International Journal of Advanced Engineering Research and Science (IJAERS) [Vol-6, Issue-10, Oct- 2019] https://dx.doi.org/10.22161/ijaers.610.21 ISSN: 2349-6495(P) | 2456-1908(O) www.ijaers.com Page | 136 Modified Momentum Euler EquationforWater Wave Modeling Syawaluddin Hutahaean Ocean Engineering Program, Faculty of Civil and Environmental Engineering,-Bandung Institute of Technology (ITB), Bandung 40132, Indonesia syawaluddin1@ocean.itb.ac.id Abstract— In this research, weighted total acceleration for a function (, , )was formulated. This total acceleration equation was done at the Euler momentum equation. Then, the Euler momentum equation was done together with free surface boundary condition equation to formulate water wave constant at the solution of Laplace equation. The velocity potential of the solution of Laplace equation actually consists of two components that were used in this research. Keywords— weighted total acceleration,convective acceleration, complete velocity potential. I. INTRODUCTION Momentum equation is an important basic equation in mathematic modeling of hydrodynamics, including water wave modeling. Momentum equation commonly used in water wave modeling is Euler momentum equation. There is a constraint in this equation, i.e. Euler momentum equation has no hydrodynamic force in the horizontal direction or convective acceleration has a value of zero when velocity potential is substituted to the term. To overcome this problem, weighted total acceleration equation was formulated where there are two weighted coefficients, i.e. at the time differential term and at the differential term of vertical-direction. Laplace equation solution consists of two velocity potential components (Dean (1991)). However, only one component that has been used. Equations from water wave constant, i.e. wave number and wave constant can be formulated using only one velocity potential component, but the value is determined by both the two velocity components. In this research, the water wave surface equation is formulated using the two velocity potential components, then the condition of the water wave surface that has been produced is studied. II. WEIGHTED TOTAL ACCELERATION Hutahaean (2019a) formulated weighted total acceleration in a function = (, ), is horizontal axis and is time, using Taylor series. The formulation of weighted total acceleration in a function = (, , ),is vertical axis, is done using similar method, therefore the formulation of weighting total acceleration in = (, , )will be preceded by reviewing the formulation of weighting total acceleration in = (, )to obtain a clearer description. 2.1. Weighted Total Acceleration for the function of= (, ) The changes in the value of a function in a function = (, )for a very smallandusing Taylor series only until the second derivative is, ( + , + ) = (, ) + Ƌ Ƌ + Ƌ Ƌ Ƌ + 2 2 Ƌ 2 Ƌ 2 + Ƌ 2 ƋƋ + 2 2 Ƌ 2 Ƌ 2 By working on the argument of Courant (1928) that in order to obtain a good result on horizontal velocity = , then weighting coefficient , is done which is a positive number, in time differential in Taylor series. ( + , + ) = (, ) + Ƌ Ƌ + Ƌ Ƌ Ƌ + 2 2 Ƌ 2 Ƌ 2 + Ƌ 2 ƋƋ + 2 2 2 Ƌ 2 Ƌ 2 .......(1) At the limit , close to zero the following equation is obtained, = Ƌ Ƌ + Ƌ Ƌ or = Ƌ Ƌ + Ƌ Ƌ ......(2) This equation is weighted total derivative equation or weighted total acceleration for the functionof= (, )where is weighting coefficient. The method of calculating weighting coeffecient will be formulated using Taylor series (1). The second derivative term can be omitted if, | 2 2 Ƌ 2 Ƌ 2 + Ƌ 2 ƋƋ + 2 2 2 Ƌ 2 Ƌ 2 Ƌ Ƌ +Ƌ Ƌ Ƌ |≤ɛ ........(3)