Abstract: In this paper the semi-analytical theory of water wave propagation through vegetation developed by Mei, Chan and Liu (2014) is extended to examine the cases where the vegetated area has a finite extent. A mathematical model for small-amplitude periodic waves propagating through a lattice-like array of vertical cylinders within a finite region is developed. Assuming periodic lattice configuration and strong contrast between the cylinder spacing and the typical wavelength, the multi-scale perturbation theory of homogenization (Mei and Vernescu, 2010) is employed to derive the effective equations governing the macro-scale wave dynamics and the boundary-value problem of micro-scale flows within a unit cell. The constitutive coefficients in the macro-scale effective equations are computed from the solution of the micro-scale boundary-value problem, which is driven by the macro-scale pressure gradients. Furthermore, a bulk eddy viscosity is determined by balancing the time-averaged rate of dissipation and the rate of work done by wave forces on the forest, integrated over the entire forest. The wave forces are modeled by the Morison-type formula (Morison et al., 1950), in which the drag coefficient formula is constructed based on experimental data by Hu, Suzuki, Zitman, Uittewaal and Stive (2014). The theory was checked with the experimental data from Hu et al. (2014) for wave decay through a forest strip, in which waves are of normal incidence. The agreement between the theory and experiment is very good. To further check the theory, a new set of experiments for periodic waves propagating through a circular shape forest was conducted. The Reynolds numbers for these experiments are in the same range as those of Hu et al. (2014). Because of the circular shape, analytical solutions of the macro-scale problem can be obtained. Again, good agreement between the theory and experimental data is observed. No additional fitting coefficient is needed in the theory/experiment comparison.

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