Water Wave Modeling Using Complete Solution of Laplace Equation

— Analytical solution of Laplace equation using variable separation method, consists of two velocity potentials. However, only one component has been used. This research used both velocity potential equation components. With the potential equation, water wave surface equation and the related wave constants were formulated using kinematic free surface boundary condition and surface momentum equation. The characteristic of water wave surface that was produced was observed, both in deep water and shallow water.


INTRODUCTION
The completion of Laplace equation using variable separation method in Dean (1991), produces two potential velocities, i.e. cosine and sine components. However, the application only exists in the cosine component in formulating various characters of water wave. This research does not discuss the use of the second component in Dean (1991), rather it studies the characteristic of water wave surface if the two velocity potential components are used simultaneously, i.e. water wave surface equation is formulated using a complete velocity potential. The formulation begins by formulating the final form of the two velocity potential components. Then, in each velocity potential, water wave surface and wave amplitude equations were formulated. Using wave amplitude equation and surface momentum equation, equations for wave number and wave constant were formulated. It is obtained that the two velocity potential components have similar wave number , wave constant , but with different water wave surface equations. As the final water wave surface is the superposition or the sum of the two water wave surface equations. The characteristic of the water wave surface equation consists of maximum wave amplitude in a wave period, wave length, correlation between wave amplitude and wave height and other produced water wave surface profile was studied. Where, (3) and (4), the values of constants A, B, C and D should be determined. Equation (3) was performed at flat bottom (Dean (1991)) ( , , ) = ℎ ( ℎ + ) (5) was obtained Similar procedure was performed in(4), ( , , ) = ℎ ( ℎ + ) As has been mentioned, the two velocity potentials have similar wave number . There should have been one wave constant, i.e. = = , but to ensure,a proof will be done in the following chapters. In the previous researches Hutahaean (2019 a, b) formulated equations for wave number and wave constant using kinematic free surface boundary condition (KFSBC)and surface momentum equation. So is the case with this research, KFSBC equation and surface momentum equation will be used to formulate equations for wave number k and wave constant . At the same time, this research is a improvementon the procedure of KFSBC integration against time , in Hutahaean (2019a,b).

III. THE FORMULATION OF WAVE NUMBER
AND WAVE CONSTANT USING .

Water wave surface equation
The first step in formulating equation for wave number and wave constant is the formulation of water wave surface equation to obtain wave amplitude equation. The formulation was performed using KFSBC. KFSBC equation using weighted total acceleration is (Hutahaean (2019 a,b,c)), Where is weighting coefficient with the value of 2.87-3.14 (Hutahaean (2019 c)). = ( , ) is water wave surfaceelevation against still water level ( = 0), is water particle velocity at horizontal-direction at the water surface ( = ), whereas is the water particle velocity at vertical direction at the surface water. Using (5), equations of particles velocity at horizontal and vertical directions were obtained, i.e. Water wave surface equation was obtained by integrating (10) against time . It's visible that (10) is a non-linear function against time which is difficult to complete its integration. However, there are two arguments to make it simple, where the two arguments produce similar conclusion.
The first argument is that the velocity potential equation was obtained using variable separation method, i.e. velocity potential that is regarded to have a form of ( , , ) =  17) and (18), where wave amplitude in (15) is as input or known number.

Equation for and .
The next step is formulating equations for wave number and wave constant . The equation to calculate the two parameters can be obtained using (13) and surface momentum equation Bearing in mind that there are two variables that need to be calculated, then two equations are needed. As the second equation is surface momentum equation, where convective velocity is ignored. , for which is not the same with zero and remembering that in deep water ℎ ( ℎ + ) = 1 Keeping in mind (14), The left side of the equation is constant number, therefore the right side should be constant, maximum value of = 1 is used This equation is an equation to calculate wave number in the deep water. This equation has a maximum wave amplitude value and at the same time is a critical wave steepness for a wave period, i.e. in a large wave amplitude, ( 1 − ) = 0 can occur, or The calculation of (22) can be done if wavelength is already known. In the case that wavelength is not known, the equation for wave amplitude maximumcan be obtained by bearing in mind that (23)  As has been stated that from the two velocity potentials and , there is only one value of wave number , therefore it can be estimated that by using the form of wave number equation that is similar with (21) will be obtained.
As an equation for , surface momentum equation (22) and water wave surface equation (15) were used and were performed at characteristic point.  (26) and (27) that was performed at = to (7), As has been performed in previous section, the right side of equation (28) can be written as, ... (29) Then, it was integrated against time . The two terms of the equation were divided with − ℎ ( ℎ + ) , and keeping in mind that in deep water ℎ ( ℎ + ) = 1 ,

International Journal of Advanced Engineering Research and Science (IJAERS)
[  (21)  Therefore the values of wave freuency and 0 parameter were absorbed in the value of . For the following calculation, = was used. The value of 0 cannot be used too large, e.g. 2.25, where ℎ ( 2.25 ) = 1, but it should take into consideration the characteristic of breaking that was produced, with the best value of 0 = 1.6 − 1.9. Hutahaean (2019b)) obtained that with 0 = 1.65, breaker depth that is in accordance with CERC (1984) was obtained.  (1) shows the result of the calculation of wave characteristic for several wave periods which includes deep water wave amplitude maximum , deep water wave length 0 anddeep water depth ℎ 0 . The wave amplitude looks small but it will produce a large wave height, where the relation of wave height that is twice wave amplitude cannot be used. The calculation was done using the value of

Water wave surface profile
The model was performed using wave period8 sec., wave amplitude 0.95 m, = 2.05and 0 = 1.75. The result of the model can be seen in Fig.1.a., Fig.1.b. and Fig.1.c.  (Table (2)).  wave profile type. The condition is very different from the ones previously known, i.e. Airy's wave type can only be formed in a wave with a very small wave amplitude. One thing that should be noticed is that there is a concavity in in the wave crest. A wave with a sharp wave crest can hardly be seen in a wave in the deep water, it always looks flat. Wave crest in Fig. 1.b can be stated as flat, which is quite in accordance with the one in the nature.  Table (3) and Table ( Table (3) 6.5. Water wave surface profile at breaker point To obtain water wave surface profile at breaker point, the values of , and are needed at breaker point. To obtain the value of the three wave parameters , shoaling and breaking analysis was performed. The shoaling and breaking model used in this research looks similar to the one in Hutahaean (2019 b), the model that was not discussed here. Bearing in mind that the two wave potentials have similar equations for wave amplitude, wave constant and wave number, then the shoaling and breaking model will also be similar to the model in Hutahaean (2019 b) that was formulated using .
a. Water wave surface profile As an example of water wave surface profile at breaker point, a wave with wave period =8 sec., wave amplitude = 0.95 m and bottom slope ℎ = −0.005was used. Water wave surface profile at the breaker point is presented in Fig.  2 a. and Fig.2b.  crest. The presence of two adjacent waves also found in the coastal water. A more vivid example is tsunami wave on the coast or land, consist of two large main wave crests.
In the profile of the breaking wave, it is also visible that there is a wave trough in fornt of wave crest. This also occurs in tsunami, where prior to the coming of the peak of the tsunami, the coastal water recedes first.  As has been stated that the adjustment of breaker height was performed by multiplying wave constant at breaker pointwith a coefficient of 0.477.

VII. CONCLUSION
Both components of velocity potential equation as the solution of Laplace equation have similar wave number and wave constant, so that both can be performed as a unity to model water wave mechanics Water wave surface equation from each velocity potential component has different form, where the total of water wave surface equation is the sum of the two water wave surface equations. However, as has been stated that both have similar wave amplitude value and equation, wave number and wave constant . Both produced similar wave profile. Therefore, both water wave surface equation are actually identical. Wave separation in the shallow water, also occurs in the nature, shows that the two velocity potentials should have been used. In addition, water wave surface resultant have different wave height with each component of water wave surface. This also strengthens that the two velocity potential components of Laplace equation should have been used all simultaneously.