Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 331 / 1 / b

Put $A=g({\rho }_1-{\rho }_2)>0$. In the deep-layer limit, the finite-depth Rayleigh-Taylor dispersion relation gives ${\omega }^2=(\gamma k^3-Ak)/({\rho }_1+{\rho }_2)$. Its sign changes atThe Rayleigh-Taylor instability band is $0<k<k_c$; $k=k_c$ is neutral and $k>k_c$ is oscillatory.

For the growing normal mode in two deep layers,It vanishes at the endpoints of the unstable band and has its unique interior maximum where $A-3\gamma k^2=0$. Using $k_c=\sqrt{A/\gamma }$ givesThis is the maximum exponential growth rate, not the real oscillation frequency.

The stationary-point condition for the squared growth rate is $A-3\gamma k_m^2=0$. Its second derivative is $-6\gamma k_m/({\rho }_1+{\rho }_2)<0$, so