{Hz}$, $c_p\simeq 0.394\mathrm{m}{\mathrm{s}}^{-1}$ and $c_g\simeq 0.580\mathrm{m}{\mathrm{s}}^{-1}$. This short wavelength is capillary dominated, but its group velocity is still slightly below the pure-capillary ratio $3c_p/2$ because gravity has not vanished.

When the smoothing width exceeds the capillary length, gravity dominates capillarity. Balancing hydrostatic flux $\rho g{h}_0^4/(\mu \Delta )$ against the Marangoni stress flux gives $\Delta \sim \rho g{h}_0^2{x}_N^2/(AM)\propto t^{2/3}$. The condition is ${\Delta }^2\gg {\gamma }_0/(\rho g)$ in physical units. The smoothing region catches up with the spreading pool at $x_N\sim AM/(\rho g{h}_0^2)$ and $t\sim \mu (AM{)}^2/[(\rho g{)}^3{h}_0^7]$, after which the sharp-front outer similarity solution ceases to apply.

Zhu, Miao, Xu and Cao's 2007 experiment drove a triangular metal frame electromagnetically with a sinusoidal generator. In a roughly $3\mathrm{cm}$-deep, temperature-regulated pool, it produced nearly plane capillary waves. A $632.8\mathrm{nm}$ laser illuminated the surface obliquely; the moving ripples acted as a reflection diffraction grating. One split beam monitored laser stability, while a CCD recorded diffraction orders on a screen $6.76\mathrm{m}$ away. An air-supported optical table reduced ambient vibration. ^[ref]

The frequency sweep was $100$–$290\mathrm{Hz}$ in $10\mathrm{Hz}$ steps, using distilled water and kerosene at $20{}^{\circ }\mathrm{C}$. For each setting they averaged at least three measurements of first-order spot separation. Known optical geometry and the grating equation yielded $k=2\pi /\lambda$; generator readings yielded $\omega =2\pi f$. Agreement with the predicted log–log slope $3/2$ tests the capillary law, and its intercept gives surface tension if mass density is known. The authors identify surface contamination as a major error source; noncontact optical detection reduces additional disturbance. ^[ref]

The optical measurement can be understood from the grating equation. Let ${\lambda }_L$ be the laser wavelength, $\alpha$ the incident grazing angle above the mean surface, and ${\beta }_m$ the outgoing grazing angle of diffraction order $m$, with the sign of $m$ chosen so thatThus calibrated spot positions determine outgoing angles, which determine the ripple wavelength $\lambda$, independently of the mechanical dispersion relation. This optical geometry treats the surface as a reflection diffraction grating at the laser wavelength. ^[ref]

The inferred surface tension in the full deep-water model isThis formula makes useful experimental controls explicit. Measure mass density at the sample's temperature; check depth with $kh$; keep the driven wave amplitude small enough that $ka\ll 1$; and distinguish $f$ from angular frequency $\omega$. For the stipulated parameters, the $100$–$290\mathrm{Hz}$ window corresponds to calculated wavelengths about $3.62$–$1.76\mathrm{mm}$ and $kh\simeq 52$–$107$ at $h=30\mathrm{mm}$. Depth corrections are negligible here, but the gravity contribution is about $4.5\%$–$1.1\%$ of the capillary contribution. These are ideal predictions, not digitized results from the experiment.

Omitting gravity would infer ${\gamma }_{\mathrm{cap}}=\rho {\omega }^2/k^3=\gamma +\rho g/k^2$, biasing surface tension upward. In the capillary limit, the differential estimate $d\gamma /\gamma \simeq d\rho /\rho +2d\omega /\omega -3dk/k$ also shows why careful spot-spacing calibration matters. A clean surface and weak viscosity are modeling assumptions: appreciable damping, surfactant stresses or large wave amplitudes call for additional physics. The experiment tests the short-wave regime; its drive window does not establish the crossover or the long-wave surface gravity wave limit.

At leading order in the lubrication approximation, surface arclength is horizontal distance. An insoluble surfactant is transported by the surface velocity, without bulk exchange or diffusion. Conservation of insoluble surfactant on a moving interface therefore becomesThe local hydrostatic pressure and capillary normal stress giveSolve $\mu u_{zz}=p_x$ with no-slip boundary condition at $z=0$ and Marangoni stress $\mu u_z(h)={\gamma }_x=-A{C}_x$. The velocity, surface velocity and volume flux per unit width areMass conservation, $h_t+q_x=0$, yields the thin-film equations with insoluble surfactant:and the corresponding surfactant equation isThe hydrostatic and capillary terms thus affect both the film flux and the surface transport, with different coefficients.

For finite-mass Marangoni spreading on a liquid film, first neglect both pressure-gradient terms. If the spread has size $\ell$, then $C\sim M/\ell$ and $C_x\sim M/{\ell }^2$, giving surface speed $u_s\sim AM{h}_0/(\mu {\ell }^2)$. Equating $\ell /t$ to this speed givesSet $\ell =(AM{h}_{0}t/\mu {)}^{1/3}$ exactly as a similarity scale, and writeOn the positive half of the pool, substitution produces two ordinary differential equations:Symmetry fixes the flux integration constant in the second equation to zero, giving $H{\Gamma }^{\prime }=-\eta /3$ wherever $\Gamma >0$. Substitution into the first equation gives $\eta H^{\prime }=H$. Thus the linear similarity profiles for surfactant spreading have the formThe concentration vanishes at the moving front. The two integral constraints are total surfactant mass and conservation of the fluid volume relative to the undisturbed layer:The first gives $k={\eta }_N^2/3$; the second gives $k{\eta }_N=2$. Therefore ${\eta }_N^3=6$. In physical variables, the solution isOutside the pool the leading outer solution has $h=h_0$, $C=0$. In particular, $h(x_{N-},t)=2h_0$: the subscript means the one-sided limit just behind the front. As a consistency check, the surface speed there is $x_N/(3t)={\dot{x}}_N$ and the fluid jump satisfies the moving-front mass conservation condition. This ideal outer solution has a height jump from $2h_0$ to $h_0$.

That jump cannot persist in a physical interface with nonzero gravity or surface tension. Across a smoothing region of width $\Delta$, $h$ changes by $O(h_0)$. The Marangoni flux scales as $AM{h}_0^2/(\mu x_N^2)$. At $g=0$, its balance with the capillary flux ${\gamma }_0{h}_0^4/(\mu {\Delta }^3)$ gives capillary smoothing of a Marangoni front:Here the surface tension approaches ${\gamma }_0$ at the surfactant-free edge. The assumed concentration-gradient scale is $M/x_N^2$ even inside the smoothing region.

When gravity dominates capillarity, the hydrostatic flux is $\rho g{h}_0^4/(\mu \Delta )$. The same balance gives gravity smoothing of a Marangoni front:The gravity-dominated condition is ${\Delta }^2\gg {\gamma }_0/(\rho g)$, the square of the capillary length. The printed condition omits $\rho$; the expression above restores dimensional consistency.

This width grows faster than $x_N$. It becomes comparable with the pool size whenThe time is a scaling estimate, so numerical factors cannot be fixed by the width balance. For $t\gg t^{\ast }$, the sharp-front similarity profile no longer describes the film: hydrostatic pressure levels the layer over the whole pool, and its thickness tends towards $h_0$ with an increasingly small depression beneath the surfactant. The surface can still spread by Marangoni stress while a hydrostatic-pressure-driven return flow nearly cancels the net film flux. In this gravity-levelled surfactant film, $q\simeq 0$ gives $h_x\simeq -3A{C}_x/(2\rho g{h}_0)$, so the fractional height variation is of order $AM/(\rho g{h}_0^2{x}_N)\ll 1$.

In a static meniscus, hydrostatic pressure variation $\rho g\ell$ balances capillary pressure $\gamma /\ell$, giving the capillary lengthThe small parameter satisfiesIt is reasonable to match to a nearly static meniscus because the bulk pool has negligible thin-film viscous resistance, while its local capillary-hydrostatic shape approaches the vertical wall almost tangentially.

Since $h=h_{0}H$ and $x=h_{0}X/ϵ$, dimensional curvature isMatching it to $(2\rho g/\gamma {)}^{1/2}=\sqrt{2}/\ell$ gives