Evanescent Bloch wave framework for water-wave propagation through an array of porous structures

Abstract

We present a theoretical framework for water-wave interaction with finite periodic arrays of porous blocks, with emphasis on the effect of dissipation on the complex band diagram of the media. A semi-analytical model based on linear potential-flow theory, coupled with porous structure, is developed to derive complex Bloch dispersion relations and finite-array scattering characteristics. For finite arrays, the wave field is represented as a superposition of left- and right-travelling Bloch eigenmodes. An orthogonality relation derived via Green's identity yields a compact scattering formulation, and the approach is compared with scattering-matrix method and published benchmarks. Complex band diagrams are presented to illustrate how dissipation shifts the real Bloch wavenumber of propagating modes in classical passbands into complex plane (indicating attenuation), although intervals of high and low attenuation persist, producing fuzzy passbands and band gaps. We demonstrate that analysing pole-zero pairs in complex frequency gives insight into finite-array resonances. As the number of porous blocks increases, resonances split and densify owing to the interference of the attenuating Bloch modes, while increasing porosity shifts the poles deeper, broadening and damping resonances to yield robust broadband attenuation. The framework provides a systematic basis for designing compact porous metamaterial arrays for coastal protection and wave energy mitigation.