We present analytical results on the propagation of a classical string in
non-zero modes through the singularity of the compactified Milne space. We
restrict our analysis to a string winding around the compact dimension of
spacetime. The compact dimension undergoes contraction to a point followed by
re-expansion. We demonstrate that the classical dynamics of the string in
excited states is non-singular in the entire spacetime.Comment: Version accepted for publication in Class. Quantum Gra