Water Wave Scattering by a Dock in Presence of Bottom
Transcript Of Water Wave Scattering by a Dock in Presence of Bottom
American Journal of Fluid Dynamics 2012, 2(4): 5560 DOI: 10.5923/j.ajfd.20120204.04
Water Wave Scattering by a Dock in Presence of Bottom Undulation
U. Basu*, S. De R. Maiti
Department of Applied M athematics, University of Calcutta, 92, A.P.C Road, Kolkata, 700009, India
Abstract The problem of water wave scattering by a semiinfin ite floating dock on the free surface of an ocean with
variable bottom topography is considered using linear theory. By employing a perturbation analysis and using Green's integral theorem, the analytical exp ression for the first order reflection coefficient is obtained in terms of a co mputable integral involving the bottom shape function. Also the zero order reflection coefficient is obtained by using residue calculus method. The first order reflection coefficient is computed numerically by considering some special types of bottom shape function and is depicted graphically against the wave number in a number of figures.
Keywords Wave Scattering, Surface Discontinuity, Bottom Undulation, Shape Function, Perturbation Technique,
Green's Integral Theorem, Reflection Coefficient
1. Introduction
The p rob lem o f wat er wav e scatt ering by o bstacles situated at the bottom of water of finite depth having a free surface have been investigated with in the framewo rk of linearized theory during last few decades. The problem of propagation of long wave along free surface over a sudden change in depth was discussed by Lamb[1], Kreisel[2] who were the earliest contributors in this context. They emp loyed the conformal transformat ion technique in the mathematical analys is. Dav ies [3] so lved th e p rob lem of reflect ion of normally incident wave by a patch of sinusoidal undulation on the seabed in a finite region using Fourier transformation technique. Mei[4] considered the problem of wave scattering by period ic sandbars at the ocean botto m. Man dal and Basu[5] ext ended the prob lem fo r an ob liquely incident surface wavetrain in the presence of surface tension at the free surface and employed a perturbation analysis directly to the govern ing partial d ifferent ial equat ion and boundary condition to obtain reflection and transmission coefficients up to the first order in terms o f integrals involving bottom shape fu nct ion d escrib ing th e geo met ry o f th e botto m topography. Martha and Bora[6] investigated water wave scattering by sma ll deformation of the ocean bo ttom in case o f ob lique incid ence o f inco min g wav e. Th ey o bt ain reflection and t ransmission coefficients analyt ically using Green's integral theorem and the nu merical results were depicted g raph ically . M andal and Basu[7], M andal and
* Corresponding author: [email protected] (U. Basu) Published online at http://journal.sapub.org/ajfd Copyright Β© 2012 Scientific & Academic Publishing. All Rights Reserved
Maiti[8] studied water wave diffraction by a s mall elevation of the bottom of an ocean with an ice cover modelled as a thin elastic plate. Mohapatra and Bora[9] investigated the problem of oblique wave scattering over a bottomundulation in the case of ice covered two layered flu id. They obtained the reflect ion and transmission coefficient by using perturbation technique.
Another class of water wave scattering problems involving a discontinuity in the surface boundary condition was considered in the literature where surface waves are propagating along an inertial surface e.g. broken ice floating mat etc. Weitz and Keller[10] considered a scattering problem involving a discontinuity in the surface boundary condition arising due to the presence of a broken ice on the half portion of the surface , the other half being free. They emp loyed the WinnerHopf technique in the mathematical analysis. Evans and Linton[11] considered a problem of water wave scattering by a discontinuity in the surface boundary condition and determined the exp licit fo rm of the reflection and transmission coefficient by employ ing residue calculus method of complex variable theory. Mandal and De[12] investigated a problem of water wave scattering by a small bottom undulation in presence of discontinu ity in the upper surface boundary condition and obtained the first order correction of the reflect ion and transmission coefficients analytically in terms of the integral involving the bottom shape function using the perturbation technique.
Problems concerning the scattering of water wave by a floating mat or a rigid dock on the free surface have a long history. Heins[13] considered interaction of water wave with a semiinfinite dock in water and used WienerHopf technique for analytical solution of the mathematical problem. Friedrich and Lewy[14] solved the problem by
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U. Basu et al.: Water Wave Scattering by a Dock in Presence of Bottom Undulation
using complex variable theory. Linton[15] solved the scattering of water wave by a dock of finite width in water of fin ite depth by using modified residue calculus technique. This problem has been formulated for the obliquely incident waves and the case of normal incidence is recovered by taking appropriate limit. Leppington[16] investigated the problem of water wave scattering due to a finite dock and obtained the amplitude of the reflected and transmitted wave train fo r short wave using integral equation technique. The dock problem was also reinvestigated by Chackrabarti, Mandal and Gayen[17] by utilizing a Fourier type analysis, giving rise to the Carleman type singular integral equation over semiinfin ite range.
In the present paper, the problem of water wave scattering due to bottomundulation in the presence of a discontinuity in the upper boundary condition is considered. Here the free surface discontinuity arises due to the presence of a floating dock which is extended infinitely on other side of the ocean. The thickness of the dock is assumed to be very small compared to the water depth and the amplitude of the incoming wave. The reflect ion of inco ming water wave occurs when a progressive wave train propagating from negative infinity is incident upon the dock in the presence of bottom undulation. Hence, it is interesting to study the reflection coefficients in the presence of a small undulation at the bottom. The solution is being obtained by employing a simp lified perturbation technique directly to the governing partial d ifferential equation and the boundary conditions. Use of the perturbation technique produces two boundary value problems up to the first order. The boundary value problem (BVP1) for zero order potential function is concerned with the problem of scattering of normally incident wave by a rigid dock floating along the surface of water of unifo rm fin ite depth. Following Evans and Linton(1994), the BVP1 is being solved by using residue calculus method and the ze ro order reflection coeffic ients are obtained. Now, without solving the second boundary value problem (BVP2), the first order correction to the reflection coeffic ient is eva luated by appropriate use of Green's integral theorem. Analytical expression for the first order reflection coefficient is obtained in terms of the integral involving the shape function of the bottom undulation and the solution of the BVP1. Considering two different types of bottom shape function the first order reflection coefficient is computed numerically. For the shape function in the form of a sinusoidal ripple of finite e xtent, a number of figures for the first order reflect ion coefficient are drawn against the wave number considering different values of the parameters. For an exponential decaying profile of bottom a few figures are also drawn.
2. Formulation of the Problem
We consider the two dimensional motion in an ocean of variable bottom topography in the form of small undulation at the bottom. A rectangular cartesian coordinate system is
chosen in which π¦axis is taken vertically downwards into
the fluid region and the undisturbed free surface corresponds
to π¦ = 0, π₯ β€ 0. A semiinfinite floating dock of negligib le
thickness occupies the region 0 β€ π₯ < β, π¦ = 0 and other
side π₯ < 0, π¦ = 0 is the open free surface. This produces a
discontinuity in the surface boundary condition in the sense
that free surface boundary condition holds for π₯ < 0 and
there is another boundary condition for π₯ > 0. Let a train of
surface water wave be incident fro m negative infinity upon
the semiinfinite dock. The water under the dock is
undisturbed and since the plate extended infin itely along the
positive π₯axis, there is no transmission of the incident wave
field. The flu id is assumed to be inco mpressible and inviscid.
The flow is irrotational and the motion is simp le harmonic in
time π‘ with angular frequency π, it can be described by a velocity potential π(π₯, π¦, π‘) = π π{π(π₯, π¦)πβπππ‘ } , where
π(π₯, π¦) satisfies the two dimensional Lap lace equation: β2π + β2π = 0, πππ‘πππππ’ππππππππ ,(2.1)
βπ₯2 βπ¦ 2
with the free surface boundary condition
πΎπ + βπ = 0 πππ¦ = 0, π₯ < 0,
(2.2)
βπ¦
where πΎ = π2 , π is the accelerat ion due to gravity. The
π
surface boundary condition on the dock region is given by
βπ = 0 πππ¦ = 0, π₯ > 0,
βπ¦
(2.3)
with the edge condition 1 π1/2βπ = π (1) ππ π = {π₯2 + π¦2 }2 β 0,
(2.4)
and the bottom boundary condition
βπ = 0 πππ¦ = π + ππ π₯ ,
βπ
(2.5)
where π is the outward normal to the sea bed. π¦ = π +
ππ(π₯) denotes the bottom of the sea of variable depth. π is a
small nondimensional positive number signify ing the
smallness of the bottom undulation and π(π₯) is a bounded,
continuous function characterizing the shape of the bottom
with the property that π(π₯) β 0 as π₯ β Β± β. This ensures
that far away fro m the undulation the sea bottom is of
uniform finite depth π below the mean free surface. The
incident wave field fro m the direction of πππππ‘ππ£ππππππππ‘π¦
is described by π0(π₯, π¦) = πππ0 π₯π01(π¦),
where
π01(π¦) = π01coshπ0(π β π¦),
π01 =
2 π0 2π0π + sinh2π0 π
and π0 is unique the positive real root of the transcendental equation
πtanhππ = πΎ.
Finally, the far field behavior of the potential function is
described by 0
π π₯, π¦ β π0 π₯, π¦ + π π0 βπ₯, π¦
ππ π₯ β β, ππ π₯ β ββ. (2.6)
Where π is the unknown reflect ion coefficient(co mplex).
Determination of π is of prime concern here.
3. Method of Solution
American Journal of Fluid Dynamics 2012, 2(4): 5560
57
The boundary condition (2.5) can be written as,
π βπ + π βπ = 0,
βπ₯
βπ¦
(3.1)
where (π, π) is the direction cosine of π to the curve
π(π₯, π¦) = π¦ β π β ππ(π₯) = 0.
Using Taylor series expansion of π(π₯, π¦) on π¦ = π +
ππ(π₯) and the equation (2.1), we get the following form:
βπ β π π π π₯ βπ + π π2 = 0 πππ¦ = π.
βπ¦
ππ₯
βπ₯
(3.2)
The form of the approximate boundary condition (3.2)
suggests that π and π have the follo wing perturbational
e xpansion in terms of the small para meter π: π π₯, π¦, π = π0 π₯ ,π¦ + ππ1 π₯, π¦ + π π2 , π π = π 0 + ππ 1 + π π2 .
(3.3) (3.4)
Substituting from (3.3) to (3.4) in the governing PDE (2.1)
and the conditions (2.2) to (2.4), (2.6), (3.2) and equating the coefficients of like powers of π (π0 and π1) on both sides
of the results, we find that, the zero and first order potentials
π0 and π1 satisfies the following two different boundary
value problems. They are named as BVP1 and BVP2
res p ectiv ely .
BVP1:
The function π0(π₯, π¦) satisfies
(i) β2π0 + β2π0 = 0 ππ 0 < π¦ < π, ββ < π₯ < β,
βπ₯2
βπ¦ 2
(ii) πΎπ0 + ββππ¦0 = 0 πππ¦ = 0, π₯ < 0,
(iii) βπ0 = 0 πππ¦ = 0, π₯ > 0,
βπ¦
(iv) βπ0 = 0 πππ¦ = π,
βπ¦
0
ππ π₯ β +β,
(v) π0 π₯, π¦ β π0(π₯, π¦) + π 0π0(βπ₯, π¦) ππ π₯ β ββ.
Where π 0 is the zero o rder reflect ion coefficient.
BVP2:
The function π1(π₯, π¦) satisfies
(i) β2π1 + β2π1 = 0 ππ 0 < π¦ < π, ββ < π₯ < β,
βπ₯2
βπ¦ 2
(ii) πΎπ1 + βπ1 = 0 πππ¦ = 0, π₯ < 0,
βπ¦
(iii) βπ1 = 0 πππ¦ = 0, π₯ > 0,
βπ¦
(iv) βπ1 = π [π(π₯)π0π₯ (π₯, π)] πππ¦ = π,
β π¦ ππ₯
0
ππ π₯ β +β,
(v) π1(π₯, π¦): π 1π0(βπ₯, π¦) ππ π₯ β ββ.
Where π 1 is the first order reflection coefficient.
The BVP1 involves wave scattering by a semi in fin ite rig id
dock floating on the free surface of an ocean of uniform
fin ite depth and is of classical nature. This has been
considered by Weitz and Keller [10] in connection with the
reflection and transmission of water waves by floating ice.
The BVP2 is a rad iation problem in presence of a free
surface discontinuity with variable bottom depth. Here,
Without solving the BVP2, the analytical exp ression for the
first order reflect ion coefficient is determined in terms of an
integral involv ing the bottom shape function π(π₯) and the
solution of the BVP1, namely π0(π₯, π¦) (Its first order partial derivative with respect to π₯).
4. The Zero Order Reflection Coefficient
In this section, the zero order reflect ion coefficient is
obtained by using residue calculus technique as used by Evans and Linton[11] for the problem of wave scattering by
a upper surface discontinuity of somewhat different type in uniform fin ite depth water. The zero order potential function π0(π₯, π¦) of the π΅ππ β 1 can be expanded in terms of orthogonal eigenfunctions for the free surface region( π₯ < 0) and the dock region(π₯ > 0) in the form g iven by
(4.1)
Here π΄π , π΅π (π = 1,2, . . . . ) are unknown constants, ππ1(π¦) , ππ2(π¦)(π = 1,2,3. . . . ) are the orthogonal depth eigenfunctions for the two regions (π₯ < 0 and π₯ > 0) given
by ππ1(π¦) = ππ1cosππ (π β π¦), ππ2(π¦) = cosπ π (π β π¦),
where
ππ1 =
2 ππ
(π = 1,2, . . . . )
2ππ π + sin2ππ π
and ππ , π π (π = 1,2,3, . . . ) are g iven by the following two
equations
ππ tanππ π + πΎ = 0, π π π = (β1)π ππ .
The matching conditions for π0(π₯, π¦) are the continuity of the velocity potential π0(π₯, π¦) and the linear velocity at
x=0:
π0(π₯β, π¦)π₯<0 = π0 (π₯+, π¦)π₯ >0, π0π₯ (π₯β, π¦)π₯ <0 = π0π₯ (π₯+, π¦)π₯ >0.
The orthogonality of the depth eigenfunctions of the two
regions together with the above matching conditions at
π₯ = 0 produces the following two systems of linear
equations
β π
=
1
β
where π΄ =
1,2, . . . . ).
β π=
1
β
π΅π
= π΄πΏ0π ,
π π βππ
(4.2)
ππ = π01coshπ0π π 0 β 1 , (4.3)
ππ βπ π
π π0 +π π π π0βπ π
β
2π π0 1
and ππ = π΄π ππ1cosππ π(π, π =
πΎπ0 cosh π π
The unknown constants π΄π , π΅π (π = 1,2, . . . . . ) can be
computed numerically fro m thesystem o f linear equations
(4.2) and (4.3) after truncation. The zero order reflection
coefficient π 0 can be determined following the procedure
used by Evans and Linton[11].
We consider the integral π½ = β π π§ ππ§, π = 0,1,2, . . . . ,
πΆπ π§βππ
where the function π(π§) has simple poles
(4.4) at π§ =
π 1, π 2, . . . . . π π , simp le zeroes at π§ = π1, π2 , . . . . ππ and π π§ β π( 1 ) as π§ β β. πΆπ β²π are the sequence of circles
π§
with radius π π increases without bound as π β β. These sequence of circles πΆπ π avoids the zeroes of the integrand and all the poles and zeroes of π(π§) are inside of it and πΆπ
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U. Basu et al.: Water Wave Scattering by a Dock in Presence of Bottom Undulation
must not pass through (0,0). Further assuming that
π(π0 ) = β1, the function π(π§) following the aforesaid
properties can be taken as
π π§ = π0
1β π§ 1βπ0
β β ππ π π .
π§ π=1 1β π§ 1βπ0
π π
ππ
(4.5)
The Cauchy's integral formu la and residue theorem gives
π π0 = 1 πΆ βπ π§ ππ§ =
2ππ π π§ βπ0
β π=
1
β(
π
ππ
(
π
(
π§
)

π§
=
π π
)
.
(4.6)
π π βπ0
Using π(π0) = β1, with πΏ denoting the kronecker delta,
we find
πΏ0π =
β π=
1
β(
π
ππ
(
π
(
π§
)

π§
=
π π
)
.
π π βπ0
Now co mparing (4.6) with (4.2), we obtain
(4.7)
π΅π = π΄π ππ (π(π§)π§ = π π ).
Again, we consider the integral
πΌ = β π π§ ππ§, π = 0,1,2, . . . . . . ,
πΆπ π§ +ππ
(4.8)
with the same p roperty of the integrand function π(π§) as
mentioned above in (4.4) . The matching conditions at
π₯ = 0 can be combined to give
β π
=
1
β
π΅π
=β
2π π0 π 0 1
.
π π +π0
πΎπ0 cosh ππ
(4.9)
The Cauchy's residue theorem gives for π = 0 and at
π§ = βπ0
β π=
1
β
π΅π
= π΄π βπ0 .
π π +π0
Co mparing (4.8) and (4.9), we obtain
π΄π βπ0 =
2π π0 π 0 1
.
πΎπ0 cos π π0 π
(4.10) (4.11)
At π§ = βπ0, using π(π§) fro m (4.5), (4.10) gives
1+ππ 1βπ0
π βπ0 = β
β π=
1
β
ππ π0
π π π0
,
1+π π 1βππ
(4.12)
and replacing
π΄ and π(βπ0 ) in (4.11), we obtain
β (1 + ππ )(1 β π0 )
π 0 = β[(1 + πππ0 )(1 β π ππ0 )].
π=1
π π
ππ
Thus
π 0 = π2ππΌ ,
where
πΌ = β β[tanβ1(π0) β tan β1(π0)], (π = 1,2, . . . . . . . ).
π =1
π π
ππ
Thus R0=1, which is expected since all the incident waves
are fu ll reflected by the rigid dock floating on water of
uniform fin ite depth.
5. The First Order Reflection Coefficient
As mentioned earlier, without solving the BVP 2, the first
order correction to the reflection coefficient π 1 can be
obtained by appropriate use of Green's integral theorem to
the functions π0(π₯, π¦) and π1(π₯, π¦) . We consider the
region bounded by the lines
y=0, πβ€π₯β€π ;y=h, πβ€π₯β€π ; π₯=Β±π , 0
(where
πππ π π£πππ¦ πππππ πππ ππ‘ππ£π ππ’ππππ )
and employ Green's integral theorem in the form
πΏ β π0π1π β π1π0π ππ = 0, (5.1)
where π is the outward normal to the line element ππ.The
surface boundary condition satisfied by the potential
functions π0(π₯, π¦) and π1(π₯, π¦) ensures that there is no contribution to the integral along the part (π¦ = 0, βπ β€ π₯ β€
π ). Fu rthermore the water under the dock region is
undisturbed and there is no transmission of incoming wave.
So there is no contribution to the integral along the part
(π₯ = π, 0 β€ π¦ β€ π). The only contribution to the integral
(5.1) arises from the line ( π₯ = βπ, 0 β€ π¦ β€ π ) and the
bottom. Finally making π β β, we obtain 2ππ0π 1 = ββββπ π₯ π02π₯ π₯, π ππ₯.
(5.2)
Thus π 1 can be obtained exp licitly once the shape function π(π₯) is known and the potential function π0(π₯, π¦)
is obtained.
6. Numerical Results and Discussions
The first order correct ion to the reflection coefficientπ 1
can be computed numerically fro m (5.2) once the shape
function π(π₯) is known. In the present section, the values of
π 1 are co mputed for different values of the wave number πΎπ by considering two types of shape function π(π₯)
characterizing the bottom undulations. It may be noted that
for convenience both the bottom shape functions are chosen
symmetric about the point of discontinuity of the surface
boundary condition namely π₯ = 0 here. The expressions for
π 1 are given in the Appendix for such bottom profiles. Fo r the numerical co mputation of π 1, the infinite series in the
appendix are truncated upto desired accuracy. The
convergence of the integral (5.2) for the chosen bottomshape
functions is assured.
Cas e1 :
The first bottom shape funcβtioπnπis chosenπaπs π π₯ = π0sinππ₯ π β€ π₯ β€ π
=0
ππ‘ππππ€ππ π .
The above π(π₯) represents a bottom profile of π
sinusoidal ripples with amplitude π0 and wave number π. The analytical expression for π 1 is given in the Appendix for the above bottom profile. For nu merical co mputation the
values of non dimensional ripple amp litude is taken as π0 = 0.1,0.2 and ripp le wave nu mber ππ = 1 . π 1 is
π
depicted graphically against the wave number πΎπ in
ππππ’ππ β 1. The graph of π 1 is observed to be oscillatory in nature against πΎπ and the oscillation peaks up to a
highest value for a certain frequency. This observation can
be explained as multip le interaction of the incident wave
with d ifferent frequencies and the sinusoidally varying
bottom topography. Occurrence of zeroes of π 1 for certain πΎπ indicates that the sinusoidal bottom topography does not
affect the reflected wave of the incident wave train up to first
order. The peak values of π 1 are found to be increasing with the values of π0. The heights of the oscillations of π 1
π
are gradually dimin ishing along with πΎπ . Th is can be
explained as wave damping due to energy dissipation when
wave propagates over uneven sea bottom and the presence of
the upper surface discontinuity.
American Journal of Fluid Dynamics 2012, 2(4): 5560
59
The ππππ’ππ β 2 is plotted by taking dimensionless ripple wave nu mber ππ = 3,5,7 and fixed amplitude π0 = 0.1. In
π
this case the oscillation also decreases and gradually
vanishes with πΎπ . A lso as the wave nu mber ππ of the
sinusoidal bottom p rofile function increases, number of
oscillations of π 1 decreases with πΎπ . Also the highest peak values of π 1 decrease with the increasing value of ππ.
Figure 3. π 1 for different ππ
Figure 1. π 1 for differentc0/π
Figure 4. π 1 for different π0/π
The ππππ’ππ β 4 depicts π 1 against πΎπ for different dimensionless ripple amplitude π0 = 0.1,0.2 and fixed value
π
of ππ = 0.3. As before, π 1 first increases with πΎπ and then rapidly decreases with πΎπ for πΎπ > 3 . Also the highest peak value of π 1 is found to be increasing with increasing value of the dimensionless ripple amp litude π0.
π
Figure 2. π 1 for different ππ
Cas e2 :
The second shape function is taken as π(π₯) = π0πβππ₯  β β < π₯ < β, π > 0.
This above π(π₯) represents exponentially decaying bottom pro file. The analytical expression of π 1 is given in the Appendix for this shape function. π 1 are depicted against πΎπ in ππππ’ππ β 3 by taking ππ = 0.3,0.5,1 and π0 = 0.1. An important feature of this figure is that for each
π
value of the parameter ππ, π 1 first increases with πΎπ and attains a maximu m peak value and then gradually decreases with πΎπ . This may be attributed due to exponentially
decaying bottom pro file as the bottom undulation vanishes far away where the water is of uniform depth π. A noticeable
feature of this figure is that this maximu m peak value is found to be decreasing with the increasing value of ππ.
7. Conclusions
A few problems involving water wave scattering by surface discontinuity in presence variable bottom topograph y has already been studied in the literature. The present study is concerned with water wave scattering by a semiinfin ite floating dock in presence of a uneven bottom. Study of such type of problem is important for quite some time due to several valuable aspects in research areas in marine science and oceanography.
A simplified perturbation analysis together with appropri ate use of Green's integral theorem is employed to obtain first order correction to the reflection coefficient. Two different type of shape functions are chosen to describe the bottom topography in case of sinusoidal patch and exponentially decaying bottom undulation. The numerical results of the firstorder correction of the reflection
60
U. Basu et al.: Water Wave Scattering by a Dock in Presence of Bottom Undulation
coefficient are being obtained by utilizing the expressions given in the appendix for both sinusoidal patches and exponentially decaying bottom undulation. These res ults are depicted graphically against the wave number of the incident wave in a number of figures. It may be noted that the discontinuity of the upper surface boundary condition may be at the another point and in that case same method can be applied.
ACKNOWLEDGEMENTS
R. Maiti thanks CSIR, New Delh i (India) for provid ing support to prepare this paper.
Appendix
For a sinusoidal bottom topography (π(π₯) = π0π ππππ₯):
π = π0 βπ2(π 1)2
π
β2iπ0π π
(β1)π e π β 1)
1 2iπ0 0 0 π2 β 4π02
ππ 02 +
2iπ0π π
(β1)π e π β 1
π2 β 4π02
+ 2π 0 (β1)π β 1
π
0
β
2
+
β βππ π΄π eππ π₯ ππ1 sinππ₯dπ₯
βππ /π π=1
β
ππ π΄π ππ1π
π π (ππ +iπ0)
+2iπ0π01 π=1 βπ2 + (ππ + iπ0)2 (β1)π e π β 1
β
ππ π΄π ππ1π
π π (ππ βiπ0)
β2iπ0π01π 0 π=1 βπ2 + (ππ β iπ0)2 (β1)π e π β 1
ππ β
2
π
+ β
βπ π π΅π eβπ π π₯ ππ1 sinππ₯dπ₯ .
0
π=1
For an exponential bottom topography (π π₯ = π0πβππ₯):
π = π0 βπ2(π 1)2
1
π 02
2π 0
+
β
1 2iπ0 0 0 2iπ0 + π π β 2iπ0 π
0β
2
+ β βππ π΄π eππ π₯ ππ1 eππ₯ dπ₯
ββ π=1
β
ππ π΄π ππ1
+2iπ0 π01
β π + (π + iπ )
π =1
π
0
β
ππ π΄π ππ1
β 2i π0π01π 0 π =1 βπ + (ππ β iπ0)
ββ
2
+ β
βπ π π΅π eβπ π π₯ ππ1 eβπ π₯ dπ₯ .
0 π=1
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[3] Davies A.G., The reflection of wave energy by undulation on the seabed. Dynamics of Atmospheres and Oceans,6, 1982, 207232.
[4] M ei C.C., Resonant Reflection of Surface Water Waves by Periodic Sandbars. Journal of Fluid Mechanics, 152, 1985, 315335.
[5] M andal B.N., Basu U.,A note on oblique waterwave diffraction by a cylindrical deformation of the bottom in the presence of surface tension. Archive of Mechanics, 42, 1990, 723727.
[6] M artha S.C., Bora S.N., Water wave diffraction by a small deformation of the ocean bottom for oblique incidence. Geophysical and Astrophysical Fluid Dynamics,101, 2007, 6580.
[7] M andal B.N., Basu U., Wave diffraction by a small elevation of the bottom on an ocean with an ice cover. Archive of Applied Mechanics,73, 2004, 812822.
[8] M aiti P, M andal B.N., Water Wave generated by disturbances at an ice cover. International Journal of Mathematics and Mathematical Science,2005, 737746.
[9] M ohapatra S, Bora S.N., Propagation of oblique waves over small bottom undulation in an icecovered twolayer fluid. Geophysical and Astrophysical Fluid Dynamics,103, 2009, 347 374.
[10] Weitz M , Keller J.B., Reflection of water waves from floating ice in water of finite depth. Communications on Pure and Applied Mathematics,3, 1950, 305318.
[11] Evans D.V., Linton CM ., On step approximations for waterwave problems. Journal of Fluid Mechanics,278,1994, 229β244.
[12] M andal B.N.,De S., Surface wave propagation over small undulation at the bottom of an ocean with surface discontinuity. Geophysical and Astrophysical Fluid Dynamics,103, 2009, 1930.
[13] Heins A.E., Water waves over a channel of finite depth with a dock. American Journal of Mathematics,70, 1970, 730748.
[14] Friedrich K.O., Lewy H., The Dock Problem. Communications on Pure and Applied Mathematics, 1, 1948,135148.
[15] Linton C.M ., The finite dock problem. ZAMP,52, 2006,640656.
[16] Leppington F.G., On the radiation of short surface waves by a finite dock. Journal of the Institute of Mathematics and Its Applications,6,1970, 319340.
[17] Chackbarti A, M andal B.N., Gayen R., The dock problem revisited. International Journal of Mathematics and Mathematical Science,21, 2006, 34593470.
Water Wave Scattering by a Dock in Presence of Bottom Undulation
U. Basu*, S. De R. Maiti
Department of Applied M athematics, University of Calcutta, 92, A.P.C Road, Kolkata, 700009, India
Abstract The problem of water wave scattering by a semiinfin ite floating dock on the free surface of an ocean with
variable bottom topography is considered using linear theory. By employing a perturbation analysis and using Green's integral theorem, the analytical exp ression for the first order reflection coefficient is obtained in terms of a co mputable integral involving the bottom shape function. Also the zero order reflection coefficient is obtained by using residue calculus method. The first order reflection coefficient is computed numerically by considering some special types of bottom shape function and is depicted graphically against the wave number in a number of figures.
Keywords Wave Scattering, Surface Discontinuity, Bottom Undulation, Shape Function, Perturbation Technique,
Green's Integral Theorem, Reflection Coefficient
1. Introduction
The p rob lem o f wat er wav e scatt ering by o bstacles situated at the bottom of water of finite depth having a free surface have been investigated with in the framewo rk of linearized theory during last few decades. The problem of propagation of long wave along free surface over a sudden change in depth was discussed by Lamb[1], Kreisel[2] who were the earliest contributors in this context. They emp loyed the conformal transformat ion technique in the mathematical analys is. Dav ies [3] so lved th e p rob lem of reflect ion of normally incident wave by a patch of sinusoidal undulation on the seabed in a finite region using Fourier transformation technique. Mei[4] considered the problem of wave scattering by period ic sandbars at the ocean botto m. Man dal and Basu[5] ext ended the prob lem fo r an ob liquely incident surface wavetrain in the presence of surface tension at the free surface and employed a perturbation analysis directly to the govern ing partial d ifferent ial equat ion and boundary condition to obtain reflection and transmission coefficients up to the first order in terms o f integrals involving bottom shape fu nct ion d escrib ing th e geo met ry o f th e botto m topography. Martha and Bora[6] investigated water wave scattering by sma ll deformation of the ocean bo ttom in case o f ob lique incid ence o f inco min g wav e. Th ey o bt ain reflection and t ransmission coefficients analyt ically using Green's integral theorem and the nu merical results were depicted g raph ically . M andal and Basu[7], M andal and
* Corresponding author: [email protected] (U. Basu) Published online at http://journal.sapub.org/ajfd Copyright Β© 2012 Scientific & Academic Publishing. All Rights Reserved
Maiti[8] studied water wave diffraction by a s mall elevation of the bottom of an ocean with an ice cover modelled as a thin elastic plate. Mohapatra and Bora[9] investigated the problem of oblique wave scattering over a bottomundulation in the case of ice covered two layered flu id. They obtained the reflect ion and transmission coefficient by using perturbation technique.
Another class of water wave scattering problems involving a discontinuity in the surface boundary condition was considered in the literature where surface waves are propagating along an inertial surface e.g. broken ice floating mat etc. Weitz and Keller[10] considered a scattering problem involving a discontinuity in the surface boundary condition arising due to the presence of a broken ice on the half portion of the surface , the other half being free. They emp loyed the WinnerHopf technique in the mathematical analysis. Evans and Linton[11] considered a problem of water wave scattering by a discontinuity in the surface boundary condition and determined the exp licit fo rm of the reflection and transmission coefficient by employ ing residue calculus method of complex variable theory. Mandal and De[12] investigated a problem of water wave scattering by a small bottom undulation in presence of discontinu ity in the upper surface boundary condition and obtained the first order correction of the reflect ion and transmission coefficients analytically in terms of the integral involving the bottom shape function using the perturbation technique.
Problems concerning the scattering of water wave by a floating mat or a rigid dock on the free surface have a long history. Heins[13] considered interaction of water wave with a semiinfinite dock in water and used WienerHopf technique for analytical solution of the mathematical problem. Friedrich and Lewy[14] solved the problem by
56
U. Basu et al.: Water Wave Scattering by a Dock in Presence of Bottom Undulation
using complex variable theory. Linton[15] solved the scattering of water wave by a dock of finite width in water of fin ite depth by using modified residue calculus technique. This problem has been formulated for the obliquely incident waves and the case of normal incidence is recovered by taking appropriate limit. Leppington[16] investigated the problem of water wave scattering due to a finite dock and obtained the amplitude of the reflected and transmitted wave train fo r short wave using integral equation technique. The dock problem was also reinvestigated by Chackrabarti, Mandal and Gayen[17] by utilizing a Fourier type analysis, giving rise to the Carleman type singular integral equation over semiinfin ite range.
In the present paper, the problem of water wave scattering due to bottomundulation in the presence of a discontinuity in the upper boundary condition is considered. Here the free surface discontinuity arises due to the presence of a floating dock which is extended infinitely on other side of the ocean. The thickness of the dock is assumed to be very small compared to the water depth and the amplitude of the incoming wave. The reflect ion of inco ming water wave occurs when a progressive wave train propagating from negative infinity is incident upon the dock in the presence of bottom undulation. Hence, it is interesting to study the reflection coefficients in the presence of a small undulation at the bottom. The solution is being obtained by employing a simp lified perturbation technique directly to the governing partial d ifferential equation and the boundary conditions. Use of the perturbation technique produces two boundary value problems up to the first order. The boundary value problem (BVP1) for zero order potential function is concerned with the problem of scattering of normally incident wave by a rigid dock floating along the surface of water of unifo rm fin ite depth. Following Evans and Linton(1994), the BVP1 is being solved by using residue calculus method and the ze ro order reflection coeffic ients are obtained. Now, without solving the second boundary value problem (BVP2), the first order correction to the reflection coeffic ient is eva luated by appropriate use of Green's integral theorem. Analytical expression for the first order reflection coefficient is obtained in terms of the integral involving the shape function of the bottom undulation and the solution of the BVP1. Considering two different types of bottom shape function the first order reflection coefficient is computed numerically. For the shape function in the form of a sinusoidal ripple of finite e xtent, a number of figures for the first order reflect ion coefficient are drawn against the wave number considering different values of the parameters. For an exponential decaying profile of bottom a few figures are also drawn.
2. Formulation of the Problem
We consider the two dimensional motion in an ocean of variable bottom topography in the form of small undulation at the bottom. A rectangular cartesian coordinate system is
chosen in which π¦axis is taken vertically downwards into
the fluid region and the undisturbed free surface corresponds
to π¦ = 0, π₯ β€ 0. A semiinfinite floating dock of negligib le
thickness occupies the region 0 β€ π₯ < β, π¦ = 0 and other
side π₯ < 0, π¦ = 0 is the open free surface. This produces a
discontinuity in the surface boundary condition in the sense
that free surface boundary condition holds for π₯ < 0 and
there is another boundary condition for π₯ > 0. Let a train of
surface water wave be incident fro m negative infinity upon
the semiinfinite dock. The water under the dock is
undisturbed and since the plate extended infin itely along the
positive π₯axis, there is no transmission of the incident wave
field. The flu id is assumed to be inco mpressible and inviscid.
The flow is irrotational and the motion is simp le harmonic in
time π‘ with angular frequency π, it can be described by a velocity potential π(π₯, π¦, π‘) = π π{π(π₯, π¦)πβπππ‘ } , where
π(π₯, π¦) satisfies the two dimensional Lap lace equation: β2π + β2π = 0, πππ‘πππππ’ππππππππ ,(2.1)
βπ₯2 βπ¦ 2
with the free surface boundary condition
πΎπ + βπ = 0 πππ¦ = 0, π₯ < 0,
(2.2)
βπ¦
where πΎ = π2 , π is the accelerat ion due to gravity. The
π
surface boundary condition on the dock region is given by
βπ = 0 πππ¦ = 0, π₯ > 0,
βπ¦
(2.3)
with the edge condition 1 π1/2βπ = π (1) ππ π = {π₯2 + π¦2 }2 β 0,
(2.4)
and the bottom boundary condition
βπ = 0 πππ¦ = π + ππ π₯ ,
βπ
(2.5)
where π is the outward normal to the sea bed. π¦ = π +
ππ(π₯) denotes the bottom of the sea of variable depth. π is a
small nondimensional positive number signify ing the
smallness of the bottom undulation and π(π₯) is a bounded,
continuous function characterizing the shape of the bottom
with the property that π(π₯) β 0 as π₯ β Β± β. This ensures
that far away fro m the undulation the sea bottom is of
uniform finite depth π below the mean free surface. The
incident wave field fro m the direction of πππππ‘ππ£ππππππππ‘π¦
is described by π0(π₯, π¦) = πππ0 π₯π01(π¦),
where
π01(π¦) = π01coshπ0(π β π¦),
π01 =
2 π0 2π0π + sinh2π0 π
and π0 is unique the positive real root of the transcendental equation
πtanhππ = πΎ.
Finally, the far field behavior of the potential function is
described by 0
π π₯, π¦ β π0 π₯, π¦ + π π0 βπ₯, π¦
ππ π₯ β β, ππ π₯ β ββ. (2.6)
Where π is the unknown reflect ion coefficient(co mplex).
Determination of π is of prime concern here.
3. Method of Solution
American Journal of Fluid Dynamics 2012, 2(4): 5560
57
The boundary condition (2.5) can be written as,
π βπ + π βπ = 0,
βπ₯
βπ¦
(3.1)
where (π, π) is the direction cosine of π to the curve
π(π₯, π¦) = π¦ β π β ππ(π₯) = 0.
Using Taylor series expansion of π(π₯, π¦) on π¦ = π +
ππ(π₯) and the equation (2.1), we get the following form:
βπ β π π π π₯ βπ + π π2 = 0 πππ¦ = π.
βπ¦
ππ₯
βπ₯
(3.2)
The form of the approximate boundary condition (3.2)
suggests that π and π have the follo wing perturbational
e xpansion in terms of the small para meter π: π π₯, π¦, π = π0 π₯ ,π¦ + ππ1 π₯, π¦ + π π2 , π π = π 0 + ππ 1 + π π2 .
(3.3) (3.4)
Substituting from (3.3) to (3.4) in the governing PDE (2.1)
and the conditions (2.2) to (2.4), (2.6), (3.2) and equating the coefficients of like powers of π (π0 and π1) on both sides
of the results, we find that, the zero and first order potentials
π0 and π1 satisfies the following two different boundary
value problems. They are named as BVP1 and BVP2
res p ectiv ely .
BVP1:
The function π0(π₯, π¦) satisfies
(i) β2π0 + β2π0 = 0 ππ 0 < π¦ < π, ββ < π₯ < β,
βπ₯2
βπ¦ 2
(ii) πΎπ0 + ββππ¦0 = 0 πππ¦ = 0, π₯ < 0,
(iii) βπ0 = 0 πππ¦ = 0, π₯ > 0,
βπ¦
(iv) βπ0 = 0 πππ¦ = π,
βπ¦
0
ππ π₯ β +β,
(v) π0 π₯, π¦ β π0(π₯, π¦) + π 0π0(βπ₯, π¦) ππ π₯ β ββ.
Where π 0 is the zero o rder reflect ion coefficient.
BVP2:
The function π1(π₯, π¦) satisfies
(i) β2π1 + β2π1 = 0 ππ 0 < π¦ < π, ββ < π₯ < β,
βπ₯2
βπ¦ 2
(ii) πΎπ1 + βπ1 = 0 πππ¦ = 0, π₯ < 0,
βπ¦
(iii) βπ1 = 0 πππ¦ = 0, π₯ > 0,
βπ¦
(iv) βπ1 = π [π(π₯)π0π₯ (π₯, π)] πππ¦ = π,
β π¦ ππ₯
0
ππ π₯ β +β,
(v) π1(π₯, π¦): π 1π0(βπ₯, π¦) ππ π₯ β ββ.
Where π 1 is the first order reflection coefficient.
The BVP1 involves wave scattering by a semi in fin ite rig id
dock floating on the free surface of an ocean of uniform
fin ite depth and is of classical nature. This has been
considered by Weitz and Keller [10] in connection with the
reflection and transmission of water waves by floating ice.
The BVP2 is a rad iation problem in presence of a free
surface discontinuity with variable bottom depth. Here,
Without solving the BVP2, the analytical exp ression for the
first order reflect ion coefficient is determined in terms of an
integral involv ing the bottom shape function π(π₯) and the
solution of the BVP1, namely π0(π₯, π¦) (Its first order partial derivative with respect to π₯).
4. The Zero Order Reflection Coefficient
In this section, the zero order reflect ion coefficient is
obtained by using residue calculus technique as used by Evans and Linton[11] for the problem of wave scattering by
a upper surface discontinuity of somewhat different type in uniform fin ite depth water. The zero order potential function π0(π₯, π¦) of the π΅ππ β 1 can be expanded in terms of orthogonal eigenfunctions for the free surface region( π₯ < 0) and the dock region(π₯ > 0) in the form g iven by
(4.1)
Here π΄π , π΅π (π = 1,2, . . . . ) are unknown constants, ππ1(π¦) , ππ2(π¦)(π = 1,2,3. . . . ) are the orthogonal depth eigenfunctions for the two regions (π₯ < 0 and π₯ > 0) given
by ππ1(π¦) = ππ1cosππ (π β π¦), ππ2(π¦) = cosπ π (π β π¦),
where
ππ1 =
2 ππ
(π = 1,2, . . . . )
2ππ π + sin2ππ π
and ππ , π π (π = 1,2,3, . . . ) are g iven by the following two
equations
ππ tanππ π + πΎ = 0, π π π = (β1)π ππ .
The matching conditions for π0(π₯, π¦) are the continuity of the velocity potential π0(π₯, π¦) and the linear velocity at
x=0:
π0(π₯β, π¦)π₯<0 = π0 (π₯+, π¦)π₯ >0, π0π₯ (π₯β, π¦)π₯ <0 = π0π₯ (π₯+, π¦)π₯ >0.
The orthogonality of the depth eigenfunctions of the two
regions together with the above matching conditions at
π₯ = 0 produces the following two systems of linear
equations
β π
=
1
β
where π΄ =
1,2, . . . . ).
β π=
1
β
π΅π
= π΄πΏ0π ,
π π βππ
(4.2)
ππ = π01coshπ0π π 0 β 1 , (4.3)
ππ βπ π
π π0 +π π π π0βπ π
β
2π π0 1
and ππ = π΄π ππ1cosππ π(π, π =
πΎπ0 cosh π π
The unknown constants π΄π , π΅π (π = 1,2, . . . . . ) can be
computed numerically fro m thesystem o f linear equations
(4.2) and (4.3) after truncation. The zero order reflection
coefficient π 0 can be determined following the procedure
used by Evans and Linton[11].
We consider the integral π½ = β π π§ ππ§, π = 0,1,2, . . . . ,
πΆπ π§βππ
where the function π(π§) has simple poles
(4.4) at π§ =
π 1, π 2, . . . . . π π , simp le zeroes at π§ = π1, π2 , . . . . ππ and π π§ β π( 1 ) as π§ β β. πΆπ β²π are the sequence of circles
π§
with radius π π increases without bound as π β β. These sequence of circles πΆπ π avoids the zeroes of the integrand and all the poles and zeroes of π(π§) are inside of it and πΆπ
58
U. Basu et al.: Water Wave Scattering by a Dock in Presence of Bottom Undulation
must not pass through (0,0). Further assuming that
π(π0 ) = β1, the function π(π§) following the aforesaid
properties can be taken as
π π§ = π0
1β π§ 1βπ0
β β ππ π π .
π§ π=1 1β π§ 1βπ0
π π
ππ
(4.5)
The Cauchy's integral formu la and residue theorem gives
π π0 = 1 πΆ βπ π§ ππ§ =
2ππ π π§ βπ0
β π=
1
β(
π
ππ
(
π
(
π§
)

π§
=
π π
)
.
(4.6)
π π βπ0
Using π(π0) = β1, with πΏ denoting the kronecker delta,
we find
πΏ0π =
β π=
1
β(
π
ππ
(
π
(
π§
)

π§
=
π π
)
.
π π βπ0
Now co mparing (4.6) with (4.2), we obtain
(4.7)
π΅π = π΄π ππ (π(π§)π§ = π π ).
Again, we consider the integral
πΌ = β π π§ ππ§, π = 0,1,2, . . . . . . ,
πΆπ π§ +ππ
(4.8)
with the same p roperty of the integrand function π(π§) as
mentioned above in (4.4) . The matching conditions at
π₯ = 0 can be combined to give
β π
=
1
β
π΅π
=β
2π π0 π 0 1
.
π π +π0
πΎπ0 cosh ππ
(4.9)
The Cauchy's residue theorem gives for π = 0 and at
π§ = βπ0
β π=
1
β
π΅π
= π΄π βπ0 .
π π +π0
Co mparing (4.8) and (4.9), we obtain
π΄π βπ0 =
2π π0 π 0 1
.
πΎπ0 cos π π0 π
(4.10) (4.11)
At π§ = βπ0, using π(π§) fro m (4.5), (4.10) gives
1+ππ 1βπ0
π βπ0 = β
β π=
1
β
ππ π0
π π π0
,
1+π π 1βππ
(4.12)
and replacing
π΄ and π(βπ0 ) in (4.11), we obtain
β (1 + ππ )(1 β π0 )
π 0 = β[(1 + πππ0 )(1 β π ππ0 )].
π=1
π π
ππ
Thus
π 0 = π2ππΌ ,
where
πΌ = β β[tanβ1(π0) β tan β1(π0)], (π = 1,2, . . . . . . . ).
π =1
π π
ππ
Thus R0=1, which is expected since all the incident waves
are fu ll reflected by the rigid dock floating on water of
uniform fin ite depth.
5. The First Order Reflection Coefficient
As mentioned earlier, without solving the BVP 2, the first
order correction to the reflection coefficient π 1 can be
obtained by appropriate use of Green's integral theorem to
the functions π0(π₯, π¦) and π1(π₯, π¦) . We consider the
region bounded by the lines
y=0, πβ€π₯β€π ;y=h, πβ€π₯β€π ; π₯=Β±π , 0
(where
πππ π π£πππ¦ πππππ πππ ππ‘ππ£π ππ’ππππ )
and employ Green's integral theorem in the form
πΏ β π0π1π β π1π0π ππ = 0, (5.1)
where π is the outward normal to the line element ππ.The
surface boundary condition satisfied by the potential
functions π0(π₯, π¦) and π1(π₯, π¦) ensures that there is no contribution to the integral along the part (π¦ = 0, βπ β€ π₯ β€
π ). Fu rthermore the water under the dock region is
undisturbed and there is no transmission of incoming wave.
So there is no contribution to the integral along the part
(π₯ = π, 0 β€ π¦ β€ π). The only contribution to the integral
(5.1) arises from the line ( π₯ = βπ, 0 β€ π¦ β€ π ) and the
bottom. Finally making π β β, we obtain 2ππ0π 1 = ββββπ π₯ π02π₯ π₯, π ππ₯.
(5.2)
Thus π 1 can be obtained exp licitly once the shape function π(π₯) is known and the potential function π0(π₯, π¦)
is obtained.
6. Numerical Results and Discussions
The first order correct ion to the reflection coefficientπ 1
can be computed numerically fro m (5.2) once the shape
function π(π₯) is known. In the present section, the values of
π 1 are co mputed for different values of the wave number πΎπ by considering two types of shape function π(π₯)
characterizing the bottom undulations. It may be noted that
for convenience both the bottom shape functions are chosen
symmetric about the point of discontinuity of the surface
boundary condition namely π₯ = 0 here. The expressions for
π 1 are given in the Appendix for such bottom profiles. Fo r the numerical co mputation of π 1, the infinite series in the
appendix are truncated upto desired accuracy. The
convergence of the integral (5.2) for the chosen bottomshape
functions is assured.
Cas e1 :
The first bottom shape funcβtioπnπis chosenπaπs π π₯ = π0sinππ₯ π β€ π₯ β€ π
=0
ππ‘ππππ€ππ π .
The above π(π₯) represents a bottom profile of π
sinusoidal ripples with amplitude π0 and wave number π. The analytical expression for π 1 is given in the Appendix for the above bottom profile. For nu merical co mputation the
values of non dimensional ripple amp litude is taken as π0 = 0.1,0.2 and ripp le wave nu mber ππ = 1 . π 1 is
π
depicted graphically against the wave number πΎπ in
ππππ’ππ β 1. The graph of π 1 is observed to be oscillatory in nature against πΎπ and the oscillation peaks up to a
highest value for a certain frequency. This observation can
be explained as multip le interaction of the incident wave
with d ifferent frequencies and the sinusoidally varying
bottom topography. Occurrence of zeroes of π 1 for certain πΎπ indicates that the sinusoidal bottom topography does not
affect the reflected wave of the incident wave train up to first
order. The peak values of π 1 are found to be increasing with the values of π0. The heights of the oscillations of π 1
π
are gradually dimin ishing along with πΎπ . Th is can be
explained as wave damping due to energy dissipation when
wave propagates over uneven sea bottom and the presence of
the upper surface discontinuity.
American Journal of Fluid Dynamics 2012, 2(4): 5560
59
The ππππ’ππ β 2 is plotted by taking dimensionless ripple wave nu mber ππ = 3,5,7 and fixed amplitude π0 = 0.1. In
π
this case the oscillation also decreases and gradually
vanishes with πΎπ . A lso as the wave nu mber ππ of the
sinusoidal bottom p rofile function increases, number of
oscillations of π 1 decreases with πΎπ . Also the highest peak values of π 1 decrease with the increasing value of ππ.
Figure 3. π 1 for different ππ
Figure 1. π 1 for differentc0/π
Figure 4. π 1 for different π0/π
The ππππ’ππ β 4 depicts π 1 against πΎπ for different dimensionless ripple amplitude π0 = 0.1,0.2 and fixed value
π
of ππ = 0.3. As before, π 1 first increases with πΎπ and then rapidly decreases with πΎπ for πΎπ > 3 . Also the highest peak value of π 1 is found to be increasing with increasing value of the dimensionless ripple amp litude π0.
π
Figure 2. π 1 for different ππ
Cas e2 :
The second shape function is taken as π(π₯) = π0πβππ₯  β β < π₯ < β, π > 0.
This above π(π₯) represents exponentially decaying bottom pro file. The analytical expression of π 1 is given in the Appendix for this shape function. π 1 are depicted against πΎπ in ππππ’ππ β 3 by taking ππ = 0.3,0.5,1 and π0 = 0.1. An important feature of this figure is that for each
π
value of the parameter ππ, π 1 first increases with πΎπ and attains a maximu m peak value and then gradually decreases with πΎπ . This may be attributed due to exponentially
decaying bottom pro file as the bottom undulation vanishes far away where the water is of uniform depth π. A noticeable
feature of this figure is that this maximu m peak value is found to be decreasing with the increasing value of ππ.
7. Conclusions
A few problems involving water wave scattering by surface discontinuity in presence variable bottom topograph y has already been studied in the literature. The present study is concerned with water wave scattering by a semiinfin ite floating dock in presence of a uneven bottom. Study of such type of problem is important for quite some time due to several valuable aspects in research areas in marine science and oceanography.
A simplified perturbation analysis together with appropri ate use of Green's integral theorem is employed to obtain first order correction to the reflection coefficient. Two different type of shape functions are chosen to describe the bottom topography in case of sinusoidal patch and exponentially decaying bottom undulation. The numerical results of the firstorder correction of the reflection
60
U. Basu et al.: Water Wave Scattering by a Dock in Presence of Bottom Undulation
coefficient are being obtained by utilizing the expressions given in the appendix for both sinusoidal patches and exponentially decaying bottom undulation. These res ults are depicted graphically against the wave number of the incident wave in a number of figures. It may be noted that the discontinuity of the upper surface boundary condition may be at the another point and in that case same method can be applied.
ACKNOWLEDGEMENTS
R. Maiti thanks CSIR, New Delh i (India) for provid ing support to prepare this paper.
Appendix
For a sinusoidal bottom topography (π(π₯) = π0π ππππ₯):
π = π0 βπ2(π 1)2
π
β2iπ0π π
(β1)π e π β 1)
1 2iπ0 0 0 π2 β 4π02
ππ 02 +
2iπ0π π
(β1)π e π β 1
π2 β 4π02
+ 2π 0 (β1)π β 1
π
0
β
2
+
β βππ π΄π eππ π₯ ππ1 sinππ₯dπ₯
βππ /π π=1
β
ππ π΄π ππ1π
π π (ππ +iπ0)
+2iπ0π01 π=1 βπ2 + (ππ + iπ0)2 (β1)π e π β 1
β
ππ π΄π ππ1π
π π (ππ βiπ0)
β2iπ0π01π 0 π=1 βπ2 + (ππ β iπ0)2 (β1)π e π β 1
ππ β
2
π
+ β
βπ π π΅π eβπ π π₯ ππ1 sinππ₯dπ₯ .
0
π=1
For an exponential bottom topography (π π₯ = π0πβππ₯):
π = π0 βπ2(π 1)2
1
π 02
2π 0
+
β
1 2iπ0 0 0 2iπ0 + π π β 2iπ0 π
0β
2
+ β βππ π΄π eππ π₯ ππ1 eππ₯ dπ₯
ββ π=1
β
ππ π΄π ππ1
+2iπ0 π01
β π + (π + iπ )
π =1
π
0
β
ππ π΄π ππ1
β 2i π0π01π 0 π =1 βπ + (ππ β iπ0)
ββ
2
+ β
βπ π π΅π eβπ π π₯ ππ1 eβπ π₯ dπ₯ .
0 π=1
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