Waves in Fluids

Surface Gravity Waves

Surface Gravity Waves Dispersion Relation

For a flow of depth $H$, the dispersion relation of surface gravity waves is$w=\sqrt{gk\tanh (kH)}.$In shallow water ($kH\ll 1⟹\tanh (kH)\approx kH$), it is non-dispersive, we have $v_g=v_p=\sqrt{gH}.$

Interfacial Waves

Thrm Interfacial Waves Dispersion Relation
The dispersion relation for waves on an interface between two fluids of density ${\rho }_1$ (top fluid) and density ${\rho }_2$ (bottom fluid), with a surface tension $T$ (N/m) at the interface, and a uniform flow in the upper layer of $U$ (m/s) is given by

$$\omega =\frac{kU{\rho }_1\pm \sqrt{({\rho }_1+{\rho }_2)(k^{3}T+({\rho }_2-{\rho }_1)gk)-{\rho }_1{\rho }_2{k}^2{U}^2}}{{\rho }_1+{\rho }_2}$$

Hydrodynamic Instabilities

Def Instability
A system such that any small initial perturbation will be exponentially amplified with time is called an unstable system.

Rayleigh-Taylor Instability

Def Rayleigh-Taylor Instability
The Rayleigh-Taylor instability happens when the uniform flow in the upper layer of doesn’t move(i.e. $U=0$). And we have the dispersion relation:

$$({\rho }_1+{\rho }_2){\omega }^2=k^{3}T+({\rho }_2-{\rho }_1)gk$$

Kelvin Helmholtz Instability

Def Kelvin Helmholtz Instability
Kelvin Helmholtz instability happens when ${\rho }_1\approx {\rho }_2=\rho$ and ${\rho }_2-{\rho }_1=\Delta \rho \ll \rho$:

$$\omega =\frac{kU}{2}\pm \sqrt{\frac{k^{3}T+\Delta \rho gk}{2\rho }-\frac{k^2{U}^2}{4}}$$