= Gravity-capillary wave
{title2=$\omega^2=gk+(\gamma/\rho)k^3$}
= Capillary-gravity wave
{synonym}
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 $ka\ll1$ and $a\ll h$, so <linearized boundary conditions> apply. Deep water means $kh\gg1$, rather than a particular depth in metres. https://www.ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/6163837c5efa5d6ed969e535ec8f890c_MIT18_354JS15_lectureNotes.pdf[Brenner, Peacock, Stocker, Reis and Dunkel, MIT Nonlinear Dynamics II, §23.1, pp. 105–106]{ref}
Let $z=0$ be the undisturbed <free surface>, with <liquid> below it. <Potential flow> gives a <velocity potential> $\phi$ with $\mathbf u=\nabla\phi$, 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>, give
$$
\phi_{xx}+\phi_{zz}=0\quad(z<0),
\qquad
\eta_t=\phi_z,
\qquad
\phi_t+g\eta-\frac{\gamma}{\rho}\eta_{xx}=0\quad(z=0).
$$
Here $\eta(x,t)$ 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> $\omega$:
$$
\eta=a\cos(kx-\omega t),
\qquad
\phi=\frac{a\omega}{k}e^{kz}\sin(kx-\omega t).
$$
The factor $e^{kz}$ decays into the deep <liquid>, satisfies the <Laplace equation>, and makes $\phi_z=\eta_t$ at $z=0$. Substituting in the dynamic <boundary condition> leaves
$$
-\frac{a\omega^2}{k}+ga+\frac{\gamma}{\rho}k^2a=0,
\qquad
\boxed{\omega^2=gk+\frac{\gamma}{\rho}k^3}.
$$
This derivation displays the two restoring contributions with the same sign. <Dimensional analysis> also checks the result: $\gamma/\rho$ has units $\mathrm{m^3\,s^{-2}}$, so both $gk$ and $(\gamma/\rho)k^3$ have units $\mathrm{s^{-2}}$. https://www.ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/6163837c5efa5d6ed969e535ec8f890c_MIT18_354JS15_lectureNotes.pdf[MIT Nonlinear Dynamics II, §23.1, equations (547)–(551): boundary conditions and dispersion]{ref}
At finite depth, a flat impermeable bottom at $z=-h$ requires $\phi_z=0$. Replacing $e^{kz}$ by the normalized vertical factor $\cosh(k(z+h))/\cosh(kh)$ makes the surface ratio $\phi_z/\phi=k\tanh(kh)$. Eliminating the mode amplitudes then gives
$$
\omega^2=\left(gk+\frac{\gamma}{\rho}k^3\right)\tanh(kh).
$$
Thus 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. https://ocw.mit.edu/courses/18-357-interfacial-phenomena-fall-2010/04759cfc15c79928ca9f72a18b3864dd_MIT18_357F10_lec_all.pdf[John W. M. Bush, Interfacial Phenomena, chapter 19, equation (19.4)]{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> gives
$$
c_p=\frac{\omega}{k}=\sqrt{\frac{g}{k}+\frac{\gamma k}{\rho}},
\qquad
c_g=\frac{d\omega}{dk}
=\frac{g+3\gamma k^2/\rho}{2\sqrt{gk+\gamma k^3/\rho}},
\qquad
\frac{c_g}{c_p}=\frac{1+3(k\ell_c)^2}{2\left(1+(k\ell_c)^2\right)}.
$$
For $k\ell_c\ll1$, the <surface gravity wave> limit has $c_g\simeq c_p/2$. For $k\ell_c\gg1$, the <deep-water capillary-wave dispersion relation> has $c_g\simeq3c_p/2$: 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>. https://ocw.mit.edu/courses/18-357-interfacial-phenomena-fall-2010/04759cfc15c79928ca9f72a18b3864dd_MIT18_357F10_lec_all.pdf[John W. M. Bush, Interfacial Phenomena, chapter 19, equation (19.5) and the two group-speed limits]{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. Therefore
$$
c_{p,\min}=\left(\frac{4g\gamma}{\rho}\right)^{1/4},
\qquad c_g(k_c)=c_p(k_c).
$$
For the stipulated parameters in <capillary length>, the minimum is $0.231\,\mathrm{m\,s^{-1}}$. At $\lambda=3\,\mathrm{mm}$, direct substitution gives $f=\omega/(2\pi)\simeq131\,\mathrm{Hz}$, $c_p\simeq0.394\,\mathrm{m\,s^{-1}}$ and $c_g\simeq0.580\,\mathrm{m\,s^{-1}}$. 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.
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