  The localized low-energy interfacial excitations, or Nambu-Goldstone modes,
of phase-segregated binary mixtures of Bose-Einstein condensates are
investigated analytically by means of a double-parabola approximation (DPA) to
the Lagrangian density in Gross-Pitaevskii theory for a system in a uniform
potential. Within this model analytic expressions are obtained for the
excitations underlying capillary waves or "ripplons" for arbitrary strength
$K\,(>1)$ of the phase segregation. The dispersion relation $\omega \propto
k^{3/2}$ is derived directly from the Bogoliubov-de Gennes equations in limit
that the wavelength $2\pi/k$ is much larger than the healing length $\xi $. The
proportionality constant in the dispersion relation provides the static
interfacial tension. A correction term in $\omega (k)$ of order $k^{5/2}$ is
calculated analytically, entailing a finite-wavelength correction factor
$(1+\frac{\sqrt{K-1} \,k\xi}{4\sqrt{2}\,(\sqrt{2}+\sqrt{K-1})})$. This
prediction may be tested experimentally using (quasi-)uniform optical-box
traps. Explicit expressions are obtained for the structural deformation of the
interface due to the passing of the capillary wave. It is found that the
amplitude of the wave is enhanced by an amount that is quadratic in the ratio
of the phase velocity $\omega/k$ to the sound velocity $c$. For generic
asymmetric mixtures consisting of condensates with unequal healing lengths an
additional modulation is predicted of the common value of the condensate
densities at the interface.
