Scattering of oblique water waves by two thin unequal barriers with non-uniform permeability

The scattering of obliquely incident water waves by two unequal permeable barriers with variable permeability is investigated for two types of barriers, namely partially immersed barriers and bottom-standing barriers, under the consideration of the theory of linear water waves. The barriers are present in water of uniform finite depth. The velocity potential is expanded by using Havelock’s expansion of water wave potential and employing Havelock’s inversion formula together with the conditions on the permeable barriers; the boundary value problem is reduced to a coupled Fredholm-type vector integral equations. The integral equations are solved using the multi-term Galerkin approximation where the unknown functions are approximated in terms of Chebyshev polynomials. The numerical results for the reflection coefficient, the transmission coefficient and the energy dissipation are depicted graphically. Known results for two identical as well as two non-identical impermeable barriers and for a single and twin permeable barriers are recovered in the limiting cases.