Consider deep water gravity waves but assume now that there are two fluids separated by the interface at $y=\eta(x,t)$ and suppose that the upper fluid extends to $y\to\infty$. Let the fluid in the lower layer have density $\rho_1$ and let the density of the upper fluid be $\rho_2\lt\rho_1$. Show that the phase speed $c$ of waves on the interface with wave number $k$ will be given by $$c^2=\frac{g}{|k|}\left[\frac{\rho_1-\rho_2}{\rho_1+\rho_2}\right].$$
Am I able to tell from this question that there is no surface tension? It's just the solution assumes this but it is not stated explicitly in the question.
That result applies to gravity waves only. In the presence of capillary effects the dispersion relation has an additional term that is proportional to $\sigma/(\rho_1 + \rho_2).$