Wave Trajectory Study on the Coast of Lhoknga, Aceh Besar, Indonesia: A Numerical Model Approach.
The coast of Lhoknga is located on the west coast of the tip of Sumatra Island, Indonesia, in Aceh Besar district of Aceh province (Figure 1). This coast is unique because of its geographical location between the two major currents, which are the Indian Ocean and the Malacca Strait current [1,2]. Beside the geographical factors of the sea, this beach is also strongly influenced by the topographic area around the coast . The topography of the Eastern and Southeastern coast of Lhoknga is a mountainous region bordering the sea . The Northern part is a valley and lowland basin, and in the West is the Indian Ocean. The factors mentioned above are features affecting the nature of the wave propagation  in the waters of Lhoknga waters.
The interaction of wave propagation with coastal topography and the profile of bathymetry of Lhoknga (Figure 2) has unique wave dynamic compared to other coastal areas in Aceh. The existence of shoaling is due to beach bathymetry profile toward the coastal area causing refraction and diffraction of wave distributions [5,6] and therefore causes the spread of energy through divergence because of the influence of forces acting on the topography of the coast [7,8].
Hence, the study of wave propagation on the coast of Lhoknga needs to be focused on how the wave propagation and wave trajectory path in this area react to the aspects of coastal geography and the dynamics of hydro-oceanography [9,10].
Shoaling occurs because of the bathymetry effects that influence the wave trajectory characteristics toward shallow water near the beach. The assumption given to describe this shoaling is by defining that wave is constant throughout its propagation, so that the shoaling coefficient may mathematically be described as:
[mathematical expression not reproducible] (1)
[K.sub.s] can be simplified in the function of its depth, as follows
[mathematical expression not reproducible] (2)
[H.sub.1] and [H.sub.0] are wave height and initial wave height
[n.sub.1] and [n.sub.0] are energy fraction and initial energy fraction
[c.sub.1] and [c.sub.0] are phase flow and initial phase flow
[K.sub.s] is shoaling coefficient
k is wave number
d is water depth
Refraction is one of the factors causing the changes of the seafloor topography through the effects of erosion and deposition of coastal sediments [11,12]. Refraction plays a role in influencing energy through divergence on the headland, which affect the forces on offshore structures . However, refraction also helps in describing on how rough the seafloor is, by looking at the phenomenon of the wave front [14,15] (Figure 3).
The equation of the deflection coefficient used in this study is derived from the theory of wave trajectory equations, which can be graphically shown in Figure 4. To describe the evolution of a wave trajectory from the addition of a length along the path of ds or the addition of time where dt = ds/c, uses a kinematic principle defined as [17,18]:
d[theta]/ds = -1 dc/c dn (3)
In Equation (3) [theta] is the wave angle, is the distance, and is the energy fraction
Equation (3) denotes the change of angle along ds that depends on the rapid change of wave propagation in the normal direction, where the components are:
* Tangential direction (offensive direction) dx = cos [theta] ds and dy = sin [theta] ds (4)
* Normal direction (perpendicular direction): dx = sin [theta] dn and dy = -cos [theta] dn (5)
Therefore Equation (3) can be reformulated into:
d[theta]/ds = 1/c ([partial derivative]c/[partial derivative]x sin [theta] - [partial derivative]c/[partial derivative]y cos [theta]) (6)
From the velocity relationship, the change of distance, the change of angle, and the conservation of wave energy flux are formulated as the equations to calculate the refraction coefficient, ie:
[[partial derivative].sup.2][beta]/[partial derivative][t.sup.2] + 1/2 [p.sub.t] [partial derivative][beta]/[partial derivative]t [q.sub.t][beta] = 0 (7)
[p.sub.t] = -[[partial derivative].sub.c]/[[partial derivative].sub.x] cos[theta] + [[partial derivative].sub.c]/[[partial derivative].sub.y] sin [theta] (8)
[q.sub.t] = c([[partial derivative].sup.2]c/[partial derivative][x.sup.2]si[n.sup.2][theta] - [[partial derivative].sup.2]c/[partial derivative]x[partial derivative]y sin 2[theta] + [[partial derivative].sup.2]c/[partial derivative][y.sup.2]si[n.sup.2][theta] (9)
Than the deflection coefficient ([K.sub.r]) can be defined as
[mathematical expression not reproducible] (10)
Where, dx and dy are the distance fraction of x and y direction, t is time, [beta] is the distance between the orthogonals, and [p.sub.t] and [q.sub.t] are wave power conservation coefficient
The numerical method used in this study is based on equations of shoaling and wave refraction, which have been developed by Koutitas  and Ippen , while the input data provided as initial input for the numerical model is obtained from field measurements made during high tide and low tide .
The numerical solution of the wave equations in this study is solved by discretizing the wave equation (6) which also comprises the shoaling and refraction coefficients of (7), (8), and (9). By using this equation the wave height can also be calculated based on the wave equation below .
H = [H.sub.o][K.sub.s][K.sub.r] (11)
In which H and are [H.sub.o] wave height and initial wave height respectively, [K.sub.s] is shoaling coefficient, and [K.sub.r] is refraction coefficient.
Using the central difference approach, the Equation of refraction coefficient (Equation (7)) can be obtained as:
[[beta].sup.n+1] = ([p.sub.t][Delta]t-2)[[beta].sup.n-1]+(4-2[q.sub.t][Delta][t.sup.2])[[beta].sup.n]/(2+[p.sub.t]) (12)
[p.sub.t] = [partial derivative][c.sup.n]/[partial derivative]x cos [[theta].sup.n] + [partial derivative][c.sup.n]/[partial derivative]y sin [[theta].sup.n] (13)
[q.sub.t] = [c.sub.n] ([[partial derivative].sup.2][c.sup.n]/[partial derivative][x.sup.2] [sin.sup.2][[theta].sup.n] - [[partial derivative].sup.2][c.sup.n]/[partial derivative]x[partial derivative]y sin 2[[theta].sup.n] + [[partial derivative].sup.2][c.sup.n]/[partial derivative][y.sup.2] [cos.sup.2][[theta].sup.n] (14)
While the governing equation used in this numerical approach is the wave form Equation (6), namely:
[[theta].sup.n+1] - [[theta].sup.n] = [[Delta].sup.s]/[c.sup.n](sin [[theta].sup.n] [dc.sup.n]/dx - cos [[theta].sup.n] [dc.sup.n]/dy) (15)
[partial derivative]c/[partial derivative]x = [c.sub.i+1,j]-[c.sub.i-1,j]/[Delta][x.sub.i]+[Delta][x.sub.i+1] and [partial derivative]c/[partial derivative]y = [c.sub.i,j+1]-[c.sub.i,j-1]/[Delta][y.sub.i]+[Delta][y.sub.i+1] (16)
Here, the position of the new wave trajectory point is governed by the tangential component of the wave path equation in equation (4) which is discretized as:
[x.sup.n+1] = [x.sup.n] + [Delta]s.cos ([[theta].sup.n+1]+[[theta].sup.n]/2) ds (17)
[y.sup.n+1] = [y.sup.n] + [Delta]s.cos ([[theta].sup.n+1]+[[theta].sup.n]/2) ds (18)
[Delta]s = [Delta]t.[c.sup.n] (19)
with point of [x.sup.n], [y.sup.n], [x.sup.n+1], and [y.sup.n+1]do not have to have a grid value.
To attain the distribution of the wave trajectory multiply (17) and (18) by interpolating the points in a grid (Figure 5). The interpolation formulation used is:
[c.sup.n] = [c.sub.i,j]([xi] - 1)([eta] - 1) - [c.sub.i+1,j]([eta] - 1)[xi] - [c.sub.i+1,j+1][xi][eta] + [c.sub.i,j+1][eta]([xi] - 1) (20)
Where [x.sup.n], [y.sup.n], [x.sup.n+1], and are the positions of x and y at n and n+1 time step respectively, [[beta].sup.n-1], [[beta].sup.n] and [[beta].sup.n+1] are orthogonal coefficients at [beta] at n-1, n, and n+1 time step, [[theta].sup.n] and [[theta].sup.n+1] are wave angle at n and n+1 time step, and i, i + 1, i - 1 and j, j + 1, j - 1 are distance position on grid x and y at distance step, [Delta]x, [Delta]y, [Delta]t, [Delta]s are the grid spaces of x, y, t and s, and [xi] and [eta] are the elevation factor toward x and y grid.
Results and Discussion
The simulation result of wave trajectory path propagation in the coast of Lhoknga can be seen in Figures 6 and 7. These results show that the pattern of wave propagation at high and low tide is similar. This is caused by the angle of the incident wave at the initial condition both at low tide and high tide is equivalent, which is 2400 from the Northwest, based on the measurement result. In the pattern of trajectory wave radiation, the wave path does not show the generated distribution of wave height, but the result of the wave distribution is shown by the contour of shading which is overlaid with the spreading of wave trajectory.
The wave height profile also shows that when the waveform has reached maximum height (when the wave will break), the distribution of wave propagation also have a maximum energy value as well. According to Goncalves et al. , this state is characterized as the relationship of energy and amplitude in the propagation of waves, where high-energy waves can be characterized by the magnitude of the amplitude of the waves. On the other hand, low wave amplitude will result in low energy of wave propagation.
The simulation results of the wave field distribution model are shown in Figures 10 and 11. The model shows that the maximum wave height at high tide condition is 1.72 m and at low tide is 1.31 m. Because of the high and low amplitude of the wave influenced by the topography of the sea-bed and the surrounding geographical profiles, it can be seen that the distribution of the wave field propagation generates a unique waveform and trajectory when it faces certain areas.
The wave height profile also shows that when the waveform has reached maximum height (when the wave will break), the distribution of wave propagation also have a maximum energy value as well. According to Goncalves et al.  this state is characterized as the relationship of energy and amplitude in the propagation of waves, where high-energy waves can be characterized by the magnitude of the amplitude of the waves. On the other hand, low wave amplitude will result low energy of wave propagation.
The results prove that the distribution of wave trajectory propagation in the cape region produces maximum wave heights whereas the bay area indicates the minimum wave height. Therefore, the propagation of wave trajectory path in this region causes divergence and convergence zones. The convergence zone occurs in the cape region that has a greater amount of energy than in the bay area (the divergence zone) (Figures 10 and 11). Zones that have large wave energy has a higher wave amplitude pattern that occurs in the divergence region while low wave amplitude distribution patterns occur in the convergence region.
The numerical model result on this research shows that the profile and pattern of wave trajectory path and propagation both at high and low tide condition are similar because of the angle of the incident wave at initial condition is equivalent. However, the distribution of wave heights at high tide and low has significant differences as it is influenced by the variance in initial incoming waves and the topography at the study area. Furthermore, the propagation of wave trajectory path and wave distribution are influenced by zone of divergent (bay region) and convergent (cape region).
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Setiawan, I. (1*) and Irham, M. (2)
(1,2) Faculty of Marine and Fisheries, Syiah Kuala University, Banda Aceh 23111, INDONESIA
(*) Corresponding author: firstname.lastname@example.org
Note: Discussion is expected before June, 1st 2018, and will be published in the "Civil Engineering Dimension", volume 20, number 2, September 2018.
Received 03 May 2017; revised 26 February 2018; accepted 08 March 2018.
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|Author:||Setiawan, I.; Irham, M.|
|Publication:||Civil Engineering Dimension|
|Date:||Mar 1, 2018|
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