  In this paper, we study representations of certain string algebras, which are
referred to as of affine type $\widetilde{C}$. We introduce minimal string
modules and apply them to explicitly describe components of the
Auslander-Reiten quivers of the string algebras and $\tau$-locally free modules
defined by Geiss-Lerclerc-Schr\"{o}er. As an application, we prove
Geiss-Leclerc-Schr\"{o}er's conjecture on the correspondence between positive
roots of type $\widetilde{C}$ and $\tau$-locally free modules of the
corresponding string algebras.
