- Email: [email protected]
A note on the wave propagation in water of variable depth
Applied Mathematics and Computation 218 (2011) 2294–2299
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
A note on the wave propagation in water of variable depth Hilmi Demiray Department of Mathematics, Isik University, 34980 Sile, Istanbul, Turkey
a r t i c l e
i n f o
Keywords: Solitary waves Water of variable depth Variable coefficient KdV equation
a b s t r a c t In the present work, utilizing the two dimensional equations of an incompressible inviscid fluid and the reductive perturbation method we studied the propagation of weakly nonlinear waves in water of variable depth. For the case of slowly varying depth, the evolution equation is obtained as the variable coefficient Korteweg–de Vries (KdV) equation. Due to the difficulties for the analytical solutions, a numerical technics so called ‘‘the method of integrating factor’’ is used and the evolution equation is solved under a given initial condition and the bottom topography. It is observed the parameters of bottom topography causes to the changes in wave amplitude, wave profile and the wave speed. Ó 2011 Published by Elsevier Inc.
1. Introduction The propagation of solitary waves in water of variable depth has attracted considerable interest since the beginning of 1970, and has become the focus of intensive study following the tsunami in the Indian Ocean in December 2004. A solitary wave of constant depth h0, is a progressive wave of permanent form consisting of a single elevation above the undisturbed 2 free surface, whose amplitude a and the effective length L0, are such that = a/L0 and h0 =L20 are comparatively small quantities. Early studies on the propagation of weakly nonlinear waves in water of variable depth had been cultivated by Grimshaw [1,2], Freeman and Johnson [3], Johnson [4–6]. Later, the reflection of shallow-water solitary waves in channels with decreasing depth by Knickerbocker and Newell [7,8], Sugimoto et al. [9], Liu Philip et al. [10], Jeng and Seymour [11], Li and Jeng [12] and Duin van [13]. Recently, Killen and Johnson [14] further established a model for Korteweg–deVries equation in the cylindrical coordinates. In the present work, utilizing the two dimensional equations of an incompressible inviscid fluid and the reductive perturbation method we studied the propagation of weakly nonlinear waves in water of variable depth. For the case of slowly varying depth, the evolution equation is obtained as the variable coefficient Korteweg–de Vries (KdV) equation. Due to the difficulties for the analytical solutions, a numerical technics so called ‘‘ the method of integrating factor’’ is used and the evolution equation is solved under a given initial condition and the bottom topography. It is observed the parameters of bottom topography causes to the changes in wave amplitude, wave profile and the wave speed. The numerical results indicate that the wave amplitude and the wave speed decrease with the increasing undulation of bottom topography. Moreover, it is observed that, the symmetrical bell shaped solitary wave profile of conventional KdV equation is distorted by the existence of the bottom undulation. 2. Basic equations We consider two dimensional incompressible non-viscous fluid in a constant gravitational field g. The space coordinates are denoted by (x⁄, z⁄) and the corresponding velocity components by (u⁄, w⁄). The gravitational force is assumed to be acting along negative z-axis, Fig. 1. The equations describing the motion of such a fluid are: E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2011 Published by Elsevier Inc. doi:10.1016/j.amc.2011.07.049
H. Demiray / Applied Mathematics and Computation 218 (2011) 2294–2299
2295
Fig. 1. Geometry of the general wave propagation problem.
@u @w þ ¼ 0 ðincompressbility conditionÞ; @x @z @u @u 1 @P @u þ w þ ¼ 0; þu @t @x @z qf @x @w @w 1 @P @w þ w þ þ g ¼ 0; þu @t @x @z qf @z
ð1Þ ð2Þ ð3Þ
where qf is the mass density and P⁄ is the pressure function of the fluid. Assuming that the flow is ir-rotational, the velocity vector can be derived from a scalar potential /⁄ (x⁄, z⁄, t⁄) as
u ¼
@/ ; @x
w ¼
@/ : @z
ð4Þ
Then, the incompressibility condition reduces to
@ 2 / @ 2 / þ 2 ¼ 0; @x2 @z
ð5Þ
and the Euler equation becomes
P P0
qf
@/ 1 ¼ 2 @t
" 2 2 # @/ @/ gz ; þ @x @z
ð6Þ
where P0 is an integration constant and can be considered as the atmospheric pressure. We consider the case of the fluid of height h⁄(x⁄) bounded by a steady atmospheric pressure P0. Let the upper surface of the fluid be described by z⁄ = w⁄. Then, the kinematic boundary condition on this surface reads
@/ @w @/ @w ¼ þ ; @z @t @x @x
on z ¼ w :
ð7Þ
From the condition (6), the dynamical boundary condition on this surface becomes
@/ 1 þ @t 2
" 2 2 # @/ @/ þ gw ¼ 0; þ @x @z
on z ¼ w :
ð8Þ
Finally, the lower boundary is supposed to be rigid. Therefore, at z⁄ = h⁄(x⁄) = h0 + f⁄(x⁄), the normal velocity component must vanish
@/ df @/ ¼ 0; @z dx @x
at z ¼ h0 þ f ðx Þ:
ð9Þ
Here f⁄(x⁄) is the profile function at the bottom of the channel. At this stage it is convenient to introduce the following non-dimensional quantities
^ / ¼ c0 h0 /; t ¼
h0 t; c0
^ w ¼ h0 w;
x ¼ h0 x; z ¼ h0 z; qffiffiffiffiffiffiffiffi f ðx Þ ¼ h0 f ðxÞ; c0 ¼ gh0 ;
ð10Þ
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H. Demiray / Applied Mathematics and Computation 218 (2011) 2294–2299
where c0 is the phase speed of the linearized wave equations. Introducing (10) into the field Eqs. (5), (7), (8), (9), the following non-dimensional equations are obtained
^ @2/ ^ @2/ þ ¼ 0; @x2 @z2 ^ @w ^ @/ ^ @w ^ @/ ^ ¼ þ ; on z ¼ w: @z @t @x @x 2 3 !2 !2 ^ 1 ^ ^ @/ @/ @/ ^ ¼ 0; 4 5þw þ þ @t 2 @x @z ^ df @ / ^ @/ ¼ 0; @z dx @x
ð11Þ ð12Þ ^ on z ¼ w:
ð13Þ
at z ¼ 1 þ f ðxÞ:
ð14Þ
3. Long-wave approximation In this section we consider the long-wave in shallow water approximation to the above equations by applying the reductive perturbation method. For that purpose, we introduce the following stretched coordinates
n ¼ 1=2 ðx tÞ;
s ¼ 3=2 t;
ð15Þ
where is the smallness parameter. For our future purposes we introduce the following new dependent variables:
^ ¼ 1=2 /; /
^ ¼ w: w
ð16Þ
Introducing (16) into the field Eqs. (11)–(14) we obtain
@2/ @2/ þ 2 ¼ 0; 2 @z @n @/ @w @w @/ @w þ 2 ¼ þ on z ¼ w; @z @n @s @n @n 2 2 @/ @/ 1 @/ 1 @/ þ þ þ þ w ¼ 0 on z ¼ w; @n @s 2 @n 2 @z
ð18Þ
@/ @f @/ ¼ 0; @z @n @n
ð20Þ
ð17Þ
ð19Þ
at z ¼ 1 þ f ðxÞ:
Now, we expand the functions / and w into a suitable power series of
as
/ ¼ /0 þ /1 þ 2 /2 þ ; w ¼ w0 þ w1 þ 2 w2 þ :
ð21Þ
Introducing (21) into the Eqs. (17)–(20), considering that f(x) is of the form f(x) = h(n), and setting the coefficients of like powers of equal to zero, we obtain the following sets of differential equations: O (1) equations:
@ 2 /0 ¼ 0; @z2
@/0 ¼ 0; @z z¼0
@/0 ¼ 0; @z z¼1
"
2 # @/0 1 @/0 þ @n 2 @z
þ w0 ¼ 0:
ð22Þ
z¼0
O()equations:
@ 2 /1 @ 2 /0 @/1 @ 2 /0 @w0 þ ¼ 0; þ þ w0 ¼ 0; at z ¼ 0; 2 @z2 @z @z2 @n @n " # 2 @/ @ 2 /0 @/0 1 @/0 @/ @/1 þ þ 0 þ 1 w0 þ w1 ¼ 0; @n @z@n @ s 2 @n @z @z z¼0 " # 2 @/1 @ / ¼ 0: þ hðnÞ 20 @z @z z¼1
ð23Þ
H. Demiray / Applied Mathematics and Computation 218 (2011) 2294–2299
2297
\O(2) equations:
@ 2 /2 @ 2 /1 þ ¼ 0; @z2 @n2 @/2 @ 2 /1 1 2 @ 3 /0 @ 2 /0 @w1 @w0 @/0 @w0 þ w0 ¼ 0; at z ¼ 0; þ w0 þ w1 þ 2 3 @z @z @z @z2 @n @s @n @n 2 @/ @ 2 /1 1 2 @ 3 /0 @ 2 /0 @/1 @ 2 /0 þ w0 2 w0 w0 2 w1 þ @n @n@z 2 @z @n @z@n @ s @z@ s !2 ! 2 2 @/ @/1 @ /0 1 @/1 @ /0 þ 0 þ w0 þ þ w0 2 @z @n @n @z@n @z2 ! @/ @/2 @ 2 /1 1 2 @ 3 /0 @ 2 /0 þ w2 ¼ 0; at z ¼ 0; w þ 0 þ þ w þ w0 1 2 0 @z3 @z @z @z2 @z2 @/2 @ 2 /1 @h @/0 þ hðnÞ ¼ 0 at z ¼ 1: @n @n @z @z2
ð24Þ
3.1. Solution of the field equations From the solution of the set of differential Eq. (22) and the associated boundary conditions we obtain
/0 ¼ uðn; sÞ;
w0 ¼
@u ; @n
ð25Þ
where u(n, s) is an unknown function whose governing equation will be obtained later. To obtain the solution to O() equations, given in (23), we introduce the solution (25) into the differential equation (23) to have
@ 2 /1 @ 2 u @/1 @2u þ ¼ 0; þ 2 ¼ 0; 2 2 @z @z z¼0 @n @n " 2 # @/ @u 1 @u 1þ þ þ w1 ¼ 0; @n @ s 2 @n z¼0
@/1 ¼ 0: @z z¼1
ð26Þ
The solution of the Eq. (26), along with the use of the boundary conditions yields
/1 ¼
1 @2u 2 ðz þ 2zÞ þ u1 ; 2 @n2
w1 ¼
2 @ u1 1 @ u @u ; 2 @n @n @s
ð27Þ
where u1(n, s) is another unknown function whose governing equation will be obtained later. To obtain the solution for O(2) equations, given in (24), we introduce the solutions (25) and (27) into the Eq. (24), to have
@ 2 /2 1 @ 4 u 2 @ 2 u1 ðz þ 2zÞ þ ¼ 0; 4 2 2 @n @z @n2 @/2 @ u @ 2 u @ 2 u1 @u þ 2 ¼ 0; at z ¼ 0; 3 2 @z @n @n2 @n@ s @n !2 @/2 @ u @ 3 u @ u @ u1 1 @ 2 u @ u1 þ þ þ w2 ¼ 0; þ þ 2 @n2 @n @n @n3 @n @n @s @/2 @ 2 u @h @ u hðnÞ 2 ¼ 0; @n @n @z @n
at z ¼ 0
at z ¼ 1:
ð28Þ
The solution of (29) along with the use of the boundary conditions yields
! 1 @4u 4 1 @ 2 u1 2 @ u @ 2 u @ 2 u1 @2u 3 z þ u2 ðn; sÞ; /2 ¼ ðz þ 4z Þ z þ 3 þ2 24 @n4 2 @n2 @n @n2 @n@ s @n2
ð29Þ
where u2(n, s) is another unknown function whose governing equation will be obtained from the solution of higher order equations. The use of the last boundary conditions of (28) yields the following evolution equation
@2u 3 @u @2u 1 @4u 1 @ 2 u 1 @h @ u hðnÞ þ þ ¼ 0; @n@ s 2 @n @n2 6 @n4 2 @n2 2 @n @n
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H. Demiray / Applied Mathematics and Computation 218 (2011) 2294–2299
or, in terms of the function w0, it reads
@w0 3 @w0 1 @ 3 w0 1 @ ½hðnÞw0 ¼ 0: þ w þ 6 @n3 2 @n @ s 2 0 @n
ð30Þ
When the function h(n) is equal to zero, the evolution equation reduces to the conventional KdV equation
@w0 3 @w0 1 @ 3 w0 þ w ¼ 0: þ 6 @n3 @ s 2 0 @n
ð31Þ
This equation has the following progressive wave solution 2
w0 ¼ asech f;
f¼
pffiffiffiffiffiffi a 3a n s ; 2 2
ð32Þ
where a is the amplitude of the solitary wave. To our knowledge, there is no analytical solution in the literature to the Eq. (30). Therefore, in what follows we shall present a numerical solution for the variable coefficient KdV equation. 3.2. Numerical results and discussion In this sub-section we shall try to give a numerical solution to the evolution Eq. (30) under the initial condition w0(n, 0) = a sech2kn for the bottom profile function h(n) = c exp(kn2), where a, k, c and k are some constants. For the numerical analysis we shall use the method of integrating factor. For that purpose we shall apply the Fourier transform to the Eq. (30) on the space variable n to get
3 ik ^ 3 1 w^0 s w0 þ ikF w20 hw0 ¼ 0; 4 2 6
ð33Þ
where F is the Fourier transform defined by
w^0 ðk; sÞ ¼
Z
1
w0 ðn; sÞeikn dn ¼ F½w0 :
ð34Þ
1 ik3 6
Multiplying (33) by e ik3 6
e
s
we obtain
3 1 ik3 ik ik3 w^0 s e 6 s w^0 þ e 6 s ikF w20 hw0 ¼ 0; 4 2 6 3
s
ð35Þ
b as Introducing the new function U 3
b ik6 s w^0 ¼ Ue
ð36Þ
and substitute (36) into the Eq. (35) we get
Fig. 2. The variation of solitary wave profile with bottom topography.
H. Demiray / Applied Mathematics and Computation 218 (2011) 2294–2299
2 1 3 3 1 b ik3 s b s þ ikeik6 s F 3 F 1 Ue b ik6 s U Ue 6 hF ¼ 0: 4 2
2299
ð37Þ
For the numerical solution of the problem, by working in Fourier space we can first discretize the equation given in (37) and then use a fourth-order Runge–Kutta method in time.pTo ffiffiffi be consistent with the exact solution of the conventional KdV equation, we have selected the parameters as a ¼ 1:0; k ¼ 3=2; k ¼ 1:5 and solved the equation for the values of the parameter c, c = 0.0, c = 0.1 and c = 0.5 and the result is depicted in Fig. 2. As is seen from this graph, the numerical result for c = 0.0 is exactly the same with the analytical result given in (32). This gives an idea about the precision of the numerical method we utilized. The numerical results for other values of c indicate that, the wave amplitude decreases with increasing bottom undulation. This heterogeneity also causes to the distortion of the symmetrical solitary wave profile. Finally, as might bo observed from the figure, the bottom undulation also decreases the wave speed. Acknowledgement This work was partially supported by the Turkish Academy of Sciences (TUBA). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
R. Grimshaw, The solitary wave in water of variable depth Part 1, J. Fluid Mech. 42 (1970) 639–656. R. Grimshaw, The solitary wave in water of variable depth Part 2, J. Fluid Mech. 46 (1971) 611–622. N.C. Freeman, R.S. Johnson, Shallow water waves on shear flows, J. Fluid Mech. 42 (1970) 401–409. R.S. Johnson, On the development of a solitary wave moving over an uneven bottom, Proc. Camb. Phil. Soc. 73 (1973) 183–203. R.S. Johnson, Water waves and Korteweg-deVries equations, J. Fluid Mech. 97 (1970) 701–719. R.S. Johnson, Solitary-wave, soliton and shelf evolution over variable depth, J. Fluid Mech. 276 (1994) 125–138. A.J. Knickerbocker, A.C. Newell, Shelves and the Korteweg-deVries equation, J. Fluid Mech. 98 (1980) 803–818. A.J. Knickerbocker, A.C. Newell, Reflection from solitary waves in channels of decreasing depth, J. Fluid Mech. 153 (1985) 1–16. N. Sugimoto, N. Nakajima, T. Kakutani, Edge-layer theory for shallow water waves over a step-reflection and transmission of solitons, J. Phys. Soc. Japan 56 (1987) 1717–1730. 0.3cm. L.-F. Liu Philip, L. Patrick, E.S. Costas, Analytical solution for forced long waves on sloping beach, J. Fluid Mech. (2003) 478:101–109. D.- S. Jeng, B.R. Seymour, A new approximation for ocean waves propagating over a beach with variable depth, Ocean Eng. 32 (2005) 1906–1916. J. Li, D.-S. Jeng, Note on the propagation of a shallow water wave of variable depth, Ocean Eng. 34 (2007) 1336–1343. C.A. Duin van, Obliquely incident surface waves in shallow water of slowly varying depth, Acta Mech. 189 (2007) 225–239. S.M. Killen, R.S. Johnson, Propagation of axi-symmetric nonlinear shallow water waves over slowly varying depth, Math. Comp. Simul. 1899 (2001) 463–472.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
A note on the wave propagation in water of variable depth Hilmi Demiray Department of Mathematics, Isik University, 34980 Sile, Istanbul, Turkey
a r t i c l e
i n f o
Keywords: Solitary waves Water of variable depth Variable coefficient KdV equation
a b s t r a c t In the present work, utilizing the two dimensional equations of an incompressible inviscid fluid and the reductive perturbation method we studied the propagation of weakly nonlinear waves in water of variable depth. For the case of slowly varying depth, the evolution equation is obtained as the variable coefficient Korteweg–de Vries (KdV) equation. Due to the difficulties for the analytical solutions, a numerical technics so called ‘‘the method of integrating factor’’ is used and the evolution equation is solved under a given initial condition and the bottom topography. It is observed the parameters of bottom topography causes to the changes in wave amplitude, wave profile and the wave speed. Ó 2011 Published by Elsevier Inc.
1. Introduction The propagation of solitary waves in water of variable depth has attracted considerable interest since the beginning of 1970, and has become the focus of intensive study following the tsunami in the Indian Ocean in December 2004. A solitary wave of constant depth h0, is a progressive wave of permanent form consisting of a single elevation above the undisturbed 2 free surface, whose amplitude a and the effective length L0, are such that = a/L0 and h0 =L20 are comparatively small quantities. Early studies on the propagation of weakly nonlinear waves in water of variable depth had been cultivated by Grimshaw [1,2], Freeman and Johnson [3], Johnson [4–6]. Later, the reflection of shallow-water solitary waves in channels with decreasing depth by Knickerbocker and Newell [7,8], Sugimoto et al. [9], Liu Philip et al. [10], Jeng and Seymour [11], Li and Jeng [12] and Duin van [13]. Recently, Killen and Johnson [14] further established a model for Korteweg–deVries equation in the cylindrical coordinates. In the present work, utilizing the two dimensional equations of an incompressible inviscid fluid and the reductive perturbation method we studied the propagation of weakly nonlinear waves in water of variable depth. For the case of slowly varying depth, the evolution equation is obtained as the variable coefficient Korteweg–de Vries (KdV) equation. Due to the difficulties for the analytical solutions, a numerical technics so called ‘‘ the method of integrating factor’’ is used and the evolution equation is solved under a given initial condition and the bottom topography. It is observed the parameters of bottom topography causes to the changes in wave amplitude, wave profile and the wave speed. The numerical results indicate that the wave amplitude and the wave speed decrease with the increasing undulation of bottom topography. Moreover, it is observed that, the symmetrical bell shaped solitary wave profile of conventional KdV equation is distorted by the existence of the bottom undulation. 2. Basic equations We consider two dimensional incompressible non-viscous fluid in a constant gravitational field g. The space coordinates are denoted by (x⁄, z⁄) and the corresponding velocity components by (u⁄, w⁄). The gravitational force is assumed to be acting along negative z-axis, Fig. 1. The equations describing the motion of such a fluid are: E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2011 Published by Elsevier Inc. doi:10.1016/j.amc.2011.07.049
H. Demiray / Applied Mathematics and Computation 218 (2011) 2294–2299
2295
Fig. 1. Geometry of the general wave propagation problem.
@u @w þ ¼ 0 ðincompressbility conditionÞ; @x @z @u @u 1 @P @u þ w þ ¼ 0; þu @t @x @z qf @x @w @w 1 @P @w þ w þ þ g ¼ 0; þu @t @x @z qf @z
ð1Þ ð2Þ ð3Þ
where qf is the mass density and P⁄ is the pressure function of the fluid. Assuming that the flow is ir-rotational, the velocity vector can be derived from a scalar potential /⁄ (x⁄, z⁄, t⁄) as
u ¼
@/ ; @x
w ¼
@/ : @z
ð4Þ
Then, the incompressibility condition reduces to
@ 2 / @ 2 / þ 2 ¼ 0; @x2 @z
ð5Þ
and the Euler equation becomes
P P0
qf
@/ 1 ¼ 2 @t
" 2 2 # @/ @/ gz ; þ @x @z
ð6Þ
where P0 is an integration constant and can be considered as the atmospheric pressure. We consider the case of the fluid of height h⁄(x⁄) bounded by a steady atmospheric pressure P0. Let the upper surface of the fluid be described by z⁄ = w⁄. Then, the kinematic boundary condition on this surface reads
@/ @w @/ @w ¼ þ ; @z @t @x @x
on z ¼ w :
ð7Þ
From the condition (6), the dynamical boundary condition on this surface becomes
@/ 1 þ @t 2
" 2 2 # @/ @/ þ gw ¼ 0; þ @x @z
on z ¼ w :
ð8Þ
Finally, the lower boundary is supposed to be rigid. Therefore, at z⁄ = h⁄(x⁄) = h0 + f⁄(x⁄), the normal velocity component must vanish
@/ df @/ ¼ 0; @z dx @x
at z ¼ h0 þ f ðx Þ:
ð9Þ
Here f⁄(x⁄) is the profile function at the bottom of the channel. At this stage it is convenient to introduce the following non-dimensional quantities
^ / ¼ c0 h0 /; t ¼
h0 t; c0
^ w ¼ h0 w;
x ¼ h0 x; z ¼ h0 z; qffiffiffiffiffiffiffiffi f ðx Þ ¼ h0 f ðxÞ; c0 ¼ gh0 ;
ð10Þ
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H. Demiray / Applied Mathematics and Computation 218 (2011) 2294–2299
where c0 is the phase speed of the linearized wave equations. Introducing (10) into the field Eqs. (5), (7), (8), (9), the following non-dimensional equations are obtained
^ @2/ ^ @2/ þ ¼ 0; @x2 @z2 ^ @w ^ @/ ^ @w ^ @/ ^ ¼ þ ; on z ¼ w: @z @t @x @x 2 3 !2 !2 ^ 1 ^ ^ @/ @/ @/ ^ ¼ 0; 4 5þw þ þ @t 2 @x @z ^ df @ / ^ @/ ¼ 0; @z dx @x
ð11Þ ð12Þ ^ on z ¼ w:
ð13Þ
at z ¼ 1 þ f ðxÞ:
ð14Þ
3. Long-wave approximation In this section we consider the long-wave in shallow water approximation to the above equations by applying the reductive perturbation method. For that purpose, we introduce the following stretched coordinates
n ¼ 1=2 ðx tÞ;
s ¼ 3=2 t;
ð15Þ
where is the smallness parameter. For our future purposes we introduce the following new dependent variables:
^ ¼ 1=2 /; /
^ ¼ w: w
ð16Þ
Introducing (16) into the field Eqs. (11)–(14) we obtain
@2/ @2/ þ 2 ¼ 0; 2 @z @n @/ @w @w @/ @w þ 2 ¼ þ on z ¼ w; @z @n @s @n @n 2 2 @/ @/ 1 @/ 1 @/ þ þ þ þ w ¼ 0 on z ¼ w; @n @s 2 @n 2 @z
ð18Þ
@/ @f @/ ¼ 0; @z @n @n
ð20Þ
ð17Þ
ð19Þ
at z ¼ 1 þ f ðxÞ:
Now, we expand the functions / and w into a suitable power series of
as
/ ¼ /0 þ /1 þ 2 /2 þ ; w ¼ w0 þ w1 þ 2 w2 þ :
ð21Þ
Introducing (21) into the Eqs. (17)–(20), considering that f(x) is of the form f(x) = h(n), and setting the coefficients of like powers of equal to zero, we obtain the following sets of differential equations: O (1) equations:
@ 2 /0 ¼ 0; @z2
@/0 ¼ 0; @z z¼0
@/0 ¼ 0; @z z¼1
"
2 # @/0 1 @/0 þ @n 2 @z
þ w0 ¼ 0:
ð22Þ
z¼0
O()equations:
@ 2 /1 @ 2 /0 @/1 @ 2 /0 @w0 þ ¼ 0; þ þ w0 ¼ 0; at z ¼ 0; 2 @z2 @z @z2 @n @n " # 2 @/ @ 2 /0 @/0 1 @/0 @/ @/1 þ þ 0 þ 1 w0 þ w1 ¼ 0; @n @z@n @ s 2 @n @z @z z¼0 " # 2 @/1 @ / ¼ 0: þ hðnÞ 20 @z @z z¼1
ð23Þ
H. Demiray / Applied Mathematics and Computation 218 (2011) 2294–2299
2297
\O(2) equations:
@ 2 /2 @ 2 /1 þ ¼ 0; @z2 @n2 @/2 @ 2 /1 1 2 @ 3 /0 @ 2 /0 @w1 @w0 @/0 @w0 þ w0 ¼ 0; at z ¼ 0; þ w0 þ w1 þ 2 3 @z @z @z @z2 @n @s @n @n 2 @/ @ 2 /1 1 2 @ 3 /0 @ 2 /0 @/1 @ 2 /0 þ w0 2 w0 w0 2 w1 þ @n @n@z 2 @z @n @z@n @ s @z@ s !2 ! 2 2 @/ @/1 @ /0 1 @/1 @ /0 þ 0 þ w0 þ þ w0 2 @z @n @n @z@n @z2 ! @/ @/2 @ 2 /1 1 2 @ 3 /0 @ 2 /0 þ w2 ¼ 0; at z ¼ 0; w þ 0 þ þ w þ w0 1 2 0 @z3 @z @z @z2 @z2 @/2 @ 2 /1 @h @/0 þ hðnÞ ¼ 0 at z ¼ 1: @n @n @z @z2
ð24Þ
3.1. Solution of the field equations From the solution of the set of differential Eq. (22) and the associated boundary conditions we obtain
/0 ¼ uðn; sÞ;
w0 ¼
@u ; @n
ð25Þ
where u(n, s) is an unknown function whose governing equation will be obtained later. To obtain the solution to O() equations, given in (23), we introduce the solution (25) into the differential equation (23) to have
@ 2 /1 @ 2 u @/1 @2u þ ¼ 0; þ 2 ¼ 0; 2 2 @z @z z¼0 @n @n " 2 # @/ @u 1 @u 1þ þ þ w1 ¼ 0; @n @ s 2 @n z¼0
@/1 ¼ 0: @z z¼1
ð26Þ
The solution of the Eq. (26), along with the use of the boundary conditions yields
/1 ¼
1 @2u 2 ðz þ 2zÞ þ u1 ; 2 @n2
w1 ¼
2 @ u1 1 @ u @u ; 2 @n @n @s
ð27Þ
where u1(n, s) is another unknown function whose governing equation will be obtained later. To obtain the solution for O(2) equations, given in (24), we introduce the solutions (25) and (27) into the Eq. (24), to have
@ 2 /2 1 @ 4 u 2 @ 2 u1 ðz þ 2zÞ þ ¼ 0; 4 2 2 @n @z @n2 @/2 @ u @ 2 u @ 2 u1 @u þ 2 ¼ 0; at z ¼ 0; 3 2 @z @n @n2 @n@ s @n !2 @/2 @ u @ 3 u @ u @ u1 1 @ 2 u @ u1 þ þ þ w2 ¼ 0; þ þ 2 @n2 @n @n @n3 @n @n @s @/2 @ 2 u @h @ u hðnÞ 2 ¼ 0; @n @n @z @n
at z ¼ 0
at z ¼ 1:
ð28Þ
The solution of (29) along with the use of the boundary conditions yields
! 1 @4u 4 1 @ 2 u1 2 @ u @ 2 u @ 2 u1 @2u 3 z þ u2 ðn; sÞ; /2 ¼ ðz þ 4z Þ z þ 3 þ2 24 @n4 2 @n2 @n @n2 @n@ s @n2
ð29Þ
where u2(n, s) is another unknown function whose governing equation will be obtained from the solution of higher order equations. The use of the last boundary conditions of (28) yields the following evolution equation
@2u 3 @u @2u 1 @4u 1 @ 2 u 1 @h @ u hðnÞ þ þ ¼ 0; @n@ s 2 @n @n2 6 @n4 2 @n2 2 @n @n
2298
H. Demiray / Applied Mathematics and Computation 218 (2011) 2294–2299
or, in terms of the function w0, it reads
@w0 3 @w0 1 @ 3 w0 1 @ ½hðnÞw0 ¼ 0: þ w þ 6 @n3 2 @n @ s 2 0 @n
ð30Þ
When the function h(n) is equal to zero, the evolution equation reduces to the conventional KdV equation
@w0 3 @w0 1 @ 3 w0 þ w ¼ 0: þ 6 @n3 @ s 2 0 @n
ð31Þ
This equation has the following progressive wave solution 2
w0 ¼ asech f;
f¼
pffiffiffiffiffiffi a 3a n s ; 2 2
ð32Þ
where a is the amplitude of the solitary wave. To our knowledge, there is no analytical solution in the literature to the Eq. (30). Therefore, in what follows we shall present a numerical solution for the variable coefficient KdV equation. 3.2. Numerical results and discussion In this sub-section we shall try to give a numerical solution to the evolution Eq. (30) under the initial condition w0(n, 0) = a sech2kn for the bottom profile function h(n) = c exp(kn2), where a, k, c and k are some constants. For the numerical analysis we shall use the method of integrating factor. For that purpose we shall apply the Fourier transform to the Eq. (30) on the space variable n to get
3 ik ^ 3 1 w^0 s w0 þ ikF w20 hw0 ¼ 0; 4 2 6
ð33Þ
where F is the Fourier transform defined by
w^0 ðk; sÞ ¼
Z
1
w0 ðn; sÞeikn dn ¼ F½w0 :
ð34Þ
1 ik3 6
Multiplying (33) by e ik3 6
e
s
we obtain
3 1 ik3 ik ik3 w^0 s e 6 s w^0 þ e 6 s ikF w20 hw0 ¼ 0; 4 2 6 3
s
ð35Þ
b as Introducing the new function U 3
b ik6 s w^0 ¼ Ue
ð36Þ
and substitute (36) into the Eq. (35) we get
Fig. 2. The variation of solitary wave profile with bottom topography.
H. Demiray / Applied Mathematics and Computation 218 (2011) 2294–2299
2 1 3 3 1 b ik3 s b s þ ikeik6 s F 3 F 1 Ue b ik6 s U Ue 6 hF ¼ 0: 4 2
2299
ð37Þ
For the numerical solution of the problem, by working in Fourier space we can first discretize the equation given in (37) and then use a fourth-order Runge–Kutta method in time.pTo ffiffiffi be consistent with the exact solution of the conventional KdV equation, we have selected the parameters as a ¼ 1:0; k ¼ 3=2; k ¼ 1:5 and solved the equation for the values of the parameter c, c = 0.0, c = 0.1 and c = 0.5 and the result is depicted in Fig. 2. As is seen from this graph, the numerical result for c = 0.0 is exactly the same with the analytical result given in (32). This gives an idea about the precision of the numerical method we utilized. The numerical results for other values of c indicate that, the wave amplitude decreases with increasing bottom undulation. This heterogeneity also causes to the distortion of the symmetrical solitary wave profile. Finally, as might bo observed from the figure, the bottom undulation also decreases the wave speed. Acknowledgement This work was partially supported by the Turkish Academy of Sciences (TUBA). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
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