Modified Momentum Euler EquationforWater Wave Modeling

In this research, weighted total acceleration for a function f(x, z, t)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.


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.

Weighted Total Acceleration for the function of = ( , )
The changes in the value of a function in a function = ( , )for a very small and using Taylor series only until the second derivative is, ( + , + ) = ( , ) + Ƌ Ƌ + Ƌ Ƌ Ƌ + 2 2 Ƌ 2 Ƌ 2 + Ƌ 2 Ƌ Ƌ + 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.
The completions of this equation requires a function form of = ( , ). And the following sinusoidal function form will be used, ( , ) = cos cos ....(5) This equation is water wave surface equation of the linear wave theory. The derivative of the function is as follows

+ | ≤ ɛ
The numerator(1 + )is a positive number, then the equation can be written as, If equals (=) relation is used, then Considering that is a positive number, the right side of the equation is a positive number. Therefore, the left side of the equation is also a positive number. The calculation of the value can be done by releasing the sign | |in the left side of the equation, i.e. using equation (5).
The calculation of the value with (5) requires an input . The value of , is obtained from the function = ( ). The approximation of Taylor series for the function is, In order to be able to be used only until the first derivative, then | is obtained . Substitution of (6) to (5) obtains = 3 ...(7) It is obtained that has a constant value, i.e. independent of wave period or the level of accuracy ɛ.   Table (2).
To simplify the writing, the followings are defined (8) to be able to be used with only the first derivate, then The substitution of differential equations in Table (2) to (9) will obtain, ... (15) There are four constants that should be determined, i.e. , , and . Hutahaean (2019b) has shown that the two equations have similar constant value, or in other words there is only one constant value in velocity potential total (12). However, in the next section it will be proven again with another method that (12) has one constant value.
Equation (12)  The constants of , , and will be formulated using (14) and (15), where it will be proven that either using (14) or (15) similar constant will be obtained. The formulation is done by doing kinematic bottom boundary condition on flat bottom, as was done by Dean (1991). a.
Alternative I The constants , , and will be determined using (14) where water particle velocity at the vertical-direction is Substitute Alternative II The constants , , and will be determined using (15), Substitute equations for and to the kinematic bottom boundary condition equation ....19) From(16) and (18) obtained that = = , so it is proven that in (1) there is only one wave constant value ,then (7)  The particle velocity in horizontal-direction is, The particle velocity in vertical-direction is, is done using velocity potential (21), where the particle velocity in horizontal direction is in equation (27), and the particle velocity in vertical-direction (28). From (28) the following is obtained, This equation is integrated against time , Then, it is differentiated against horizontal-axis This equation is a surface momentum equation that will be used in the calculation of and .

V. THE FORMULATION OF AN EQUATION FOR THE CALCULATION OF AND
As has been mentioned in the previous section that the calculation of and is done in the point of characteristic where = . Therefore, (27) is used as the particle velocity in horizontal direction and (28) is particle velocity equation in vertical direction.

The formulation of wave amplitude function
The weighted total acceleration equation (2) The form was selected becauseit has been determined that the velocity potential component that was used is componentIt is defined Substitute wave amplitude function, The equation is divided by cosh (ℎ + 2 ), Wave amplitude equation is written as an equation for , i.e. The value of can be calculated using (49). In the calculations that will be done in this section, the value of = 3.0and = 1.630are used and the calculation is done in the deep water. Deep water depth ℎ 0 is obtained with the following equation

VI. THE FORMULATION OF WATER WAVE
Where is calculated using (52).  The comparison was done by calculating in (58) usingthe wave height which is the result of a calculation using the model, where the input in the model is wave period and wave amplitude calculated using (52), so that the wave height that is obtained is the wave height maximum in the related wave period. Table (5) shows that for = 3, the obtained is almost similar with the that is a wave period to calculate with the model. Whereas in = 2.97102, it can be said that the obtained is equal with . The result of this calculation concludes that the values of , and equations formulated in this research are in line with the result of Wiegel research (1949Wiegel research ( -1964 which is the result of an observation.

VIII. CONCLUSION
If the characteristic of ideal fluid i.e. irrotational flow is done at Euler momentum equation, and the velocity potential as the product of Laplace equation solution is substituted, the hydrodynamic force or convective equation in the horizontal direction becomes zero. This problem can be solved using weighted total acceleration where there is weighting coefficient at the differential term against vertical-axis and the resulted model producewave height that corresponds to Wiegel equation. Another finding that should be noticed is that the value of wave height is not twice the value of wave amplitude.