THE MAXIMUM RUN UP PROPAGATION OF WAVE ON A SLOPING BEACH

  rd Proceedings of the 3

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

THE MAXIMUM RUN UP PROPAGATION OF WAVE

ON A SLOPING BEACH

  1

  2

  1

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

Jurusan Ilmu Kelautan

FMIPA, Universitas Syiah Kuala, Banda Aceh, Indonesia

Contact person : smunzir2001@yahoo.com

  

Abstract. An integrated code for wave propagation (tsunami) caused by bottom motion such as underwater land

slides or tectonic earth quakes toward the beach should contain three essential elements. The first element is the code

that can represent the dynamic motion of the bottom in the source region, the code for the free flow region, and the

run-up code suitable in the coastal and inland region. Most of existing codes can predict well the arrival time of the

waves from the source region to the coast but are not very reliable for predicting wave height near the coastal

region. This is because such codes do not take the effect of dispersion into account; such effect is not negligible in the

generation region, with consequences for the free flow region and for the arrival wave near the coast. For given

bathymetry and topography, the wave height near the coast is important information to further predict the run-up

height and so to compute the flooded area inland. The objective of this paper is to present the essential third element

to a currently developed code for water wave propagation caused by moving bottoms. The code is based on shallow

water equation (SWE model) to include dynamic bottom variations suitable for computation in the source region that

are significant in the deep water of the free flow region. The additional element to the code is the run-up code. The

existing 1-D SWE code will be connected to a run-up code for which the SWE code will provide influx to the last one.

  

Extension to 2-D code will be considered and executed rudimentarily giving direction to further in depth

investigation Keywords : Tsunami waves , shallow water equation , run-up waves, sloping beach.

  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. Abbrasion problem and coastal damage triggered by water wave, which is an integrated part of this research, has been studied by many experts. This problem is a complex problem as it involve many parameters, both hydrodynamic parameters from fluid motion as well as parameters related to material transport in the sea bottom [9,13,14]. However, the study of maximum run-up of a water wave traveled on a sloping beach has not been discussed. It should be emphasized that the abrasion and desedimentasion of beaches is strongly related to the wave run up. In this paper, the wave maximum run up when it travels to the coastal area will be presented. The relation between this run up and the beach slope will also be shown. Understanding this formula is important, to give development direction and model application in the specific study which connect the real condition of abrasion and desedimentation in the coastal area.

  This paper is organized in the following order. In the section 2, the mathematical model describing the wave evolution near the beach is presented. The model solution followed by the maximum run-up is presented in the section 3. Finally, this paper will be ended with concluding remarks.

  2 Mathematical Model

  Water wave evolution as ideal fluid is represented by a complete equation. This equation consists of Laplace equation for velocity potential in the interior domain under the water surface. It complimented by free boundary conditions in the form of kinematics and dynamics boundary condition on the free surface and the kinematics boundary condition on the moving plate. The mathematical model describing the wave evolution on a shallow water surface can be written as [12]

  ∂ = −∇ ∇ h . h φ

  ( ) t

  , (1)

  ⎡

  1 2 ⎤

• ∂ = −∂ gh gb ∇

  φ φ

  t x

  2 ⎢⎣ ⎥⎦

  Where b denote dynamical sea bottom and φ is fluid velocity potential, while h is the water depth. The equation can also be written as

  ∂ h ∂ (( h η + + ) u ) =

  t x _,

  (2)

  u u u g ∂ ( h η ) gb =

• ∂ ∂ + +

  t x x x

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

  Here, the SWE is solved using the condition occurred in the Burger equation. This is done to explain the problem of wave breaking phenomena. The solution of the equation (1) is obtained by initially transforming the equation into Burger equation which can be written as

  ∂ u ∂ u = + u . (3) ∂ t ∂ x

  One of the solutions of the equation (3) can be written as (4)

  

u = ( x , t ) = u ( x − u ( x , t ) t )

  With the boundary condition

  u ( x ) = u ( x , ) . (5)

  This solution represents the event of the wave breaking due to nonlinearity effect. The breaking of the wave is an important information for the wave run up.

3 The solution of the model and run up wave

3.1 Case of b ( x ) =

  The equation (1) for b = can be written as ∂

  ∂ h ( hu ) = +

  ∂ t ∂ x . (6)

  ∂ u ∂ u ∂ h

• u g = ∂ t ∂ x ∂ x

2 Then, using the transformations h = ( h ) , a = gh and c =

  2 gh , give

  ∂ ( u ± c ) ∂ ( u ± c ) ± + ( u a ) = (7) ∂ t ∂ x dx

• This mean that u c has a constant value when = u ± a , or

  dt ∂ ( u ± c ) =

  (8)

  ∂ t

  d x

  when = u ± c . For the positive sign (+), equation (7) can also be written as

  dt

2 a ) ∂ ( u

+ + ∂ ( u 2 a )

• = (9)

  ( u a )

  ∂ t ∂ x

  or

  ∂ K ∂ K

• ( u a ) = , (11) ∂ t ∂ x
• with K = u 2 a . Mean while, for the negative sign (-) ∂ ( u − c ) ∂ ( u − c( u − a ) = (12) ∂ t ∂ x

  or

  

∂ − ∂ −

( K

4 a ) (K

4 a )

• + ( K −

  3 a ) = (13)

  ∂ t ∂ x

• Using the Riemann Invariant u c = u 2 a = K = constant, which is derived on the initial condition

  u ( x ) = u ( x , ) and h ( x ) = h ( x , ) , giving q = u − a = ( K − 2 a ) − a = K − 3 a (14)

  and

  u − c = u −

2 a = ( K −

2 a ) − 2 a = K − 4 a . (15)

  Hence, it gives

  

∂ q ∂ q

• q = , (16) ∂ ∂

  

t x

  which is a Burger equation. The solution of this equation is

  q ( x , t ) = q ( x − q ( x , t ), t ) , (17) With the initial condition q ( x ) = q ( x , ) .

  3.2 b ( x ) = x . Case of h max

  θ=45 x smax x

  Figure 1 Topography for the case of b ( x ) = x For b ( x ) = x , the corresponding form of equation (1) is

  ∂ h ∂ ( hu )

• = ∂ ∂

  t x

  . (18)

  ∂ u ∂ u ∂ h

• u − + g =

  1 ∂ ∂ ∂ t x x

  1 t

  2 + = ξ = + It can be simplified using the transformations h H , x t , ε − x = , τ = t , A = a and U = u t .

  2

  2 Using the transformations, the equation (18) is transformed into the new shallow water equation which is written

  as

  ∂ H ∂ ( HU ) = + ∂ ∂ t ξ

  . (19)

  ∂ U ∂ U ∂ H

• U g = ∂ ∂ ∂

  τ ξ ξ It can be observed that the structure of the equation (19) is similar to that of the equation (6). Then, it can be

3 A , A = H , U ( ξ , τ ) = K −

  2 A =

  2 A together with l

  2 A and K = U + further transformed using q ( ξ , τ ) = k −

  the initial condition for q ( ξ , τ ) in the form of

  ⎧ − A ; ξ − ξ < − A τ

l l

  ⎪

  ξ − ξ

  ⎪ q ξ , = ; − A τ ≤ ξ − ξ ≤

  2 A τ (20) ( )

  ⎨ l l

  τ

  ⎪ ⎪

2 A ; ξ − ξ >

  2 A τ

l l

  ⎩

  Hence, it gives

• 2 A

  2 q

  2 1 ⎛ ξ − ξ ⎞

  l

  = . (21) = + = +

  U A q ⎜ A ⎟ ( l ) l

  3

  3

  3 τ

  ⎝ ⎠ Consequently, it yields

  2

  2

  2

  1 ⎛ ⎞

• x t − x ⎛ ⎞ ⎛ − ξ ξ ⎞

  1

  

1

⎜ 2 ⎟ h ( ξ , τ ) =

2 A − =

  2 A − , (22) l l

  ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ 9 τ 9 ⎜⎜ t ⎟⎟ ⎝ ⎠

  ⎝ ⎠ ⎝ ⎠

  or

  2

  1

  1

  2

  2

( ( ) )

  h ( , ) =

2 A t − x t − x = x − x . (23)

• +

  1

  ξ τ

  l ( s )

  2

  2

  2

  9

  9 t t

3.3 Case of b ( x ) = α x

  h max θ x x smax

  = = Figure 2. Topography for the case of b ( x ) α x where α tan θ .

  For this case, the equation (1) which correspond to the case is ∂ h ∂ ( hu )

  = + ∂ ∂

  t x

  . (24) ∂ u ∂ u ∂ h

• u g = − g α ∂ ∂ ∂

  t x x L

  = = = = = Then, the transformations that can be used are defined as h α L h ' , u u u ' , x L x ' , t T t ' , T

  g α

  = and u gL α . Using these transformations, the equation (24) can also be written as ∂ h ' ∂ ( h ' u ' )

  = + ∂ ∂

  t ' x '

  . (25) ∂ u ' ∂ u ' ∂ h '

• u ' = − +

  1 ∂ ∂ ∂

  t ' x ' x '

  It can be observed that the equation (25) is similar to the equation (18). Using the same method as was used the equation (18), it gives

  1

  2

  = −

  h x , t x x , (26)

( ) ( )

s

  

2

  9 gt with

  1

  x = x

• 2

  2 gh t − g α t . (27)

  s l

  2 This equation represent the distance reached by the wave on the beach having the slope of α . Here, x represent the coastal line and h is the wave amplitude before reaching the coastal line. Form the equation (27),

  l

  the formula for maximum to reach maximum run up of the wave is obtained. This formula can be written as 2 h

  l

  =

  t . (28)

max

g

  α Hence, the maximum run up of the wave coming to the beach having the slope α can also be found. This maximum run up formula is written as

  2 h

  l

• = . (29)

  x x

max

  α It can be observed that the maximum run up as well as the time needed to reach the maximum distance are both proportional to incoming wave amplitude and reverse proportional to the beach slope. This fact can be obviously seen in the Figure 3. The figure shows the relationship between x and t with

  α . It shows that the

  max max

  bigger the value of x and t . Conversely, the smaller the value of α the smaller the value of

  α the

  

max max bigger the value of x and t . This fact fits and confirms the real situation. max max Figure 3. Graph describing the relation between _max

  x

  (left) and _max

  t

  (right) with α

4 Concluding Remarks

  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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