Suppose that a heavy fluid of constant density \(\rho_1\) is placed above a light fluid of another constant density \(\rho_2\),
and separated by a surface S in a vertically downward gravitational field of acceleration \(g\).
In the unperturbed state, the surface S was a horizontal plane located at \(y = 0\)
(with the \(y\) axis taken vertically upward) and the fluid was at rest, and the density was
\[
\rho_1 \quad \text{for } y > 0; \qquad \rho_2(<\rho_1) \quad \text{for } y < 0
\]
Suppose that the surface S is deformed in the form
\[
y = \zeta(x,t)
\]
Both above and below S, the flow is assumed to be irrotational and the velocity potential \(\phi\) is expressed as
\[
\phi = \phi_1(x,y,t) \quad \text{for } y > \zeta; \qquad \phi_2(x,y,t) \quad \text{for } y < \zeta
\]
(1) Derive linear perturbation equations for small perturbations \(\zeta\), \(\phi_1\), and \(\phi_2\) from the dynamic boundary condition and kinematic boundary condition.
(2) Expressing the perturbations in the following forms with a growth rate \(\sigma\) and a wavenumber \(k\),
\[
\zeta = Ae^{\sigma t} e^{ikx}, \qquad \phi_i = B_i e^{\sigma t} e^{ikx - ky}, \qquad (i = 1, 2)
\]
derive an equation to determine the growth rate \(\sigma\), where \(A, B_1, B_2\) are constants. State whether the basic state is stable or unstable.
(3) Apply the above analysis to the case where a lighter fluid is placed above a heavy fluid, and derive a conclusion that there exists an interfacial wave (the internal gravity wave). State what is the frequency.