Oscillating Droplet — Rayleigh Modes

Capillary oscillation modes of a liquid sphere · ω² = σ n(n-1)(n+2)/(ρR³)

0.20
1.00
0
ωₙ² = σ·n(n−1)(n+2) / (ρR³)  |  r(θ,φ,t) = R[1 + A·Yₙᵐ(θ,φ)·cos(ωₙt)]
Rayleigh (1879) showed that surface-tension-driven oscillations of a liquid droplet have frequency ω² ∝ n(n−1)(n+2). Mode n=2 is the fundamental (prolate↔oblate). Higher modes oscillate faster. n=1 is translation (zero frequency). The droplet shape is R(θ) = R₀·[1 + A·Pₙ(cos θ)·cos(ωₙt)].