International Journal of Scientific & Engineering Research, Volume 4, Issue 4, April-2013 294

ISSN 2229-5518

Scattering of Water Waves in an Ocean with Uniform Porous Bed by a Surface Discontinuity Due to Inertial Surfaces in Presence of Surface Tension

Subhabrata Gangopadhyay, Uma Basu

Abstract – The phenomenon of scattering of water waves by a surface discontinuity over uniform porous sea-bottom of finite depth is investigated in the framework of linearized water wave theory. The surface discontinuity is thought of as originating due to two vast floating inertial surfaces of different materials having different densities. The inertial surfaces are considered to be subject to surface tension. The eigen function matching technique is used as the method of solution. Hydrodynamic parameters of interest such as reflection and transmission coefficients are obtained by employing residue calculus method. These coefficients are computed numerically by considering different values of the porosity parameter and for a fixed value of surface tension and the results are depicted graphically.

Keywords – eigen function matching, inertial surface, porous sea bed, reflection coefficient, residue calculus method, surface discontinuity, surface tension, transmission coefficient, water wave scattering.

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1 INTRODUCTION

The problem of water wave scattering by obstacles situated at the bed in a finite depth of water has been studied in the framework of linearised theory by several scientists over a last few decades. Some of the earliest contributors in this subject were Lamb[1], Stoker[2], Kreisel[3] and Davies[4]. Basu and Mandal [5] studied the water wave scattering problem in presence of bottom undulation and surface tension in the free surface using perturbation expansion in terms of bed undulation parameter.
Another class of problem is that of water wave scattering in presence of a discontinuity at the upper surface of water. A discontinuity in the upper surface or elsewhere may occur when there is a difference of wave number of the incoming waves of certain frequency from a sudden change in the constant width of the region. Therefore, there will be two different boundary conditions on the either side of the ocean. The upper surface of the ocean may be covered by two vast sheets of floating ice plate or mat of different thickness or materials, broken ice- cover, semi-infinite floating dock etc. The presence of

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Subhabrata Gangopadhyay is currently an Associate Professor in the Department of Mathematics, The Bhawanipur Education Society College, 5,L.L.R.Sarani, Kolkata – 700 020, India.

E-mail: subhagan31@gmail.com

Uma Basu, is currently a Professor in the Department of Applied

Mathematics, University of Calcutta, 92, A.P.C. Road, Kolkata – 700

009, India.

obstacles or materials of different densities or properties yields a change in the boundary condition due to difference in wave number. Evans and Linton [6] considered the problem of water wave scattering by a surface discontinuity in an uniform finite depth of water. They obtained the reflection and transmission coefficients by employing technique of residue calculus. Mandal and De[7] considered the problem of water wave scattering by a small undulation at the bottom in presence of upper surface discontinuity and arrived at the energy coefficients using perturbation method and Green’s integral theorem.
The problem of water wave interaction over a
permeable bed or porous bottom is also investigated by several researches in recent times. If the bottom is composed of some specific type of porous materials, rigid or non-rigid, the effect of porosity on the hydrodynamic coefficients is another important aspect of study. A porous sea bottom or structure may be rigid or non-rigid. The only difference is that a rigid or impermeable type porous structure or bottom does not allow the fluid to penetrate into it. The water wave interaction with the porous sea bed was studied by Chakrabarti [8], Mase and Takeba [9], Silva, Salles and Palacio [10], Jeng [11], and many others. Martha and Bora [12] applied Fourier transform to analyse scattering waves by small undulation on a porous sea-bed. The flow of fluid into the porous media or the presence of porous substances in the bottom leads to different

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International Journal of Scientific & Engineering Research, Volume 4, Issue 4, April-2013 295

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phenomena like energy dissipation, wave damping or decaying of wave height reaching towards the coast etc.
In the present paper, we consider the problem of
scattering of an incoming wave train in presence of a discontinuity at the upper surface of the ocean. The upper surface is assumed to be covered by two semi-infinite inertial surfaces subject to surface tension. The inertial surfaces are considered to be of negligible thickness and are composed of different materials and densities. Hence, there will be a difference of wave number of the incoming wave
with angular frequency w, it can be described by a velocity potential l/J(x, y, t) = Re{cp(x, y)e -iwt }, where cp(x, y)
satisfies the two dimensional Laplace equation:
2 cp = 0 in the entire fluid region (1)
The upper surface boundary conditions taking surface
tension parameter M into account, are given by:
K1 cp + cpy + Mcpyyy = 0 on y = 0, x < 0 (2)
K2 cp + cpy + Mcpyyy = 0 on y = 0, x > 0 (3)
This produces a discontinuity in the upper surface
boundary conditions at the point (0,0) where
train. If the problem is formulated mathematically, there
K1 = , K
= , E E

w2

< and K = .
will be two different boundary conditions on either side of

1-E1 2

1-E1 1 2 w2

the discontinuity of the upper surface of the ocean. The
The edge condition is given by:

1 1

water is of uniform finite depth and the bed is composed of some specific kind of rigid porous material which is characterized by a known porosity parameter G’. The incoming wave train is partially reflected and partially


r2 ∇cp = 0(1) as r = {x2 + y2 }2 → 0 (4)
The sea-bottom boundary condition is given by:
cpy − G t cp = 0 on y = ℎ (5) The far field behavior of cp(x, y) is described by:
transmitted through the ocean. The method of residue
Te
cp(x, y)~

is0 x

l/J2 (y) as x → ∞

1(y) as x → −∞

(6)
calculus of the complex variable theory (cf.[13]) is
employed here to determine the reflection and transmission
where

(e ik0 x + Re -ik0 x )l/JO


1 (y) = N1 cosℎk (ℎ − y) − G sinℎk (ℎ − y) ,

coefficients. Evans and Linton [6] also followed the same
technique to obtain these hydrodynamic coefficients for
l/JO O O

k0 O

2 (y) = N2 cosℎs (ℎ − y) − G sinℎs (ℎ − y)

uniform non-porous bottom of finite depth. Here, the effect
of porosity on the reflection and transmission coefficients are investigated numerically and corresponding graphs are
l/JO O
and

NO =

O s0

2k0 .jk0

2

O

2 2

plotted against the wave number of the incident wave for

2k0 G -G 2 h+k

h -2G k0

osh2k0 h+(k0 +G

)sinh2k0 h

different values of the porous parameter and for a fixed
value of the surface tension parameter.
NO =

2s0 G -G 2h+s2

2s0 .js0

-2G s0 0

2 2

0

0 h osh2s h+(s0 +G

)sinh2s h

2 MATHEMATICAL FORMULATION

We consider the two dimensional motion in case of uniform finite depth of water. A rectangular cartesian
coordinate system is chosen in which y −axis is taken
vertically downwards in the fluid region. The discontinuity

Here, e ik0 x l/JO (y) represents the incident wave field, R

and T are respectively the unknown reflection and
transmission coefficients to be determined. kO and sO are the
real positive roots (cf. McIver [14]) of the following two
transcendental equations in terms of A :
is taken at the original by assuming that the upper surface

A + MA3 + 1 Gt tanℎAℎ = K

+ G t + G′MA2 ,
of the ocean is covered by two vast inertial surfaces of
different materials and of different densities E1 , p and E2 p
A + MA3

+ 2 Gt tanℎAℎ = K + G t

+ G′MA2 .
respectively. Moreover, the inertial surfaces are subject to a
fixed value of the surface tension with parameter M given by M = T/pg where g is the acceleration due to gravity, T is the coefficient of surface tension, p is the density of water.
The ocean bottom is composed of some specific kind of
porous materials characterized by the porosity parameter

G′ where G t a , the quantity a is dimensionless constant

√v

which depends on the structure of the porous medium and
v is the permeability of the porous medium. Let a train of
surface water wave be incident from negative infinity,
which is partially reflected and partially transmitted through the ocean. Assuming that the fluid flow is
irrotational and the motion is simple harmonic in time t

3 SURFACE DISCONTINUITY : ENERGY IDENTITY

RELATION

As mentioned earlier, the difference in the wave number could arise due to change in the constant width of the region or sudden change in the boundary condition. The
energy identity |R|2 + |T|2 = 1 is not followed in this form
because of the presence of discontinuity at the upper
surface boundary condition at the junction of the two
inertial surfaces, i.e. at x = 0. However, the modified
energy identity has been formulated using Green’s integral
theorem by Evans and Linton [6]. Here we reproduce the same energy identity in case of uniform porous bottom. Two distinct types of solutions can be considered

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International Journal of Scientific & Engineering Research, Volume 4, Issue 4, April-2013 296

ISSN 2229-5518

3 2 Gt

tans ℎ + (K
+ G t − G t Ms2 ) = 0.
describing waves incident from either x → −∞ or x → ∞
respectively and these waves are partially reflected and
partially transmitted from x = 0.
When the wave train is incident from negative infinity
sn − Msn n 2 n sn
The potential function cp(x, y) can now be expanded for
two different regions in terms of orthogonal depth eigen
functions in the form given by
direction,
oo

B e -snx l/J2(y) as x > 0

Te is0 x l/JO

cp(x, y) =

n=O n

ik x 1

n

oo k x 1

(16)
cp(x, y)~

(e ik0 x + Re -ik0 x )l/J1(y) as x → −∞

(7)
e
where
0 l/JO (y) + ∑n=O An n

n y) 0,

And when the wave train is incident from x → +∞

te is0 x l/JO

AO = R
x(x, y)~

2 (y) as x → −∞

1

(8)
BO = T

(e ik0 x + re -ik0 x )l/JO (y) as x → ∞

and A , B n
are the unknown constants.

n n (

= 1,2, … )
Employing Green’s integral theorem for the two
functions x(x, y) and cp(x, y), along a contour L bounded by
the lines
The matching conditions at x = 0 for cp(x, y) and the
orthogonality of the depth eigen functions produce the
following two systems of linear equations:

y = 0, −X ≤ x ≤ X; x = ±X, 0 ≤ y ≤ ℎ; y = ℎ, −X ≤ x ≤ X (X > 0)

oo
= Ao
we get,

n=1 sn -km

O , (17)

oo

= N1 cosℎk (ℎ − y) − sinℎk (ℎ − y) −

L cpxT − cpT x)d = 0
(9)

n=1




kn-sm O O

k0 O

ik0+sm

ik0-sm

where rJ is the outward normal to the line element d .
There is no contribution to the integral from the part
where

2 G

(18)
0 < y < ℎ, x = 0 and y = ℎ, −X ≤ x ≤ X (X > 0). Now using

Vn = Bn Nn
cossn (ℎ − y) −

n

sinsn (ℎ − y ,
the far field conditions (7) and (8), we get,

2i k0

1 oshk (h-y)-G sinhk (h-y)

a|t| = |T|. (10)

( 2 - 1 )N0 0

k0 0

The following relations can be obtained by choosing the
functions cp, cp- ; x, x and cp, x in turn, in place of cp, x in (9)
respectively.
and Un = An Nn [coskn (ℎ − y) −
(m, n = 1,2 … )

sinkn (ℎ − y),

n

a(1 − |R|2) = |T|2, (11)
The unknown constants An , Bn (n = 1,2, … ) can be

2 2 estimated numerically from the above system of linear

a(1 − |r| ) = |t| , (12)
a|R||r| + |T||t| = 0, (13)
equations (17) and (18) after truncating the infinite sum upto desired accuracy. We now proceed to determine the

where a k0 . Now eliminating

s0

|r|, we obtain,

|t|

reflection and transmission coefficients by appropriate use

|R|2 + 1 |T|
= 1. (14)
of residue calculus method.

a

The above relation holds good in absence of the
discontinuity at x = 0 in the form
|R|2 + |T|2 = 1 (15)

5 REFLECTION AND TRANSMISSION COEFFICIENTS:

We consider the following integral
f(z)
which is the well known energy identity.

] = z − k
dz, (m = 0,1,2, … )

4 METHOD OF SOLUTION: EIGEN FUNCTION MATCHING TECHNIQUE

We consider the orghogonal depth eigen functions for
the two regions (x < 0 and x > 0) respectively as:

1 1 G

CN

Here the function f(z) has simple poles at z =

s , s , … , s , simple zeros at z = k , k , … , k and f(z): 0 1

1 2 n 1 2 n √z

as z → ∞. Here, CN are the sequence of circles with radius

RN which increases without bound as N → ∞ avoiding the
l/Jn (y) = Nn
coskn (ℎ − y) −

n

sinkn (ℎ − y) ,
zeroes of the integral and all the poles and zeros are inside

2 2 G

l/Jn (y) = Nn
where

N1 =

cossn (ℎ − y) −

n

2kn .jkn

2

sinsn (ℎ − y)

2

of it. Furthermore CN must not pass through (0,0).
The Cauchy’s integral formula and residue theorem
gives:

2 2 1

f(z)

(Res(f(z))|z=s )

2kn G -G

2

h+knh -2G kn os2knh+(kn+G

2sn .jsn

)sinkn h

f(kO ) =

2zi

CN z-k0

dz = ∑oo

n

sn -k0

(19)
Nn = 2 2


Assuming that f(kO ) = −1, we find

2sn G -G

h+snh -2G sn os2snh+(sn+G

)sinsn h

oO = ∑oo

(Res(f(z))|z=sn)

(20)
and kn , sn (n = 1,2,3 … . ) are given by the following two
equations:

sn -k0

Now comparing (19) and (17), we obtain
kn − Mk3

1 Gt kn

tankn ℎ + (K1 + G t − G t Mk2 ) = 0

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Vn = A Res(f(z)|z = sn ).

International Journal of Scientific & Engineering Research, Volume 4, Issue 4, April-2013 297

ISSN 2229-5518

Again, we consider the integral
f(z)
The expressions of the reflection and transmission coefficients given by (26) and (30) are computed

I =
z + k

CN

dz, (m = 0,1,2, … ),
numerically against the wave number of the incident wave
for a fixed value of surface tension. The effect of bottom
with the same property of the integrand function f(z) as above. The matching conditions at x = 0 can be combined
to give:
porosity is investigated on the values of the reflection and transmission coefficients by considering different values of
the dimensionless porosity parameter G′ℎ. The absolute

oo Vn

n=1

2ik0 R

1 G

(21)
values of the reflection and transmission coefficients are

sn +k0

( 2 - 1 )N0

oshk0 (h-y)-

k0sinhk0 (h-y)

given by:
The Cauchy’s residue theorem gives for m = 0 and at
z = −kO

R kO − sO
=

O + sO

oo
= Af(−k ). (22)
2kO

n=1 sn +k0 O T =

Comparing (21) and (22), we obtain

2ik0 R

kO + sO
These absolute values of the hydrodynamics coefficients
Af(−kO) =

( 2 - 1 )N1


oshk0 (h-y)-G sinhk0 (h-y)

0

(23)
are depicted in the fig.1 and fig.2 respectively for different
The function f(z) can be taken as
values of the dimensionless porosity parameter G t ℎ =

k0

k0 oo

1-kn 1-sn

0.00, 0.3, 0.5 respectively. The surface tension parameter M



f(z) = z n=1 1-

sn

1-k0

kn

(24)
is fixed at 0.1. In fig.1, it is seen that as the value of the
At z = −kO, from (24), f(z) gives:
dimensionless porosity parameter G′ℎ increases, the values
f(−kO) = − ∏oo

1+k0 kn

1+k0 sn

1-k0 sn

1-k0 kn

, (25)
of |R| decrease rapidly. This may be attributed due to the
presence of specific porous material at the bottom of the
and replacing A and f(−kO) in (22) we obtain:


k0 k0

ocean which resists the wave field reflection by the
discontinuity at the upper surface. The absolute values of

R k0 -s0

k +s

oo

n=1


1+kn 1-sn .

the reflection coefficient is decreased due to the

0 0

Thus,

1+k0

sn

1-k0

kn

characteristic of the porous materials at the bottom. The
reverse phenomena is observed in the fig.2 for the case of

R k0 -s0

k +s

e 2ia
, (26)
|T|. As the value of the porosity parameter G′ℎ increases,

0 0

where,
a = ∑oo
tan-1 k0
− tan-1 k0
, (n = 1,2, … )
the values of |T| also increase. This fact can be explained by
the energy identity relation given by (14) in case of a

n=1 sn kn

To obtain the transmission coefficient, we consider the following relation:
Vn = A Res(f(z)|z = sn ). (27) Since we have BO = T,
difference in wave number. Hence the effect of porosity in
the ocean bed does not violate the energy identity relation
Vn = Bn N2
obtain,

T


coss (ℎ − y) − G

n


sinsn (ℎ − y) and hence we

= 1 2 Gt

2kO P (sO + kO )

Gt

(K2 − K1 )NO NO

where,

cosℎkO (ℎ − y) − kO sinℎkO (ℎ − y) cosℎsO (ℎ − y) − sO sinℎ sO (ℎ − y)

(28)

oo n =1


1+ s0 kn

1+k0 kn

1+s0

sn

1+k0

kn

(29)
An alternative form of T can be obtained by using relation (14) and the expressions given by (28)-(29) as:

2k0


|T| = k +s
(30)

0 0 as given by (14) in presence of upper surface discontinuity.

We also note that, in absence of the upper surface

6 GRAPHS

discontinuity, we have kO = sO and in that case R = 0, T = 1.

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ISSN 2229-5518


Fig.1: Reflection coefficient for different values of porosity parameter at a fixed M
Fig.2: Transmission coefficient for different values of porosity parameter at a fixed M

7 CONCLUSION

The present study is concerned with scattering of surface wave by a discontinuity at the upper surface in a finite depth of water with porous bottom. The discontinuity arises due to presence of two types of semi-infinite inertial surfaces on either side of the origin. The inertial surfaces are also subject to surface tension. The eigen function matching technique and the residue calculus method have been made use of to determine analytical expressions of reflection and transmission coefficients. The magnitude of the hydrodynamic coefficients are plotted against wave number for different values of the porous parameter and for a fixed value of the surface tension parameter. From the analytical and numerical results, it is observed that the porous bottom has an effect on the absolute value of the hydrodynamic coefficients.
The present investigation is significant in its applications to ocean engineering, marine sciences and coastal dynamics.

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