Breaking Index Study on Weighted Laplace Equation

This study serves as an extension of prior research focusing on weighting coefficients within the context of weighted Taylor series. The primary objective is to determine the weighting coefficients' values in the weighted Taylor series for the purpose of modeling water waves based on velocity potential. Utilizing the weighted Taylor series, we derive both the weighted continuity equation and the weighted Laplace equation. The latter is addressed using the variable separation method over a sloping bottom, leading to the formulation of the velocity potential equation, wave constant equations, and energy conservation equations. Within the wave constant equations, a breaking equation is incorporated. Leveraging both the breaking equation and the energy conservation equations, breaking indexes equations are formulated. These equations encompass breaker length, breaker depth, and breaker height indexes, with weighting coefficients prominently featured. Calibrating the results of the breaking indexes equations against findings from earlier studies provides suitable values for the weighting coefficients. Additionally, this research introduces a shoaling-breaking model and a refraction-diffraction model to explore the phenomena of shoaling-breaking within the solution of the weighted Laplace equation.