Interface Dispersion Relation (KH / RT / Surface Waves)

Equation

For a perturbation \(h \sim e^{ikx + st}\) of the interface between stream 1 (\(\rho_1, U_1\), above) and stream 2 (\(\rho_2, U_2\), below), with gravity \(g\) and surface tension \(\gamma\):

\[\boxed{\;s = -ik\,\frac{\rho_1 U_1 + \rho_2 U_2}{\rho_1 + \rho_2} \pm \sqrt{\frac{k^2\rho_1\rho_2(U_1 - U_2)^2}{(\rho_1+\rho_2)^2} + \frac{kg(\rho_1 - \rho_2)}{\rho_1+\rho_2} - \frac{\gamma k^3}{\rho_1+\rho_2}}\;}\]

Instability ⇔ the radicand is positive ⇔ \(\operatorname{Re}(s) > 0\).

Physical meaning

One formula, four phenomena, depending on which term wins under the square root:

Dominant term	Phenomenon
shear \(k^2\rho_1\rho_2(U_1-U_2)^2\)	Kelvin–Helmholtz instability
gravity with \(\rho_1 > \rho_2\)	Rayleigh–Taylor instability
gravity with \(\rho_2 > \rho_1\)	stable interfacial gravity waves (\(s = \pm i\omega\))
surface tension \(\gamma k^3\)	capillary waves; stabilizes short wavelengths

The real part of the first term is a Doppler shift: ripples ride the density-weighted mean stream.

Variables

\(s\) — complex growth rate (s⁻¹): \(\operatorname{Re}(s)\) = growth, \(\operatorname{Im}(s)\) = oscillation · \(k\) — wavenumber (m⁻¹) · \(\rho_{1,2}\) — densities · \(U_{1,2}\) — stream speeds · \(g\) — gravity · \(\gamma\) — surface tension (N/m).

Assumptions

Inviscid · incompressible · irrotational away from the interface (vortex sheet at \(y=0\)) · linearized: infinitesimal amplitude, \(O(\epsilon^2)\) dropped · deep layers (perturbations decay as \(e^{\mp ky}\)).

Derivation outline

Base state: uniform potentials \(\bar\phi_{1,2} = U_{1,2}\,x\); interface \(y = 0\).
Perturb: \(\phi_i = U_i x + \epsilon\phi_i'\), interface \(y = \epsilon h(x,t)\); each \(\phi_i'\) solves Laplace, decaying away from the interface: \(\tilde\Phi_1 \propto e^{-ky}\), \(\tilde\Phi_2 \propto e^{+ky}\).
Kinematic condition (interface moves with the fluid): \(\partial_y\phi_i' = \partial_t h + U_i\,\partial_x h\) at \(y = 0\), giving \(\tilde\Phi_1(0) = -\frac{h_0}{k}(s + ikU_1)\), \(\tilde\Phi_2(0) = \frac{h_0}{k}(s + ikU_2)\).
Dynamic condition (pressure continuity, with surface tension \(P_1 = P_2 + \gamma\kappa\)) via unsteady Bernoulli: \(\rho_1[(s+ikU_1)\tilde\Phi_1(0) + gh_0] = \rho_2[(s+ikU_2)\tilde\Phi_2(0) + gh_0] - \gamma k^2 h_0\).
Nontrivial \(h_0\) ⇒ the quadratic in \(s\) above.

Key special cases

Pure KH (\(g = \gamma = 0\)): \(s_+ = \frac{k|U_1-U_2|\sqrt{\rho_1\rho_2}}{\rho_1+\rho_2}\) — all wavelengths unstable, fastest at large \(k\).
Pure RT (\(U_1 = U_2 = 0\), \(\rho_1 > \rho_2\)): \(s = \sqrt{gk\,\mathcal{A}}\) with Atwood number \(\mathcal{A} = \frac{\rho_1-\rho_2}{\rho_1+\rho_2}\).
Deep-water waves (\(\rho_1 \to 0\)): \(\omega^2 = gk + \gamma k^3/\rho_2\) — gravity–capillary dispersion; minimum phase speed ≈ 23 cm/s for water at \(\lambda \approx 1.7\) cm.
Wind over water: onset at \((U_1 - U_2)^2 = \frac{2(\rho_1+\rho_2)}{\rho_1\rho_2}\sqrt{\gamma g(\rho_2-\rho_1)}\) → critical wind ≈ 6.6 m/s.
3-D perturbations (\(e^{ikx+ilz}\), \(\kappa = \sqrt{k^2+l^2}\)): shear drive keeps \(k^2\) but restoring terms use \(\kappa\) — 2-D modes are the most dangerous.

Limitations

Linear (onset only, no saturation or billow roll-up) · inviscid (no damping of short waves except by tension) · sharp interface (finite shear layers cap the growth rate at \(k \sim 1/\text{thickness}\)).

Laplace's equation — governs the perturbation potentials
Bernoulli's equation — the dynamic boundary condition

Quiz

Q1 (computational). Deep-water gravity wave, \(\lambda = 100\) m. Phase speed?

Answer

\(c = \sqrt{g/k} = \sqrt{9.8\times100/2\pi} \approx 12.5\) m/s — ocean swell moves at highway speed.

Q2 (conceptual). Why does surface tension stabilize short waves but not long ones?