International Journal of Advanced Engineering Research and Science (IJAERS) Peer-Reviewed Journal ISSN: 2349-6495(P) | 2456-1908(O) Vol-12, Issue-1; Jan, 2025 Journal Home Page Available: https://ijaers.com/ Article DOI: https://dx.doi.org/10.22161/ijaers.121.3 www.ijaers.com Page | 27 New Weighted Taylor Series for Water Wave Energy Loss and Littoral Current Analysis Syawaluddin Hutahaean Ocean Engineering Program, Faculty of Civil and Environmental Engineering-Bandung Institute of Technology (ITB), Bandung 40132, Indonesia. syawalf1@yahoo.co.id Received: 15 Dec 2024, Receive in revised form: 13 Jan 2025, Accepted: 19 Jan 2025, Available online: 25 Jan 2025 ©2025 The Author(s). Published by AI Publication. This is an open-access article under the CC BY license (https://creativecommons.org/licenses/by/4.0/). Keywords— weighted Taylor series, weighting coefficients, littoral current. Abstract—This paper presents a more systematic formulation of the weighted Taylor series, resulting in a more accurate determination of the weighting coefficient. The weighted Taylor series is derived by truncating the Taylor series to the first order and assigning weighting coefficients to the first-order terms, which reflect the contribution of higher-order terms. The resulting weighted Taylor series is applied to the analysis of wave constant equations in deep water, including wavelength and wave period, which are primarily governed by the Kinematic Free Surface Boundary Condition. The input for these wave constant equations is the wave amplitude. Using these wave constant equations, a shoaling-breaking model is developed, accounting for wave energy loss. The lost wave energy is then utilized to derive the radiation current equation, which subsequently leads to the formulation of the littoral current equation. I. INTRODUCTION The fundamental equations of hydrodynamics are often formulated using truncated Taylor series, which retain only the first-order terms. The justification for truncation lies in the assumption that, for sufficiently small intervals in both time and space, the contributions of second-order and higher-order terms become negligible. However, this reasoning is not entirely accurate. As the interval size decreases, the value of the first-order term also diminishes, rendering the higher-order terms relatively significant. Consequently, neglecting these terms can lead to a loss of important characteristics of the underlying function, as higher-order differentials carry specific physical meanings. For instance, second-order differentials are associated with identifying maxima or minima, while third-order differentials convey additional information about the curvature and behavior of the function. Excluding these terms compromises the accuracy and completeness of the representation, as the first-order approximation alone is insufficient to capture the essential properties of the system. Despite this limitation, incorporating higher-order terms into the formulation of the basic equations of hydrodynamics poses considerable challenges, particularly in terms of complexity and computational feasibility. To address this issue, it is necessary to develop a modified truncated Taylor series that retains the influence of higher- order terms indirectly. This research introduces such a formulation, termed the weighted Taylor series, in which the effects of higher-order terms are embedded into the first- order term through the use of weighting coefficients. The accurate determination of these weighting coefficients requires careful consideration of the interval size at which the Taylor series can be truncated to a first-order approximation. Consequently, this research also formulates an appropriate interval size for numerical modeling, ensuring that the weighted Taylor series captures the essential dynamics of the system while remaining computationally efficient. Numerical methods, such as the Finite Difference Method (FDM) and the Finite Element Method (FEM), are