On the Stokes Nonlinear Waves in 2D
NINO KHATIASHVILI
I. Vekua Institute of Applied Mathematics
Iv. Javakhishvili Tbilisi State University
2., University St., 0186Tbilisi
GEORGIA
Abstract: - The Stokes nonlinear waves associated with the nonlinear problem of a free boundary with peaks
in incompressible heavy fluid are studied in 2D. In the early works of the author by using the conformal
mapping method this problem was reduced to the nonlinear integral equation with the weakly singular kernel.
In this paper one parameter of the mapping is chosen sufficiently small and the equation is linearized. The
approximate solution of the linearized equation is obtained. The profile of the free boundary is plotted by
means of Maple-12.
Key-Words: - Incompressible heavy fluid, Stokes waves, Non-linear integral equation
1 Introduction
In the incompressible heavy fluid nonlinear
peaked waves are originated under certain
conditions[1-5, 8, 9]. These waves are known as
Stokes waves. Here we investigate the
nonlinear integral equation associated with this
phenomena .This equation was obtained by the
author [see 4, 5] by means of the conformal
mapping method from the initial problem which
will be stated below. As three parameters of the
mapping can be chosen arbitrarily, we choose
one parameter sufficiently small. The nonlinear
integral equation is simplified and the
approximate solution is obtained. This solution
represents peaked symmetric Stokes wave. The
profile of the wave is constructed by using
Maple. The approximate solution of this
equation depending on some parameters is
obtained. The profile of symmetric wave is
constructed.
2 Problem Formulation
In the coordinate system
Oxy
of the Euclidian
space
,
2
R
the initial problem is stated as follows
[7]
PROBLEM ST. Find the periodic
curve
)(: xyy =Γ
such that , if
f
is a conformal
mapping of the area
{ }
)(0 xytD <<=
on the
strip
{ }
∞=±∞=<< )(,,0 fconstqq
ψ
, then
the following condition holds
,,)('
2
12constAAgyzf ==+
(1)
where
)(' zf
,
, is a complex potential,
ϕ
is a speed potential,
ψ
is a stream function,
)(' zf
is a complex speed,
A
and
q
are the
definite constants,
g
is a gravity acceleration.
Here we assume, that the bottom of the reservoir is
planar and filled with incompressible heavy fluid,
the wave moves with the constant speed c. We
choose the mobile coordinate system moving with
the wave, with the axis
oy
passing through the
maximum point of the wave and the axis
ox
passing along the bottom. We consider the Problem
ST for the symmetric periodic waves with the period
ω
in case of
,...2,1,0,0)(' ±±==+ nniqf
ω
i.e.
the Stokes waves [1-9]. The case
0)(' ≠zf
was
considered by different authors [1, 2, 3, 7, 9].
PROOF
DOI: 10.37394/232020.2022.2.3
Nino Khatiashvili
E-ISSN: 2732-9941
14
Volume 2, 2022
By means of the conformal mapping method in the
previous works of the author [4,5] Problem ST was
transformed to the following nonlinear integral
equation with the weakly singular kernel
,),()(ln
3
2
)('ln)('
4
3
)(
011
dttKtutztz
g
u
a
ξ
π
ξ
∫
−−
=
(2)
where
[ ]
.,0
,lnln2),( 22
22
2222
2222
a
t
ta
tbab
tbb
tK
∈
−
−
+
−+−
−+−
=
ξ
ξ
ξ
ξ
,),()( 1
11
ηξζζζ
izz +== − is a conformal
mapping of one period
OABC
of the area
D
on
the rectangle
1111 CBAO
of the complex plane
ζ
∫++
−−
=
ζ
ω
ζ
02222
1,
2
))((
1
)( iqdt
tbta
Cz
constCzfz == ),(
1
, with the following
correspondence of points
),0,(
),0,(),0,(),0,(
1
111
aC
aBbAbO
↔
−↔−↔↔
where
ba,
are the definite constants,
a
is
sufficiently small.
Having found the solution of the equation (2) the
profile of Stokes wave will be given by [5]
−)(2
2
13
2
ξ
uA
g
. (3)
Our purpose is to find the approximate solution
of the equation (2) and to construct profile of Stokes
wave.
3 Problem Solution
In [5] it is proved that the function
)(
ξ
u
could
be represented in the form
( )
,)(ln)()()( 22222
0
ξξξξ
−−= aauu (4)
where )(
0
ξ
u is bounded function of the class
),(max
,0)(,)(],,0[
0
00
1
ξ
ξξ
uM
uMumaC
=
≠≤≤
By using (4) the equation (2) can be rewritten in
the form
( )
( )
]
)5(.),()(ln
3
2
)(ln)(ln
3
2
)('ln)('
4
3
)(ln)()(
0
0
22222
1
22222
0
dttKtu
tatatztz
g
aau
a
ξ
π
ξξξ
∫−
−−−−
=−−
We admit, that
2
01
−
M
u
is sufficiently small .
Taking into the account the formula
,1ln 00 −≈ M
u
M
u
the equation (5) takes the form
( )
[
( )
]
.),()(
3
2
)(ln)(ln
3
2
)6(
3
2
ln
3
2
)('ln)('
4
3
)(ln)()(
0
22222
01
22222
0
dttKtu
M
tata
Mtztz
g
aau
a
ξ
π
ξξξ
−−−
−+−−
=−−
∫
We represent the right hand side of (6) in the
form
( )
]
( )
]
,,...,,0
,),()(
3
2
3
2
)(ln)(ln
3
2
ln
3
2
)('ln)('
),()(
3
2
3
2
)(ln)(ln
3
2
ln
3
2
)('ln)('
1010
0
22222
1
0
0
0
22222
1
1
Nnaaaaaaa
dttKtu
M
tataMtztz
dttKtu
M
tataMtztz
nn
a
a
n
i
a
i
i
∈<<<<==
−
+−−−−
=−
+−−−−
+
=
∫
∑
∫
+
ξ
ξ
and use the approximation
;,...,1,0
),,(,
2
)(
1
1
00
ni
aaC
aa
uu
iii
ii
=
∈≡
+
≈
+
+
ξξ
then from (6) we obtain
PROOF
DOI: 10.37394/232020.2022.2.3
Nino Khatiashvili
E-ISSN: 2732-9941
15
Volume 2, 2022
( )
( )
]
.,...,,0
)7(,),(
3
2
)(ln)(ln
3
2
ln
3
2
)('ln)('
4
3
),()('
2
3
)(ln)()(
1010
22222
11
0
1
0
22222
0
1
1
Nnaaaaaaa
dttKtata
Mtztz
g
dttKtzC
M
g
aau
nn
a
a
n
i
a
a
n
ii
i
i
i
i
∈<<<<==
+−−
−−
−
=−−
+
=
=
∫
∑
∫
∑
+
+
ξ
π
ξ
π
ξξξ
Hence, for the definition of ;,...,1,0, niCi=from
(7) we obtain the system of algebraic equations
( )
( )
]
.;,...,1,0,...,,0
,
2
)8(,),(
3
2
)(ln)(ln
3
2
ln
3
2
)('ln)('
4
3
),()('
2
3
)(ln)(
1010
1
22222
11
0
1
0
2
22
2
2
1
1
Nnniaaaaaaa
aa
dttKtata
Mtztz
g
dttKtzC
M
g
aaC
nn
ii
i
i
a
a
n
i
i
a
a
n
ii
iii
i
i
i
i
∈=<<<<==
+
=
+−−
−
−
−
=−−
+
+
=
=
∫
∑
∫
∑
+
+
ξ
ξ
π
ξ
π
ξξ
(8) is the system of algebraic equations with respect
to ;,...,1,0,
niC
i=Having found
i
C
by using (7)
we can construct the graph of (3) by means of
Maple 12 .
Below the graphic of (3) is plotted for the
parameters ;1;1;10;1 3==== −nMab (Fig.1).
Fig. 1. The graph of
−)(2
2
13
2
ξ
uA
g, one period
of the Stokes wave.
NOTE . In the work [4] the Problem ST is reduced
to the nonlinear integral equation with the
Weierstrass kernel. In the work of the author [6] the
solutions of this equation are obtained in the
linearized case.
4 Conclusion
The approximate solution of the nonlinear
integral equation (2) is given by (7), where the
constants
;,...,1,0, niC
i
=
are the solutions of the
system of the algebraic equations (8). The
function given by the formula (7) represents
periodic symmetric Stokes wave with peaks.
References:
[1] TB. Benjamin, PJ.Olver, New Hamiltonian
structure, symmetries and conservation lows
for water waves, Journal of Fluid Mechanics,
Vol.125, 1982, pp. 137-185.
[2] EV. Buldakov, PH.Taylor, New asymptotic
description of nonlinear water waves in
Lagrangian coordinates, Journal of Fluid
Mechanics, Vol.562, 2006, pp. 431-185.
[3] SK. Chakrabarti, Handbook of Offshore
Engineering, Elsevier, 2005.
[4] N.Khatiashvili, On the nonlinear plane
boundary value problem, Reports of VIAM,
Vol.X, No.I, 1995, pp. 46-48.
[5] N.Khatiashvili, On Stokes nonlinear integral
wave equation, Integral Methods in Science
and Engineering (editors B. Bertram, C.
Constanda, A. Struthers), Research Notes in
Mathemathcs Series, CRC , 2000, pp. 200-204.
[6] N.Khatiashvili, On the singular integral
equation with the Weierstrass kernel, Complex
Variables and Elliptic Equations, Vol.53,
No.10, 2008, pp. 915-943.
[7] MA. Lavrent’ev and BV Shabat, The Problems
of Hydrodynamics and their Mathematical
Models, Nauka, Moscow, 1980.
[8] G. Stokes, On the Theory of Ocsillatory
Waves, Transactions of the Cambridge
Philosophical Society, Vol.VIII, No.10, 1847,
pp. 197-229.
[9] GB. Whitman, Linear and Nonlinear Waves,
Wiley-Interscience, Moscow, 1974.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the Creative
Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en_US
PROOF
DOI: 10.37394/232020.2022.2.3
Nino Khatiashvili
E-ISSN: 2732-9941
16
Volume 2, 2022