  Based on a tight-binding approximation, we present analytical solutions for
the wavefunction and propagation velocity of an electron in armchair graphene
ribbons. The derived expressions are used for computing the transmission
coefficients through step-like and barrier-like potentials. Our analytical
solutions predict a new kind of transmission resonances for one-mode
propagation in semiconducting ribbons. Contrary to the Klein paradox in
graphene, this approach shows that backscattering for gapless mode is possible.
In consistence with a higher order $\bf{k\cdot p}$ method, the backscattering
probabilities vary with the square of the applied potential in the low-energy
limit. We also demonstrate that gapless-mode propagation through a potential
step in armchair ribbons can be described by the same through-step relation as
that for an undimerized 1D chain of identical atoms.
