  A family of effective equations for wave propagation in periodic media for
arbitrary timescales $\mathcal{O}(\varepsilon^{-\alpha})$, where
$\varepsilon\ll1$ is the period of the tensor describing the medium, is
proposed. The well-posedness of the effective equations of the family is
ensured without requiring a regularization process as in previous models [A.
Benoit and A. Gloria, arXiv:1701.08600, 2017], [G. Allaire, A. Lamacz, and J.
Rauch, arXiv:1803.09455, 2018]. The effective solutions in the family are
proved to be $\varepsilon$ close to the original wave in a norm equivalent to
the $L^\infty(0,\varepsilon^{-\alpha}T;L^2(\Omega))$ norm. In addition, a
numerical procedure for the computation of the effective tensors of arbitrary
order is provided. In particular, we present a new relation between the
correctors of arbitrary order, which allows to substantially reduce the
computational cost of the effective tensors of arbitrary order. This relation
is not limited to the effective equations presented in this paper and can be
used to compute the effective tensors of alternative effective models.
