  We pursue a low-wavenumber, second-order homogenized solution of the
time-harmonic wave equation at both low and high frequency in periodic media
with a source term whose frequency resides inside a band gap. Considering the
wave motion in an unbounded medium $\mathbb{R}^d$ ($d\geqslant1$), we first use
the (Floquet-)Bloch transform to formulate an equivalent variational problem in
a bounded domain. By investigating the source term's projection onto certain
periodic functions, the second-order model can then be derived via asymptotic
expansion of the Bloch eigenfunction and the germane dispersion relationship.
We establish the convergence of the second-order homogenized solution, and we
include numerical examples to illustrate the convergence result.
