  In this paper, we investigate the influence of coherent pion emission on
long-range azimuthal correlations in relativistic proton-proton collisions. We
study the pion momentum distribution for a coherent source with both transverse
and Bjorken longitudinal expansions, and calculate the two-particle correlation
function $C(\Delta\eta,\Delta\phi)$. A "ridge" structure is observed in the
correlation function $C(\Delta\eta,\Delta\phi)$ of the coherent pion emission,
whether the coherent source is expanding or static. The onset of this
long-range azimuthal correlation can be traced back to the asymmetric initial
transverse profile of the coherent source, owing to the interference in
coherent emission. We further construct a partially coherent pion-emitting
source by incorporating a chaotic emission component. From the experimental
data of the two-pion Hanbury-Brown-Twiss correlations in pp collisions at
$\sqrt{s}\!=\!7\,$TeV, we extract a coherent fraction of pion emission, which
increases with increasing charged-particle multiplicity, as an input of the
partially coherent source model. The ridge structure is observed in the
correlation function $C(\Delta\eta,\Delta\phi)$ for the partially coherent
source. The correlation becomes stronger at higher multiplicities because of
the larger degrees of coherence. The results in this work are meaningful for
fully understanding the collectivity in the small system of pp collisions.
