  We describe the second (generalized) Feng-Rao distance for elements in an Arf
numerical semigroup that are greater than or equal to the conductor of the
semigroup. This provides a lower bound for the second Hamming weight for one
point AG codes. In particular, we can obtain the second Feng-Rao distance for
the codes defined by asymptotically good towers of function fields whose
Weierstrass semigroups are inductive. In addition, we compute the second
Feng-Rao number, and provide some examples and comparisons with previous
results on this topic. These calculations rely on Ap\'{e}ry sets, and thus
several results concerning Ap\'ery sets of Arf semigroups are presented.
