Water Wave Scattering by a Dock in Presence of Bottom

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Water Wave Scattering by a Dock in Presence of Bottom

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American Journal of Fluid Dynamics 2012, 2(4): 55-60 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 semi-infin 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 sea-bed 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 wave-train 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 Winner-Hopf 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 semi-infinite dock in water and used Wiener-Hopf 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 re-investigated by Chackrabarti, Mandal and Gayen[17] by utilizing a Fourier type analysis, giving rise to the Carleman type singular integral equation over semi-infin 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 (BVP-1) 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 BVP-1 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 (BVP-2), 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 BVP-1. 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 𝑦-axis is taken vertically downwards into

the fluid region and the undisturbed free surface corresponds

to 𝑦 = 0, π‘₯ ≀ 0. A semi-infinite floating dock of negligib le

thickness occupies the region 0 ≀ π‘₯ < ∞, 𝑦 = 0 and other

side π‘₯ < 0, 𝑦 = 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 semi-infinite 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 𝑑 with angular frequency πœ”, it can be described by a velocity potential πœ“(π‘₯, 𝑦, 𝑑) = 𝑅𝑒{πœ™(π‘₯, 𝑦)π‘’βˆ’π‘–πœ”π‘‘ } , where

πœ™(π‘₯, 𝑦) satisfies the two dimensional Lap lace equation: βˆ‚2πœ™ + βˆ‚2πœ™ = 0, π‘–π‘›π‘‘π‘•π‘’π‘“π‘™π‘’π‘–π‘‘π‘Ÿπ‘’π‘”π‘–π‘œπ‘› ,(2.1)
βˆ‚π‘₯2 βˆ‚π‘¦ 2

with the free surface boundary condition

πΎπœ™ + βˆ‚πœ™ = 0 π‘œπ‘›π‘¦ = 0, π‘₯ < 0,

(2.2)

βˆ‚π‘¦

where 𝐾 = πœ”2 , 𝑔 is the accelerat ion due to gravity. The

𝑔

surface boundary condition on the dock region is given by

βˆ‚πœ™ = 0 π‘œπ‘›π‘¦ = 0, π‘₯ > 0,
βˆ‚π‘¦

(2.3)

with the edge condition 1 π‘Ÿ1/2βˆ‡πœ™ = 𝑂 (1) π‘Žπ‘ π‘Ÿ = {π‘₯2 + 𝑦2 }2 β†’ 0,

(2.4)

and the bottom boundary condition

βˆ‚πœ™ = 0 π‘œπ‘›π‘¦ = 𝑕 + πœ€π‘ π‘₯ ,
βˆ‚πœˆ

(2.5)

where 𝜈 is the outward normal to the sea bed. 𝑦 = 𝑕 +

πœ€π‘(π‘₯) denotes the bottom of the sea of variable depth. πœ€ is a

small non-dimensional positive number signify ing the

smallness of the bottom undulation and 𝑐(π‘₯) is a bounded,

continuous function characterizing the shape of the bottom

with the property that 𝑐(π‘₯) β†’ 0 as π‘₯ β†’ Β± ∞. This ensures

that far away fro m the undulation the sea bottom is of

uniform finite depth 𝑕 below the mean free surface. The

incident wave field fro m the direction of π‘›π‘’π‘”π‘Žπ‘‘π‘–π‘£π‘’π‘–π‘›π‘“π‘–π‘›π‘–π‘‘π‘¦

is described by πœ‘0(π‘₯, 𝑦) = π‘’π‘–π‘˜0 π‘₯πœ“01(𝑦),

where

πœ“01(𝑦) = 𝑁01coshπ‘˜0(𝑕 βˆ’ 𝑦),

𝑁01 =

2 π‘˜0 2π‘˜0𝑕 + sinh2π‘˜0 𝑕

and π‘˜0 is unique the positive real root of the transcendental equation
π‘˜tanhπ‘˜π‘• = 𝐾.

Finally, the far field behavior of the potential function is

described by 0
πœ™ π‘₯, 𝑦 β†’ πœ‘0 π‘₯, 𝑦 + π‘…πœ‘0 βˆ’π‘₯, 𝑦

π‘Žπ‘ π‘₯ β†’ ∞, π‘Žπ‘  π‘₯ β†’ βˆ’βˆž. (2.6)

Where 𝑅 is the unknown reflect ion coefficient(co mplex).

Determination of 𝑅 is of prime concern here.

3. Method of Solution

American Journal of Fluid Dynamics 2012, 2(4): 55-60

57

The boundary condition (2.5) can be written as,

𝑙 βˆ‚πœ™ + π‘š βˆ‚πœ™ = 0,

βˆ‚π‘₯

βˆ‚π‘¦

(3.1)

where (𝑙, π‘š) is the direction cosine of 𝜈 to the curve

𝜁(π‘₯, 𝑦) = 𝑦 βˆ’ 𝑕 βˆ’ πœ€π‘(π‘₯) = 0.

Using Taylor series expansion of πœ™(π‘₯, 𝑦) on 𝑦 = 𝑕 +

πœ€π‘(π‘₯) and the equation (2.1), we get the following form:

βˆ‚πœ™ βˆ’ πœ€ 𝑑 𝑐 π‘₯ βˆ‚πœ™ + 𝑂 πœ€2 = 0 π‘œπ‘›π‘¦ = 𝑕.

βˆ‚π‘¦

𝑑π‘₯

βˆ‚π‘₯

(3.2)

The form of the approximate boundary condition (3.2)

suggests that πœ™ and 𝑅 have the follo wing perturbational

e xpansion in terms of the small para meter πœ€: πœ™ π‘₯, 𝑦, πœ€ = πœ™0 π‘₯ ,𝑦 + πœ€πœ™1 π‘₯, 𝑦 + 𝑂 πœ€2 , 𝑅 πœ€ = 𝑅0 + πœ€π‘…1 + 𝑂 πœ€2 .

(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œ€ (πœ€0 and πœ€1) on both sides

of the results, we find that, the zero and first order potentials

πœ™0 and πœ™1 satisfies the following two different boundary

value problems. They are named as BVP-1 and BVP-2

res p ectiv ely .

BVP-1:

The function πœ™0(π‘₯, 𝑦) satisfies

(i) βˆ‚2πœ™0 + βˆ‚2πœ™0 = 0 𝑖𝑛 0 < 𝑦 < 𝑕, βˆ’βˆž < π‘₯ < ∞,

βˆ‚π‘₯2

βˆ‚π‘¦ 2

(ii) πΎπœ™0 + βˆ‚βˆ‚πœ™π‘¦0 = 0 π‘œπ‘›π‘¦ = 0, π‘₯ < 0,

(iii) βˆ‚πœ™0 = 0 π‘œπ‘›π‘¦ = 0, π‘₯ > 0,
βˆ‚π‘¦

(iv) βˆ‚πœ™0 = 0 π‘œπ‘›π‘¦ = 𝑕,
βˆ‚π‘¦

0

π‘Žπ‘ π‘₯ β†’ +∞,

(v) πœ™0 π‘₯, 𝑦 β†’ πœ‘0(π‘₯, 𝑦) + 𝑅0πœ‘0(βˆ’π‘₯, 𝑦) π‘Žπ‘ π‘₯ β†’ βˆ’βˆž.

Where 𝑅0 is the zero o rder reflect ion coefficient.

BVP-2:

The function πœ™1(π‘₯, 𝑦) satisfies

(i) βˆ‚2πœ™1 + βˆ‚2πœ™1 = 0 𝑖𝑛 0 < 𝑦 < 𝑕, βˆ’βˆž < π‘₯ < ∞,

βˆ‚π‘₯2

βˆ‚π‘¦ 2

(ii) πΎπœ™1 + βˆ‚πœ™1 = 0 π‘œπ‘›π‘¦ = 0, π‘₯ < 0,
βˆ‚π‘¦

(iii) βˆ‚πœ™1 = 0 π‘œπ‘›π‘¦ = 0, π‘₯ > 0,
βˆ‚π‘¦

(iv) βˆ‚πœ™1 = 𝑑 [𝑐(π‘₯)πœ™0π‘₯ (π‘₯, 𝑕)] π‘œπ‘›π‘¦ = 𝑕,
βˆ‚ 𝑦 𝑑π‘₯

0

π‘Žπ‘ π‘₯ β†’ +∞,

(v) πœ™1(π‘₯, 𝑦): 𝑅1πœ‘0(βˆ’π‘₯, 𝑦) π‘Žπ‘ π‘₯ β†’ βˆ’βˆž.

Where 𝑅1 is the first order reflection coefficient.

The BVP-1 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 BVP-2 is a rad iation problem in presence of a free

surface discontinuity with variable bottom depth. Here,

Without solving the BVP-2, the analytical exp ression for the

first order reflect ion coefficient is determined in terms of an

integral involv ing the bottom shape function 𝑐(π‘₯) and the

solution of the BVP-1, namely πœ™0(π‘₯, 𝑦) (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 πœ™0(π‘₯, 𝑦) of the 𝐡𝑉𝑃 βˆ’ 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 𝐴𝑛 , 𝐡𝑛 (𝑛 = 1,2, . . . . ) are unknown constants, πœ“π‘›1(𝑦) , πœ“π‘›2(𝑦)(𝑛 = 1,2,3. . . . ) are the orthogonal depth eigenfunctions for the two regions (π‘₯ < 0 and π‘₯ > 0) given

by πœ“π‘›1(𝑦) = 𝑁𝑛1cosπ‘˜π‘› (𝑕 βˆ’ 𝑦), πœ“π‘›2(𝑦) = cos𝑠𝑛 (𝑕 βˆ’ 𝑦),

where

𝑁𝑛1 =

2 π‘˜π‘›

(𝑛 = 1,2, . . . . )

2π‘˜π‘› 𝑕 + sin2π‘˜π‘› 𝑕

and π‘˜π‘› , 𝑠𝑛 (𝑛 = 1,2,3, . . . ) are g iven by the following two

equations

π‘˜π‘› tanπ‘˜π‘› 𝑕 + 𝐾 = 0, 𝑠𝑛 𝑕 = (βˆ’1)𝑛 π‘›πœ‹ .

The matching conditions for πœ™0(π‘₯, 𝑦) are the continuity of the velocity potential πœ™0(π‘₯, 𝑦) and the linear velocity at

x=0:

πœ™0(π‘₯βˆ’, 𝑦)|π‘₯<0 = πœ™0 (π‘₯+, 𝑦)|π‘₯ >0, πœ™0π‘₯ (π‘₯βˆ’, 𝑦)|π‘₯ <0 = πœ™0π‘₯ (π‘₯+, 𝑦)|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

∞ 𝑛

=

1

‍

where 𝐴 =

1,2, . . . . ).

∞ 𝑛=

1

‍

𝐡𝑛

= 𝐴𝛿0π‘š ,

𝑠𝑛 βˆ’π‘˜π‘š

(4.2)

π‘ˆπ‘› = 𝑁01coshπ‘˜0𝑕 𝑅0 βˆ’ 1 , (4.3)

π‘˜π‘› βˆ’π‘ π‘š

𝑖 π‘˜0 +π‘ π‘š 𝑖 π‘˜0βˆ’π‘ π‘š

βˆ’

2𝑖 π‘˜0 1

and π‘ˆπ‘› = 𝐴𝑛 𝑁𝑛1cosπ‘˜π‘› 𝑕(π‘š, 𝑛 =

𝐾𝑁0 cosh π‘˜ 𝑕

The unknown constants 𝐴𝑛 , 𝐡𝑛 (𝑛 = 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 𝑅0 can be determined following the procedure

used by Evans and Linton[11].

We consider the integral 𝐽 = ‍ 𝑓 𝑧 𝑑𝑧, π‘š = 0,1,2, . . . . ,
𝐢𝑁 π‘§βˆ’π‘˜π‘š
where the function 𝑓(𝑧) has simple poles

(4.4) at 𝑧 =

𝑠1, 𝑠2, . . . . . 𝑠𝑛 , simp le zeroes at 𝑧 = π‘˜1, π‘˜2 , . . . . π‘˜π‘š and 𝑓 𝑧 β†’ 𝑂( 1 ) as 𝑧 β†’ ∞. 𝐢𝑁 ′𝑠 are the sequence of circles
𝑧
with radius 𝑅𝑁 increases without bound as 𝑁 β†’ ∞. These sequence of circles 𝐢𝑁 𝑠 avoids the zeroes of the integrand and all the poles and zeroes of 𝑓(𝑧) are inside of it and 𝐢𝑁

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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

𝑓(π‘˜0 ) = βˆ’1, the function 𝑓(𝑧) following the aforesaid

properties can be taken as

𝑓 𝑧 = π‘˜0

1βˆ’ 𝑧 1βˆ’π‘˜0
∞ ‍ π‘˜π‘› 𝑠𝑛 .

𝑧 𝑛=1 1βˆ’ 𝑧 1βˆ’π‘˜0

𝑠𝑛

π‘˜π‘›

(4.5)

The Cauchy's integral formu la and residue theorem gives

𝑓 π‘˜0 = 1 𝐢 ‍𝑓 𝑧 𝑑𝑧 =
2πœ‹π‘– 𝑁 𝑧 βˆ’π‘˜0

∞ 𝑛=

1

‍(

𝑅

𝑒𝑠

(

𝑓

(

𝑧

)

|

𝑧

=

𝑠𝑛

)

.

(4.6)

𝑠𝑛 βˆ’π‘˜0

Using 𝑓(π‘˜0) = βˆ’1, with 𝛿 denoting the kronecker delta,

we find

𝛿0π‘š =

∞ 𝑛=

1

‍(

𝑅

𝑒𝑠

(

𝑓

(

𝑧

)

|

𝑧

=

𝑠𝑛

)

.

𝑠𝑛 βˆ’π‘˜0

Now co mparing (4.6) with (4.2), we obtain

(4.7)

𝐡𝑛 = 𝐴𝑅𝑒𝑠 (𝑓(𝑧)|𝑧 = 𝑠𝑛 ).

Again, we consider the integral

𝐼 = ‍ 𝑓 𝑧 𝑑𝑧, π‘š = 0,1,2, . . . . . . ,
𝐢𝑁 𝑧 +π‘˜π‘š

(4.8)

with the same p roperty of the integrand function 𝑓(𝑧) as

mentioned above in (4.4) . The matching conditions at

π‘₯ = 0 can be combined to give

∞ 𝑛

=

1

‍

𝐡𝑛

=βˆ’

2𝑖 π‘˜0 𝑅0 1

.

𝑠𝑛 +π‘˜0

𝐾𝑁0 cosh π‘˜π‘•

(4.9)

The Cauchy's residue theorem gives for π‘š = 0 and at

𝑧 = βˆ’π‘˜0

∞ 𝑛=

1

‍

𝐡𝑛

= 𝐴𝑓 βˆ’π‘˜0 .

𝑠𝑛 +π‘˜0

Co mparing (4.8) and (4.9), we obtain

𝐴𝑓 βˆ’π‘˜0 =

2𝑖 π‘˜0 𝑅0 1

.

𝐾𝑁0 cos 𝑕 π‘˜0 𝑕

(4.10) (4.11)

At 𝑧 = βˆ’π‘˜0, using 𝑓(𝑧) fro m (4.5), (4.10) gives

1+π‘˜π‘œ 1βˆ’π‘˜0

𝑓 βˆ’π‘˜0 = βˆ’

∞ 𝑛=

1

‍

π‘˜π‘› π‘˜0

𝑠𝑛 π‘˜0

,

1+𝑠𝑛 1βˆ’π‘˜π‘›

(4.12)

and replacing

𝐴 and 𝑓(βˆ’π‘˜0 ) in (4.11), we obtain

∞ (1 + π‘˜π‘œ )(1 βˆ’ π‘˜0 )

𝑅0 = ‍[(1 + π‘˜π‘˜π‘›0 )(1 βˆ’ π‘ π‘˜π‘›0 )].

𝑛=1

𝑠𝑛

π‘˜π‘›

Thus

𝑅0 = 𝑒2𝑖𝛼 ,

where

𝛼 = ∞ ‍[tanβˆ’1(π‘˜0) βˆ’ tan βˆ’1(π‘˜0)], (𝑛 = 1,2, . . . . . . . ).

𝑛 =1

𝑠𝑛

π‘˜π‘›

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 𝑅1 can be

obtained by appropriate use of Green's integral theorem to

the functions πœ™0(π‘₯, 𝑦) and πœ™1(π‘₯, 𝑦) . We consider the

region bounded by the lines

y=0, 𝑋≀π‘₯≀𝑋 ;y=h, -𝑋≀π‘₯≀𝑋 ; π‘₯=±𝑋 , 0
(where

𝑋𝑖𝑠 π‘Ž π‘£π‘’π‘Ÿπ‘¦ π‘™π‘Žπ‘Ÿπ‘”π‘’ π‘π‘œπ‘ π‘–π‘‘π‘–π‘£π‘’ π‘›π‘’π‘šπ‘π‘’π‘Ÿ )

and employ Green's integral theorem in the form

𝐿 ‍ πœ™0πœ™1𝜈 βˆ’ πœ™1πœ™0𝜈 𝑑𝑙 = 0, (5.1)

where 𝜈 is the outward normal to the line element 𝑑𝑙.The

surface boundary condition satisfied by the potential

functions πœ™0(π‘₯, 𝑦) and πœ™1(π‘₯, 𝑦) ensures that there is no contribution to the integral along the part (𝑦 = 0, βˆ’π‘‹ ≀ π‘₯ ≀

𝑋 ). 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‘₯ = 𝑋, 0 ≀ 𝑦 ≀ 𝑕). The only contribution to the integral

(5.1) arises from the line ( π‘₯ = βˆ’π‘‹, 0 ≀ 𝑦 ≀ 𝑕 ) and the

bottom. Finally making 𝑋 β†’ ∞, we obtain 2π‘–π‘˜0𝑅1 = βˆ’βˆžβˆžβ€π‘ π‘₯ πœ™02π‘₯ π‘₯, 𝑕 𝑑π‘₯.

(5.2)

Thus 𝑅1 can be obtained exp licitly once the shape function 𝑐(π‘₯) is known and the potential function πœ™0(π‘₯, 𝑦)

is obtained.

6. Numerical Results and Discussions

The first order correct ion to the reflection coefficient𝑅1

can be computed numerically fro m (5.2) once the shape

function 𝑐(π‘₯) is known. In the present section, the values of

|𝑅1| are co mputed for different values of the wave number 𝐾𝑕 by considering two types of shape function 𝑐(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

𝑅1 are given in the Appendix for such bottom profiles. Fo r the numerical co mputation of |𝑅1|, 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 e-1 :

The first bottom shape funcβˆ’tioπ‘šnπœ‹is chosenπ‘šaπœ‹s 𝑐 π‘₯ = 𝑐0sinπœ†π‘₯ πœ† ≀ π‘₯ ≀ πœ†

=0

π‘œπ‘‘π‘•π‘’π‘Ÿπ‘€π‘–π‘ π‘’ .

The above 𝑐(π‘₯) represents a bottom profile of π‘š

sinusoidal ripples with amplitude 𝑐0 and wave number πœ†. The analytical expression for 𝑅1 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 𝑐0 = 0.1,0.2 and ripp le wave nu mber πœ†π‘• = 1 . |𝑅1| is
𝑕
depicted graphically against the wave number 𝐾𝑕 in

π‘“π‘–π‘”π‘’π‘Ÿπ‘’ βˆ’ 1. The graph of |𝑅1| is observed to be oscillatory in nature against 𝐾𝑕 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 |𝑅1| for certain 𝐾𝑕 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 |𝑅1| are found to be increasing with the values of 𝑐0. The heights of the oscillations of |𝑅1|
𝑕
are gradually dimin ishing along with 𝐾𝑕 . 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): 55-60

59

The π‘“π‘–π‘”π‘’π‘Ÿπ‘’ βˆ’ 2 is plotted by taking dimensionless ripple wave nu mber πœ†π‘• = 3,5,7 and fixed amplitude 𝑐0 = 0.1. In
𝑕
this case the oscillation also decreases and gradually
vanishes with 𝐾𝑕 . A lso as the wave nu mber πœ†π‘• of the
sinusoidal bottom p rofile function increases, number of
oscillations of |𝑅1| decreases with 𝐾𝑕 . Also the highest peak values of |𝑅1| decrease with the increasing value of πœ†π‘•.

Figure 3. |𝑅1| for different πœ‡π‘•

Figure 1. |𝑅1| for differentc0/𝑕

Figure 4. |𝑅1| for different 𝑐0/𝑕
The π‘“π‘–π‘”π‘’π‘Ÿπ‘’ βˆ’ 4 depicts |𝑅1| against 𝐾𝑕 for different dimensionless ripple amplitude 𝑐0 = 0.1,0.2 and fixed value
𝑕
of πœ‡π‘• = 0.3. As before, |𝑅1| first increases with 𝐾𝑕 and then rapidly decreases with 𝐾𝑕 for 𝐾𝑕 > 3 . Also the highest peak value of |𝑅1| is found to be increasing with increasing value of the dimensionless ripple amp litude 𝑐0.
𝑕

Figure 2. |𝑅1| for different πœ†π‘•
Cas e-2 :
The second shape function is taken as 𝑐(π‘₯) = 𝑐0π‘’βˆ’πœ‡|π‘₯ | βˆ’ ∞ < π‘₯ < ∞, πœ‡ > 0.
This above 𝑐(π‘₯) represents exponentially decaying bottom pro file. The analytical expression of 𝑅1 is given in the Appendix for this shape function. |𝑅1| are depicted against 𝐾𝑕 in π‘“π‘–π‘”π‘’π‘Ÿπ‘’ βˆ’ 3 by taking πœ‡π‘• = 0.3,0.5,1 and 𝑐0 = 0.1. An important feature of this figure is that for each
𝑕
value of the parameter πœ‡π‘•, |𝑅1| first increases with 𝐾𝑕 and attains a maximu m peak value and then gradually decreases with 𝐾𝑕 . 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 𝑕. A noticeable
feature of this figure is that this maximu m peak value is found to be decreasing with the increasing value of πœ‡π‘•.

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 semi-infin 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 first-order 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 (𝑐(π‘₯) = 𝑐0π‘ π‘–π‘›πœ†π‘₯):

𝑅 = 𝑐0 βˆ’π‘˜2(𝑁 1)2

πœ†

βˆ’2iπ‘˜0π‘š πœ‹
(βˆ’1)π‘š e πœ† βˆ’ 1)

1 2iπ‘˜0 0 0 πœ†2 βˆ’ 4π‘˜02

πœ†π‘…02 +

2iπ‘˜0π‘š πœ‹
(βˆ’1)π‘š e πœ† βˆ’ 1

πœ†2 βˆ’ 4π‘˜02

+ 2𝑅0 (βˆ’1)π‘š βˆ’ 1

πœ†

0

∞

2

+

‍ β€π‘˜π‘› 𝐴𝑛 eπ‘˜π‘› π‘₯ 𝑁𝑛1 sinπœ†π‘₯dπ‘₯

βˆ’π‘šπœ‹ /πœ† 𝑛=1

∞

π‘˜π‘› 𝐴𝑛 𝑁𝑛1πœ†

π‘š πœ‹ (π‘˜π‘› +iπ‘˜0)

+2iπ‘˜0𝑁01 𝑛=1 β€πœ†2 + (π‘˜π‘› + iπ‘˜0)2 (βˆ’1)π‘š e πœ† βˆ’ 1

∞

π‘˜π‘› 𝐴𝑛 𝑁𝑛1πœ†

π‘š πœ‹ (π‘˜π‘› βˆ’iπ‘˜0)

βˆ’2iπ‘˜0𝑁01𝑅0 𝑛=1 β€πœ†2 + (π‘˜π‘› βˆ’ iπ‘˜0)2 (βˆ’1)π‘š e πœ† βˆ’ 1

π‘šπœ‹ ∞

2

πœ†
+ ‍

‍𝑠𝑛 𝐡𝑛 eβˆ’π‘ π‘› π‘₯ 𝑁𝑛1 sinπœ†π‘₯dπ‘₯ .

0

𝑛=1

For an exponential bottom topography (𝑐 π‘₯ = 𝑐0π‘’βˆ’πœ‡|π‘₯|):

𝑅 = 𝑐0 βˆ’π‘˜2(𝑁 1)2

1

𝑅02

2𝑅0

+

βˆ’

1 2iπ‘˜0 0 0 2iπ‘˜0 + πœ‡ πœ‡ βˆ’ 2iπ‘˜0 πœ‡

0∞

2

+ ‍ β€π‘˜π‘› 𝐴𝑛 eπ‘˜π‘› π‘₯ 𝑁𝑛1 eπœ‡π‘₯ dπ‘₯

βˆ’βˆž 𝑛=1

∞

π‘˜π‘› 𝐴𝑛 𝑁𝑛1

+2iπ‘˜0 𝑁01

‍ πœ‡ + (π‘˜ + iπ‘˜ )

𝑛 =1

𝑛

0

∞

π‘˜π‘› 𝐴𝑛 𝑁𝑛1

βˆ’ 2i π‘˜0𝑁01𝑅0 𝑛 =1 β€πœ‡ + (π‘˜π‘› βˆ’ iπ‘˜0)

∞∞

2

+ ‍

‍𝑠𝑛 𝐡𝑛 eβˆ’π‘ π‘› π‘₯ 𝑁𝑛1 eβˆ’πœ‡ π‘₯ dπ‘₯ .

0 𝑛=1

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DockPresenceWater WaveSurfaceWave