hypersingular integral equation approach to the porouslate problem
. Gayen ∗, Arpita Mondalepartment of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur 721 302, West Bengal, India
r t i c l e i n f o
rticle history:eceived 27 March 2013eceived in revised form 30 January 2014ccepted 31 January 2014vailable online 20 March 2014
a b s t r a c t
Many problems of mathematical physics are investigated by reducing them to hypersingular integralequations. Most of them are formulated in terms of first kind hypersingular integral equations. Here wereduce the problem of wave scattering by a porous plate to solving a second kind hypersingular integralequation in terms of the difference in the unknown potential function across the plate. Exploiting theconditions at the two tips of the plate, the discontinuity is approximated by expanding it in terms of a finite
eywords:ater wave scattering
inear theoryorous plateypersingular integral equationeflection coefficient
series involving Chebyshev polynomials of the second kind, multiplied by an appropriate weight functionand then solved numerically by a collocation method. Using the solution, the reflection coefficient (R) andthe transmission coefficient (T) are computed numerically. In absence of the porous effect parameter G,the results show good agreement with those for an impermeable plate. It is observed that if the absolutevalue of the porous-effect parameter G be zero, the conservation of energy holds, that means, |R|2 + |T|2 = 1.However, when |G| > 0, |R|2 + |T|2 < 1 and a relevant energy identity has been determined in support of it.
. Introduction
Hypersingular integral equations play an important role intudying many research problems in various fields of Applied Math-matics. Seminal investigations include Martin and Rizzo [1], Chant al. [2] for elasticity, Krishnasamy et al. [3], Davydov et al. [4]or diffraction of electromagnetic and acoustic waves, Linkov and
ogilevskaya [5], Boström et al. [6] for elasticity, Vainikko et al.7] for aerodynamics and Farina and Martin [8] and Renzi and Dias9–11] for fluid dynamics. Generally hypersingular integral equa-ions are obtained as a result of reducing Neumann boundary valueroblems for the Laplace or Helmholtz equation to integral equa-ions. Integrals in this type of equations do not exist in ordinary,mproper or even Cauchy principal value sense. These are evaluateds Hadamard finite part integrals [12,13].
Parsons and Martin [14,15] first exploited the idea of applicationf hypersingular integral equation to study wave scattering by flatnd curved plates within the framework of linear water wave the-ry. Later, the technique was employed by Kuznetsov et al. [16] tonvestigate the existence of trapped modes, by Mandal and Gayen
17] and Gayen and Mandal [18] to study wave scattering by twoymmetric circular-arc shaped and two symmetric inclined plates
respectively, by Maiti and Mandal [19] to analyze the effect of anice-cover in wave scattering by a thin inclined barrier.
All the problems cited so far are formulated in terms of a firstkind hypersingular integral equation. Indeed, very few researchpapers are available in the literature wherein the problems arereduced to a second kind hypersingular integral equation (HSIE II).Ingber and Mondy [20] investigated Stokes flow with the veloc-ity components prescribed on the boundary of the domain byformulating the problem in terms of a hypersingular Fredholmintegral equation of the second kind. The Prandtl’s integro – dif-ferential equation can also be cast into an HSIE II [21,22]. Thetwo-dimensional problem of inverse scattering of acoustic wavesby sound – hard obstacle was reduced to an HSIE II by Mönch [23].
Our main aim of the present study is to explore the scope ofapplication of the second kind hypersingular integral equation inthe water wave problem. To the authors’ knowledge no problemin the water wave literature prior to the present work has beenreduced to a hypersingular integral equation of the second kind.Here, using linear theory, we have transformed the problem of scat-tering of water waves by a thin porous plate submerged beneath thefree surface by formulating the problem in terms of a hypersingularintegral equation of the second kind.
Breakwaters of various geometrical shapes and sizes are usedto protect a coast or activities along the coastline (e.g. ports, ship
wharves) from wave action. They are also used to prevent beacherosion. Breakwaters are generally designed to dissipate waveenergy so that the littoral drift can be slowed, sediment deposi-tion and a shoreline bulge or salient feature in the sheltered area
ehind the breakwater can be produced. A thin vertical rigid bar-ier is perhaps the simplest model of a breakwater for its simplicityn the engineering design. Dean [24] studied the problem of water
ave scattering by a thin vertical plate completely submerged andxtending infinitely downwards in deep water under the assump-ion of the linearized theory. Later, Ursell [25] obtained explicitolutions when the obstacle is in the form of a thin vertical plateartially immersed in deep water or a submerged thin vertical planearrier extending infinitely downwards. Evans [26] used a complexariable theory to study the problem of scattering of water wavesy a submerged vertical plate present in deep water. Porter [27]btained explicit solutions for scattering problems involving a sub-erged thin wall with a gap. Porter [28] investigated the reflection
f surface waves normally incident upon an infinite uniform hori-ontal cylinder of arbitrary cross-section totally immersed beneathhe free surface of a fluid of either finite or infinite depth by for-
ulating the problem in terms of a first-kind integral equationor an unknown function related to the tangential velocity of theuid around the cylinder. More recently, Chakraborty and Mandal29,30] investigated the problem of water wave scattering by a thinertical elastic plate submerged in deep and finite depth water byeducing the problem to the solution of three simultaneous integralquations.
Porous structures such as porous plates, rubble-mounds andoncrete armors are often used in coastal engineering for dissipat-ng wave energy from open sea. Numerous analytical studies on
ater wave interaction with porous mediums have been conductedo gain scientific insights into the hydrodynamic performance oftructures.
The topic of wave scattering by porous structures gained con-iderable importance since the analytical development of waveotion through porous media by Sollitt and Cross [31]. Macaskill
32] obtained the explicit solution of the two dimensional lin-arized problem of water-wave reflection by a thin barrier ofrbitrary permeability by formulating the problem in terms of aet of integral equations, which are solved using a special decom-osition of the finite depth source potential. Yu [33] employed anpproximate method to solve the problem of diffraction of surfaceater waves by a semi-infinite porous breakwater using a bound-
ry condition based on the formulation of Sollitt and Cross [31].cIver [34] employed Wiener–Hopf technique to show that the
roblem considered by Yu [33] has an exact solution for all anglesf wave incidence. McIver [35] later considered the diffraction of anncident wave by gaps between thin structures of porous breakwa-er using an integral equation formulation. The two-dimensionalroblems of scattering and radiation of small-amplitude wateraves by thin vertical porous plates were investigated by Lee andhwang [36] by applying the method of eigenfunction expansionnd converting those boundary value problems to certain dualeries relations, which are solved using the least square method.vans and Peter [37] solved the problem of water-wave reflec-ion by a semi-infinite and a finite submerged horizontal porouslate by the Wiener–Hopf technique and residue - calculus methodespectively. Recently, Tsai and Young [38] have solved the waveiffraction problem involving a thin porous vertical breakwater ofemi-infinite extent. They have used the method of fundamentalolution along with domain decomposition method.
In the current problem, we consider the scattering of wateraves by a thin porous plate submerged beneath the free surface.ppropriate use of the Green’s integral theorem in the fluid regionrovides representation of the potential function in terms of inte-rals involving discontinuity in the potential function across the
late. Use of the boundary condition on the porous plate producesn integral equation. This yields, when interpreted as a Hadamardnite part integral for the purpose of interchange of the orderf integration and differentiation, a second kind hypersingular
Fig. 1. Geometry of the porous plate.
integral equation for the discontinuity of the potential across theplate. This hypersingular integral equation is solved numericallyby approximating the unknown discontinuity in terms of a finiteseries involving Chebyshev polynomial of the second kind fol-lowed by a collocation method. The reflection and transmissioncoefficients are then computed numerically using the numericalsolution of this integral equation. The reflection coefficient isdepicted graphically against the wave number for different depthof submergence of the plate. It is observed that when the value ofthe porous effect parameter G is zero, the graph for |R| obtainedhere matches exactly with the corresponding graphs in Evans [26]for a single rigid vertical barrier in deep water. Although here wemainly consider a vertical plate, some results for a inclined plateare also given. The figures manifest that the reflection coefficientdecreases and thus the transmission coefficient increases whenthe absolute value of the porous effect parameter G increases forany fixed depth of submergence. This is due to the fact that someenergy is dissipated by friction as the fluid is pushed through theporous plate. Also this has been experimentally and numericallyshown in literature (cf. [36]). In this paper, we endeavor to provethis phenomenon by deriving an appropriate energy identityinvolving porous plate at the end of section 3.
2. Formulation of the problem
A rectangular Cartesian co-ordinate system is chosen in whichthe y-axis is taken vertically downwards into the fluid region andthe plane y = 0 denotes the position of the undisturbed free surface.A submerged thin porous plate in water occupies the position in (x,y) plane as illustrated in Fig. 1.
The fluid occupies the region y ≥ 0 or 0 ≤ y ≤ h according as itis of infinite depth or of uniform finite depth. The fluid region isdivided into two sub-regions: a negative sub-region (x < 0) and apositive sub-region (x > 0). The velocity potentials in these regionsare �1 and �2 respectively. The length of the porous plate is 2band d(> b) is the depth of submergence of its mid-point below themean free surface. In our discussion, we denote the barrier by Lb,and the gap by Lg, so that Lb
⋃Lg is [0, ∞) or [0, h] according as the
water is deep or of uniform finite depth. The fluid is assumed to beinviscid, incompressible and the flow is irrotational. Assuming thelinear theory, let the incident surface wave train be described bythe potential function Re{�inc(x, y)e−i�t} where
⎧ −Ky+iKx
�inc(x, y) =⎨⎩e for deep water (DW)
cosh k0(h − y)cosh k0h
eik0x for finite depth water (FDW)
(2.1)
7 Ocea
wattT
∇
t
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t
t
r
wm
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tm
aftac
�
bsdadn
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G
2 R. Gayen, A. Mondal / Applied
here � is the angular frequency and K = �2/g, g being the acceler-tion due to gravity and k0 is the unique positive real root of theranscendental equation k tanh kh = K. Let the resulting motion inhe fluid be described by the potential function Re
{�(x, y)e−i�t
}.
hen �(x, y) satisfies
2� = 0 in the fluid region, (2.2)
he free surface boundary condition
� + �y = 0 on y = 0, (2.3)
he bottom boundary condition,
∇� → 0 as y → ∞ for DW,
∂�∂y
= 0 on y = h for FDW,(2.4)
he end condition
1/2∇� is bounded as r → 0 (2.5)
here r is the distance of a fluid particle from either of the sub-erged ends of the plate.The boundary condition on the porous plate surface is given by,
∂�1
∂n= ∂�2
∂n= −ik1G[�](y) on Lb, (2.6)
here k1 = K or k0 according as the water is deep or of uniform finiteepth, [�](y) = �2(y) − �1(y), ∂
∂ndenoting the normal derivative at a
oint on Lb and G(= Gr + iGi) is the porous-effect parameter definedy Yu and Chwang [39] as
= �(f + iS)k1d(f 2 + S2)
. (2.7)
Here � is the porosity; f is the resistance force coefficient; S ishe inertial force coefficient and d is the thickness of the porous
edium.Let a train of regular, small-amplitude progressive waves prop-
gate towards the barrier from the direction of x =− ∞. When italls on the plate, some part of it is transmitted above or below orhrough the pores of the plate and rest of it is reflected back. If Rnd T denote the reflection and transmission coefficients, then theonditions at infinity are given by
(x, y) →{�inc(x, y) + R�inc(−x, y) as x → −∞,T�inc(x, y) as x → ∞.
(2.8)
In the next section we describe a method through which theoundary value problem as given in Eqs. (2.2)–(2.8) is reduced toolving a hypersingular integral equation of the second kind in theiscontinuity function [�](y). This integral is solved numerically byn expansion-collocation method, taking into account the end con-ition (2.5). The numerical solutions are then utilized to compute
umerical values of R and T.
. Method of solution
The fundamental potential function G(x, y; �, �) due to a lineource situated at (�, �)(� > 0) is given by
(x, y; �, �) = lnr
r′− 2
∫C
e−k(y+�)
k − Kcos k(x − �)dk for DW (3.1a)
n Research 46 (2014) 70–78
and
G(x, y; �, �)
= lnr
r′− 2
∫ ∞
0
e−kh sinh k� sinh ky
k cosh khcos k(x − �)dk
−2
∫C ′
cosh k(h − �) cosh k(h − y)k sinh kh − K cosh kh
cos k(x − �)cosh kh
dk for FDW.
(3.1b)
Here r, r′ = {(x − �)2 + (y ∓ �)2}1/2and the path C is along the
positive real axis in the complex k-plane indented below the poleat k = K and C′ is indented below the unique real positive root k0 ofthe transcendental equation k tanh kh = K.
We now apply the Green’s integral theorem∫V
(G∇2 − ∇2G)dV =∫�
(G∂ ∂n
− ∂G∂n
)ds (3.2)
to the functions
(x, y) =
⎧⎨⎩�(x, y) − e−Ky+iKx for DW,
�(x, y) − cosh k0(h − y)cosh k0h
eik0x for FDW
and G(x, y; �, �). Here � is the boundary of the region enclosed bythe lines y = 0, − X ≤ x ≤ X; x = X, 0 ≤ y ≤ Y; y = Y, − X ≤ x ≤ X; x = − X,0 ≤ y ≤ Y; a small circle of radius with the center at (�, �) anda contour enclosing Lb for deep water. For finite depth water, wetake the same region with Y replaced by h. We ultimately make X,Y→ ∞ (for DW), X→ ∞ (for FDW) and → 0 and shrink the contourenclosing Lb to the two sides of Lb, to obtain
(�, �) = − 12
∫Lb
{ (0+, y) − (0−, y)}∂G∂x
(0, y; �, �)dy. (3.3)
However,
(0+, y) − (0−, y) = �(0+, y) − �(0−, y)
= �2(y) − �1(y) = [�](y), y ∈ Lg.
This gives an integral representation for the velocity potential �(�,�) as
�(�, �) = e−K�+iK� − 12
∫Lb
[�](y)∂G∂x
(0, y; �, �)dy for DW, (3.4a)
�(�, �) = cosh k0(h − �)cosh k0h
eik0�
− 12
∫Lb
[�](y)∂G∂x
(0, y; �, �)dy for FDW. (3.4b)
It should be noted that the unknown function [�](y) vanishes atthe end points of Lb. In order to get an integral equation in [�](y),we use the boundary condition (2.6) on the porous plate. For this,we differentiate both sides of (3.4) with respect to �. This yields thehypersingular integral equation of the second kind:
(3.5a)
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a
t(t
cnL
n
tp
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K
a
K
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ffp
R. Gayen, A. Mondal / Applied
nd
(3.5b)
Here the sign ‘=’ on the integral sign indicates that the integral iso be interpreted as a Hadamard finite part integral. Eqs. (3.5a) and3.5b) are to be solved subject to the condition that [�] vanishes athe end points of Lb.
Here we present the detailed analysis for a vertical plate. Thisan be generalized further for an inclined plate. Thus, let np′ and
q′ denote the unit normals at the points p′ and q′ respectively onb. Then according to the geometry of our problem
p′ = (1, 0) = nq′; (3.6)
he co-ordinates of the points p′ ≡ (x, y) and q′ ≡ (�, �) on Lb beingarametrically expressed as
x = 0, y = d − bu, −1 ≤ u ≤ 1,
� = 0, � = d − bs, −1 ≤ s ≤ 1.(3.7)
Using this parametrization, the kernel of the integral equationsn (3.5a) and (3.5b) can be rewritten as
∂2G∂x∂�
= − 1
(s − u)2+ K(s, u)
here
(s, u) = b2
[X2 − Y2
(X2 + Y2)2− 2KYX2 + Y2
− 2K2˚0(X, Y)
]for DW
(3.8a)
nd
(s, u) = b2
[Y2 − X2
(X2 + Y2)2− 2
∫ ∞
0
ke−kh sinh k� sinh ky
cosh khcos k(x − �)dk
−4i
{k0
2 cosh k0(h − �) cosh k0(h − y)eik0 |x−�|
2k0h + sinh 2k0h
+∞∑n=1
ikn2 cos kn(h − �) cosh kn(h − y)e−kn |x−�|
2knh + sin 2knh
}]for FDW (3.8b)
with
= x − �, Y = y + �,
0(X, Y) =∫C
e−kY
k − Kcos kXdk = ie−KY+iK |X|
−∫ ∞
0
K sin kY − k cos kY
k2 + K2e−k|X|dk. (3.9)
It may be noted that K(s, u) in (3.8a) and (3.8b) are smoothunctions of s and u. If we introduce a new unknown function(u) = [�(p′(u))], representing the discontinuity in � across the
orous plate at the point p′ ≡ (x, y), Eq. (3.5) becomes
(3.10)
n Research 46 (2014) 70–78 73
where −1 < s < 1 and
h(s) =
⎧⎨⎩
2iKbe−K(d−bs) for DW,
2ik0bcosh k0(h − (d − bs))
cosh k0hfor FDW.
(3.11)
We now solve the hypersingular integral equations (3.10), keep-ing in mind that
f (±1) = 0. (3.12)
To solve Eq. (3.10), we now approximate f(u) as (cf. [40])
f (u) = (1 − u2)1/2
N∑n=0
anUn(u) (3.13)
where Un(u) is the Chebyshev polynomial of the second kind andan(n = 0, 1, . . ., N)’s are unknown constants to be found. The squareroot factor in (3.13) ensures that f(u) i.e. [�](y), has the correctbehavior at the end points of the porous plate. Using the expansions(3.13) in (3.10), we get
N∑n=0
anAn(s) = h(s) (3.14)
where
An(s) = {(n + 1) − 2ibk1G(1 − s2)1/2}Un(s)
+∫ 1
−1
(1 − u2)1/2K(s, u)Un(u)du. (3.15)
To find the unknown constants an(n = 0, 1, . . ., N), we puts = sj(j = 0, 1, . . ., N) in (3.14) to obtain the linear system of equationsas
N∑n=0
anAn(sj) = h(sj) (3.16)
for j = 0, 1, . . ., N where sj’s are the collocation points chosen as
sj = cos2j + 12N + 2
, j = 0, 1, . . ., N. (3.17)
These are the zeros of the Chebyshev polynomial of the first kind.If G = 0, Eq. (3.10) reduces to the first kind hypersingular integral
equation
(3.18)
Golberg [41,42] has shown that sj’s as given in Eq. (3.17) area good choice: he has proven that it yields uniformly-convergentmethods,
max−1≤u≤1
|g(u) − gN(u)| → 0 as N → ∞
where
f (u) = (1 − u2)1/2g(u) (3.19)
and
g(u) ∼=N∑anUn(u) ≡ gN(u). (3.20)
n=0
The convergence of the second kind hypersingular integralequation (3.10) can be proven in a similar manner. So we skip thedetails of it.
7 Ocean Research 46 (2014) 70–78
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Table 1Values of |R| and |T| when the plate is impermeable (d/p = 0.625, b/p = 0.375, G = 0).
Kp |R| |T| |R|2 + |T|2
0.6 0.0814 0.9967 1.0000
for three different values of depths of submergence d/b of the mid-point of the plate. It shows the effect of different depth parameterson |R| for a fixed and real value of the porous-effect-parameter(G = 1). It is observed that |R| reduces with the increase in the
Table 2Values of |R|, |T| and 1 − J for a permeable barrier (d/p = 0.625, b/p = 0.375, G = 1).
Kp |R| |T| |R|2 + |T|2 1 − J
4 R. Gayen, A. Mondal / Applied
Eq. (3.16) is a linear system of (N + 1) equations in (N + 1)nknowns, which can be solved by any standard method to deter-ine an’s numerically. Here we use Gauss–Jordan eliminationethod.
.1. Reflection and transmission coefficients
The reflection and the transmission coefficients R and T cane determined approximately in terms of series involving theonstants an’s defined in §3. This is achieved by making �→ ∓ ∞iven in (3.4) and comparing with the infinity conditions (2.8) (withx, y) replaced by (�, �)). For this, we require the asymptotic result
(x, y; �, �)
→
{−2ie−K(y+�)+iK |x−�| as |x − �| → ∞ for DW,
−4i cosh k0(h − y) cosh k0(h − �)2k0h + sinh 2k0k
eik0 |x−�| as |x − �| → ∞ for FDW.
(3.21)
Using the above result, we find the reflection and transmissionoefficients as
= −b∫ 1
−1
f (u)L(u)du (3.22)
here L(u) = M(y(u)),
(y) =
⎧⎨⎩Ke−Ky for DW,
2k0 cosh k0h
2k0h + sinh 2k0hcosh k0(h − y) for FDW.
nd
= 1 − R. (3.23)
he integrals (3.22) and (3.23) can be computed numerically forifferent values of the parameters Kb, Kh, d/h, b/h and d/b. Thus,nce the an’s are found by solving the linear system (3.16), |R| and |T|an be computed numerically from (3.22) and (3.23), respectively.f the plate is impermeable, i.e. when G = 0, |R| and |T| must satisfyhe identity
R|2 + |T |2 = 1. (3.24)
However, this is not the case when |G| > 0. This is established inhe following subsection.
.2. The energy identity
Due to porosity of the plate, some energy will be dissipated byt [36]. So the value of |R|2 + |T|2 will be less than unity. To provet mathematically, we once again apply a modified form of Green’sntegral theorem. We apply this theorem to the functions � and
in the fluid region bounded by the lines y = 0, − X ≤ x ≤ X; x = X, ≤ y ≤ Y; y = Y, − X ≤ x ≤ X; x = − X, 0 ≤ y ≤ Y and a contour enclosingb for deep water. For finite depth water, we take the same regionith Y replaced by h. Finally making X, Y→ ∞ (for DW), X→ ∞ (for
DW), we obtain the energy identity as
R|2 + |T |2 = 1 − 2KGr
∫Lb
|(�2 − �1)|2ds (3.25)
here Gr(= (�f)/(k1d(f2 + S2))) is the real part of G. After using theelations (3.7) and (3.13), Eq. (3.25) finally takes the form
In the expression of J, Gr as well as the integrand is positive.Therefore, from Eq. (3.26) it is clear that |R|2 + |T|2 < 1 . In the nextsection, we present |R| and |T| in a number of figures. While takingthe data, it has been observed that the values of |R| and |T| satisfy theenergy identities (3.24) and (3.26) for G = 0 and G /= 0 respectively.
4. Numerical results
For all the numerical computations, we take N = 10. We firstpresent two tables, which validate the correctness of the numericalresults computed by the present method. For the data given in thetables we make the different physical quantities non-dimensionalwith respect to the parameter p. Here p(= d + b), is the depth of thelower end of the vertical plate. In Table 1, a representative set ofvalues of |R|, |T| and |R|2 + |T|2 against Kp for d/p = 0.625, b/p = 0.375,G = 0 are given. This table shows that for an impermeable barrier,|R|2 + |T|2 is exactly equal to unity. Table 2 displays the numer-ical results for |R|2 + |T|2 and 1 − J for different values of Kp ford/p = 0.625, b/p = 0.375, G = 1. |R| and |T| have been calculated from(3.22) and (3.23) whereas 1 − J is the left hand side of Eq. (3.26). Thisverifies the fact that for non-zero G, |R| and |T| satisfy the energyidentity as given in (3.25).
Fig. 2 stands for comparing the values of |R| determined fromour method with the data available in Evans [26] for a rigid verticalplate. This has been accomplished by introducing a new parameter� = d−b
d+b which is the ratio of submergence of the top end to thebottom end of the plate and taking the value of G to be zero. We plot|R| against Kp for three different values of �(= 0.01, 0.05, 0.25). Thegraph shows that the agreement between our results with those inEvans [26] for an impermeable barrier is highly satisfactory.
Fig. 3 shows the results for |R| and |T| against Kp for � (=0.01,0.05, 0.25) and G = 1. We observe that as in Fig. 2, here also |R|decreases with an increase in the value of �. However, the key dif-ference between the two figures (Figs. 2 and 3) is that the heightsof the reflection curves are less in Fig. 3 i.e. less energy is reflectedby a permeable breakwater. The curves for |T| describes oppositebehavior to that of |R| mentioned here.
For the next figures, the parameters have been made non-dimensional with respect to the half plate-length b for the caseof deep water. Fig. 4 depicts the results for |R| and |T| against Kb
R. Gayen, A. Mondal / Applied Ocean Research 46 (2014) 70–78 75
Fa
dai
tmp
wtpefll
Fw
0 0.5 1 1.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Kb
|R|,|T|
0.75
0.851.0
0.75
0.851.0
ig. 2. Reflection coefficient for vertical porous plate in deep water, for various �nd fixed value G = 0.
epth of submergence of the plate. This is likely to happen as were considering surface waves whose effect diminishes with thencrease in the depth of submergence of a barrier.
The same phenomenon is observed in Figs. 5 and 6 where weake G = 2 and G = 1 +0.5i respectively. Here we mention that for a
edium in which the resistance dominates the inertial effect, theorous-effect-parameter G becomes purely real [36].
The influence of permeability of the plate on the propagation ofater waves is graphically presented in Fig. 7. This figure reveals
hat the amount of reflections decreases with the increase in theermeability of the plate. This phenomenon is important in coastalngineering, because the porous structures cause smaller surface
uctuation due to the low reflection which is important for ship
oading and unloading.
0 0.5 1 1.5 2 2.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Kp
|R|,|T|
µ=0.01µ=0.05
µ=0.25
µ=0.01µ=0.05
µ=0.25
ig. 3. Reflection and Transmission coefficients for vertical porous plate in deepater, for various � and fixed value G = 1.
Fig. 4. Reflection and transmission coefficients for vertical porous plate in deepwater, for various submergence depths d/b and fixed value G = 1.
Fig. 8 displays |R| for three different depths of submergenceof the porous plate in water of uniform finite depth h. Here |R|has been plotted against non-dimensional wave number Kb forfixed values of G and d/b as G = 1 and d/b = 0.85; h is varied takingd/h = 0.5, 0.3, 0.1. It is noticed that when the depth of the mid-pointof the vertical plate is one-tenth of the bottom depth (i.e. d/h = 0.1),the results almost coincide with those for deep water as given inFig. 4. From this figure, it is evident that the bottom effect appears
to be significant in the lower wavenumber range, e.g. for Kb < 0.7for d/h = 0.5 and Kb < 0.4 for d/h = 0.3. This happens because in thelower wavenumber range, the wavelength of the incident wave
0 0.5 1 1.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Kb
|R|,|T|
0.75
0.851.0
0.75
0.851.0
Fig. 5. Reflection and transmission coefficients for vertical porous plate in deepwater, for various submergence depths d/b and fixed value G = 2.
76 R. Gayen, A. Mondal / Applied Ocean Research 46 (2014) 70–78
0 0.5 1 1.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Kb
|R|,|T|
0.75
0.851.0
0.75
0.851.0
Fw
ttttd
pw
Fw
0 0.5 1 1.50
0.1
0.2
0.3
0.4
0.5
Kb|R|
d/h=0.3
d/h=0.1
d/h=0.5
ig. 6. Reflection and transmission coefficients for vertical porous plate in deepater, for various submergence depths d/b and fixed value G = 1 +0.5i.
rain is relatively large so that it can penetrate sufficiently belowhe free surface. In this situation not only the porous plate, but alsohe bottom plays a significant role in reflecting the incident waverain. However, with the increase in wavenumber, bottom effect
ecreases thereby reducing in the amount of reflections.
The next two figures depict the effect of porosity on |R| when thelate is inclined. The variation of |R| against the non-dimensionalave number Kb for different angles of inclinations of the plate
0 0.5 1 1.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Kb
|R|,|T|
G=2
G=1
G=0
G=2
G=1
G=0
ig. 7. Reflection and transmission coefficients for vertical porous plate in deepater, for various values of G and fixed value d/b = 0.85.
Fig. 8. Reflection coefficient for vertical porous plate in finite depth water, for var-ious submergence depths and fixed values G = 1, d/b = 0.85.
with the vertical has been plotted in Fig. 9, by taking the angles as0◦, 5◦ and 10◦ with the vertical, G = 1 and d/b = 0.75. In this case theupper end of the plate deflects from its initial position towards theleft so that the surface waves get enough space to transmit towardsthe other side. This causes reduction in the amount of reflectionswith the increase in the inclination of the plate, which is evidentfrom Fig. 9.
The results for a rigid plate (taking G = 0) inclined at an angleof 45◦ to the vertical, and three different depths of submergencehave been plotted in Fig. 10. These curves show a good agreement
Fig. 9. Reflection coefficient for porous plate, for various inclinations and fixed valueG = 1.
R. Gayen, A. Mondal / Applied Ocea
Fig. 10. Reflection coefficient for a plate inclined at 45◦ , for various submergencedepths d/b and fixed value G = 0.
0 0.5 1 1.5 2 2.5 30
0.1
0.2
0.3
0.4
0.5
J
G=2
G=1G=1.5
b[|t
efdfpe
5
tbtchld
[
[
[
[
[
[
[
[
[
[
[
[
Kp
Fig. 11. Dissipated energy for a vertical porous plate, for various G.
etween our results with those obtained by Parsons and Martin14] for a rigid inclined barrier. From this graph, it is observed thatR| reduces with depth of submergence and approaches to zero ashe waves become shorter.
Finally, the effect of porosity on the proportion of dissipatednergy, J, has been shown in Fig. 11. This figure has been plottedor the three different values of G (=1, 1.5, 2) and the fixed values of/p = 0.625 and b/p = 0.375. It may be noticed from the figure that,or Kp < 1.3, energy dissipation increases with the increase in theorous-effect parameter G. However, for larger wave numbers, lessnergy is dissipated as the permeability increases.
. Conclusion
The method of hypersingular integral equation has been appliedo consider the reflection and transmission of the surface wavesy a thin porous plate. Application of Green’s integral theoremo suitable functions followed by utilization of the boundary
ondition on the porous breakwater has led to a second kindypersingular integral equation, whose occurrence is rare in the
iterature. An energy identity is also derived, which confirms thatue to porous effect of the plate, some energy is dissipated. The
[
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n Research 46 (2014) 70–78 77
reflection and the transmission coefficients are computed numer-ically using the solution of the integral equation. In absence of theporous-effect-parameter, numerical results of |R| completely agreewith the benchmark paper of Evans [26]. Also, it has been shownthat computed values of |R| and |T| satisfy the energy identities forpermeable and impermeable breakwaters.
The present investigation can be further extended to analyzethe corresponding radiation problem. This will explore the scopeto derive different hydrodynamic parameters such as added iner-tia, radiation damping, exciting forces, wave-drift forces etc. Thedifferent mathematical relations like Haskind–Hanaoka relation,Bessho–Newman relation can also be determined. It can be men-tioned that the Haskind relation for a porous plate is identical tothat for an impermeable plate. However, the relation between thedamping and the corresponding wave exciting forces includes anadditional term arising from the porosity of the plate. This is alsothe case for Bessho–Newman relation, where a correction term fordissipation due to porosity will arise.
Acknowledgements
The work is supported by DST, India. We thank Mr. Ashesh Paulfor his help in numerical computations.
References
[1] Martin PA, Rizzo FJ. On the boundary integral equations for crack problems.Proceedings of the Royal Society London A 1989;421:341–55.
[2] Chan YS, Albert CF, Glaucio HP. Integral equations with hypersingular ker-nels – theory and applications to fracture mechanics. International Journal ofEngineering Science 2003;41:683–720.
[3] Krishnasamy G, Schmerr LW, Rudolphi TJ, Rizzo FJ. Hypersingular boundaryintegral equations: some applications in acoustic and elastic wave scattering.Journal of Applied Mechanics 1990;57:404–14.
[4] Davydov AG, Zakharov EV, Pimenov YV. Hypersingular integral equations forthe diffraction of electromagnetic waves on homogeneous magneto-dielectricbodies. Computational Mathematics and Modeling 2006;17:97–104.
[5] Linkov AM, Mogilevskaya SG. Hypersingular integrals in plane problems ofelasticity. Journal of Applied Mathematics and Mechanics 1990;54:116–22.
[6] Boström A, Grahn T, Niklasson AJ. Scattering of elastic waves by a rectangularcrack in an anisotropic half-space. Wave Motion 2003;38:91–107.
[7] Vainikko GM, Lebedeva NV, Lifanov IK. Numerical solution of a singular and ahypersingular integral equation on an interval and the delta function. Sbornik:Mathematics 2007;193:1397–410.
[8] Farina L, Martin PA. Scattering of water waves by a submerged disc using ahypersingular integral equation. Applied Ocean Research 1998;20:121–34.
[9] Renzi E, Dias F. Resonant behaviour of an oscillating wave energy converter ina channel. Journal of Fluid Mechanics 2012;701:482–510.
10] Renzi E, Dias F. Relations for a periodic array of flap-type wave energy convert-ers. Applied Ocean Research 2013;39:31–9.
11] Renzi E, Dias F. Hydrodynamics of the oscillating wave surge converter in theopen ocean. European Journal of Mechanics B: Fluids 2013;41:1–10.
12] Hadamard J. Lectures on Cauchy Problem in Linear Partial Differential Equa-tions. London: Oxford University Press; 1923.
13] Kaya AC, Ergodan F. On the solution of integral equations with strongly singularkernels. Quarterly of Applied Mathematics 1987;45:105–22.
14] Parsons NF, Martin PA. Scattering of water waves by submerged plates usinghypersingular integral equations. Applied Ocean Research 1992;14:313–21.
15] Parsons NF, Martin PA. Scattering of water waves by submerged curved platesand by surface-piercing flat plates. Applied Ocean Research 1994;16:129–39.
16] Kuznetsov N, McIver P, Linton CM. On uniqueness and trapped modes in thewater-wave problem for vertical barriers. Wave Motion 2001;33:283–307.
17] Mandal BN, Gayen R. Water wave scattering by two symmetric circular-arc-shaped thin plates. Journal of Engineering Mathematics 2002;44:297–309.
18] Gayen R, Mandal BN. Water wave scattering by two thin symmetric inclinedplates submerged in finite depth water. International Journal of AppliedMechanics and Engineering 2003;8:589–601.
19] Maiti P, Mandal BN. Wave scattering by a thin vertical barrier submergedbeneath an ice-cover in deep water. Applied Ocean Research 2010;32:367–73.
24] Dean WR. On the reflection of surface waves by a submerged plane barrier.Proceedings of Cambridge Philosophical Society 1945;41:231–8.
25] Ursell F. The effect of a fixed vertical barrier on surface waves in deep water.Proceedings of Cambridge Philosophical Society 1947;43:374–82.
26] Evans DV. Diffraction of surface waves by a submerged vertical plate. Journalof Fluid Mechanics 1970;40:433–51.
27] Porter D. The transmission of surface waves through a gap in a vertical barrier.Proceedings of the Cambridge Philosophical Society 1972;71:411–21.
28] Porter R. Surface wave scattering by submerged cylinders of arbitrary cross-section. Proceedings of the Royal Society A 2002;458:581–606.
29] Chakraborty R, Mandal BN. Scattering of water waves by a sub-merged thin vertical elastic plate. Archive of Applied Mechanics 2013,http://dx.doi.org/10.1007/s00419-013-0794-x (in press).
30] Chakraborty R, Mandal BN. Water waves scattering by an elastic thin verticalplate submerged in finite depth water. Journal of Marine Science and Applica-tions 2013;12, http://dx.doi.org/10.1007/s11804-009-7078-4.
31] Sollitt CK, Cross RH. Wave transmission through permeable breakwa-ters. In: Proceedings of 13th Conference on Coastal Engineering. 1972.p. 1827–46.
32] Macaskill C. Reflexion of water waves by permeable barrier. Journal of FluidMechanics 1979;95:141–57.
[
[
n Research 46 (2014) 70–78
33] Yu X. Diffraction of water waves by porous breakwaters. Journal of WaterwayPort Coastal and Ocean Engineering 1995;121:275–82.
34] McIver P. Water-wave diffraction by thin porous breakwater. Journal of Water-way Port Coastal and Ocean Engineering 1999;125:66–70.
35] McIver P. Diffraction of water waves by a segmented permeable breakwater.Journal of Waterway Port Coastal and Ocean Engineering 2005;131:69–76.
36] Lee MM, Chwang AT. Scattering and radiation of water waves by permeablebarriers. Physics of Fluids 2000;12:54–65.
37] Evans DV, Peter MA. Asymptotic reflection of linear water waves by submergedhorizontal porous plates. Journal of Engineering Mathematics 2011;69:135–54.
38] Tsai CH, Young DL. The method of fundamental solutions for water-wavediffraction by thin porous breakwater. Journal of Mechanics 2011;27:149–55.
39] Yu X, Chwang AT. Wave induced oscillation in harbour with porousbreakwaters. Journal of Waterway Port Coastal and Ocean Engineering1994;120:125–44.
40] Linton CM, McIver P. Handbook of Mathematical Techniques for
Wave/Structure Interactions. CRC Press; 2010.
41] Golberg MA. The convergence of several algorithms for solving integral equa-tions with finite-part integrals. Journal of Integral Equations 1983;5:329–40.
42] Golberg MA. The convergence of several algorithms for solving integral equa-tions with finite-part integrals II. Journal of Integral Equations 1985;9:267–75.
