EFFECT OF WAVE BREAKER ON AMPLITUDE REDUCTION OF THE INCOMING WAVE

  rd Proceedings of the 3

  IMT-GT Regional Conference on Mathematics, Statistics and Applications Universiti Sains Malaysia

EFFECT OF WAVE BREAKER ON AMPLITUDE REDUCTION

OF THE INCOMING WAVE

  1

  1

  2

  1 Marwan, Said Munzir, Ichsan Setiawan, Rasudin _{1 2} Jurusan Matematika

Jurusan Ilmu Kelautan

FMIPA, Universitas Syiah Kuala, Banda Aceh, Indonesia

Contact person : marwan.ramli@math-usk.org

  Abstract.

  We consider a bar type of wave breaker. In this paper effect of such wave breaker on incoming wave is

presented. The relationship formulation between height and width of the wave breaker with the amplitude of the wave

is derived. The formula is derived for two cases of wave, i.e linear wave and dispersive wave. The formula is found by

using the same analogy as well as grating phenomena in optic waves. The comparison of amplitude reduction of both

waves is also presented.

  Keywords : wave breaker, dispersive wave, amplitude reduction.

  1 Introduction The general characteristic of water wave is mainly influenced by the nonlinear behavior of the water medium.

  This characteristic causes the changes of water wave form as a function of position and time. In the wave travel, due to this characteristic, the wave amplitude is increasing as reported by [1,2,3,5,6,10]. This surely causes certain consequences over the beach condition if the wave travels toward the coastal area [7,8,9]. The damage caused by water wave is related to how far, big, and quick the wave motion when it reach the coast area. The damage is worse when the incoming wave is big and with high speed, like the tsunami wave occurred in December 2004 in Aceh and the area around it. This killing wave has taken 175.000 lives of the victims, while millions of others have to live in temporary shelters. Besides, Aceh coastal area along with its beautiful natural scene is totally damaged by the giant wave. Some references have estimated that the tsunami wave occurred in Aceh at the December 26, 2004, has the amplitude of 9 meters with the speed of 750 km/hr (see [19]). This kind of wave can destroy anything on it way. It is suspected that one of the factors that causes the severe damage by this giant wave is the unavailability of the wave breaker which can reduce the amplitude of the incoming wave at the beach. It can be obviously proven that in the more populated area (more building) the distance reached by the tsunami wave is relatively shorter than that of the empty area.

  This research focus on the problem related to wave breaker design at the bottom of the sea which is directed to reduce the impact of the coming wave along the coastal line. These wave breakers, located at the bottom of the sea, functioned as an instrument to decrease the amplitude of the wave moving above them. This result is important to know, to give development direction and model application in the specific study which connect the real condition of abrasion and dissedimentation in the coastal area. The study here is conducted using the analogy of the grating phenomena in the study of optics. Here, two waves modeled are used: linear wave and dispersion wave. For dispersion wave, the model used is the Boussinesq equation with the exact dispersion relation.

  This paper is organized as follows. In the section 2 the mathematical model used is presented. Monochromatic wave and the construction of wave reflector is presented in the section 3. Then, several graphical illustrations are presented in the section 4. Finally, the paper is ended with concluding remarks.

  2 Mathematical Model

  The mathematical model describing the wave evolution on the surface of sufficiently shallow water can be written as [12]

  ∂ h ∂ h ∂ u

• u h = ∂ t ∂ x ∂ x

  . (1) ∂ u ∂ u ∂ h

• u g = ∂ t ∂ x ∂ x

  This equation is known as shallow water equation (SWE). Here x represent the spatial variable in the horizontal direction and z represent the spatial variable in vertical direction. h ( t x , ) is the water depth and η ( t x , ) is the wave elevation at the position x and the time t . Mean while, u ( t x , ) denote the water particle flow velocity component in the horizontal direction and g is the gravity accelerations.

  Equation (1) is a simple form of the complete equation. After linearization, it gives [12]

  ∂ η ∂ u

• H = ∂ t ∂ x

  , (2)

  ∂ u ∂ η

• g = ∂ t ∂ x

  η x where H is the water depth measured from the bottom of the sea up to the equilibrium point, and ( t , ) represent the height of the water surface measured from the surface of the water at rest. By eliminating u , this equation can be written as

  2

  

2

  ∂ η ∂ η

  2

  = c , (3)

  2

  

2

  ∂ ∂

  t x

  2

  where c = g H . Equation (3) is known as the one dimension wave equation having the

• solution η + ( x , t ) = f ( x − ct ) g ( x ct ) . This solution represent the wave motion in two directions without

  changes in its form and having the velocity c . Such waves are called linear waves as it travels with a constant velocity gH . If the velocity does not only depend on the water depth but also depend on the wave length, the wave is also called dispersive wave. This type of wave can be modeled using the following Boussinesq equation [11]

  2

  2

  ∂ η ∂ = R

  ( η ) , (4)

  2

  2

  ∂ t ∂ x where R denotes the differential operator having the Fourier symbol

  g kH

tanh( )

  R ˆ = . (5) k

  As this equation applied to the wave amplitude reduction as the result of installing a bar at the bottom, then the solution of equation (3) and (4) will be determined on three areas described in the Figure 1

  v (x) ₂ v (x) v (x) _{1 3} Figure 1. The construction of the wave breaker building of the bar form

3 Monochromatic solution and reflector contruction

  Assuming the solution of equation (3) can be written as

  ω − i t

  =

  η ( x , t ) v ( x ) e , (4)

  ∈ ℜ v

  ω , then (x ) have to satisfy

  2

  2 ∂ v ω = + v ( x ) .

  2

  2 ∂ x c

  Hence, the general solution of the equation (3) can be written as

  ω ω

  t i x i ω t ( − ) − ( ) c c

• i x ω

  η + ( x , t ) = Ae Be , (5)

  A B ∈ ℜ

  where , . Equation (5) represents a monochromatic wave traveling in two direction having the constant velocity c. Dispersion relation is a relation that must be satisfied between the wave number k and the angular frequency ω . Dispersion relation of the equation (3) can be obtained by substitution of the monochromatic solution

  ω

  i ( kx − ) ω t

  η + ( x , t ) = e cc to the equation (3) which give = ± c . Since c = gH , then it gives

  k

  ω = ± gH (6)

  k

  Hence, it can be concluded that the solution of the equation (3) is a monochromatic wave traveling at the = velocity c gh if it move in the water having the flat bottom with the depth h .

  Furthermore, from the dispersion relation, it is well understood that the phase velocity depends on the water depth. The installation of a bar as a wave reflector will cause the differences of the incoming wave phase velocity. Assuming that the sea bottom as well as the wave reflector is flat, the water depth can be represented as the stair function. As the angular frequency ω is a fix given input, although the wave passing through the depth changing water, k is also a stair function. The wave number k has piece wise constant value where discontinuities occurred at the points of the depth changes. For this piece wise constant k , the solution on each area having uniform depth of the water can be easily found. Then, adjusting and joining all the solutions in the whole interval will be conducted. Natural phenomena show that water wave surface is always smooth although its bottom is not flat. In the other words, the smooth water surface is denoted by a continuous function. Meanwhile, slippery surface is represented by a continuous first derivative. However, according to the dispersion relation, if the wave reflector is installed, the wave number k is a stair function which is discontinue. Therefore, for the points of discontinuities, the following conditions should be added: lim η ( x , t ) = lim η ( x , t ) and η x t = η x t

  lim ( , ) lim ( , )

• − −

  x → x → x → L x → L

  Similarly, as the travelling water wave surface is always slippery or does not have the slope change due to the changes of water depth, then at the discontinuities, the solution function must have the continuous first derivative with respect to x , i.e.

  = lim η lim η and lim η = lim η .

  

x x x x

• − −

  x x x L x L → → → →

  For a wave reflector having the width of then discontinuities will occur at x = and x = L , as shown in the

  L k

  Figure 1. Wave number (x ) will have a constant value in the interval between,

  

⎧ k , x ∉ [ , L ]

k x =

  ( ) , (7) ⎨

k , x ∈ [ , L ]

  1 ⎩

  ω ω

  k = k = where , and .

  1 gh gh

1 Here, it can be assumed that in the area of x < , there is an incoming wave having the amplitude A that travel

  x ∈

  to the right. Since there is a depth changing as the result of wave reflector installation at the position [ L , ] , then part of the waves will be reflected to the left and part of them will be transmitted to the right. This is similar

  x = L

  to the grating phenomena in the study of optics. Furthermore, at the position the depth change is occured again and this mean there is another wave reflected to the left and also transmitted wave traveling to the right. Therefore, with the wave traveling as above, the solution of equation (3) can be determined based on the area given in the Figure 1. The solution is given as

  ik x ik x

−

  ⎧

• &lt;

  Ae re , x ⎪⎪ ik x ik x

  

−

  1 = &lt; &lt; v ( x ) ae be , x L , (8)

• 1

  ⎨ ⎪ ik ( x − L )

t e , x &gt; L

  ⎪ r ⎩

  where

  A x &lt;

  : Incoming wave amplitude traveling to the right for

  r : Reflected wave amplitude traveling to the left for x &lt; a : Wave amplitude traveling to the right for x ∈

  [ L , ] b x ∈

  : Wave amplitude traveling to the left for [ L , ]

  A

  : Transmitted wave amplitude traveling to the right for x &gt; L

  r x ∉ h : Water depth when [ L , ] x ∈ h : Water depth when [ L , ] .

1 If the incoming wave amplitude is given, coefficient a , b , r , A can be determined through the condition at

  r x = x = L

  and . With the requirement that the solution and its first derivative must be continue, the condition

  =

  at x gives

  ⎛

  1

1 ⎞ ⎛ A ⎞ ⎛

  1 1 ⎞ ⎛ a ⎞ = ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ − − k k r k k b . (9)

  ⎝ ⎠ ⎝ ⎠ ⎝

  1 1 ⎠ ⎝ ⎠

  In a simiar way, using the condition at x = L gives

  ⎛ E 1 / E ⎞ ⎛ a ⎞ ⎛ 1 ⎞ = A (10) r

  ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ − k

k E k / E b

  ⎝

  1 1 ⎠ ⎝ ⎠ ⎝ ⎠ ik L

  1 where E = e .

  According to the equation (9) and (10), it can be obviously seen that there are four equations with four unknown, i.e. a , b , r and A . Using the elimination method, it gives

  r

  

2

  2

  −

  S D A = A , (11) r i δ i δ

  2 −

  2

  −

  

S e D e

  where δ = k L , S = + k h k h and D = k h − k h . Furthermore, for

  1

  1

  1

  1

  1

  π δ = m π + ,

  2 m =

  with ,

  1 , 2 ,... , it yields the smallest transmitted wave amplitude, i.e.

  

2

  2 S − D

A = A

  | | . (10)

  r min

  

2

  2

• S D = k L Since δ , then

  1

  ⎛ +

  1

2 m ⎞

2 π L =

  ⎜ ⎟ , (11)

  

4 k

  ⎝ ⎠

  1 =

  with m ,

  1 , 2 ,... . This mean that there is a combination of width and height of the bar that yield the smallest transmitted wave amplitude.

  Using the same approach, this can be derived also for the case of dispersion wave.

  4 Graphical Illustration

  In this section, several graphical representations based on the result obtained in the previous sec tion is presented.

  Figure 2. Wave elevation at the surface having the bar at the bottom (top) and without the bar (bottom). (Left) for linear wave and (right) for dispersion wave

  A = =

  Figure 2 show the wave traveling with amplitude

1 .

  5 m and frequency ω 2 .

  2 Hz. This wave travels at the h = d =

  water depth of

  3 m. For this case, the height of the bar is chosen as 1 . 75 m and the width is chosen as L =

  2 m. It shows that the incoming wave (blue) amplitude is reduced after passing the bar (black). This

  occurred as part of the incoming wave is reflected (pink) and part of it is transmitted due to the bar placement (red). This can be compared to the wave elevation on the water surface without the placement of bar at the bottom. Here, it shows that there is no amplitude reduction.

  Furthermore, Figure 3 show the relation between wave amplitude reduction with the height d and the width

  L of bar.

  Figure 3. Relation between wave amplitude reduction with the height (left) and the width (right) of bar. The figure shows that the dependency of wave amplitude reduction over the height and the width of the reflector bar is having a similar pattern. The larger and the higher the reflector bar, the bigger reduction in the wave amplitude. Contrarily, The smaller and the shorter the reflector bar, the smaller reduction in the wave amplitude. In addition, it shows that the calculation using the dispersion model (red) gives larger amplitude reduction compared to using the linear model (blue). This gives indication that in addition to the depth factor, dispersion effect also influences the wave amplitude reduction.

5 Concluding Remarks

  The formula that relates the wave amplitude reduction with the wave breaker placed at the sea bottom has been derived. Based on the study conducted, it show that there is a combination between the height and the width of the wave breaker that give the optimum amplitude reduction. In addition, it has been found that the wave amplitude reduction computed using dispersion model is bigger than using linear model.

  Acknowledgments

  The authors would like to express their grattitude to Dr. Andonowati for her collaboration and cooperation on conducting this research. We would also like to thank Mathematics Department, Syiah Kuala University for the facilities we used in the Modeling and Simulation Lab. Finally, special honour for BRR NAD-Nias for financing this research in the financial year 2007.

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