Parabolic focusing of water waves via a straight reflector based on the space transformation method

Water-wave focusing is a significant technology for enhancing wave energy density, which can effectively improve the efficiency of the wave energy converter. To achieve parabolic focusing of water waves using a straight boundary, the space transformation method (STM) was employed to map a parabolic reflector onto a straight configuration. Initially, a straight reflector was designed using the STM by spatially compressing the domain of a parabolic reflector. The spatial transformation relationship between the parabolic and straight reflectors was established. Subsequently, leveraging the form invariance of the governing equation for water waves, the ideal effective parameters, including anisotropic water depth and gravitational acceleration within the transformed region, were derived. To circumvent the impracticality of altering gravitational acceleration, a set of simplified anisotropic parameters was obtained by incorporating the scale factor of gravity acceleration into the adjusted water depth. Finally, the Helmholtz equation was solved via the finite element method to analyze the focusing performance of three configurations: the original parabolic reflector, the ideal straight reflector, and the simplified straight reflector. The results demonstrate that the proposed straight reflector successfully focuses water waves at designated points corresponding to those of the parabolic reflector. Moreover, the wave distribution of the ideal straight reflector in the unmodified region beyond the transformed reflector area is identical to that generated by the parabolic reflector.