= Solution
A local thermal imbalance can amplify a gas perturbation even without an initial displacement of an interface. Let $\mathcal L(\rho,T)$ denote net cooling per unit <mass>, including heating with a negative sign, and suppose $\mathcal L(\rho_0,T_0)=0$. When the <sound-crossing time> is short compared with the thermal evolution time, <pressure> equilibrates and an <ideal gas> perturbation is approximately isobaric. The linearized thermal equation is
$$
c_p\frac{d\delta T}{dt}
=-\left(\frac{\partial\mathcal L}{\partial T}\right)_p\delta T.
$$
Thus a negative constant-pressure derivative produces <isobaric thermal instability>: a cooler parcel experiences a net loss that cools it further, while a warmer parcel heats further. For example, <isobaric thermal stability of a heated ideal gas> with $\mathcal L=A\rho T^\alpha-H$ gives $\sigma=(1-\alpha)H/(c_pT_0)$ and instability for $\alpha<1$. Different heating laws and fixed-density perturbations give different criteria; <thermal conduction> damps sufficiently short-scale <temperature> contrasts. Such feedback helps a gas separate into denser cool and rarer warm regions when stable thermal branches exist.
The <Rayleigh-Taylor instability> instead releases <gravitational potential energy> when a heavier fluid lies above a lighter fluid in a downward gravitational field. For two inviscid incompressible half-spaces with no <surface tension> or <magnetic field>, a horizontal perturbation of <wavenumber> $k$ has
$$
\sigma^2=\mathcal A gk,\qquad
\mathcal A=\frac{\rho_h-\rho_l}{\rho_h+\rho_l}>0.
$$
The <Atwood number> measures the density contrast. A descending heavy-fluid finger and an ascending light-fluid bubble both lower the total <gravitational potential energy>, increasing the original displacement. Interchanging the layers gives an oscillatory stable interface. An accelerating or decelerating astrophysical shell has an analogous effective acceleration, allowing mixing at suitable ejecta interfaces in a <supernova remnant>. Finite interface width, <viscosity>, <surface tension> and <magnetic tension> modify the short-wavelength behavior; the ideal sharp-interface formula does not select a finite fastest-growing wavelength.
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