Scattering of water waves by a pair of vertical porous plates

The two-dimensional problem of scattering of small-amplitude surface water-waves by two symmetric vertical thin porous plates is investigated here assuming the linear theory. The problem is formulated in terms of two hypersingular integral equations of the second kind involving the discontinuities in the unknown symmetric and antisymmetric potential functions describing the motion in the fluid across one of the plates. Exploiting the conditions at the two tips of the plate, the discontinuity is approximated by expanding it in terms of a finite series involving Chebyshev polynomials of the second kind multiplied by an appropriate weight function, and then the integral equations are solved numerically by a collocationmethod. Using the solutions, the reflection and the transmission coefficient are computed numerically. The numerical results for the reflection and the transmission coefficients, amount of energy dissipation and the hydrodynamic forces are depicted graphically against the wave number. Known results for the reflection coefficient for two impermeable plates are recovered when the porous-effect parameter G is set to be equal to zero. An energy identity for the permeable plates is determined.