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Simulation for linear oblique waves in 2D
L. Yuliawati, W. S. Budhi, and J. M. Tuwankotta
Citation: AIP Conference Proceedings 1677, 030017 (2015);
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Published by the American Institute of Physics
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Simulation For Linear Oblique Waves in 2D
L. Yuliawatia), W.S. Budhi and J.M. Tuwankotta
Department of Mathematics, Institut Teknologi Bandung, Bandung, Indonesia 40132
Corresponding author:
Abstract. Accurate model for uni-directional surface waves in 2D has been introduced by van Groesen et al [1, 2],
known as AB2-equation. This equation has been improved to generate a wave by Lie et al [3] based on the inhomogenous
boundary condition from MARIN experiments. In this note we will verify the model using the simple harmonic waves as
an inhomogenous boundary condition or an influx. In order to use spectral method to solve the problem, we will use
Duhamel principle to replace the inhomogenoues boundary condition with the homogenenous boundary condition and
external force. A preliminary study of this problem gives a promising result that this approach serves as a good model for
simulation for linear oblique waves in 2D.
Keywords : AB2 equation, linear wave, oblique wave
PACS: 04.30 Nk,02.30.Jr
The results of a recent study in 2010, van Groesen [1, 2] found the new water waves equation, known as
AB2-equation, which describes the nonlinear dispersive waves which travel mainly in one direction. The AB2equation is variational derivation of improved Kadomtsev-Petviashvili equation and valid for the waves on deep
water. We will solve numerically the equation by spectral methods. The advantages of this equation that has the
exact dispersion relation and is exact up to second order of the wave height.
We interest to learn the AB2-equation to investigate the wave generation problem by using this equation. In
AB2-equation, the waves generated on the boundary be considered as an inhomogeneous boundary conditions or an
influx problems. Using spectral methods, it is not easy to perform the problem with inhomogeneous boundary
conditions. However, this problem has been studied by Lie in 2010 [3]. In his study, Lie using Duhamel principle to
convert the inhomogeneous boundary condition into external force on the equation with homogeneous boundary
conditions. In Lie's study, the source function for the wave influxing is derived base on the linear theory. Then, Lie
has done the simulation by taking the data from MARIN as the influx and compared his simulation with the MARIN
Based on research conducted by Lie, we interest to verify the solution of AB2-equation with special profile as
the solution. To understand this equation, we begin the verification by doing the simulation. We take previously the
linear part of AB2-equation and the simple harmonic waves as an influx. After that, we use Lie's model to generate
this oblique waves with an angle. In the simulation, first we take a simple harmonic wave as the influx with the
angle is zero. Secondly, we generate a combination of two simple harmonic wave to see the effect of dispersive
properties. Then, we do the simulation for a simple harmonic waves with the angle are -30 degrees and 30 degrees.
Furthermore, we generate two waves with different angle and we can see the collision of two waves. In addition, we
try to generate a wave that has soliton profile.
The 5th International Conference on Mathematics and Natural Sciences
AIP Conf. Proc. 1677, 030017-1–030017-4; doi: 10.1063/1.4930639
© 2015 AIP Publishing LLC 978-0-7354-1324-5/$30.00
For completeness, in this section we describe the model for the simulation. We consider surface waves on a layer
of irrotational, inviscid and incompressible fluid that propagate in the ( x, y ) direction over the depth h. The wave
elevation is denoted by η ( x, y ) . Hence, we define Ae as the pseudo-differential operator that acts in Fourier space as
multiplication: Ae =ˆ iΩ 2 (k ) with iΩ 2 (k ) = sign (k ⋅ e) g | k | tanh(| k | h) with the wave vector k = (kx, ky) and e is
the direction vector of the wave. The AB2-equation is explicitly given by:
∂ tη = − g A2 ⎢η − ( A2η )2 + A2 (ηA2η ) + (B2η )2 + B2 (ηB2η ) + (γ 2η )2 + γ 2 (ηγ 2η )⎥
with A2 =
, B2 = ∂ x A2 and γ 2 = ∂ y A2 .
Wave Generation for Linear Wave Model
For note that the linear part of the AB2-equation can be used to model the generation of oblique wave in 2D. We
consider the influx along the y -axis below:
= − Aeη
= s( y, t ).
Based on the Duhamel principle [4], the problem is equivalent to the following problem
∂ tη
= − Aeη + S ( x, y, t )
η (0, y, t ) =
with S ( x, y, t ) is the external force relation to the influx s( y, t ) If S ( x, y, t ) is separated source with the form
S ( x, y, t ) = g ( x) ⋅ f ( y, t ) then for a given function g (x) , we can choose the function f ( y, t ) as the inverse Fourier
transform of
∂ tη
η (0, y, t )
( (
( (
1 Vg K k y , ω
f k y ,ω =
2π g K x k y , ω
)) K x (k y ,ω ) s(k ,ω ).
)) K (k y ,ω ) y
Suppose the wave generated along y -axis with the angle θ 0 . If the influx is s( y, t ) = ae
i k y y −ω0t
k y = k 0 sin θ 0 and k 0 is wave number that correspond with ω 0 where ω0 > 0 , then we have the following source
function for any function g (x)
S ( x, y, t ) = g ( x)ae
i k y y −ω0t
) 1 Vg (K (k y , ω ))cosθ .
( )
gˆ k x
The following figures shown a simulation of harmonic waves. In Figure 1, we show the influx wave for a single
sine harmonic wave and a combination of two sine harmonic waves respectively. We performed the simulations of
the generation of sine waves using AB2-equation with the angle θ 0 = 0 . We can see the simulation in Figure 2 and
Figure 3. The effect of the dispersive appears on Figure 3 when the long wave move faster than the other wave.
FIGURE 1. The influx, (a) s ( y, t ) = sin t (b)
s( y, t ) = 0.5(sin t + sin 2t ) .
FIGURE 2. Simulation for s ( y, t ) = sin t with
FIGURE 3. Simulation for
θ0 = 0 .
s( y, t ) = 0.5(sin t + sin 2t ) with θ 0 = 0 .
In Figure 4 shows the oblique wave generation for sine wave, left figure with the angle θ 0 = −300 and the right
figure for the angle θ 0 = 300 .
FIGURE 4. Oblique wave generation for sine wave: (a)
θ 0 = −30
, (b)
θ 0 = 300 .
We can see the collision of two waves in the Figure 5. From this figure, the amplification of collision between
two linear waves was only two-fold. This happens because only the linear part of the AB2-equation is taken.
In Figure 6, we can see the simulation of the travelling wave that has KdV-soliton profile that is a square of
secant hyperbolic. We can see in Figure 6 that finally the wave is not soliton. This is happen because the effect of
dispersive properties and the nonlinear part of the AB2-equation is not included. Therefore, the study will be
proceed with taking the whole AB2-equation.
FIGURE 5. The collision two waves (for
FIGURE 6. Simulation for
θ 0 = −300
θ 0 = 300 ).
s( y, t ) = sech 2 (0.2(t − 30)).
The generation of the wave by using AB2-equation is very interesting. Based on the simulation has been done,
we can conclude that interesting things can be obtained if the whole of AB2-equation is included. Therefore, the
next study is to model the wave generation for nonlinear wave especially for tsunami wave.
The author wishes to thank to Prof. E. van Groesen from LabMath-Indonesia for suggesting and discussing the
problem. Moreover the author feel very proud for the opportunity to work in LabMath-Indonesia.
E. van Groesen, and Andonowati, “Variational derivation of KdV-type of models for surface waves,”
Phys.Lett.A 366 195-201 (2007).
E. van Groesen, and L. She Liam, “Variational derivation of improved KP-type of equations,” Phys.Lett.A 374
411-415 (2010).
L. She Liam, D. Adytia, & E. van Groesen, “Embedded wave generation for dispersive surface models,”
Ocean Engineering. (submitted 2013).
E. Zauderer, Partial Differential Equations of Applied Mathematics, John Wiley & Sons, Canada, 1989.
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