Integral equation technique for water wave interaction by an array of vertical flexible porous wave barriers

Abstract

Scattering of obliquely incident surface gravity waves by an array of partial flexible vertical porous wave barriers of varied configurations is studied in finite water depth and deep water cases. For the mathematical modeling purpose, linearized water wave theory is considered. Firstly, the problem associated with double vertical barriers is handled for a solution. To solve the problem, the associated BVP (boundary value problem) is transformed into a system of Fredholm integral equations of the second kind. By using suitable quadrature formulae, these integral equations are converted to a system of linear algebraic equations. These are solved to compute various important physical quantities associated with wave scattering. The energy identities for both finite water depth and deep water cases are derived. These derived energy identities are used to check the computational accuracy and to quantify the loss of wave energy by the barrier system. Subsequently, the study is extended to deal with wave scattering by multiple non-identical unequally spaced barriers using wide-spacing approximation and results for equally spaced identical barriers are obtained as a special case. The methodology is further generalized to deal with Bloch problem to incorporate a periodic array of infinite barriers. Occurrences of Bragg resonance in case of multiple barriers are analyzed. From the results associated with the linearized water wave theory, the second-order mean wave forces are computed and compared with the first-order wave forces. The phenomena of cloaking for wave scattering by multiple impervious barriers are discussed for a wide variety of wave and structural parameters. The methodology used in this present study can easily be generalized to solve similar problems that arise in mathematical physics.