  Slow flows of an ideal compressible fluid (gas) in the gravity field in the
presence of two isentropic layers are considered, with a small difference of
specific entropy between them. Assuming irrotational flows in each layer [that
is ${\bf v}_{1,2}=\nabla\phi_{1,2}$], and neglecting acoustic degrees of
freedom by means of the conditions
${div}(\bar\rho(z)\nabla\phi_{1,2})\approx0$, where $\bar\rho(z)$ is a mean
equilibrium density, we derive equations of motion for the interface in terms
of the boundary shape $z=\eta(x,y,t)$ and the difference of the two boundary
values of the velocity potentials: $\psi(x,y,t)=\psi_1-\psi_2$. A Hamiltonian
structure of the obtained equations is proved, which is determined by the
Lagrangian of the form ${\cal L}=\int \bar\rho(\eta)\eta_t\psi \,dx dy -{\cal
H}\{\eta,\psi\}$. The idealized system under consideration is the most simple
theoretical model for studying internal waves in a sharply stratified
atmosphere, where the decrease of equilibrium gas density with the altitude due
to compressibility is essentially taken into account. For planar flows, a
generalization is made to the case when in each layer there is a constant
potential vorticity. Investigated in more details is the system with a model
density profile $\bar\rho(z)\propto \exp(-2\alpha z)$, for which the
Hamiltonian ${\cal H}\{\eta,\psi\}$ can be expressed explicitly. A long-wave
regime is considered, and an approximate weakly nonlinear equation of the form
$u_t+auu_x-b[-\hat\partial_x^2+\alpha^2]^{1/2}u_x=0$ (known as Smith's
equation) is derived for evolution of a unidirectional wave.
