A family of effective equations for wave propagation in periodic media for arbitrary timescales \(\mathcal {O}(\varepsilon ^{-\alpha })\), where \(\varepsilon \ll 1\) 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 (Benoit and Gloria in Long-time homogenization and asymptotic ballistic transport of classical waves, 2017, arXiv:1701.08600; Allaire et al. in Crime pays; homogenized wave equations for long times, 2018, arXiv:1803.09455). The effective solutions in the family are proved to be \(\varepsilon \) close to the original wave in a norm equivalent to the \({\mathrm {L}^{\infty }}(0,\varepsilon ^{-\alpha }T;{{\mathrm {L}^{2}}(\varOmega )})\) 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.

一族有效波在周期介质中任意时间尺度\（\ mathcal {O}（\ varepsilon ^ {-\ alpha}）\）的有效方程组，其中\（\ varepsilon \ ll 1 \）是张量的周期建议描述介质。无需像以前的模型那样进行正则化过程即可确保该族有效方程的适定性（Benoit和Gloria在经典波的长期均质化和渐近弹道运输中，2017，arXiv：1701.08600; Allaire等。犯罪支付;长期均质化的波动方程，2018年，arXiv：1803.09455）。该族中的有效解证明是\（\ varepsilon \）接近于原始波，其范数等于\（{\ mathrm {L} ^ {\ infty}}（0，\ varepsilon ^ {-\ alpha} T; {{\ mathrm {L} ^ {2}}（\ varOmega}}）\）范数。另外，提供了用于计算任意阶的有效张量的数值过程。尤其是，我们提出了任意阶校正器之间的新关系，从而可以大大降低任意阶有效张量的计算成本。这种关系不仅限于本文介绍的有效方程，还可以用于计算替代有效模型的有效张量。