Richtmyer–Meshkov instability

Remarks

Let $\rho_2$ be the density on the side towards which the impulse points, and $A = (\rho_2 - \rho_1) / (\rho_2 + \rho_1)$ the Atwood number. The linear equation of the Rayleigh–Taylor instability for a mode $\eta = a(t) \cos kx$ is $a'' = A k g(t) a$. Richtmyer's impulsive model uses it with $g(t) = \Delta u \, \delta(t)$. This gives, for $t > 0$, $$a(t) = a_0 + A k \, \Delta u \, a_0 \, t,$$ so the amplitude grows linearly in time, where $a_0$ is the amplitude just after the impulse. Richtmyer proposed this model, with post-shock values of $A$ and $a_0$, and Meshkov later observed the instability in experiments.

Example: when a shock passes from a light gas into a heavy gas, $A > 0$ and the perturbation grows with the same phase. When it passes from heavy to light, $A < 0$: the amplitude first decreases through zero, so the interface inverts its phase, and then grows. Non-example: an unperturbed flat interface, $a_0 = 0$, stays flat in this model; the instability needs an initial perturbation.