  In a recent article, we described how the microscopic structure of
density-density correlations in the fluid interfacial region, for systems with
short-ranged forces, can be understood by considering the resonances of the
local structure factor occurring at specific parallel wave-vectors $q$. Here,
we investigate this further by comparing approximations for the local structure
factor and correlation function against three new examples of analytically
solvable models within square-gradient theory. Our analysis further
demonstrates that these approximations describe the correlation function and
structure factor across the whole spectrum of wave-vectors, encapsulating the
cross-over from the Goldstone mode divergence (at small $q$) to bulk-like
behaviour (at larger $q$). As shown, these approximations are exact for some
square-gradient model potentials, and never more than a few percent inaccurate
for the others. Additionally, we show that they very accurately describe the
correlation function structure for a model describing an interface near a
tricritical point. In this case, there are no analytical solutions for the
correlation functions, but the approximations are near indistinguishable from
the numerical solutions of the Ornstein-Zernike equation.
