  The quasistatic limit of the antiplane shear-wave speed ('effective speed')
$c$ in 2D periodic lattices is studied. Two new closed-form estimates of $c$
are derived by employing two different analytical approaches. The first
proceeds from a standard background of the plane wave expansion (PWE). The
second is a new approach, which resides in $\mathbf{x}$-space and centers on
the monodromy matrix (MM) introduced in the 2D case as the multiplicative
integral, taken in one coordinate, of a matrix with components being the
operators with respect to the other coordinate. On the numerical side, an
efficient PWE-based scheme for computing $c$ is proposed and implemented. The
analytical and numerical findings are applied to several examples of 2D square
lattices with two and three high-contrast components, for which the new PWE and
MM estimates are compared with the numerical data and with some known
approximations. It is demonstrated that the PWE estimate is most efficient in
the case of densely packed stiff inclusions, especially when they form a
symmetric lattice, while in general it is the MM estimate that provides the
best overall fitting accuracy.
