A Polar Method using cubic spline approach for obtaining wave resonating quadruplets

A Polar Method based on parametric Cubic Spline technique (PM-CS) is presented for obtaining wave resonating quadruplets {K1,K2,K3,K4} in the calculation of the nonlinear source term of the wave model, with results for both deep and finite depths. This polar method is applied in the WRT method for the nonlinear term, and may be considered in addition to the Polar Method with Adaptive Stepping (PM-AS) by Van Vledder (2000). The PM-CS is used to obtain an uniform distribution of points on the locus of K2, for input pair (K1,K3), similar to PM-AS. However, it differs from PM-AS in two ways: (i) no assumption has been made regarding the shape of the locus, and (ii) iterative procedure is avoided, except for locus points on the symmetry in finite depths. For a smaller set of points on the locus, the present PM-CS with variable spacing can be employed. This is illustrated through the convergence of the nonlinear source term by varying points on the locus and the transfer integral evaluated using the Gauss-Legendre quadrature method. Results presented for 1-D nonlinear source term using WRT method are shown comparable with existing WRT results of Resio and Perrie (1991) and Vledder and Hashimoto (2013).