Solution of the problem of propagation of water waves over a pair of asymmetrical rectangular trenches

Applied Ocean Research 93 (2019) 101946

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Solution of the problem of propagation of water waves over a pair of asymmetrical rectangular trenches

T

Amandeep Kaura, S.C. Martha ,a, A. Chakrabartib ⁎

a b

Department of Mathematics, Indian Institute of Technology Ropar, Rupnagar 140001, Punjab, India Department of Mathematics, Indian Institute of Science, Bangalore 560012, Karnataka, India

ARTICLE INFO

ABSTRACT

Keywords: A pair of trenches System of singular integral equations of first kind Polynomial approximation Reflection and transmission coefficients

The problem of propagation of obliquely incident surface water waves over a pair of asymmetrical rectangular trenches in a channel of finite depth is examined for its solution. The mathematical problem is handled for its numerical solution with the aid of a system of singular integral equations of first kind. The resulting integral equations are solved numerically by using suitably designed polynomial approximations of the unknown functions. System of linear algebraic equations is obtained by utilizing the zeros of Chebyshev polynomial of first kind as collocation points and hence the occurrence of ill-conditioned matrix is avoided. The effectiveness of the pair of trenches is studied by analyzing the physical quantities namely the reflection and transmission coefficients. As a special limiting case, the results for a single trench problem are derived and are found to be in excellent agreement with the results available in the literature. The effect of various parameters are analyzed through different graphs for a pair of asymmetrical trenches. The energy balance relation is derived and used to check the accuracy of numerical results.

1. Introduction The study of propagation of surface water waves over irregular bottom topography is of fundamental interest due to their significant applications in the field of coastal and marine engineering. Different kinds of bottom topography have been investigated in the last few decades to examine the characteristics of transformation of wave energy. In this direction, the problems of propagation of surface waves over finite depth of water and bottom undulation were studied by various workers using different solution techniques. In this context, Lee and Ayer [1] employed transform method to obtain the solution of propagation of normally incident waves over a symmetric rectangular trench. They divided the fluid domain into two regions along the mouth of the trench and the solution was obtained by matching the solutions in each sub-region along the common boundary. Further, the problem involving diffraction of obliquely incident surface waves by an asymmetric trench was investigated by Kirby and Dalrymple [3] with the aid of numerical solution which was constructed by matching particular solutions for each sub-region of constant depth along vertical boundaries. Moreover, they obtained the long-wave asymptote of the planewave solution and compared their results with Miles [2] for normal and oblique incidence. Further, Bender and Dean [4] proposed the slope method for wave propagation over a trench or shoal with linear ⁎

transitions. However, in their method, the final formula of reflection coefficient was given by employing numerical method rather than presenting in an explicit form. To improve the slope method by Bender and Dean [4], Liu and Lin [6] derived analytical solution, where a variable transform was introduced to convert the quasi-idealized depth profile (i.e. a constant plus a power function) into an idealized profile, thus the reflection coefficient in a simple closed form was presented. Moreover, Lin and Liu [5] obtained the closed-form analytical solution for wave interaction with an obstacle of general trapezoidal shape based on the shallow water theory. Jung et al. [7] derived an analytical solution to the long-wave equation in terms of a Taylor series for waves propagation over an asymmetrical trench. Jung and Cho [8] gave an approximate analytical solution for long-wave reflection by an arbitrarily varying topography. Further, Xie et al. [9] also presented a closed-form analytical solution for long-wave reflection by a rectangular obstacle with two scour trenches. Liu et al. [10] developed an analytical solution for linear longwave reflection by a submerged breakwater or trench with curvilinear slopes. Based on the mild-slope equation theory, Liu et al. [11] derived the recursive formulae to calculate the arbitrary order derivatives of bottom curvature term and the slope square term for wave propagation over two dimensional piecewise smooth topographies. Further, this exact analytical solution technique was utilized by Xie and Liu [12] to

Correspondence author. E-mail address: [email protected] (S.C. Martha).

https://doi.org/10.1016/j.apor.2019.101946 Received 13 June 2019; Received in revised form 25 August 2019; Accepted 23 September 2019 0141-1187/ © 2019 Elsevier Ltd. All rights reserved.

Applied Ocean Research 93 (2019) 101946

A. Kaur, et al.

solve the modified mild-slope equation for wave propagation over a trench with various shapes. Moreover, an analytical solution to the modified mild slope equation in terms of Taylor series was constructed for wave transformation by a trench or breakwater with general trapezoidal shape by Xie and Liu [13]. Liu et al. [14] used similar analytical technique as in Liu et al. [11] for solving the modified mild-slope equation to study the wave reflection by a rectangular breakwater with two scour trenches. They investigated the influences of scour trench dimensions on the reflection through computational results. The problem involving linear wave response of a two-dimensional floating plate on water of variable depth was investigated by Wang and Meylan [15]. Further, the problems involving wave structure interaction over finite depth of water were studied by various workers (ref. Liu et al. [16,17], Bhattacharjee and Soares [18] and Meng and Lu [19]). Further, Chakraborty and Mandal [20,21] investigated the normally and obliquely incident water wave scattering problem involving a symmetric rectangular trench with the aid of multi-term Galerkin approximation technique. Due to the geometrical symmetry of the rectangular trench, they split the problem into two separate problems involving symmetric and anti-symmetric potential functions. Maiti et al. [22] used the integral equation approach by a suitable application of Green’s integral theorem to solve the scattering of water waves by thin vertical plate submerged below the ice cover surface in infinite depth of water. Moreover, Roy et al. [23] applied the multi-term Galerkin approximation technique to solve the problem of propagation of surface waves over an asymmetric trench. Further, Roy et al. [25] used the same aforesaid technique to solve the problem of wave scattering by multiple thin vertical barriers. In this paper, the method of polynomial approximation to solve the resulting singular integral equations is developed towards the analysis of transformation of wave energy by a pair of trenches. The associated mixed boundary value problem arising here is solved by using matched eigenfunction expansion method. It is shown that the presently considered problem can be solved by the aid of a system of integral equations. These equations are solved by using suitably designed polynomial approximations of the unknown functions, giving rise to numerical values of the reflection and transmission coefficients associated with the reflected and transmitted waves. These coefficients are analyzed through different graphs to study the transformation of wave energy. The energy balance relation is derived and checked numerically. As a special case, the results for a single trench profile are derived and validated with the known results available in the literature.

fluid R1:

R3:

domain is divided into five regions such < x < b , 0 y h1, R2: b < x < a, 0 y < h2, .

a < x < a, 0

y < h3, R 4 : a < x < b, 0

as

y < h4,

R5: b < x < , 0 y < h5 The width of each trench is l = b a and the distance between a pair of trenches is a1 = 2a. Let a train of monochromatic surface waves with angular frequency ω is propagating from left side and obliquely incident at an angle θ1 to the pair of asymmetrical trenches. Assuming linear theory and irrotational flow, the incident velocity potential can be inc (x , y , z , t )=Re{ inc (x , y ) e i ( z t ) }, written as where inc (x ,

y) =

cosh k 0(1) (y

h1) iµ (x + b) e

cosh k 0(1) h1

with µ = k 0(1) cos

= k0(1) sin

1,

1.

k 0(1) is

2

the real positive root of k tanh kh1 = K with K = g , g is acceleration due to gravity. The phase speed along rays of the incident wave leads to Snell’s law for refraction across discontinuities in the water depth, resulting = k 0(1) sin 1 = k 0(2) sin 2 = k0(3) sin 3 = k 0(4) sin 4 = k 0(5) sin 5 as mentioned by Kirby and Dalrymple [3]. Here, k 0(2), k 0(3), k 0(4) and k 0(5) are the wave numbers in regions R2, R3, R4 and R5 respectively and are the positive real root of the transcendental equations k tanh khj = K , j = 2, 3, 4, 5 respectively. The velocity potential Φj(x, y, z, t) in each region Rj (j = 1, 2, …, 5) can be written as i ( z t ) } where the function ϕ (x, y) satisfies j (x , y , z , t )=Re{ j (x , y ) e j the governing equation

(

2

2)

j

= 0 in the fluid region Rj , j = 1, 2, …, 5,

(1)

along with boundary conditions j

y 1

y 2

y 3

y 4

y 5

2. Physical problem and mathematical formulation

y

We use Cartesian coordinate system to explain the geometry of the problem, where y-axis is positive vertically downward and xz-plane represents the undisturbed free surface of water (see Fig. 1). The whole

+K

j

= 0 on

= 0 on y = h1,

= 0 on

y = h2 ,

= 0 on y = h3 ,

= 0 on



, j = 1, 2, …, 5,

b,

b
a,

(4)

a < x < a,

= 0 on y = h5, b < x <

(2) (3)

(5)

y = h4, a < x < b,

(6)

,

(7)

1x (

b , y) =

2x (

b+, y ) = f1 (y ), 0

y

h1,

(8)

2x (

a , y) =

3x (

a+, y ) = f2 (y ), 0

y

h3,

(9)

Fig. 1. Schematic of physical problem. 2

y = 0,

Applied Ocean Research 93 (2019) 101946

A. Kaur, et al. 3x (a

4x (b

, y) =

4x (a+,

, y) =

5x (b+,

y ) = f3 (y ), 0

y

y ) = f4 (y ), 0

(10)

h3,

y

(11)

h5,

1 ( b , y) =

2 ( b+ , y ), 0

y

h1,

(12)

2(

3(

a+, y ), 0

y

h3,

(13)

y ), 0

y

h3,

(14)

y

h5,

(15)

a , y) =

3 (a

, y) =

4 (a+,

4 (b , y ) =

2

x 4

x

5 (b+, y ), 0

x= x=

= 0 on

b, h1 < y < h2, a, h3 < y < h2,

1 (y )

An sn(1) ^1 (y )

+ n= 1

[

Bn sinh(sn(2) b)

2 (y )

Cn cosh(sn(2) b)] ^2 (y ),

+

n=1

0 < y < h1, (25)

f2 (y ) = p0(2) [B0 sin(p0(2) a) + C0 cos(p0(2) a)] + n=1

2 (y )

sn(2) [ Bn sinh(sn(2) a) + Cn cosh(sn(2) a)] ^2 (y )

= p0(3) [E0 sin(p0(3) a) + F0 cos(p0(3) a)] (17)

+ n=1

3 (y )

sn(3) [ En sinh(sn(3) a) + Fn cosh(sn(3) a)] ^3 (y ),

0 < y < h3, (26)

1 (x ,

y)

(eiµ (x + b) + R e

5 (x ,

y)

Te iµ (x

b)

sn(2)

+

far field condition:

^

R)

= p0(2) [B0 sin(p0(2) b) + C0 cos(p0(2) b)]

(16)

x = a, h3 < y < h4 , x = b, h5 < y < h 4 ,

= 0 on

f1 (y ) = iµ (1

cosh k 0(1) (y

iµ (x + b) )

h1)

cosh k 0(1) h1

cosh k0(5) (y

h5)

as x

cosh k 0(5) h5

as x

,

(18)

f3 (y ) = p0(3) [ E0 sin(p0(3) a) + F0 cos(p0(3) a)] +

,

n=1

(19)

sn(3) [En sinh(sn(3) a) + Fn cosh(sn(3) a)] ^3 (y )

= p0(4) [ G0 sin(p0(4) a) + H0 cos(p0(4) a)]

Here, R & T are the reflection and transmission coefficients, respec2}1/2 . tively, to be determined and µ^ = {(k 0(5) )2

+ n=1

3 (y )

4 (y )

sn(4) [Gn sinh(sn(4) a) + Hn cosh(sn(4) a)] ^4 (y ),

0 < y < h3, (27)

3. Method of solution Our aim is to find ϕj satisfying Eqs. (1)–(19). Using (1)–(7) & (18), (19) the Havelock’s expansion of velocity potentials j , (j = 1, 2, 5) can be written as 1 (x ,

y) =

(eiµ (x + b)


y) =

iµ (x + b) )

+ Re

1 (y )

+

An

(1) e sn (x + b)

n=1

+

+

+ n=1


(20)

+

Cn sinh(sn(2) x )] ^2 (y ),

n=1

1

[En cosh(sn(3) x ) + Fn sinh(sn(3) x )] ^3 (y ),

n= 1

5 (x ,

^

b)

5 (y )

+

Dn e

sn(4) [Gn sinh(sn(4) b) + Hn cosh(sn(4) b)] ^4 (y ), (28)

4ik 0(1) cosh k 0(1) h1 µ (2k 0(1) h1

+

h1

sinh 2k0(1) h1)

4kn(1) cos kn(1) h1

h1

sn(1) (2kn(1) h1 + sin 2kn(1) h1)

0

(22) B0 [sin(p0(2) b) + sin(p0(2) a)] =

[Gn cosh(sn(4) x ) + Hn sinh(sn(4) x )] ^4 (y ),

y ) = Te iµ (x

R=

An =

a

(4) (4) 4 (x , y ) = [G 0 cos(p0 x ) + H0 sin(p0 x )] 4 (y )

+

sn(5) (x b )

4 (y )

0

f1 (y )cosh k 0(1) (y

h1) dy ,

(29)

3 (y )

a, 0 < y < h3,


Dn sn(5) ^5 (y )

Now, applying Havelock’s inversion formula in Eqs. (25)–(28) and further using conditions (16) and (17) wherever required, we get (21)

(3) (3) 3 (x , y ) = [E0 cos(p0 x ) + F0 sin(p0 x )]

n=1

0 < y < h5.

b

a, 0 < y < h2,

+

n=1

C0 sin(p0(2) x )] 2 (y )

[Bn cosh(sn(2) x )

5 (y )

= p0(4) [ G0 sin(p0(4) b) + H0 cos(p0(4) b)]

^ (y ), 1

b, 0 < y < h1,

[B0 cos(p0(2) x )

f4 (y ) = T (iµ^)

a
^ (y ), 5

b

b, 0 < y < h4,

x<

4k 0(2) [2k 0(2) h2

p0(2)

h1

(23)

f1 (y )cos kn(1) (y

0

+ sinh 2k 0(2) h2]

f1 (y )cosh k 0(2) (y

cosh k 0(2) (y

, 0 < y < h5,

h1) dy ,

(30)

×

h2 ) dy +

h3 0

f2 (y )

h2) dy ,

n= 1

(24) with

j (y )

=

cosh k 0(j ) (y cosh k 0(j) hj

hj )

, ^j (y ) =

cos kn(j ) (y cos kn(j ) hj

hj )

, j = 1, 2, …, 5

(31)

and

C0 [cos(p0(2) b) + cos(p0(2) a)] =

kn(j ), n = 1, 2… are positive real roots of the transcendental equation 2 ) 12 , j = 2, 3, 4 k tan khj = K , j = 1, 2, …, 5; p0(j) = ({k 0(j) } 2 and

4k 0(2) p0(2)

[2k 0(2) h2 h1

1

sn(j) = ({kn(j) }2 + 2) 2 , j = 1, 2…5. Here, An, B0, C0, Bn, Cn, E0, F0, En, Fn and G0, H0, Gn, Hn, Dn , n = 1, 2, … and R, T are the unknowns to be determined. Now, by using the expressions for velocity potentials from Eqs. (20)–(24) into the Eqs. (8)–(11), we get

0

+ sinh 2k 0(2) h2]

f1 (y )cosh k 0(2) (y

cosh k 0(2) (y

×

h2) dy +

h3 0

f2 (y )

h2 ) dy ,

(32) 3

Applied Ocean Research 93 (2019) 101946

A. Kaur, et al.

4k n(2)

Bn [sinh(sn(2) b) + sinh(sn(2) a)] =

(sn(2) )[2k n(2) h2 h1 0

(y

+

sin 2kn(2) h2]

f1 (y )cos kn(2) (y

4kn(4)

Hn [cosh(sn(4) a) + cosh(sn(4) b)] =

× h3

h2) dy +

0

sn(4)

[2kn(4) h4 h3

f2 (y )cos k n(2)

+ sin 2k n(4) h 4]

f3 (y )cos kn(4) (y

0

cos kn(4) (y

h2 ) dy ,

×

h4 ) dy +

h5 0

f4 (y )

h 4 ) dy ,

(33) 4k n(2)

Cn [cosh(sn(2) b) + cosh(sn(2) a)] =

(sn(2) )[2k n(2) h2 + sin 2kn(2) h2] h1 0

f1 (y )cos kn(2) (y

cos kn(2) (y

(42)

h2) dy +

h3 0

f2 (y )

E0 sin(p0(3) a )=

p0(3) [2k 0(3) h3 + sinh 2k 0(3) h3]

f3 (y )]cosh k 0(3) (y

[f2 (y )

0

F0 cos(p0(3) a) =

p0(3) [2k 0(3) h3 + sinh 2k 0(3) h3] (y

0

f4 (y )cos kn(5) (y

h5) dy,

f4 (y )cosh k 0(5) (y

h5) dy.

0

h5 0

1 (y )

(44)

b (Eq. (12)) gives

An ^1 (y ) = [B0 cos(p0(2) b)

+

(43)

C0 sin(p0(2) b)]

2 (y )

n=1

Cn sinh(pn(2) b)] ^2 (y ),

[Bn cosh(pn(2) b)

+

(35) h3

4ik 0(5) cosh k0(5) h5 ^ µ (2k 0(5) h5 + sinh 2k 0(5) h5)

(1 + R)

h3) dy

,

2k 0(3)

h5

(2kn(5) h5 + sin 2kn(5) h5)

Now, the continuity of pressure at x = (34)

h3

sn(5)

T=

h2) dy ,

2k 0(3)

4kn(5) cos kn(5) h5

Dn =

×

0 < y < h1.

n=1

(45) The continuity of pressure at x =

[f2 (y ) + f3 (y )]cosh k 0(3)

[B0 cos(p0(2) a)

a (Eq. (13)) gives

C0 sin(p0(2) a)] 2 (y ) +

Cn sinh(sn(2) a)] ^2 (y )

[Bn cosh(sn(2) a) n=1

h3) dy ,

(36)

= [E0 cos(p0(3) a)

F0 sin(p0(3) a)] 3 (y ) +

[En cosh(sn(3) a)

Fn sinh(sn(3) a)] ^3 (y ),

n= 1

En sinh(sn(3) a)

=

h3

sn(3) [2kn(3) h3 + sin 2kn(3) h3] (y

Fn cosh(sn(3) a)=

0 < y < h3.

2kn(3) 0

[f3 (y )

h3) dy ,

[E0 cos(p0(3) a) + F0 sin(p0(3) a)] +

h3

sn(3) [2kn(3) h3 + sin 2kn(3) h3]

0

n=1

[f2 (y ) + f3 (y )]cos kn(3)

h3) dy ,

G0 [sin(p0(4) a) + sin(p0(4) b)] =

The continuity of pressure at x = a (Eq. (14)) gives

(37)

2kn(3)

(y

(46)

f2 (y )]cos kn(3)

p0(4) [2k 0(4) h 4 + sinh 2k 0(4) h 4] h3 0

f3 (y )cosh k 0(4) (y

+

cosh k 0(4) (y

n=1

[Gn cosh(sn(4) a) + Hn sinh(sn(4) a)] ^4 (y ),

+

cos(p0(4) b)]

=

h5 0

T

f4 (y )

5 (y )

Dn ^5 (y ) = [G0 cos(p0(4) b) + H0 sin(p0(4) b)]

+

+ n=1

f3 (y )cosh k 0(4) (y

cosh k 0(4) (y

0 < y < h5. After substitution of Eqs. (30)–(43) into the Eqs. (45)–(48), we get the following set of integral relations:

×

h4 ) dy +

h5 0

f4 (y )

(1 + R)

h 4 ) dy ,

=

(40) Gn [sinh(sn(4) a) + sinh(sn(4) b)] =

4kn(4) sn(4) [2kn(4) h 4 + sin 2kn(4) h4 h3 0

f3 (y )cos kn(4) (y

cos k n(4) (y

[Gn cosh(sn(4) b) + Hn sinh(sn(4) b)] ^4 (y ),

(48)

p0(4) [2k 0(4) h 4 + sinh 2k 0(4) h 4] h3

4 (y )

n=1

h 4 ) dy ,

4k 0(4)

0

0 < y < h3.

The continuity of pressure at x = b (Eq. (15)) gives

(39) H0 [cos(p0(4) a)

4 (y )

(47)

×

h4 ) dy +

[En cosh(sn(3) a) + Fn sinh(sn(3) a)] ^3 (y )

= [G0 cos(p0(4) a) + H0 sin(p0(4) a)] (38)

4k 0(4)

3 (y )

h1 0 h5 0

h1)

cosh k 0(1) h1 h1 0

b, a (N11 (y, u) + K22 (y , u)) f1 (u) du +

h3 0

b, a K22 (y, u) f2 (u) du ,

0

y

h1,

(49)

×

h 4 ) dy +

cosh k 0(1) (y

a, b K22 (y , u) f1 (u) du +

+

f4 (y ) h3

h4 ) dy ,

0

4

h5 0

a, b (K22 (y , u) + M33 (y, u)) f2 (u) du

^ (y, u) f (u) du = 0, M 33 3

^ (y , u) f (u) du + M 33 2 +

(41)

h3 0

h3 0

h3 0

0

y

h3,

(50)

a, b (M33 (y, u) + K 44 (y, u)) f3 (u) du

a, b K 44 (y, u) f4 (u) du = 0,

0

y

h3,

(51)

Applied Ocean Research 93 (2019) 101946

A. Kaur, et al.

T

cosh k 0(5) (y

h5)

h3

cosh k 0(5) h5 h3

=

0

0 h5

b, a K44 (y , u) f3 (u) du

0

b, a (N55 (y , u) + K 44 (y , u)) f4 (u) du ,

0

y

^ (y , u ) M 33

(u

g2 (u) du + h3)1/3

h3

g3 (u)

h5

where Njj (y, u) =

hj )cos kn(j) (u

hj )

sn(j) [2kn(j) hj + sin 2kn(j ) hj]

n=1

4k 0(l) cosh k 0(l) (y p0(l)

hl )cosh k 0(l) (u

[2k 0(l) hl

= 0,

sin p0(l) r

hl)

4kn(l) cos kn(l) (y

cos p0(l) r

+

hl)cos kn(l) (u

×

cosh sn(l) r

+

sinh sn(l) r + sinh sn(l) t

h3

,

cosh sn(l) r + cosh sn(l) t

l = 2, 4,

h3)cosh k 0(3) (u

y

0

h3)

2k 0(3) (tan(p0(3) a) + cot(p0(3) a))cosh k 0(3) (y h3)cosh k 0(3) (u p0(3) [2k 0(3) h3 + sinh 2k 0(3) h3] coth(sn(3) a))cos kn(3) (y

= h1 0

h3)cos kn(3) (u

h3)

h3 0

,

h3)

sn(3) [2kn(3) h3 + sin 2kn(3) h3]

n=1

h3)1/3

1l

b, a K22 (y , u )

cosh k 0(1) (y

(u

h1)

cosh k 0(1) h1

a, b K22 (y , u )

(u

,

f j (y ) =

f2 (y ) =

0

hj , j = 1, 3,

1 as y h3)1/3

h3, f4 (y ) = O

1 as y h5)1/3

(y

(y

(y

1 g (y ), & f4 (y ) = h3)1/3 2 (y

^ (y, u) M 33

h3 0

0

y h1 0

h3

y

(u

P2l (u) du + h3)1/3

h3

b, a (N11 (y, u) + K22 (y , u))

(u

h1)1/3

du +

0

b, a K22 (y , u)

(u

(u

g1 (u) du + h1)1/3

h3 0

h5 0

0

h3)1/3

,

= 0,

0

y

g3 (u) (u

h3)1/3 h3,

N11

P4l (u) du = 0, h5)1/3

h3, l = 1, 2,

h5 0

P4l (u) b, a (y , u)) (N55 (y , u) + K 44 du (u h5)1/3 cosh k 0(5) (y

h5)

cosh k 0(5) h5

,

0

y

h5, l = 1, 2,

( 1

du

= (60)

(

)h

)

b, a K22

1

2

=

dv+

(w + 1) h1 (v + 1) h3 P2l , (v 2 2

(

(

(w + 1) h1 2 cosh k 0(1) h1

cosh k 0(1) 1l

ij

(w + 1) h1 (v + 1) h1 b, a (w + 1) h1 (v + 1) h1 , + K22 , 2 2 2 2

(v + 1) h1 2 (v 1) h1 1/3 2

P1l

(59)

1

^ (y , u ) M 33

(u

P3l (u) du h3)1/3

(58)

g2 (u) du (u h3)1/3 +

y

a, b K 44 (y , u )

(u

0 if i j . 1 if i = j A suitable transformation from (y, u) to (w, v) is used to convert the integral relations (64)–(67) into a system of integral equations of first kind over [−1,1]:

a, b (K22 (y , u ) + M33 (y, u))

h3

h3, l = 1, 2,

(67)

1

h1,

a, b K22 (y , u )

P3l (u) b, a (y , u) K 44 du (u h3)1/3

1

g2 (u)

P2l (u) du h3)1/3

P3l (u) du = 0, h3)1/3

(u

where δ represents the Kronecker delta defined as

1 g (y ), h5)1/3 4

h3

^ (y , u ) M 33

(u

(57)

C , j = 1, 2, 3, 4 . Hence, Eqs. (49)–(52) become

g1 (u)

(64)

a, b (M33 (y, u) + K 44 (y, u))

0

= 2l

j = 1, 3,

h1, l = 1, 2,

(66)

h5.

cosh k 0(1) (y h1) (1 + R) cosh k 0(1) h1 h1

(62)

(63)

a, b (K22 (y , u) + M33 (y, u))

0

+

1 g (y ), hj )1/3 j

where gj (y )

0

h3 0

0

(y

du ,

(65)

So, f j (y ), j = 1, 2, 3, 4 can be expressed as

=

y

0

and

f2 (y ) = O

h5)1/3

P1l (u) du h1)1/3

(u

+

h3

(y

g4 (u) (u

j = 1, 2, 3, 4.

0

P1l (u) du + h1)1/3

.

The flow near the edge of the trenches reveals that

f j (y ) = O

b, a (N55 (y , u) + K 44 (y , u))

0

P2l (u) du h3)1/3

(56)

1 as y hj )1/3

h5

du

b, a (N11 (y, u) + K22 (y , u))

+ (55)

2kn(3) (tanh(sn(3) a)

g3 (u) (u

Hence, Pjl, j = 1, 2, 3, 4, l = 1, 2 satisfy the following set of integral relations:

p0(3) [2k 0(3) h3 + sinh 2k 0(3) h3]

+

(61)

h3,

y

h5.

h3)

2kn(3) (tanh(sn(3) a) + coth(sn(3) a))cos kn(3) (y h3)cos kn(3) (u + sn(3) [2kn(3) h3 + sin 2kn(3) h3] n=1

^ (y , u)= M 33

b, a K44 (y , u)

gj (u) = (1 + R) Pj1 (u) + TPj2 (u),

sinh sn(l) t

cot(p0(3) a))cosh k 0(3) (y

du

h5)

0

h1

2k 0(3) (tan(p0(3) a)

h5)1/3

To obtain a set of integral relations with unknown functions only, we use

hl )

(54) M33 (y , u)=

0

g 4 (u ) (u

cosh k 0(5) h5

0

cos p0(l) t

sn(l) [2k n(l) hl + sin 2k n(l) hl]

n=1

cosh k 0(5) (y

=

sin p0(l) t

sin p0(l) t

+

T

du

a, b K 44 (y , u )

0

, j = 1, 5,

+ sinh 2k 0(l) hl]

cos p0(l) r

×

h5

+

(53) Kllr , t (y, u) =

h3)1/3

(u

(52)

4kn(j) cos kn(j ) (y

a, b (M33 (y, u) + K 44 (y, u))

0

h1

),

(

)h

(v + 1) h3 2 1) h 1/3 3

2

1

)

w

3

2

dv

1, l = 1, 2, (68)

5

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1 1

1 1

(w + 1) h3 (v + 1) h1 P1l , (v 2 2

a, b K22

(w + 1) h3 (v + 1) h3 , 2 2

^ M 33

1

(

(

(

)h

(v + 1) h3 2 (v 1) h3 1/3 2

P2l

)h

(v + 1) h1 2 1) h 1/3 1

2

(

)

2 3

)

size 8M + 8 which needs to be solved to determine the unknown conn = 1, 2, …, M : stants an(jl) , j = 1, 2, 3, 4; l = 1, 2,

dv+

M

)

3

2

dv = 0,

w

1

2

j=0

dv+

P ^ (w + 1) h3 , (v + 1) h3 2l M 33 1 (v 2 2

1

(

(

1

a, b K 44

1

M33

1

(

)

3

2

(

(w + 1) h3 (v + 1) h5 P4l , (v 2 2

(

2

5

2

)

(

) h dv = 0,

)

1

2

w

1 1

N55

1

=

(

5

2

where

(

)h

)

5

)h

3

2

)

3

K j(23) (wi) =

2

Lj(21) (wi) =

dv

2

Lj(23) (wi) =

dv

(

h5

^ (33) (w ) = M i j

),

1

w

K j(43) (wi) = K j(45) (wi) =

4. Direct polynomial approximation The singularity of fluid velocity at each edge of each trench is handled using suitably designed polynomial as a basis function, which is discussed in this section. Approximating Pjl, (j = 1, 2, 3, 4; l = 1, 2) in Eqs. (68)–(71), using suitably designed polynomial as given by

P1l

P2l

P3l

P4l

(v + 1) h1 (v 1)1/3 = h 2 ( 21 ) 2/3 (v + 1) h3 (v 1)1/3 = h 2 ( 23 )2/3

M j=0 M

aj(2l) v j ,

l = 1, 2,

j=0

(v + 1) h3 (v 1)1/3 = h 2 ( 23 )2/3

(v + 1) h5 (v 1)1/3 = h 2 ( 25 )2/3

aj(1l) v j , l = 1, 2,

M

aj(3l) v j,

l = 1, 2,

j=0

M j =0

aj(4l) v j,

l = 1, 2,

2l

d (5) (wi ), (79)

1

Qj(43) (wi ) = Qj(45) (wi ) =

(80)

b, a K22

(wi + 1) h1 (v + 1) h3 j , v dv , 2 2

(82)

a, b K22

(wi + 1) h3 (v + 1) h1 j , v dv, 2 2

(83)

a, b K22

(wi + 1) h3 (v + 1) h3 j , v dv, 2 2

(84)

(wi + 1) h3 (v + 1) h3 j , v dv , 2 2

(85)

^ (wi + 1) h3 , (v + 1) h3 v j dv, M 33 2 2

(86)

a, b K 44

(wi + 1) h3 (v + 1) h3 j , v dv , 2 2

(87)

a, b K 44

(wi + 1) h3 (v + 1) h5 j , v dv , 2 2

(88)

b, a K 44

(wi + 1) h5 (v + 1) h3 j , v dv , 2 2

(89)

b, a K 44

(wi + 1) h5 (v + 1) h5 j , v dv , 2 2

(90)

(wi + 1) h5 (v + 1) h5 j , v dv , 2 2

(91)

1

1

(wi + 1) h1 (v + 1) h1 j , v dv , 2 2

(81)

1

1

N11

(wi + 1) h1 (v + 1) h1 j , v dv , 2 2

1

1

1

b, a K22

1

1 1 1 1

M33

1, l = 1, 2. (71)

1

N j(11) (wi ) =

1, l = 1, 2,

(v + 1) h3 2 1) h 1/3

aj(4l) [N j(55) (wi) + Qj(45) (wi )] =

i = 0, 1, …, M ,

M j(33) (wi ) =

(w + 1) h5 2 cosh k 0(5) h5

cosh k 0(5) 2l

(

(78)

i = 0, 1, …, M ,

j =0

l = 1, 2,

dv+

aj(4l) K j(45) (wi ) j =0

M

j =0

(w + 1) h5 (v + 1) h5 b, a (w + 1) h5 (v + 1) h5 , + K 44 , 2 2 2 2

(v + 1) h5 2 (v 1) h5 1/3 2

P4l

(

(w + 1) h5 (v + 1) h3 P3l , (v 2 2

b, a K 44

l = 1, 2,

M

(70) 1

j =0

K j(21) (wi) =

3

M

aj(3l) [M j(33) (wi ) + K j(43) (wi )] +

j=0

dv+

(77)

i = 0, 1, …, M , M

^ (33) (w ) + aj(2l) M i j

^ (33) (w ) aj(3l) M i j

j=0

l = 1, 2,

aj(3l) Qj(43) (wi )

)h

(v + 1) h5 2 1) h 1/3

j=0

= 0,

= 0,

3

M

aj(2l) [M j(33) (wi ) + Lj(23) (wi )] +

j=0

(w + 1) h3 (v + 1) h3 a, b (w + 1) h3 (v + 1) h3 , + K 44 , 2 2 2 2

(v + 1) h3 2 (v 1) h3 1/3 2

P3l

)h

(v + 1) h3 2 1) h 1/3

d (1) (wi ), (76)

M

aj(1l) Lj(21) (wi ) +

M

1l

i = 0, 1, …, M ,

M

1, l = 1, 2,

aj(2l) K j(23) (wi ) = j =0

l = 1, 2,

(69)

1

M

aj(1l) [N j(11) (wi ) + K j(21) (wi )] +

)h

(v + 1) h3 P3l 2 (v 1) h3 1/3 2

(

1

(w + 1) h3 (v + 1) h3 (w + 1) h3 (v + 1) h3 + M33 , , 2 2 2 2

a, b K22

1

(

1 1 1 1

1 1 1 1 1

(72)

N j(55) (wi) =

(73)

d (l) (wi ) =

(74)

Determination of reflection and transmission coefficients:In order to determine the coefficients R and T, we use Eqs. (29) and (44) along with relations (57) and (63), then we obtain:

(75)

(1 + R) S11 + TS12 = iµ

and collocating at M + 1 points w = wi, i = 0, 1, …, M (these are chosen as zeros of Chebyshev polynomial of first kind to avoid the occurrence of ill-conditioned matrix), we get the following system of

1

N55

cosh k0(l)

(

(wi

1) hl 2

cosh k0(l) hl

)

l = 1, 5.

2k 0(1) h1 + sinh(2k 0(1) h1) 4k 0(1) cosh k 0(1) h1

(92)

(1

2k (5) h5 + sinh(2k 0(5) h5) (1 + R) S41 + TS42 = Tiµ^ 0 (5) , 4k 0 cosh k 0(5) h5 6

R),

(93) (94)

Applied Ocean Research 93 (2019) 101946

A. Kaur, et al. hj

where Sjl = hj + 1

Sjl =

0

cosh k0(j) (y

0

cosh k 0(j) (y

hj )

hj + 1)

Pjl (y ) hj )1/3

(y Pjl (y )

hj + 1)1/3

(y

dy,

dy,

considered (H1 = H5 = 1.0 gives symmetrical single trench profile). In Table 1, |R| and |T| are tabulated for different values of K1 for a pair of symmetrical trenches. The last column of the Table 1 reveals that the energy balance relation (96) is verified.

j = 1, l = 1, 2,

j = 4, l = 1, 2.

6.1. Validation

Once the unknowns an(jl) , j = 1, 2, 3, 4, l = 1, 2 are obtained after solving the system of equations, then Pjl, j = 1, 2, 3, 4, l = 1, 2 from relations (72)–(75) can be obtained. Hence, R and T can be determined by solving the Eqs. (93) and (94).

As a special limiting case, the results of |R| for a single trench profile (Fig. 2, where H2 = H3 = H4 and b^ = a^) are derived and compared with the results of Kirby and Dalrymple [3] in Fig. 3 and Roy et al. [23] in Fig. 4. Both the figures show that the present results have been good agreement with the known results for a single trench profile.

5. Energy balance relation The energy balance relation for the present problem is derived using the Green’s integral theorem which yields

n

dS = 0

n

x < X; x = X, 0

x = b, h5 y = h3,

y a

y

h5; y = h5, b

h 4 ; y = h4 , a x

a; x =

In this section, the convergence on M (number of unknowns in polynomial approximation in Eqs. (72) and (75)) and N (number of evanescent modes) is examined. For convergence of N, we fix M = 45 and the values of |R| are tabulated against K1 for various values of N = 2, 8, 15, 20 and 25 (see Table 2). Here, the values of other parameters H2 = H3 = H4 = 2.0, H1 = H5 = 1.0, a^ = b^ = 1.5 and 1 = 45 are kept fixed. This table shows that the value of |R| is same up to four decimal places for N = 15 & 20 and 25 for all values of K1. Hence, N = 15 is taken throughout the study. For convergence on M, the number of evanescent modes N is fixed as 15 and |R| is tabulated against K1 for different values of M = 5, 25, 45, 90 and 130 in Table 3 which shows that same values of |R| is obtained up to four decimal places for M = 45, 90 and 130 for all values of K1. Hence, for all numerical calculations, M = 45 and N = 15 are considered throughout the study unless otherwise stated.

(95)

where is the complex conjugate of ϕ and is n normal derivative to the boundary Γ and Γ represents of the region R^ bounded by the lines y = 0, y = 0, b x a ; y = 0, a x a ; y = 0, a x

b

6.2. Convergence study for M and N

x

a, h3

x < X;

b; x = a, h3 y

the outward the boundary X
h2 ; y = h2 ,

x

h4;

b

x

a;

x = b , h1 y h2; y = h1, X < x b ; x = X , 0 y h1 Then we take limit as X → ∞. y = 0, X < x The contribution from the lines y = 0, b x a ; y = 0, a x a ; y = 0, a x b; y = 0,

b; is

b x
6.3. Influences of various parameters for a single trench profile

y = h2 , b x a; y = h1, X < x b to conditions (3)–(7). Further, the contribution from the line x = b, h5 y h4 ; x = a, h3 x h4 ; x = a, h3 y h2 ; is zero x = b, h1 y h2 due to conditions (16)–(17). The contribution from the line x = h1

x

0

x

iµ (|R|2

dy =

Here, our aim is to investigate the behaviour of |R| and |T|: (i) as a function of angle of incidence (θ1) for different values of trench height (ii) as a function of trench width (b/λ) for different values for trench height (iii) as a function of trench width (b/λ) for different values of angle of incidence. The influences of trench heights (H2 = H3 = H4 = 2.0, 2.5, 3.5) on |R| and |T| versus θ1 are shown in Fig. 5. From this figure, it is noticed that as θ1 increases, |R| decreases for θ1 < 35∘ but increases for θ1 > 35∘. Consequently, transmission decreases for the same. This may happen due to the mutual interaction between incident and reflected waves within the trench. Moreover, |R| increases as trench height (H2 = H3 = H4 ) increases. |R| and |T| are plotted against width (b/λ) of single trench for symmetric and asymmetric trench profile (Fig. 6(a)) and for different heights H2 = H3 = H4 = 1.3, 1.6, 1.9 (Fig. 6(b)). In both the figures, the periodically oscillatory pattern for |R| and |T| are observed as trench width b/λ increases. This is due to the constructive and/or destructive interference of incident and reflected waves within the trench. From

X , 0 < y < h1 is

1)

2k 0(1) cosh2 (k 0(1) h1)

[2k0(1) h1 + sinh(2k0(1) h1)]

The contribution from the line x = X , 0 < y < h5 is 0 h5

x

x

dy =

iµ^ |T|2 2k0(5) cosh2 (k 0(5) h5)

[2k 0(5) h5 + sinh(2k 0(5) h5)]

Finally, summing all the contribution and substituting in Eq. (95), the energy balance relation can be obtained as

|R|2 +

(96)

|T|2 = 1

where

^ 0(1) (2k 0(5) h5 µk µk 0(5) (2k 0(1) h1

=

+ +

sinh(2k 0(5) h5)) sinh(2k 0(1) h1))

cosh2 (k0(1) h1) cosh2 (k0(5) h5)

.

Table 1 Verification of energy balance relation for a pair of symmetrical trenches.

6. Results and discussion In this section, the physical quantities namely, reflection and transmission coefficients are numerically computed and plotted through different graphs for various values of parameters. The non-dimensional parameters are given as a^ = a/ h1; b^ = b / h1; l^ = l/ h1; K = Kh ; H = h / h , j = 1, 2, , 5. The values of a^ = 1.0, b^ = 4.0, 1

1

j

j

1

H2 = 2.0, H3 = 1.5, H4 = 2.0, H5 = 1.0 and K1 = 0.3, 1 = 45 are kept fixed throughout this section unless otherwise mentioned. We denote = 2 /k 0(1) . Here, H2 = H4, H1 = H5 = 1.0 gives rise to a pair of symmetrical trenches. To obtain the results related to single trench profile, the values of parameters H = H = H = 2.0, a^ = b^ = 1.5 are 2

3

4

7

K1

|R|

|T|

|R|2 + |T|2

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

0.3841686 0.2264667 0.0815371 0.2500313 0.1716754 0.0197106 0.0927548 0.0220671 0.0489453 0.0411460

0.9232629 0.9740188 0.9966703 0.9682377 0.9851535 0.9998057 0.9956889 0.9997564 0.9988014 0.9991531

1.0000000 0.9999999 0.9999999 0.9999999 1.0000000 1.0000000 0.9999999 0.9999999 1.0000000 0.9999999

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Table 2 |R| versus K1 for different values of N = 2, 8, 15, 20 & 25. K1

|R| (N = 2)

|R| (N = 8 )

|R| (N = 15)

|R| (N = 20)

|R| (N = 25)

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

0.196342 0.209125 0.154728 0.043566 0.101304 0.233935 0.310909 0.314737 0.250043 0.140311

0.248987 0.281786 0.255311 0.190149 0.106051 0.027110 0.027299 0.051000 0.049900 0.035335

0.257777 0.293802 0.270242 0.207688 0.124843 0.044624 0.013749 0.043096 0.047822 0.037913

0.257785 0.293816 0.270263 0.207717 0.124879 0.044662 0.013717 0.043075 0.047816 0.037920

0.257784 0.293814 0.270261 0.207714 0.124876 0.0446598 0.013719 0.043076 0.047816 0.037919

Table 3 |R| versus K1 for different values of M = 5, 25, 45, 90 & 130 . Fig. 2. Schematic of single trench.

K1

|R| (M = 5)

|R| (M = 25)

|R| (M = 45)

|R| (M = 90)

|R| (M = 130)

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

0.264638 0.302973 0.281093 0.219477 0.136161 0.053636 0.008445 0.041855 0.049957 0.042074

0.262451 0.300084 0.277769 0.216038 0.133121 0.051568 0.009192 0.041431 0.048546 0.040267

0.257778 0.293802 0.270242 0.207688 0.124843 0.044624 0.013749 0.043096 0.047822 0.037913

0.257742 0.293754 0.270184 0.207622 0.124774 0.044563 0.013793 0.043118 0.047823 0.037898

0.257731 0.293742 0.270167 0.207602 0.124754 0.044546 0.013806 0.043124 0.047823 0.037893

et al. [9] and confirmed by Liu et al. [14]. Moreover, it is noticed that the amplitude of |R| decreases as H5 increases. Hence, less wave energy is reflected back causing more transmission. From Fig. 6(b), it is also noticed that local maxima in |R| increases as the height of trench (H2 = H3 = H4 ) increases. Hence, local minima in transmission coefficient |T| decreases for the same. This may happen due to the same fact which is mentioned towards Fig. 5. Moreover, less number of zeros in |R| and significant phase shift in the oscillatory pattern of |R| are observed as the height of the trench increases. This phase shift may happen due to the mutual interaction of waves of different frequencies generated within the trench of varying depths. The effect of angle of incidence θ1 on |R| and |T| against normalized trench width b/λ is analyzed in Fig. 7. It is observed that amplitude of |R| increases as θ1 increases. Moreover, as θ1 increases, significant phase shift and less number of zeros in |R| are also observed which may happen due to the same fact as mentioned for Fig. 6(b).

Fig. 3. |R| versus k 0(1) h1 for fixed H2 = H3 = H4 = 7.625, H1 = H5 = 1.0, 2b^ = 5.28 and 1 = 0 .

Fig. 6(a), it is noticed that zeros in reflection coefficient |R| are obtained for symmetric trench but this is not the case for asymmetric trench when H5 = 1.3, 1.6, this phenomenon has been revealed by Xie

8

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Fig. 5. |R| and |T| versus θ1 for H2 = H3 = H4 = 2.0, 2.5, 3.5 and b^ = 1.5 for symmetric profile.

6.4. Influences of various parameters for a pair of trenches

|R| decreases as H5 increases. This is due to the same fact as mentioned in Fig. 6(a). Further, the behaviour of |R| and |T| versus K1 for different values of H2 = 2.0, 2.5, 3.0 (keeping height H4 of second trench fixed) are depicted in Fig. 9(b). Here, amplitude of |R| increases as H2 increases and hence, less transmission is possible for the same. This is due to the same fact as mentioned in Fig. 5(a). In Fig. 10, |R| & |T| are plotted against K1 for different values of trench heights (H2 = H4 = 2.0, 2.5, 3.0 ) for a pair of trenches. This figure reveals that global maxima of |R| increases as the heights (H2 & H4 ) of both trenches increase, hence, producing less transmission for the same. This is due to the same fact as mentioned towards Fig. 5(a). Further, it is noticed that more reflection is observed (Fig. 10) when both H2 & H4 increase as compared to increasing the height of one trench only (Fig. 9(b)). The influence of angle of incidence ( 1 = 30 , 45 , 60 ) is shown in Fig. 11(a) and (b) where the global maxima of |R| increases (Fig. 11(a)) and global minima of |T| decreases (Fig. 11(b)) as the angle of incidence increases. Hence, less energy is transmitted to sea side against the angle of incidence. Also, significant phase shift in |R| is observed against θ1 due to mutual interference between the incident and

The differences of |R| and |T| between the symmetrical single trench and a pair of symmetrical trenches are shown in Fig. 8 where H1 = H5 = 1.0, H2 = H4 = 3.0, H3 = 1.5, a^ = 3.0, b^ = 6.0, 1 = 45 . The figure reveals that a pair of trenches produces more reflection in comparison to the single trench, hence less transmission to the left side by a pair of trenches if imposed on the bottom. Consequently, wave impact on seashore is reduced due to a pair of trenches. Moreover, for a pair of symmetrical trenches, the phenomenon of zero reflection is observed as revealed by Mei [24]. Hence, symmetrical design may be avoided in breakwater construction to avoid zero reflection i.e. full transmission. In Fig. 9(a), |R| & |T| are plotted against K1 for symmetrical and asymmetrical pair of trenches. It is noticed that oscillatory pattern in |R| & |T| reduces as the value of K1 increases and |T| achieves the value one for higher value of K1. This may happen because for higher value of K1 shorter waves almost confined near the free surface, hence, fully transmitted to seaside. Also, |R| has zeros at some wave numbers for symmetrical pair of trenches, however, this is not the case for asymmetrical pair of trenches. Also, it is noticed that amplitude of

Fig. 6. |R| and |T| versus b/λ for (a) H5 = 1.0, 1.3, 1.6 and H2 = H3 = H4 = 2.0,

1

= 45 9

(b) H2 = H3 = H4 = 1.3, 1.6, 1.9 and

1

= 45 for symmetric profile.

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Fig. 7. |R| and |T| versus b/λ for

1

= 30 , 45 , 60 and H2 = H3 = H4 = 1.5 for symmetric profile.

Fig. 8. |R| and |T| for single trench and a pair of trenches.

Fig. 9. |R| and |T| versus K1 for (a) H5 = 1.0, 1.4, 1.6 with H2 = H4 = 2.0, H3 = 1.5 (b) H2 = 2.0, 2.5, 3.0 with H5 = 1.4, H3 = 1.5, H4 = 2.0 . 10

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Fig. 10. |R| and |T| versus K1 for H2 = H4 = 2.0, 2.5, 3.0 with H5 = 1.4, H3 = 1.5.

Fig. 11. For

1

= 30 , 45 , 60 and H2 = H4 = 4.0, H3 = 1.5, H5 = 1.2 (a) |R| versus K1 (b) |T| versus K1.

reflected waves. The effects of trench heights (H2 & H4 ) on |R| and |T| versus θ1 are depicted in Fig. 12(a). This figure reveals that for smaller values of θ1, |R| decreases, then for higher values of θ1, |R| increases and tends to one which is due to the mutual interaction between incident and reflected waves within the trenches. Moreover, |R| increases as the heights of trenches increase. This is due to the same fact which is mentioned towards Fig. 5(a). |R| and |T| against θ1 for symmetrical and asymmetrical pair of trenches are depicted in Fig. 12(b). For θ1 > 40∘, |R| increases as angle of incidence increases and achieves the value one for lager angle of incidence. Further, zeros in |R| are observed for θ1 < 40∘ for a pair of symmetrical trenches. |R| and |T| as a function of trench width (l/λ) are depicted for symmetrical and asymmetrical pair of trenches in Fig. 13(a) and for different values of trench heights (H2 & H4 ) in Fig. 13(b). From these figures, it is found that |R| and |T| have oscillatory pattern as l/λ increases and the local maxima in |R| is not same. This oscillatory pattern is due to the constrictive/ destructive interference of the incident and reflected waves between the pair of trenches as well as within each

trench. From Fig. 13(a), it is clear that |R| decreases as H5 increases which is also noticed in Fig. 6(a). In Fig. 13(b), amplitude in |R| increases as heights (H2 & H4 ) increase. Moreover, significant phase shift in |R| is observed as the height of each trench increases. This is due to the same fact as mentioned towards Fig. 6(b). The effect of H2 on |R| and |T| versus l/λ is shown in Fig. 14(a). From this figure it can be revealed that global maxima of |R| increases as H2 increases. Hence, global minima of |T| decreases for the same. The peaks in |R| and |T| are due to the mutual interaction of incident and reflected waves within the trenches. Similarly, same behaviour towards the effect of H4 on |R| and |T| can be analyzed. The effect of H3 on |R| and |T| versus l/λ is shown in Fig. 14(b). From this figure, it is clear that the amplitude of |R| decreases as H3 increases. Hence less energy is reflected back, consequently, more energy is transmitted to sea side. The effect of angle of incidence ( 1 = 25 , 40 , 50 ) on |R| and |T| versus l/λ is shown in Fig. 15. The figure reflects that |R| and |T| have same behaviour as a function of l/λ as noticed in Fig. 13(b). Further, the amplitude of |R| increases as θ1 increases, hence, transmission 11

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Fig. 12. |R| and |T| versus θ1 (a) for H2 = H4 = 3.0, 4.0, 5.0 and H5 = 1.4, H3 = 1.5 (b) for H5 = 0.7, 1.0 and H2 = H4 = 2.0, H3 = 1.5 .

Fig. 13. |R| and |T| versus trench width l/λ for (a) H5 = 1.0, 1.4, 1.6 and H2 = H4 = 2.0, H3 = 1.5,

decreases for the same. Moreover, significant phase shift in |R| is observed as angle of incidence increases. |R| and |T| versus a1/λ are plotted for symmetrical and asymmetrical pair of trenches in Fig. 16(a) and different values of trench heights in Fig. 16(b). It is observed that |R| and |T| have periodically oscillatory behavior as the distance (a1/λ) between the trenches increases in both the figures. It is also noticed that |R| and |T| have same local optima, but this is not the case when |R| and |T| are function of l/λ (see Figs. 13(a) and (b), 14(a) and (b), 15). This is due to the constructive/ destructive interference between incident and reflected waves. Further, zeros in reflection coefficient |R| are observed in Fig. 16(a) for a pair of symmetrical trenches as noticed in Fig. 9(a). Moreover, the Fig. 16(a) reveals that |R| decrease as H5 increases. From Fig. 16(b), it is noticed that amplitude of |R| increases as the height of each trench increases. Further, significant phase shift is observed in |R| as a function of a1/λ

1

= 45 (b) H2 = H4 = 1.7, 1.9, 2.1 and H3 = 1.5, H5 = 1.4,

1

= 45 .

when the height of each trench increases. This is due to the same fact as mentioned in Fig. 6(b). For symmetrical pair of trenches, the effect of angle of incidence 1 = 15 , 25 , 45 on |R| and |T| versus a1/λ is depicted in Fig. 17. From this figure, it is clear that |R| and |T| have same pattern as a function of a1/λ as noticed in Fig. 16(a). Further, it is seen that amplitude of |R| increases, hence amplitude of |T| decrease when angle of incidence increases which is due to the same fact mentioned for Fig. 7. 7. Conclusions In the present study, the problem of diffraction of obliquely incident waves by a pair of asymmetric trenches is examined for its solution by the aid of a system of singular integral equations of first kind. The resulting integral equations are solved numerically by using suitably 12

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Fig. 14. |R| and |T| versus H2 = H4 = 2.0, H5 = 1.4, 1 = 45 .

trench

width

l/λ

for

(a)

H2 = 1.7, 1.9, 2.1

Fig. 15. |R| and |T| versus l/λ for

1

and

H4 = 2.0, H3 = 1.5, H5 = 1.4,

1

= 45

(b)

H3 = 1.2, 1.5, 1.8

and

= 25 , 40 , 50 and H2 = H4 = 2.0, H3 = 1.5, H5 = 1.4 .

designed polynomial approximations.The zeros of Chebyshev polynomial of first kind are chosen as collocation points to avoid the occurrence of ill-conditioned matrix in the system of linear algebraic equations. The numerical values of reflection and transmission coefficients are plotted through different graphs to analyse the effects of various parameters. Reflection and transmission coefficients have nonuniform oscillatory pattern for a pair of trenches as a function of trench width whereas they have uniform periodically oscillatory pattern when they are functions of distance. Increasing the heights of trenches, produces more reflection consequently, less transmission to seaside, yielding less impact on seashore by a pair of trenches if imposed at the bottom. For symmetrical trenches, with respect to the angle of

incidence, the zeros in reflection coefficient are observed for θ1 < 40∘ and reflection increases for θ1 > 40∘ and achieves the value one. In addition, the differences in the results between the single trench and a pair of trenches are analysed. More energy is reflected back by a pair of trenches as compared to single trench profile, hence, less wave energy is transmitted to seaside, yielding less impact on seashore. The behaviour of zeros in reflection coefficient is observed for symmetrical pair of trenches which shows that the symmetrical designing of trenches in construction must be avoided. Further, this paper provides a simple straightforward method for solving a set of integral relations in the given water wave problem. Moreover, this study will be helpful to solve integral relations arising in the other areas of mathematical physics.

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Fig. 16. |R| and |T| versus H3 = 1.5, H5 = 1.4, 1 = 45 .

gap

a1/λ

for

(a)

H5 = 1.0, 1.4, 1.6

Fig. 17. |R| and |T| versus gap a1/λ for

1

and

H2 = H4 = 2.0, H3 = 1.5,

1

= 45 , K1 = 1.0

(b)

H2 = H4 = 1.8, 2.0, 2.2

and

= 15 , 25 , 45 and H2 = H4 = 2.0, H3 = 1.5 for symmetrical case.

Acknowledgments

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The authors thank the reviewers for their comments and suggestions to improve the article in the present form. Amandeep Kaur is thankful to the Department of Science and Technology, Govt. of India, for inspire grant for pursuing Ph.D. degree at the Indian Institute of Technology Ropar, India. A. Chakrabarti is grateful to NASI for financial support in the form of NASI Honorary-Scientist (Reference Number: NAS/1022/1/ 2015-16). References [1] J.J. Lee, R.M. Ayer, Wave propagation over a rectangular trench, J. Fluid Mech. 110 (1981) 335–347. [2] J.W. Miles, On surface-wave diffraction by a trench, J. Fluid Mech. 115 (1982)

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