Why does the Rayleigh-Taylor instability growth rate increase with wave number?

It can be easily derived that the Rayleigh-Taylor instability growth rate on the boundary of two fluids (i.e. denser fluid supported by a lighter fluid) under gravity, $g$, is given by $gk\eta$, where $k$ is the wave vector (i.e. reciprocal of wavelength) and $\eta$ is the Atwood number. What seems counterintuitive is that the growth rate is independent of the amplitude of perturbation or the length of the boundary (e.g. it could be infinite) yet the precise wavelength of the initial perturbation appears important instead. One can easily show that a small sinusoidal displacement given by $\xi sin(kx)$ along the boundary for any interval of length $0 < x < L = 2\pi/k$ results in a reduction of gravitational potential energy equal to $\xi^2/2$ which is second-order and independent of $k$. But it is this conversion of potential energy that drives the kinetic energy for the instability in the first place. That simply indicates the (in)stability of the problem, but how does $k$ affect the kinetics? Why does the growth rate physically depend on $k$ if the boundary is sufficiently long?