A Continuity Equation For Time Series Water Wave Modeling Formulated Using Weighted Total Acceleration

Continuity equation for wave modeling is still being developed. There are quite a lot of versions of this equation. This research formulates continuity equation in a simple form to simplify its numerical and analytical solution. The formulation of the continuity equation is done by performing mass conservation law in a water column with free surface and by performing weighted total acceleration. Then, the continuity equation is performed along with the surface momentum equation and completed numerically to modeling one-dimensional wave dynamism. The equation is capable of modeling shoalingand breaking. Keywords— Continuity Equation, Weighted total acceleration equation.

Both governing equations are done using numerical method where spatial differential is done using finite difference method, whereas time differential is done using corrector predictor method. is particle velocity in horizontal-direction and is particle velocity in vertical-direction. Weighted total acceleration, was actually formulated for the function = ( , ). However, in this research it is performed at = ( , , ), because the wave being discussed is a wave moving to horizontal-direction and vertical dimension is eliminated with the integration process, so the equation becomes a function of = ( , ). = ( , )is water surface elevation against still water level (Fig. 1). In (1) and (2) there is time coefficient or time scale at time differential i.e. with a value of 2.87-3.14 in Hutahaean (2019a,b). The value of γ is very much is the velocity of horizontal-direction at the surface. ℎis water depth against still water level, = ( , )is the water surface elevation also against water level..For incompressible fluid, in (4) is the same as in (5),

III. CONTINUITY EQUATION
Both sides are divided by , and , For a very small , Substitute (2) to the left side of the equation The integration of the second term of the right side is done and substituted kinematic free surface bondary condition and kinematic bottom boundary condition, The integration of the first term right side is performed with Leibniz integration (Protter (1985)), Integration in the right side in (7) is done using velocity equation from Dean (1991) and the result of integration is expressed as a function of surface horizontal velocity in order to correspond with momentum equation that produces surface velocity . Velocity potential equation as the result of Laplace equation solution (Dean (1991) wave number and angular frequency. Particle velocity in horizontal-direction is , then integration in (7) becomes, Completing the integration will obtain From the wave-number conservation equation (Hutahaean (2019a)), ℎ (ℎ + ) = ℎ 0 (ℎ 0 + 0 ) = 1, where 0 is wave number in deep water, ℎ 0 is deep water depth and 0 is water surface elevation in deep water, can have a value of 0 2 or others, 0 is wave amplitude in deep water. Therefore, the result of the integration becomes, Substitute the result of integration to (7), Equation (10) is a continuity equation that will be used in this researchor water wave surface equation in the form of differential equation. In (10), there is wave number parameter that should be known, and some other characteristics that should also be known, among other is deep water depth 0 , i.e. maximumwater depth if the equation was done in water depth which is bigger than 0 , so the calculation is done using 0 .Next is wave amplitude maximum , i.e. maximum amplitude in a wave period that can be inputted to the model.

The calculation of
and 0 . It's been known that there is a relation between water depth and wave number , then the calculation will be easier if in (10) wave number is substituted with water depth . Whereas the equation for wave number in deep water 0 can be calculated using the following equation, the formulation of an equation outside the scope of this research, will be written in the next paper. (11) 0 is wave amplitude which is an input, 0 is deep water wave number, σ is angular frequency, = 2 , is wave period.   In (17)

Numerical Solution
In this research, water surface equation and momentum equation are done with finite difference method for spatial differential, whereas time differential is done using predictorcorrector method based on Newton-Cote numerical integration (Abramowitz (1972)). Whereas the predictorcorrector method is as follows. As an example water surface equation (15)  In constant depth of 11.0 m In this section the model is done in a channel with a constant water depth of = 11.00 m, with wave period of 8 sec., wave amplitude 0 = 0.794 m, where actually that does not mean that wave height is twice that of wave amplitude, Hutahaean(2019 c).Deep water depth for this wave is 0 = 8.762m. In the case that is bigger than 0 then the calculation of( + )in (15),( 0 + ) is used. The model is done using two boundary conditions, i.e. closed-end boundary condition where horizontal velocity = 0, whereas in the opened-endthe model was given an input, i.e. sinusoidal wave 0 = 0 . The input is done only for 1 time wave period. The result of the execution for 8, 24, and 40 sec. is presented in (Fig.2.). In the execution for 8 sec., the wave profile is still in the form of sinusoidal, but the wave trough elevation is smaller than the elevation. In the execution for 24 sec, the formed wave trough is getting smaller and farther away, similarly with execution for 40 sec, the wave trough is getting smaller and farther away where water ripple is formed and the form of the main wave is a perfect cnoidal wave or more accurately it is called solitary wave. As a conclusion of the model execution in this constant water depth is that in the deep water, the equation used produced perfect cnoidaltype wave or also can be called as solitary wavetype, even though the input of sinusoidal wave, the wave trough part disappears. In a changing depth With the phenomenon of the evolution of sinusoidal wave into cnoidal wave, in the model execution at the an uneven bottom, before the wave enters the water with slopping bottom, the wave is given evolution zone, i.e. in front of the water in the form of water with constant depth.

VI. CONCLUSION
As has been shown that model can produce two main phenomena that occur at the water wave on its way to shallow water, i.e. shoaling and breaking. At the deep water, at a constant depth the profile of perfect cnoidal wave is formed which is also called solitary wave. Behind the main wave, wave ripple is formed which is also known as undular wave. Therefore, it can be said that the equation that was produced in this research can model several phenomena at water wave found in the nature. Further research that needs to be done is to study the phenomenon at the equation by producing analytical solution. Considering the simple form of the equation, the analytical solution of the water wave surface equation can be obtained easily, i.e. using velocity potential equation from Laplace solution equation. By studying analytical solution, it is expected that an explanation will be obtained on the appearance of coefficient 2.5 at the second term of the water wave surface equation. x (m)