  We use the quasiclassical Eilenberger theory to study the topological
superconducting proximity effects between a segment of a nanowire with a p-wave
order parameter and a metallic segment. This model faithfully represents key
qualitative features of an experimental setup, where only a part of a nanowire
is in immediate contact with a bulk superconductor, inducing topological
superconductivity. It is shown that the Eilenberger equations represent a
viable alternative to the Bogoliubov-de Gennes theory of the topological
superconducting heterostructures and provide a much simpler quantitative
description of some observables. For our setup, we obtain exact analytical
solutions for the quasiclassical Green's functions and the density of states as
a function of position and energy. The correlations induced by the boundary
involve terms associated with both p-wave and odd-frequency pairing, which are
intertwined and contribute to observables on an equal footing. We recover the
signatures of the standard Majorana mode near the end of the superconducting
segment, but find no such localized mode induced in the metallic segment.
Instead, the zero-bias feature is spread out across the entire metallic part in
accordance with the previous works. In shorter wires, the Majorana mode and
delocalized peak split up away from zero energy. For long metallic segments,
non-topological Andreev bound states appear and eventually merge together,
giving rise to a gapless superconductor.
