Resonance of water waves propagating over a uniform and a graded line array of rectified submerged cosinoidal bars

In this work, a Boussinesq model is adopted to investigate wave behaviors, and thus the consequent phenomena of Bragg resonance/rainbow reflection, when propagating over a uniform and a graded line array of rectified submerged cosinoidal bars. The capability of the model is validated by comparing to the experiments carried out in a wave flume and three typical published results. It is found that the widely used Bloch band theory, which is initially derived for an infinite periodic medium, provides correct occurrence conditions but inaccurate magnitudes of Bragg resonant reflected waves for a uniform finite periodic array. It has been demonstrated that the number of component bars remaining in the travelling path rather than the widely acknowledged total number is the key for incurring this discrepancy. In addition, the theory also works well for rainbow reflection; both occurrence conditions and spatial locations of resonant reflections can be captured. The results indicate that the increasing interval in spacing of a graded array is of primary importance for tuning the frequency range at which the rainbow reflection occurs, i.e. enlarging the effective frequency bandwidth.