14 May 2025

Ripples and Gravity Waves

ripples gravity-waves surface-waves dispersion

Consider a small wave on the free surface of an incompressible, inviscid liquid of density $\rho$. Take the undisturbed surface as $z=0$, with the liquid in $z<0$. If the depth is $h$, the deep-water condition is $kh\gg1$.

Write the surface displacement as the real part of

\[\zeta(x,t)=\zeta_0e^{i(kx-\omega t)},\]

where $k=2\pi/\lambda$.

Velocity potential

For irrotational flow, the fluid velocity is

\[\mathbf v=\boldsymbol\nabla\phi.\]

Incompressibility requires

\[\boldsymbol\nabla\cdot\mathbf v=\nabla^2\phi=0.\]

The solution with the same $x,t$ dependence as the surface wave and which vanishes as $z\to-\infty$ is

\[\phi(x,z,t)=\phi_0e^{kz}e^{i(kx-\omega t)}.\]

Because $k>0$ and $z<0$, the factor $e^{kz}$ shows that the fluid motion decreases exponentially with depth.

Boundary conditions at the surface

The surface moves with the liquid. To first order, the vertical velocity of the surface equals that of the fluid at $z=0$:

\[\frac{\partial\zeta}{\partial t} =\left.\frac{\partial\phi}{\partial z}\right|_{z=0}.\]

Substitution gives

\[-i\omega\zeta_0=k\phi_0, \qquad \boxed{\phi_0=-\frac{i\omega}{k}\zeta_0}.\]

The linearized unsteady Bernoulli equation gives the liquid pressure at the displaced surface:

\[p-p_0=-\rho\left( \frac{\partial\phi}{\partial t}+g\zeta \right).\]

For the weakly curved graph $z=\zeta(x,t)$, the surface-tension pressure is

\[p-p_0=-\gamma\frac{\partial^2\zeta}{\partial x^2} =\gamma k^2\zeta.\]

This sign is physically consistent: at a crest, $\partial^2\zeta/\partial x^2<0$, so the liquid pressure exceeds the air pressure.

Equating the two pressure expressions at $z=0$ gives

\[-\rho(-i\omega\phi_0+g\zeta_0) =\gamma k^2\zeta_0.\]

Since

\[-i\omega\phi_0 =-\frac{\omega^2}{k}\zeta_0,\]

the nonzero amplitude cancels and

\[\frac{\rho\omega^2}{k}-\rho g=\gamma k^2.\]

Thus the deep-water gravity-capillary dispersion relation is

\[\boxed{\omega^2=gk+\frac{\gamma}{\rho}k^3}.\]

Gravity waves and ripples

The phase speed $c=\omega/k$ satisfies

\[\boxed{ c^2=\frac{g}{k}+\frac{\gamma k}{\rho} =\frac{g\lambda}{2\pi} +\frac{2\pi\gamma}{\rho\lambda} }.\]

For $k\ll k_c$ while still satisfying $kh\gg1$, the gravity term dominates:

\[\omega^2\simeq gk, \qquad \boxed{c_{\rm grav}\simeq\sqrt{\frac{g}{k}} =\sqrt{\frac{g\lambda}{2\pi}}}.\]

These are gravity waves; gravity is the principal restoring agency.

For $k\gg k_c$, the surface-tension term dominates:

\[\omega^2\simeq\frac{\gamma}{\rho}k^3, \qquad \boxed{c_{\rm cap}\simeq\sqrt{\frac{\gamma k}{\rho}} =\sqrt{\frac{2\pi\gamma}{\rho\lambda}}}.\]

These capillary waves are ripples; surface tension is the principal restoring agency.

The two contributions to $c^2$ are equal at

\[\frac{g}{k_c}=\frac{\gamma k_c}{\rho}.\]

Therefore

\[\boxed{k_c=\sqrt{\frac{\rho g}{\gamma}}}, \qquad \boxed{\lambda_c=2\pi\sqrt{\frac{\gamma}{\rho g}}}.\]

This is also the minimum of the phase-speed curve. Differentiating $c^2$ with respect to $k$ gives

\[\frac{d(c^2)}{dk} =-\frac{g}{k^2}+\frac{\gamma}{\rho}=0 \quad\text{at}\quad k=k_c.\]

At the minimum,

\[\boxed{c_{\min}=\left(\frac{4g\gamma}{\rho}\right)^{1/4}}.\]

The editable wave and phase-speed diagram is available as a TikZ file. The boundary-condition algebra, limiting speeds and crossover are checked in the Maxima worksheet; every displayed residual is zero.