J Eng Math (2014) 89:101–112 DOI 10.1007/s10665-014-9705-6

Water wave scattering by a rectangular trench Rumpa Chakraborty · B. N. Mandal

Received: 7 September 2013 / Accepted: 11 April 2014 / Published online: 22 October 2014 © Springer Science+Business Media Dordrecht 2014

Abstract Assuming linear theory, the two-dimensional problem of water wave scattering by a rectangular submarine trench is reinvestigated here employing the multiterm Galerkin approximations involving ultraspherical Gegenbauer polynomials for solving the integral equations arising in the mathematical analysis. Because of the geometrical symmetry of the rectangular trench about the y-axis, the problem is split into two separate problems involving symmetric and antisymmetric potential functions. Very accurate numerical estimates for the reflection and transmission coefficients for various values of different parameters are obtained, and these are seen to satisfy the energy identity. These coefficients are computed numerically and depicted graphically against the wave number in a number of figures. Some figures available in the literature drawn using different mathematical methods and laboratory experiments are also recovered following the present analysis, thereby confirming the correctness of the results presented here. It is also observed that the reflection and transmission coefficients depend significantly on the width of the trench. Keywords Galerkin approximation · Reflection and transmission coefficients · Submarine trench · Water wave scattering

1 Introduction The problem of water wave scattering by obstacles of various geometrical shapes present in infinitely deep water or in uniform finite-depth water has been studied in the literature on the linearized theory of water waves for more than six decades using a variety of interesting mathematical techniques. The method of Galerkin approximations has been widely used to investigate water wave scattering problems involving thin vertical barriers (cf. Porter and Evans [1], Evans and Fernyhough [2], Banerjea et al. [3], Das et al. [4]) or thick vertical barriers with rectangular cross sections [5,6]. There is another important class of wave scattering problems involving water of variable depth in which the depth is constant except for variations over a finite interval. Kreisel [7] first investigated wave propagation over variable-depth geometries by reducing the fluid domain to a rectangular strip using an appropriate conformal mapping and then converting the boundary value problem of the potential function into an integral equation solved by iteration. Mei and Black [8] employed a variational formulation to obtain numerical estimates R. Chakraborty · B. N. Mandal (B) Physics and Applied Mathematics Unit, Indian Statistical Institute, 203, B. T. Road, Kolkata 700108, India e-mail: [email protected]

123

102

R. Chakraborty, B. N. Mandal

for the reflection coefficient for water wave scattering by a bottom-standing thick rectangular barrier. Lassiter [9] studied the complementary problem of scattering of water waves normally incident on a rectangular trench where the water depths before and after the trench are constants but not necessarily equal. He obtained reflection and transmission coefficients after formulating the problem in terms of complementary variational integrals of Schwinger’s type using the condition that the velocity potential and horizontal velocity must be continuous along the vertical lines before and after the trench. Lee and Ayer [10] employed a matching procedure to solve the rectangular trench problem by writing solutions in two subregions comprised of an infinite rectangular region of constant depth and a finite rectangular region of the trench and obtained numerically the reflection and transmission coefficients. They have also conducted a series of laboratory experiments in a wave tank and compared the experimental with the theoretical results. Shortly thereafter, Miles [11] employed a conformal mapping algorithm to solve the trench problem for normal incidence of the wave train after formulating it for long waves. He also solved the problem for oblique incidence through variational formulation and obtained long wave limits for the reflection and transmission coefficients. Kirby and Dalrymple [12] also investigated the trench problem for obliquely incident waves over an asymmetric trench for which the water depths in its two sides are unequal but constant. They numerically solved the problem by solving a set of integral equations derived by matching the truncated eigenfunction expansions for each subregion of constant depth along two vertical boundaries and compared the numerical results with the data obtained from small-scale wave-tank experiments. Recently, different types of trench problems were considered by Bender and Dean [13], Xie et al. [14], and Liu et al. [15]. Bender and Dean investigated the problem of wave propagation over two-dimensional trenches and shoals. They employed three solution methods: the step method, valid in arbitrary-depth of water, and the slope method and the numerical method, which are valid in shallow water regions. Xie et al. [14] studied the problem of long-wave reflection by a rectangular obstacle with two scour trenches of power function profile and obtained the reflection coefficient in closed form involving first- and second-kind Bessel functions. In addition, Liu et al. [15] investigated wave reflection by a rectangular breakwater with two scour trenches by formulating the problem in terms of a modified mild-slope equation (MMSE), and the reflection coefficient was obtained analytically. In the present study, we reinvestigate the problem of water wave scattering by a rectangular submarine trench employing the multiterm Galerkin approximation method. Exploiting the geometrical symmetry of the rectangular trench about its center line taken as the y-axis, the problem is split into two separate problems involving the symmetric and antisymmetric potential functions describing the resultant motion in the fluid region, as was done by Kanoria et al. [5] for the problem of water wave scattering by a thick vertical barrier of rectangular cross section having four different geometrical shapes. The use of eigenfunction expansions of the potential functions, along with the Havelock inversion formula followed by a matching process, produces integral equations for the corresponding unknown horizontal velocity components across the vertical line through the corner point of the trench. The integral equations are approximated using multiterm Galerkin approximations involving ultraspherical Gegenbauer polynomials. Numerical estimates for the reflection and transmission coefficients are then obtained for various values of different parameters involved in the problem. These are seen to satisfy the energy identity. These coefficients obtained by the present method are also depicted graphically against the wave number. Some of the curves for these coefficients are compared with those found in the literature obtained using other methods and by laboratory experiments (cf. Lee and Ayer [10]). In addition, numerical results for the reflection and transmission coefficients are tabulated to compare with the corresponding results obtained by Kirby and Dalrymple [12]. Very good agreement is seen to have been achieved in both cases. These establish the correctness of the numerical results obtained here. It is also found that the width of the trench affects the reflection and transmission coefficients significantly, and there exists an infinite number of discrete wave frequencies at which waves are completely transmitted, as was also observed by Lee and Ayer [10]. 2 Mathematical formulation A rectangular submarine trench of width 2b is present at the bottom of an ocean of uniform finite depth h, and the depth of the trench from the mean free surface is c (Fig. 1). The water is assumed to be a homogeneous, inviscid,

123

Water wave scattering by rectangular trench

103 Free surface

x (x,y)

y (x,y)

h

c (-x,y)

2b

Fig. 1 Schematic diagram of a submarine trench

and incompressible fluid, and the motion in the fluid is two-dimensional and under the action of gravity only. Let (x, y) denote a Cartesian coordinate system where the y-axis is taken vertically downward into the fluid region and y = 0 is the undisturbed free surface. Under the assumption of the linearized theory of water waves, a normally −iσ t }, where incident wave train is described by the velocity potential function Re{φ inc (x, y)e 2 cosh k0 (h − y)e−ik0 (x−b) , (2.1) cosh k0 h with k0 being the real positive root of the transcendental equation σ2 , (2.2) k tanh kh = K , K = g with σ being the circular frequency of the incoming wave train and g the acceleration due to gravity. Let the resulting motion in the fluid be described by the velocity potential Re{φ(x, y)e−iσ t }; then φ(x, y) satisfies the boundary value problem φ inc =

∇ 2 φ = 0 in the fluid region

(2.3)

K φ + φ y = 0 on y = 0, |x| < ∞,

(2.4)

φx = 0 on x = ±b, y ∈ (h, c) (c > h),

(2.5)

r

1/3

∇φ is bounded as

r → 0,

(2.6)

with r the distance from a submerged edge of the trench, φ y = 0 on y = h, |x| > b,

(2.7)

φ y = 0 on y = c, |x| < b,

(2.8)

and finally φ(x, y) ∼



φ inc (x, y) + Rφ inc (−x, y) as x → ∞, T φ inc (x, y)

as x → −∞,

(2.9)

where R and T are the unknown reflection and transmission coefficients, respectively, and are to be determined. 3 Solution method Due to the geometrical symmetry of the rectangular trench about x = 0, we split the potential function φ(x, y) into symmetric and antisymmetric parts φ s (x, y) and φ a (x, y), respectively, so that φ(x, y) = φ s (x, y) + φ a (x, y),

(3.1)

123

104

R. Chakraborty, B. N. Mandal

where φ s (−x, y) = φ s (x, y) and φ a (−x, y) = −φ a (x, y).

(3.2)

Thus, we may restrict our analysis to the region x ≥ 0 only. Now From (3.2) we note that

φ s (x,

y) and

φ a (x,

y) satisfy Eqs. (2.3)–(2.8).

−φxs (0, y) = φxs (0, y), so that φxs (0, y) = 0, 0 < y < c,

(3.3a)

and φ a (0, y) = −φ a (0, y), so that φ a (0, y) = 0, 0 < y < c.

(3.3b)

Let the behavior of φ s,a (x, y) for large x be represented by  cosh k0 (h − y)  −ik0 (x−b) e φ s,a (x, y) ∼ + R s,a eik0 (x−b) as x → ∞, cosh k0 h where R s and R a are unknown constants. Using (2.9) we find that R s and R a are related to R and T by

(3.4)

1 s (R ± R a )e−2ik0 b . (3.5) 2 Now the eigenfunction expansions of φ s,a (x, y) satisfying Eqs. (2.3)–(2.5), (2.7), (2.8), (3.3), and (3.4) (for x > b) are given in what follows. Region I (x > b, 0 < y < h): R, T =

∞

φ s,a (x, y) =

  s,a cosh k0 (h − y)  −ik0 (x−b) e + R s,a eik0 (x−b) + An cos kn (h − y)e−kn (x−b) , cosh k0 h

(3.6)

n=1

where kn (n = 1, 2, . . .) are the real positive roots of the equation k tan kh + K = 0.

(3.7)

Region II (0 < x < b, 0 < y < c):   s    s ∞  B0 cos α0 x cosh α0 (c − y)  Bns cosh αn x φ (x, y) = cos αn (c − y) + φ a (x, y) B0a sin α0 x Bna sinh αn x cosh α0 c

(3.8)

n=1

where ±α0 and ±iαn (n = 1, 2, . . .) are the roots of the equation α tanh αc = K .

(3.9)

Now we have the matching conditions φxs,a (b + 0, y) = φxs,a (b − 0, y), 0 < y < h.

(3.10)

We define φxs,a (b + 0, y) = f s,a (y) 0 < y < h,

(3.11)

φxs,a (b

(3.12)

− 0, y) = g (y) 0 < y < c;

then g s,a (y) =

123

s,a



f s,a (y) 0 < y < h, 0 h < y < c.

(3.13)

Water wave scattering by rectangular trench

105

Due to the edge condition (2.6), we find that

f s,a (y) = O |y − h|−1/3 as y → h.

(3.14)

Using the expansions of φ s,a (y) in (3.6) in (3.11), we obtain ∞   cosh k0 (h − y)  s,a s,a R −1 + As,a (y), 0 < y < h. ik0 n (−kn ) cos kn (h − y) = f cosh k0 h

(3.15)

n=1

Using Havelock’s inversion formula in (3.15), we find 1 − R s,a =

4i cosh k0 h δ0

h f s,a (y) cosh k0 (h − y)dy,

(3.16)

0

with δ0 = 2k0 h + sinh 2k0 h, and As,a n

4 =− δn

h f s,a (y) cos kn (h − y)dy, 0

with δn = 2kn h + sin 2kn h (n = 1, 2, . . .).

(3.17)

Now, using (3.8) in (3.12) we obtain (− sin α0 b, cos α0 b)B0s,a α0

∞

cosh α0 (c − y)  (sinh αn b, cosh αn b)Bns,a αn cos αn (c − y) + cosh α0 c n=1

= g s,a (y), 0 < y < c.

(3.18)

Applying Havelock’s inversion formula and noting (3.13), we obtain B0s,a

4 cos hα0 c = (− sin α0 b, cos α0 b)γ0

h f s,a (y) cosh α0 (c − y)dy, 0

with γ0 = 2α0 c + sinh 2α0 c,

(3.19)

and Bns,a

4 = (sinh αn b, cosh αn b)γn

h f s,a (y) cos αn (c − y)dy, 0

with γn = 2αn c + sinh 2αn c.

(3.20)

Now matching φ s,a (x, y) across the line x = b gives ∞

 cosh k0 (h − y) As,a {1 + R s,a } + n cos kn (h − y) cosh k0 h 0

= (cos α0 b, sin α0 b)B0s,a

∞

cosh α0 (c − y)  (cosh αn b, sinh αn b)Bns,a cos αn (c − y), 0 < y < h, + cosh α0 c

(3.21)

n=1

123

106

R. Chakraborty, B. N. Mandal

which ultimately produces the first-kind integral equations h F s,a (u)Ms,a (y, u)du = 0

cosh k0 (h − y) , 0 < y < h, cosh k0 h

(3.22)

where F s,a (y) =

4 cosh2 k0 h s,a f (y), 0 < y < h, δ0 (1 + R s,a )

and Ms,a (y, u) = ×

(3.23)

∞  cos kn (h − y) cos kn (h − u) δ0 + (coth αn b, tanh αn b) 2 δn cosh k0 h n=1

cosh α0 (c − y) cosh α0 (c − u) cos αn (c − y) cos αn (c − u) , 0 < y, u < h. + (− cot α0 b, tan α0 b) γn γ0 (3.24) Ms,a (y, u)(0

It is obvious that < y, u < h) are real and symmetric in y and u. Now, if we define 1 − R s,a C s,a = −i , 1 + R s,a then, using (3.16) in (3.22), we obtain h C

s,a

=

F s,a (y)

cosh k0 (h − y) dy, cosh k0 h

(3.25)

(3.26)

0 s,a F (y) and C s,a

and are real quantities. Thus, if the integral equation (3.22) are solved, then these solutions can be used to evaluate C s,a from the relations (3.26), and these produce the actual reflection and transmission coefficients |R| and |T | using |C s − C a | |1 + C s C a | and |T | = , (3.27) |R| =   with  1/2  = 1 + (C s )2 + (C a )2 + (C s C a )2 , (3.28) which are obtained from Eqs. (3.25) and (3.5). Now we consider the Galerkin approximation method to solve the integral equation (3.22). The unknown functions F s,a (y) are approximated as F s,a (y) ≈ F s,a (y), 0 < y < h, where

F s,a (y)

F s,a (y) =

N 

(3.29)

have multiterm Galerkin expansions in terms of suitable basis functions given by ans,a f ns,a (y), 0 < y < h,

(3.30)

n=0

with ans,a being unknown constants. Since the horizontal component of velocity near the corner points (±b, h) of the trench has a cube-root singularity, following arguments similar to those in Kanoria et al. [5], the basis functions f ms,a (y) are found to be ⎤ ⎡ h d ⎣ −ky s,a  f m (y) = − (3.31) f m (t)dt ⎦ , 0 < y < h, e dy y

123

Water wave scattering by rectangular trench

107

where  f m (y) is chosen in terms of ultraspherical Gegenbauer polynomials of order 1/6 as   27/6 ( 16 )(2m)! 1/6 y  , 0 < y < h. C f m (y) = 2m h π (2m + 13 )h 1/3 (h 2 − y 2 )1/3

(3.32)

Now, using (3.32) in (3.31) and substituting these in (3.30), we get the approximate forms of F s,a (y). Using these approximate forms in (3.22), multiplying both sides by f ms,a (y), and integrating over (0, h), we obtain the linear systems N 

s,a ans,a K mn = dms,a , m = 0, 1, . . . , N ,

(3.33)

n=0

where h h s,a K mn

=

Ms,a (y, u) f ns,a (u) f ms,a (y)du dy, m, n = 0, 1, 2, . . . , N , 0

(3.34)

0

and h dms,a

= 0

cosh k0 (h − y) s,a f m (y)dy, m = 0, 1, . . . , N . cosh k0 h

(3.35)

Integrals (3.34) and (3.35) can be evaluated explicitly, as in Kanoria et al. [5], using the different properties and standard results on Gegenbauer polynomials. Thus, we find  ∞   cos2 kr h δ0 s,a m+n J2m+1/6 (kr h)J2n+1/6 (kr h) 4(−1) K mn = 2 δr (kr h)1/3 cosh k0 h r =1

(coth αr b, tanh αr b) + cos2 αr c J2m+1/6 (αr h)J2n+1/6 (αr h) γr (αr h)1/3 (− cot α0 b, tan α0 b) cosh2 α0 c I2m+1/6 (α0 h)I2n+1/6 (α0 h) + γ0 (α0 h)1/3

 , m, n = 0, 1, . . . , N , (3.36)

and dms,a =

I2m+1/6 (k0 h) , m = 0, 1, . . . , N . (k0 h)1/6

(3.37)

The constants ans,a (n = 0, 1, . . . , N ) are now obtained by solving the linear systems (3.33), and then relations (3.26) produce C s,a = so that

N  n=0 C s,a

ans,a dns,a ,

(3.38)

are now found, and |R| and |T | are evaluated from relations (3.27).

4 Numerical results s,a To solve the linear system (3.33) numerically, we need to truncate the infinite series involving K mn [Eq. (3.36)]. s,a . Also, As in Kanoria et al. [5], here also a six-figure accuracy is achieved by taking 200 terms in each series of K mn the accuracy can be further increased by taking more terms in the series (3.36). A representative set of numerical

123

108

R. Chakraborty, B. N. Mandal

Table 1 Reflection coefficient against K h for b/ h = 1.2, c/ h = 1.5 Kh

N =1

N =2

N =3

N =4

N =5

0.2

0.530422

0.530441

0.530591

0.530461

0.530463

0.6

0.238629

0.238631

0.238694

0.238705

0.238707

1.0

0.159541

0.159554

0.159564

0.159569

0.159572

1.4

0.106001

0.106050

0.106107

0.106119

0.106127

1.8

0.050003

0.050011

0.050012

0.050051

0.050005

Table 2 Energy identity for b/ h = 1.2, c/ h = 1.2 Kh

|T |

|R|

|R|2 + |T |2

0.1

0.872057

0.489404

1.000

0.3

0.997393

0.0721599

1.000

0.5

0.989156

0.146869

1.000

0.7

0.996946

0.0780884

1.000

0.9

0.999952

0.00984322

1.000

Table 3 Transmission and reflection coefficient for c/ h = 3, b/ h = 5 Kirby and Dalrymple k0 h

T

Present method R

T

R

0.341

0.8881

0.4596

0.88926

0.469061

0.723 1.296

0.9552 0.9995

0.2960 0.0306

0.9560 0.9985

0.2956 0.0312

estimates of |R| is given in Table 1, taking N = 1, 2, 3, 4, and 5 in the (N + 1)-term Galerkin approximations for different values of the nondimensional wave number (K h) and for some particular values of nondimensional parameters b/ h and c/ h. From this table it is seen that the results for |R| converge very rapidly with N . For N ≥ 2, an accuracy of almost five decimal places is achieved. Thus, the present procedure for the numerical computation of |R| (and |T |) is quite efficient. The reflection and transmission coefficients |R| and |T | are evaluated numerically for different values of the parameters b/ h and c/ h and the wave number K h. It is observed that the energy identity |R|2 + |T |2 = 1 is always satisfied numerically. A representative set of values of |R| and |T | and |R|2 + |T |2 for b/ h = 1.2, c/ h = 1.2, and K h = 0.1, 0.3, 0.5, 0.7, and 0.9 are given in Table 2. This table provides a partial check on the correctness of the results obtained by the present method. The numerical values of |R| and |T | estimated by the present method with those given by Kirby and Darlymple [12] for c/ h = 3, b/ h = 5 and k0 h = 0.341, 0.723, 1.296 (k0 h = k1 h 1 in Kirby and Darlymple) obtained by using the eigenfunction matching method are displayed in Table 3 for the purpose of direct comparison. From this table, it is observed that the results coincide upto 2–3 decimal places. This also provides another check on the correctness of the results. We have also compared our results with those given by Lee and Ayer [9] obtained by a different method. Lee and Ayer plotted |R| and |T | against the ratio of the depth of water and incident wavelength (h/λ) (≡ k0 h/2π here) in their Figs. 2–7 for different values of d (the depth of the trench below the uniform finite depth h, here d(= (c − h)), and l (length of the trench, here l = 2b). In their paper they considered dimensional quantities (dimension of length

123

Water wave scattering by rectangular trench

Fig. 2 |R| against k0 h/2π , c/ h = 2, b/ h = 2.5 (solid line Lee and Ayer, dashed line present method)

109

Fig. 3 a |T | against k0 h/2π , c/ h = 7.625, b/ h = 2.640625 (solid line Lee and Ayer, dashed line present method). b |R| against k0 h/2π , c/ h = 7.625, b/ h = 2.640625 (solid line Lee and Ayer, dashed line present method)

in inches). In their Fig. 2, Lee and Ayer presented |R| against h/λ(≡ k0 h/2π ) for h = 4 , d(≡ c − h) = 4 , l(≡ 2b) = 20 , so that c/ h = 2, b/ h = 2.5. In Fig. 2 here, |R| is depicted against k0 h/2π for different values of the ratios c/ h and b/ h obtained from Lee and Ayer’s data. For better comparison the curve for |R| given in Fig. 2 of Lee and Ayer is displayed in Fig. 2 here, together with the curve for |R| drawn by the present method. These two curves of |R| match quite well with each other. Again, in Fig. 3 of Lee and Ayer, |T | is depicted against h/λ for h = 4 , c − h = 26.5 , 2b = 21.125 . This is also displayed in Fig. 3a here wherein |T |(calculated by the present method) is also depicted against k0 h/2π for c/ h = 7.625, b/ h = 2.640625, and the two curves match quite well. Also, the curve in Fig. 4 of Lee and Ayer depicting |R| against h/λ for the same values of h, c − h, and 2b is displayed in Fig. 3b, along with |R| (calculated by the present method) against k0 h/2π for the same values of c/ h and b/ h as in Fig. 3a. The two curves also match very well. Finally, the curve and the experimental results in Fig. 5 of Lee and Ayer [9] depicting |T | against h/λ for h = 4 , c − h = 26.5 , 2b = 42.375 are displayed in Fig. 4a, in which |T |, calculated by the present method, is also depicted against k0 h/2π for values of c/ h and b/ h obtained from Lee and Ayer’s data for their Fig. 5, i.e., c/ h = 7.625, b/ h = 5.296875. The two curves and the experimental data of Lee and Ayer [9] match more or less satisfactorily. The idea behind drawing different figures, keeping h and c − h fixed and increasing the trench length, is to find the effect of the trench length on |R| and |T |. As the trench length increases, both |R| and |T | become more oscillatory, and the number of zeros of |R| increases. Figure 4b here depicts |R| against k0 h/2π for the same values of c/ h and b/ h as in Fig. 4a (the corresponding figure was, however, not given in Lee and Ayer), and this

123

110

R. Chakraborty, B. N. Mandal

(a)

0.7

(b) 0.7

0.6

0.5

0.5

0.4

0.4

|R|

|R|

0.6

0.3

0.3

0.2

0.2

0.1

0.1

0

0

0.05

0.1

0.15

0.2

0

0.25

0

0.1

0.2

0.3

0.4

koh/2π

Fig. 4 a |T | against k0 h/2π , with c/ h = 7.625, b/ h = 5.296875 (solid line Lee and Ayer, dotted line present method). b |R| against k0 h/2π , with c/ h = 7.625, b/ h = 5.296875

0.8

0.6

0.6

0.6

|R|

0.8

0.4

0.4

0.4

0.2

0.2

0.2

0

0.2

0.4

0.6

Kh

0.8

0.7

0.8

0.9

1

(c) 1

0.8

0

0.6

Fig. 5 |R| against k0 h/2π , with c/ h = 7.625, b/ h = 5.296875

(b) 1

1

|R|

|R|

(a)

0.5

k0h/2π

1

1.2

0

0

0.2

0.4

0.6

0.8

1

1.2

0

0

0.2

Kh

0.4

0.6

0.8

1

1.2

Kh

Fig. 6 |R| against K h, with a c/ h = 1.5, b/ h = 1.2, b c/ h = 1.5, b/ h = 1.5, c c/ h = 1.5, b/ h = 2

figure clearly demonstrates the phenomenon of occurrence of more zeros of |R| and more oscillatory behavior of |R| against k0 h/2π as the trench length increases. Figure 5 is the same as Fig. 4b with an enlarged range of k0 h/2π , i.e., (0, 1.0), and it demonstrates that the maxima of the amplitude of |R| become small as k0 h/2π increases. This means that for waves of small wavelengths, most of the incident wave energy is transmitted over the trench since the waves are mostly confined near the free surface. In Fig. 6a–c, |R| is depicted against the wave number K h for c/ h = 1.5 and three different values of b/ h, viz., b/ h = 1.2, 1.5, 2.0. From these figures it is obvious that the number of zeros of |R| increases as the trench length increases. This was also demonstrated by Lee and Ayer [9] theoretically as well as experimentally. It may be noted that the zero-reflection phenomenon holds for symmetric trenches only, i.e., when the two sides of the trench are

123

Water wave scattering by rectangular trench 1

(c) 1

(b) 1 0.8

0.8

0.6

0.6

0.6

|R|

0.8

|R|

|R|

(a)

111

0.4

0.4

0.4

0.2

0.2

0.2

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Kh

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

Kh

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Kh

Fig. 7 |R| against K h, with a c/ h = 1.5, b/ h = 2, b c/ h = 5, b/ h = 2, c c/ h = 10, b/ h = 2

of the same depth, the reflection coefficient becomes zero for some discrete values of K h, and the number of zeros increases with the depth and length of the trench. The phenomenon of zero reflection was studied in some detail in papers by Xie et al. [14] and Liu et al. [15] showed that zero reflection at a long-wave range (small values of K h or k0 h/2π here) occurs only if the trench is symmetrical. This phenomenon was also observed here. However, the results obtained here are valid for all wave ranges in the case of a symmetric trench only. The periodic oscillatory behavior of the reflection coefficient against k0 c proved by Liu et al. [15] in the whole wave range was also observed for the reflection coefficient against K h here. In Fig. 7a–c, |R| is drawn against K h, keeping b/ h = 2 fixed and taking c/ h = 1.5, 5, 10 respectively to visualize the effect of the trench depth. It is observed that the amplitude of |R| increases to some extent as the trench depth increases.

5 Conclusion The problem of water wave scattering by a rectangular symmetric trench is reinvestigated here by employing the multiterm Galerkin approximation method involving ultraspherical Gegenbauer polynomials of order 1/6. Very accurate numerical estimates for the reflection and transmission coefficients for different values of the wave number and other parameters involved in the physical problem were obtained; these satisfy the energy identity. The numerical results are illustrated in a number of figures. Some of the figures match quite well with those given in the literature drawn using different mathematical methods and by laboratory experiments. The length and depth of the trench affect the reflection and transmission coefficients significantly, as is evident from the various figures depicting the reflection coefficient. Xie et al. asserted that the phenomenon of zero reflection in the long wave range occurs if and only if the trench is symmetrical. It is observed here that zero reflection also occurs in a medium wave range. In fact, for a short wave range the reflection will be negligible because the waves will be confined near the free surface and the presence of the trench will not have much effect on incident waves of large wave number. Most of the curves depicting |R| in the various figures demonstrate this behavior since the maxima of |R| become small gradually as the wave number increases. Acknowledgments The authors thank the reviewers and Professor A. A. Korobkin for their comments and suggestions for improving the paper in the present form. This work is supported by a NASI Senior Scientist Fellowship and a DST research project (Sr/S4/MS:521/08).

References 1. Porter R, Evans DV (1995) Complementary approximations to wave scattering by vertical barriers. J Fluid Mech 294:155–180 2. Evans DV, Fernyhough M (1995) Edge waves along periodic coastlines. Part 2. J Fluid Mech 297:307–325 3. Banerjea S, Kanoria M, Dolai DP, Mandal BN (1996) Oblique wave scattering by a submerged thin wall with gap in finite depth water. Appl Ocean Res 18:319–327

123

112

R. Chakraborty, B. N. Mandal

4. Das P, Dolai DP, Mandal BN (1997) Oblique wave diffraction by two parallel thin barriers with gaps. J Waterw Port Coast Ocean Eng 123:163–171 5. Kanorai M, Dolai DP, Mandal BN (1999) Water wave scattering by thick vertical barriers. J Eng Math 35:361–384 6. Mandal BN, Kanoria M (2000) Oblique wave-scattering by thick horizontal barrier. J Offshore Mech Arctic Eng 122(2):100–108 7. Kreisel H (1949) Surface waves. Q Appl Math 7:21–44 8. Mei CC, Black JL (1969) Scattering of surface waves by rectangular obstacles in water of finite depth. J Fluid Mech 38(3):499–511 9. Lassiter JB (1972) The propagation of water waves over sediment pockets. Ph.D. Thesis, Massachusetts Institute of Technology 10. Lee JJ, Ayer RM (1981) Wave propagation over a rectangular trench. J Fluid Mech 110:335–347 11. Miles JW (1982) On surface wave diffraction by a trench. J Fluid Mech 115:315–325 12. Kirby JT, Dalrymple RA (1983) Propagation of obliquely incident water waves over a trench. J Fluid Mech 133:47–63 13. Bender CJ, Dean RG (2003) Wave transformation by two dimensional bathymetric anomalies with sloped transitions. Coast Eng 56:61–84 14. Xie J-J, Liu H-W, Lin P (2011) Analytical solution for long wave reflection by a rectangular obstacle with two scour trenches. ASCE J Eng Mech 137:919–930 15. Liu H-W, Fu D-J, Sun X-L (2013) Analytic solution to the modified mild-slope equation for reflection by a rectangular breakwater with scour trenches. ASCE J Eng Mech 139:39–58

123