Hybrid functions for nonlinear energy transfers at finite depths

Abstract

A numerical method using Hybrid functions which are the orthogonal polynomials, formed from the combination of Block pulse function and Lagrange basis polynomial (HBL) are employed for the estimation of nonlinear energy transfers (NLT) occurring between set of four waves at finite water depths. The advantages and properties of HBL functions provide an easy way for estimating the quadruplets arising in NLT. The manuscript focuses on two aspects, namely sensitivity and efficiency of NLT at finite depths due to HBL. The sensitivity of NLT at finite depths to number of points on the locus curve, for various directional spectra, are studied in detail. With decreasing water depth (i) the locus curve grows and thus magnitude of NLT increases (ii) increase the number of points on the locus curve to achieve reliable (or) accurate results and (iii) the peak frequency starts shifting towards the left side of the spectrum. The efficiency of the HBL method to NLT is substantiated by comparison with the currently used Trapezoidal rule. Convergence results of NLT are established in both frequency and direction. The method is tested for its suitability in both circular and sector grids and the results confirm its adaptability to NLT. The method is validated against the methods implemented in the Wave models such as WRT and Discrete Interaction Approximation.