Analytic solutions of the rayleigh equation for linear density profiles
We consider the Rayleigh-Taylor instability in linear density profiles and we derive the exact analytic expressions of the growth rates and associated eigenfunctions. We study the behavior of the multiple eigenvalues in both the short- and the long-wavelength limit. As the largest eigenvalue gamma(max) reduces to the classical Rayleigh growth rate; the other eigenvalues vanish as the front thickness tends to zero. Furthermore, the simple expression of gamma(max) exact to first order in the long-wavelength limit differs from the widely used estimate sqrt[Akg/(1+AkL0)], where g is the acceleration, A the Atwood number, k the wave number of the perturbation, and L0 the minimum density gradient scale length.
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