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Effect of Bragg scattering due to bottom undulation on a floating dock
P. Kar, T. Sahoo, H. Behera
PII: S0165-2125(18)30435-9DOI: https://doi.org/10.1016/j.wavemoti.2019.04.011Reference: WAMOT 2367
To appear in: Wave Motion
Received date : 10 October 2018Revised date : 11 March 2019Accepted date : 28 April 2019
Please cite this article as: P. Kar, T. Sahoo and H. Behera, Effect of Bragg scattering due to bottomundulation on a floating dock, Wave Motion (2019),https://doi.org/10.1016/j.wavemoti.2019.04.011
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Effect of Bragg scattering due to bottom undulation on a floating dock
P. Kara, T. Sahooa,∗, H. Beherab
aDepartment of Ocean Engineering and Naval Architecture, Indian Institute of Technology Kharagpur, IndiabSRM Research Institute & Department of Mathematics, SRM Institute of Science and Technology, Kattankulathur,
Chennai, India
Abstract
A numerical model is developed using the boundary element method to analyze gravity wave trans-formation by a finite floating dock in the presence of bottom undulation such as trenches, breakwatersand a combination of both in two-dimensions. The role of bottom undulation on the sea side of thefloating dock has a significant role in wave scattering compared to that of bottom undulation belowthe dock. The study depicts that Bragg reflection occurs in the case of an array of trenches, breakwa-ters or a combination of both in the absence of a floating dock. Bragg resonance occurs four times inthe case of an array of rectangular breakwaters or a combination of trenches and breakwaters, whilsttwice occurs in case of parabolic breakwaters and once occurs in case of triangular breakwaters inwater of finite depth. On the other hand, in the case of submerged trenches, Bragg resonance doesnot occur in deep water. However, in the presence of a floating dock, wave reflection increases in anoscillatory manner with the occurrence of harmonic and sub-harmonic peaks and depressions. Withintwo consecutive harmonic peaks/depressions, n−1 sub-harmonic peaks/depressions occur in the caseof n trenches/breakwaters with floating dock unlike the occurrence of n−2 sub-harmonic peaks withintwo consecutive harmonic peaks without the floating dock. Certain changes in phase in the sinusoidalpattern of wave reflection by the floating dock occurs due to the change in the position of the trenchesand the breakwaters, whilst the amplitude of wave reflection is independent of the position of thetrenches and the breakwaters.
Keywords: Surface gravity waves, Bragg resonance, Submerged trench, Breakwater, Floating dock,Boundary element method, Reflection coefficients.
1. Introduction
The use of dock is well known from the early days of human civilization. The earliest dockdiscovered is the ancient Egyptian harbor dating from 2500 BCE located on the Red Sea coast, whilethe other being the dock from Lothal in India dates from 2400 BCE which was a thriving trade centerreaching the far corners of West Asia and Africa (see Rao [1]). During the last 4500 years, thereis a significant growth in the design and use of docks for various humanitarian activities including
∗Corresponding authorEmail addresses: [email protected] (P. Kar), [email protected],
[email protected] (T. Sahoo), [email protected] (H. Behera)
Preprint submitted to Elsevier March 11, 2019
global trade and ocean space utilization. However, one of the greatest challenges is the wave-inducedhydrodynamic forces acting on the dock. In the past, many docks have collapsed due to wave loadsduring extreme events like storm surges. Moreover, the rise in global trade has put pressure on the useof large vessels which requires deep sea docks for cargo handling and operation. In this context, tofind an environmentally friendly and sustainable technological solution to meet the demand for landspace and address some of the challenges due to sea level rise, there is a rise in the construction offloating docks for cargo handling operation and management. These floating structures are plannedto be kept in position with the help of appropriate mooring system that will restrict the horizontalmovement of the floating docks. They are often open to the marine environment and are subjectedto a wave-induced load which often damage these floating structures. A case study of the impact ofwind and wave action on floating dock anchorage systems on Lake Brownwood, Texas is studied byJones et al. [2]. In the literature, several studies have been performed for understanding the effectof ocean waves on these floating docks. The study of surface gravity wave interaction with a finiterigid dock started since Haskind [3]. Heins [4] studied the effect of surface waves on a semi-infinitefloating dock in a channel of uniform depth. Ursell [5] studied the two-dimensional problem of wavestransmitted under a single fixed body and presented in detail when the cross-section is circular withits center being on the mean free surface. Some of the major study on wave interaction with floatingdocks in deep water or water of uniform depth were performed by Holford [7, 8], Leppington [9],Dorfmann and Savvin [10], Linton [11], Hermans [12] and Garrett [13].
On the other hand, surface gravity wave propagation in the presence of bottom undulation is sig-nificant under various situations such as wave transformations over continental shelves and wavescattering by submerged breakwaters and trenches of different configurations as has been mentionedby Newman [14]. Surface water wave scattering due to the abrupt change in water depth (finite toinfinite) has been studied earlier by Bartholomeusz [15] using long wave approximation. Rhee [16]derived the first and second order solutions associated with the oblique wave transmission over astep. One of the interesting physical phenomena related to surface wave propagation over periodicsinusoidal sand bars or ripples is Bragg resonance which occurs when the gap between two adjacentbars is approximately half of the wavelength of the incident waves. During Bragg resonance, thereflected waves return in phase reinforcing each other. Several studies on Bragg scattering revealedthat an array of small bars is as effective as a single massive breakwater (Davies & Heathershaw [17];Mei [18]; Dalrymple & Kirby [19]) and O’Hare and Davies [20]). Motivated by the phenomenon ofBragg resonance developed by sand bars/ripples, Mei et al. [21] developed the concept of submergedartificial bars for protecting the drilling platform from storm wave attack in the Ekofisk oil fields ofthe North Sea. These artificial bars are made up of a series of shore-parallel low-height artificial barslocated outside the surf zone and are popularly known as Bragg breakwaters. During the last threedecades, the efficiency of Bragg breakwaters of various configurations such as rectangular, triangular,trapezoidal, semicircular and parabolic have been studied (Naciri and Mei [22]; Kirby and Anton[23]; Hsu et al. [24]; Jeon and Cho [25]; Tsai et al. [26]). Recently, Tsai et al. [27, 28] studied thescattering of weakly viscous waves over an undulated topography of finite length.
Apart from abrupt changes in water depth and submerged breakwaters, wave scattering by sub-
2
merged trenches plays a prominent role in the creation of calm zone and protection of coastal infras-tructures. Lee et al. [29] analyzed the scattering of gravity waves by a rectangular trench and revealedthat full transmission occurs for some wave frequencies. Kirby and Dalrymple [30] analyzed scatter-ing of gravity waves by a submerged trench and demonstrated the occurrence of full transmission inthe case of a symmetric rectangular trench when the trench width is an integer time of half of the wavelengths. Bender and Dean [31] studied the wave transformation over trenches and shoals and collatethe patterns of reflection and transmission coefficients due to abrupt changes in trench slope. Lee etal. [32] analyzed the diffraction of random waves in three dimensions by an array of submarine pitsand depicted that the change in wave height is less prominent for random waves than that of regular-waves. Liu et al. [33] studied the reflection of surface gravity waves by a rectangular breakwater withtwo scoured trenches and depicted that wave reflection increases as the steepness of the trench wallincreases, whilst zero reflection occurs for symmetrical trenches. Tsai et al. [34] studied the scatteringof water wave over steep or undulated bottom topography. A recent study by Kar et al. [35] on wavetransformation by an array of submerged trenches revealed the occurrence of Bragg reflection for anarray of submerged trenches whose amplitude increases as the number of trenches increases.
The study above depicts that effect of various geometrical configurations of the seabed profilescan be best understood by investigating wave-structure interaction problems in finite water depthhaving bottom undulation of varied nature. However, there is negligible progress in the literatureto understand wave scattering by a floating dock due to variation in bottom bed. Bhattacharjee andSoares [36] investigated wave scattering by a floating structure located near a wall in the presenceof the step-type bottom. Recently, Dhillon et al. [37] studied the scattering of surface gravity wavesby a floating dock of finite extent due to an abrupt change in water depth. In both the studies, thebottom variation was considered below the floating structure. Thus, it will be interesting to examinethe effectiveness of various combinations of bottom undulations such as submerged trenches andbreakwaters on a floating dock.
In the present study, surface gravity wave scattering by a finite floating dock in the presence ofbottom undulation of varied nature is analyzed under the assumption of linear water wave theoryusing boundary element method (BEM). The undulated bed profiles include both the cases of multiplesubmerged trenches and breakwaters of three different configurations; namely rectangular, parabolicand triangular shapes and a suitable combination of these trenches and breakwaters. Moreover, thefloating dock is assumed to be rigid in nature which is kept in position with the appropriate stationkeeping arrangement. Effects of bottom undulation below the floating dock and at a finite distanceon the seaside from the floating dock are studied for understanding the process of surface gravitywave transformation. Known results for wave scattering by a floating dock due to an abrupt changein the bottom bed below the floating dock are reproduced to validate the developed numerical model.Moreover, energy relation is used to check the accuracy of the computed results in special cases. Roleof submerged symmetric and asymmetric trenches and breakwaters on gravity wave transformationby the floating dock are investigated by analyzing the reflection and transmission coefficients fordifferent values of wave and structural parameters namely; non-dimensional wave number, structuralconfigurations of trenches and breakwaters, dock length and gap between the floating dock and the
3
adjacent bottom undulation. Further, the role of the position of multiple trenches and breakwaters andtheir combinations on wave transformation with and without floating dock are analyzed.
2. Mathematical Formulation
Scattering of surface gravity waves by a single as well as multiple trenches and breakwaters ofvarious configurations in the presence of floating dock are studied in finite water depth based on inthe linearized theory of water waves. The physical problem is analyzed in the two-dimensional x − ycoordinate system as discussed in Kar et al. [35]. The water depths on the front and lee side ofthe floating dock at the far-fields are assumed to be of depths h1 and h4 respectively, whilst h(x) inFigs. 1(a) and 1(b) represents the position of the bottom profile of the undulated seabed below thedock and at a finite distance from dock respectively. It is assumed that the fluid is homogeneous,inviscid, incompressible with the motion being irrotational. Further, it is assumed that flow is simpleharmonic in time t with angular frequency ω. Thus, there exists a velocity potential of the formRe
[ϕ1(x, y)e−iωt
], where Re [.] denotes the real part, i =
√−1 is the complex unit, and ϕ1 is thespatial component of the velocity potential which satisfies
∇2ϕ1(x, y) = 0, in the fluid region. (1)
(a) (b)
Figure 1 Schematic diagram of wave motion due to bottom undulation (a) below the dock and (b) at a finitedistance from the dock.
The linearized boundary condition on the mean free surface y = 0 satisfies
∂ϕ1
∂y− Kϕ1 = 0, (2)
with K = ω2/g and g being the acceleration due to gravity. The no flow condition on the sea bedyields
∂ϕ1
∂n= 0 on y = h(x), (3)
where h(x) is the known bottom bed profile which will be defined subsequently for specific configu-rations of the sea bed and the subscript n is in the outward normal direction to the sea bed. Finally,the radiation conditions at far fields are of the forms
ϕ1(x, y) = ϕ1
0(x, y) + Rϕ10(−x, y) as x→ −∞,
ϕ1(x, y) = Tϕ40(x, y) as x→ +∞,
(4)
4
with R and T being unknown constants corresponding to the amplitudes of the reflected and transmit-ted waves respectively. Moreover,
ϕj0(x, y) = I j
0(k0, y)eik0x, I jn(kn, y) =
−igAω
cosh kn(y + h j)cosh knh j
,−h j < y < 0, (5)
and A is the incident wave amplitude which is of unit magnitude and k0 corresponds to the the positivereal root of the open water region dispersion relation ω2 = gk0 tanh k0h j for j = 1, 4.
3. Integral equation formulation
The aforementioned boundary value problem is handled for a solution using the boundary elementmethod which is similar to the one used by Kar et al. [35] for analyzing scattering of water waves overan array of submerged trenches. Here the computational fluid domain is confined by the boundaryΓ consisting of the line segments which is defined as Γ = Γ f1 ∪ Γ f2 ∪ Γ f3 ∪ ΓC1 ∪ Γb ∪ ΓC2 as inFig. 2. Further, Γb, Γ f1∪ f3 (i.e. Γ f1 ∪ Γ f3) and Γ f2 denote the total bottom boundary, the free surfaceboundary and boundary along the floating dock respectively. Here the solution of the integral equationis based on two different approaches. In the first approach, two auxiliary boundaries ΓC1 and ΓC2 areintroduced at sufficiently large distances from the undulated seabed namely; trenches/breakwatersfor making the computational domain closed which will satisfy the far-field condition as in Eq. (4).On the other hand, in the second approach, the computational domain is divided into two regionsnamely; the outer region and the inner region which are differentiated by two auxiliary boundaries.The boundary element method is applied to the inner region and the from of the solution in the outerregion is obtained by the eigenfunction expansion method. Both the inner and outer region solutionsare matched on the auxiliary boundary to obtain the full solution. A similar approach was used byKoley et al. [40]. In both the cases, the inner region solution is denoted with subscript whilst, in thesecond approach, the outer region solutions on the sea side of the trench is denoted with subscript 2and lee side of the floating dock is denoted with subscript 3. The detail solution procedure will bediscussed in two different subsections for clarity.
(a) (b)
Figure 2 Computational domain of wave motion due to bottom undulation (a) below the dock and (b) at a finitedistance from the dock.
5
Applying Green’s second identity to the velocity potential ϕ1(x, y) and the two-dimensional freespace Green’s function G(x, y; x0, y0) bounded by Γ, it is easily derived that
ϕ1(x, y)12ϕ1(x, y)
=∫
Γ
(ϕ1∂G∂n−G∂ϕ1
∂n
)dΓ,
if (x, y) ∈ Ω \ Γif (x, y) ∈ Γ
. (6)
In Eq.(6), G(x, y; x0, y0) satisfies(∂2
∂x2 +∂2
∂y2
)G = δ (x − x0) δ (y − y0) (7)
with G(x, y ; x0, y0) = (ln r)/2πwhere r is the Euclidean distance between field point (x, y) and sourcepoint (x0, y0). Using Eqs. (2), (3), (6) is converted into the following integral equation as given by
Cϕ1 +
∫
Γ f1∪ f3
(∂G∂n− KG
)ϕ1dΓ +
∫
ΓC1
(ϕ1∂G∂n−G∂ϕ1
∂n
)dΓ +
∫
Γb∪ f2
ϕ1∂G∂n
dΓ
+
∫
ΓC2
(ϕ1∂G∂n−G∂ϕ1
∂n
)dΓ = 0 , (8)
where C = −1/2. Next, the boundary Γ is discretized into finite number of line segments referred asboundary elements and using the constant element approach as discussed in kar et al. [35], Eq. (8) istransformed into a system of linear equation which yields
N f1∪ f3∑
j=1
(Hi j − KGi j
)ϕ1 j
∣∣∣∣Γ f+
NC1∑
j=1
Hi jϕ1 j
∣∣∣∣ΓC1
+
NC2∑
j=1
Hi jϕ1 j
∣∣∣∣ΓC2
+
Nb∪ f2∑
j=1
(Hi j
)ϕ1 j
∣∣∣∣Γb
=
NC1∑
j=1
∂ϕ1 j
∂xGi j
∣∣∣∣ΓC1
+
NC2∑
j=1
∂ϕ1 j
∂xGi j
∣∣∣∣ΓC2
, (9)
where Hi j = Cδi j +∫Γ j
∂G∂n dΓ and Gi j =
∫Γ j
GdΓ are known as the influence coefficients. These co-efficients are calculated analytically if source and field points are on the same boundary segment, andestimated numerically using Gauss quadrature formula otherwise. The number N and the superscriptsN f1∪ f3 ,Nb∪ f2 ,NC1 , NC2 as in Eq. (9) corresponds to the total number of boundary elements over wholeboundary and the corresponding sub-boundaries respectively. It may be noted that the integrals overthe auxiliary boundaries Γc j are computed using two different approaches which are discussed belowseparately in two different subsections.
3.1. Solution in the outer region using far-field condition
In this approach, the auxiliary boundary is located at the far field. The equivalent differential formsof the far field conditions as in Eq. (4) are rewritten in the form (as in Kim and Kee [39] and Koley etal. [40])
lim|x| → ∞
(∂
∂x± ik
) ϕ1 − ϕ1
0
ϕ1
= 0 on ΓC1 ∪ ΓC2 , (10)
where ϕ10 is defined as in Eq. (5). Substituting the differential form of the far-field condition in Eq.
(10) into Eq. (9) and then using the collocation method as discussed in Pozrikidis [38], which wasrecently used in [35], an algebraic system of equations is obtained which is solved to determine thevelocity potential in the computational domain.
6
3.2. Solution in the outer region using eigenfunction expansion method
In this approach, the solution in the outer region is presented using eigenfunction expansionmethod satisfying the Laplace equation as well as free-surface, bottom and radiation boundary condi-tions. In this method, the far-field condition is located near the structure. The continuity of pressureand normal velocity at the auxiliary boundaries ΓC1 and ΓC2 yield
ϕ2 = ϕ1 = ϕ2,∂ϕ2
∂n=∂ϕ1
∂n=
˜∂ϕ2
∂nfor ΓC1 ,
ϕ3 = ϕ1 = ϕ3,∂ϕ3
∂n=∂ϕ1
∂n=
˜∂ϕ3
∂nfor ΓC2 .
. (11)
where the upper bar means the value newly defined at the matching boundary ΓC1 and ΓC2 . In theouter regions of uniform depths h1 and h4, using the eigenfunction expansion method, ϕ2 and ϕ3 arepresented in the form
ϕ2(x, y) = e−ik0(x+l)I10(k0, y) +
∞∑
n=0
Rneikn(x+l)I1n(kn, y) (12)
ϕ3(x, y) =∞∑
n=0
Tneikn(x−r)I4n(kn, y) (13)
where I jn(kn, y) being the same as defined in Eq. (4) and Rns are unknown constants which will be
obtained as a part of the solution. Using the orthogonality criteria of I jn(kn, y) for j = 1, 4, from Eq.
(12-13), the unknown constants Rn and T are obtained as
Rn + δn0 =1
An1
∫ 0
−hϕ2(y)I1
n(kn, y)dy, (14)
Tn =1
An2
∫ 0
−hϕ3(y)I4
n(kn, y)dy (15)
with
An1 = − g2
ω2
knh1 + sinh(knh1) cosh(knh1)2kn cosh2(knh1)
, and An2 = − g2
ω2
knh4 + sinh(knh4) cosh(knh4)2kn cosh2(knh4)
,
where ϕ2(y) = ϕ2(−l, y) and ϕ3(y) = ϕ3(r, y). Substituting for Rn and Tn from Eq. (14-15) into Eq.(12-13) and truncating the infinite series after N terms, it can be easily derived as
∂ϕ2
∂x
∣∣∣∣x=−l= Ql[ϕ1] + 2ik0I1
0 (16)
∂ϕ3
∂n
∣∣∣∣x=r= Qr[ϕr], (17)
where Ql[ϕ1] and Qr[ϕr] are given by
Ql[ϕ1] = −N∑
n=0
ikn
An
∫ 0
−hϕ2(s)I1
n(kn, s)dsI1n(kn, y) (18)
7
and
Qr[ϕr] = −N∑
n=0
ikn
An
∫ 0
−hϕ3(s)I4
n(kn, s)dsI4n(kn, y). (19)
The integral∫ 0
−hϕ2(s)I1
n(kn, s)ds and∫ 0
−hϕ3(s)I4
n(kn, s)ds are computed over same boundary elementsas used in the evaluation of Greens function and its normal derivative on the auxiliary boundary. Inthe present study, constant element method is used in which the ϕ j for j = 2 and 3 are assumed tobe constants over each boundary element. Thus, the integral in the expression for Ql, and Qr areevaluated exactly, which yields the matrix factorization of Ql, Qr. By using Eqs. (18) and (19) inEq. (9), the system of linear equations in Eq. (9) are solved to determine the velocity potential inthe computational domain. Subsequently, the vertical exciting force F and moment M on the floatingdock are computed using the formulae (as in [11])
F =
∣∣∣∣∣∣1a
∫ a
−aϕ1(x, y)
∣∣∣∣y=0
dx
∣∣∣∣∣∣ and M =
∣∣∣∣∣∣1a2
∫ a
−axϕ1(x, y)
∣∣∣∣y=0
dx
∣∣∣∣∣∣. (20)
4. Results and Discussion
Using MATLAB software, a code is written for computing various quantities of physical interestusing the boundary element method as discussed in the aforementioned Section. For the purpose ofnumerical computation in the present study, the sea bed profile h(x) below the floating dock as shownin Fig. 2(a) is given by (as in Kar et al. [35])
h(x) =
−h1 for x < −b,−h2 − (h1 − h2) (−x/b)m1 for − b < x < 0,−h2 − (h1 − h2) (−x/b)m2 for 0 < x < b,−h4 for x > b,
(21)
where 2b is width of the trench. On the other hand, the sea bed profile h(x) as shown in Fig. 2(b), inwhich the bottom undulation is at a finite distance from a floating dock, is given by
h(x) =
−h1 for x < −(b1 + p) ∪ Lg ∪ (q + a),−h2 − (h1 − h2) (−2(x + b1/2 + p)/−b1)m1 for − (b1 + p) < x < −p + b1/2,−h2 − (h1 − h2) (−2(x + b1/2 + p)/b1)m2 for − p + b1/2 < x < −p,−h3 − (h2 − h3) (−2(x + b2/2 + q)/−b2)m1 for − (b2 + q) < x < −q + b2/2,−h3 − (h2 − h3) (−2(x + b2/2 + q)/b2)m2 for − q + b2/2 < x < −q,−h4 for x > a,
(22)
where p = b2 + a + L1 + Lg, q = a + L1, b1 and b2 correspond to the widths of the first and secondtrench respectively. Further, it may be noted that in Eqs. (21) and (22), m1,m2 are positive numbersattributing to the curvature of the trenches with m1 = m2 = 1 corresponding to a triangular trench,m1 = m2 = 2 corresponding to a parabolic trench and m1 = m2 = ∞ corresponding to a rectangulartrench. From the profiles of h(x),with suitable choice h2 < h1, h3, associated profiles of the break-water can be obtained. Finally, by choosing different combination of m1 and m2, various shapes of
8
trenches and breakwaters can be developed. The values of physical parameters such as h1 = 5m,wave period T = 8 sec, g = 9.8 m/sec2, a/h1 = 3, b/h1 = 4, b1/h1 = 2.64, b2/b1 = 1, Lg/L1 = 1,L1 = 60 m, h2/h1 = 0.5, h3/h1 = 1 and h4/h1 = 1 are kept fixed unless otherwise mentioned. Toinvestigate the role of various parameters on wave scattering by the floating dock in presence of un-dulated bottom bed, several results on reflection and transmission coefficients, force and moments onthe floating dock are computed and analyzed for various physical parameters related to waves andseabed undulation. To check the accuracy of the computational result, the energy identity connectingthe reflection and transmission coefficients is used which is given by (see Karmakar and Sahoo [41],Dingemans [42])
K2r + γK
2t = 1, (23)
where γ is given by
γ =
(k1 sinh 2k1h1
k4 sinh 2k4h4
)×
(2k4h4 + sinh 2k4h4
2k1h1 + sinh 2k1h1
).
It may be noted that an equivalent relation was derived by [26] in case of wave scattering by a floatingdock in the presence of stepped bottom. Moreover, for h4/h1 = 1, the energy identity in Eq. (17) issatisfied with γ = 1.
4.1. Convergence of BEM solutions based on the two approaches
Triangular Circular Step type Rectangularκ Kr Kt Kr Kt Kr Kt Kr Kt
10 0.9524 0.4849 0.9614 0.4367 0.9066 0.4781 0.9418 0.282620 0.9523 0.4893 0.9613 0.4412 0.9053 0.4828 0.9397 0.289030 0.9524 0.4893 0.9615 0.4412 0.9053 0.4830 0.9401 0.288340 0.9524 0.4893 0.9615 0.4412 0.9053 0.4831 0.9403 0.2880
Table 1 Convergence of solution for various shapes of a breakwater below the floating dock with h2/h1 =
0.5, h3/h1 = 1, a/h1 = 3 and b/h1 = 4.
The convergence of solutions based on the boundary element method depends on the panel size,the location of the auxiliary boundary and the number of evanescent modes. In the 1st approach, theconvergence study is performed for determining the panel size as discussed in Kar et al. [35] assumingthat the location of auxiliary boundary from the floating dock is five times the water depth. Moreover,the panel size ps is given by ps = 1/κk1 where κ is the proportionality constant which is computedfrom the convergence study as in Wang and Meylan [43]. Table 1 demonstrates the convergence studyof solution for wave scattering by a single breakwater of different configurations below the floatingdock. Table 1 reveals that the reflection and transmission coefficients converge up to three decimalplaces for κ ≥ 30. Thus, in the subsequent analysis of numerical results, panel size is chosen in anappropriate manner with κ = 30 for convergence of various numerical results.
Table 2 exhibits the location of auxiliary boundary in the numerical computation based on 1stapproach for convergence of the solution. The auxiliary boundaries located at x = −l and r with
9
Triangular Circular Rectangularn1, n2 Kr Kt E Kr Kt E Kr Kt E3, 3 0.9922 0.1354 1.0029 0.9925 0.1310 1.0024 0.9906 0.0564 0.98464, 4 0.9937 0.1352 1.0057 0.9941 0.1307 1.0054 0.9932 0.0561 0.98965, 5 0.9912 0.1362 1.0011 0.9917 0.1317 1.0008 0.9922 0.0562 0.98776, 6 0.9913 0.1359 1.0012 0.9916 0.1315 1.0007 0.9902 0.0566 0.9838
Table 2 Convergence of solution for three types of breakwaters positioned below the floating dock with l =n1h1 + g1, r = n2h3 + g2, a = 3h1, b/h1 = 4, h1 = h3 and h1 = 5m.
l = n1h1 + g1, r = n2h3 + g2, a = 3h1, b/h1 = 4, h1 = h3, n1, n2 ∈ Z and h1 = 5m where g1 andg2 are the left and right end points of breakwater respectively. Table 2 reveals that K2
r + γK2t is very
close to one for n1, n2 ≥ 5. Various results are presented for three types of breakwater configurations.Thus, for all computational purposes, values of n1 and n2 are considered to be five while choosingthe locations of the auxiliary boundaries. The same procedure is adopted for other configurations oftrenches and breakwater in the subsequent computation. In the ensuing discussion, various resultsrelated to trenches, breakwaters and their combination in the presence of floating dock are analyzed.
1st approach 2nd approachk1h1 n1 = n2 = 5 N = 1 N = 5 N = 10 N = 15 N = 200.5 0.0276 0.0275 0.0274 0.0274 0.0275 0.02751.0 0.4349 0.4362 0.4360 0.4359 0.4358 0.43581.5 0.0836 0.0836 0.0826 0.0823 0.0823 0.08232.0 0.1159 0.1140 0.1141 0.1144 0.1148 0.11482.5 0.1267 0.1259 0.1263 0.1264 0.1264 0.1264
Table 3 Convergence study of Kr for different values of N with Lg/h1 = 12,b1/h1 = 2.64, b2/b1 = 1, h1 = 5mand κ = 30 in case of double rectangular breakwaters in the absence of floating dock.
The convergence of BEM solution using the second approach as discussed in the above Section isdemonstrated by computing the reflection coefficient Kr for different evanescent modes N by keepingκ fixed as mentioned in Table 1. It is observed that the computed results for Kr converge up to threedecimal places for N ≥ 15. Further, a comparison of the computed values of Kr for different valuesof k1h1 in Table 3 by the two approaches reveals that the computed results for κ = 30 along withlocation of auxiliary boundary five times the water depth in the first approach agrees well with thatof the second approach with N = 15 and location of auxiliary boundary three times that of waterdepth. Thus, in the subsequent study, various computations are carried out for analysis using the firstapproach.
4.2. Validation of BEM results with known analytic results
In Fig. 3, variations of the (a) reflection Kr and transmission Kt coefficients with a/h1 = 1, d/h1 =
0.2 and (b) force F and moment M with a/h1 = 3, d/h1 = 0 in case of a step type bottom are plotted
10
0.5 1 1.5 2 2.5 3 3.5 4Kh
1
0
0.2
0.4
0.6
0.8
1
1.2
Kr
Kt
h2/h
1=0.5
h2/h
1=1
h2/h
1=1.5 (Present model)
h2/h
1=1.5 (Dhillon et al., 2016)
(a)
h1
2a
d
h2
0.5 1 1.5 2 2.5 3 3.5 4
Kh1
0
0.5
1
1.5
2
2.5
M
F
h2/h
1=0.5
h2/h
1=1
h2/h
1=1.5 (Present model)
h2/h
1=1.5 (Dhillon et al, 2016)
2a
h1
dh
2
(b)
Figure 3 Variation of (a) reflection Kr, transmission Kt coefficients with a/h1 = 1, d/h1 = 0.2 and (b) force Fand moment M with a/h1 = 3, d/h1 = 0 versus Kh1 in case of step type bottom.
against Kh1. Both the figures depicts that the computed results of the present study agree well withthat of the analytic results of Dhillon et al. [37]. Figure 3(a) depicts that the transmission coefficientdecreases gradually, whilst the reflection coefficient increases as Kh1 increases. Moreover, both thereflection and transmission coefficients follow the steady oscillatory pattern for higher values of Kh1.The study reveals that for smaller values of Kh1, transmission coefficient becomes more than onefor h2/h1 = 0.5 which is due to the effect of wave shoaling for higher values of water depth. Thepattern is similar to the scattering of long waves by a step as discussed in Dean and Dalrymple [44].Further, the reflection coefficient increases with the reverse trend in case of transmission coefficientas the depth ratio h2/h1 increases for smaller values of Kh1 and a reverse trend is observed for highervalues of Kh1 corresponding deep water. Figure 3(b) demonstrates that both forces and moments areacting on the floating dock decrease as water depth h2/h1 increases, which is due to the redistributionof wave energy across larger depth. Moreover, a comparison of Figs. 3(a) and 3(b) demonstrates thatthe trend of wave reflection is complementary to that of wave forces exerted on the floating dock dueto a change in Kh1.
4.3. Bottom undulation below the floating dock
In this subsection, the effects of trenches and breakwaters of different configurations as in Fig. 2are studied on the scattering of surface gravity waves by a floating dock.
4.3.1. Single Trench
In Fig. 4, reflection Kr and transmission Kt coefficients are plotted against wave number k1h1 forvarious (a) water depth h3/h1 in case of rectangular type bottom and (b) trench configuration witha/h1 = 3, b/h1 = 4 and h2/h1 = 1.5 and (c) variation of error associated with energy relation. Figure4(a) reveals that the transmission coefficient Kt is more than one in shallow water depth in the caseof an asymmetric trench in the presence of a floating dock as observed in Fig. 3(a). However, dueto change in water depth h3/h1, there is no variation in wave reflection and transmission in waterof intermediate depth or case of deep water. The observation is almost identical to that of wave
11
0 0.5 1 1.5 2 2.5 3 3.5 4
k1h
1
0
0.2
0.4
0.6
0.8
1
1.2
Kr
Kt
h3/h
1=0.75
h3/h
1=1
h3/h
1=1.5
h1 h
2h
3
2b
(a)
2a
0 0.5 1 1.5 2 2.5 3 3.5 4
k1h
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Kr
Kt
m1=m
2=1
m1=m
2=2
m1=m
2=
(b)
0 0.5 1 1.5 2 2.5
k1h
1
0
0.005
0.01
0.015
0.02
0.025
E=
Kr2 +
Kt2 -1
m1=m
2=1
m1=m
2=2
m1=m
2=
(c)
Figure 4 The reflection and transmission coefficients versus wave number k1h1 with (a) water depth h3/h1 incase of rectangular type bottom, (b) trench shapes with a/h1 = 3, b/h1 = 4 and h2/h1 = 1.5 and (c) variation oferror associated with energy relation.
scattering by a single trench in the absence of a floating dock as discussed in [35]. Figure 4(b) depictsthat variation of shapes of the trench below the dock does not affect wave reflection and transmissionwhich is because the change in configurations of the submerged trench has a negligible role on wavemotion. Figure 4(c) reveals that the wave energy relation about wave number k1h1 is satisfied up tothree decimal places for different configuration of the trench. However, the minor variation in waveenergy relation for higher values of k1h1 is due to the impact of short progressive waves propagatingin deeper water whose effect is negligible in shallow water depth corresponding to smaller values ofk1h1.
0 0.5 1 1.5 2 2.5
2a/1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Kr
K
t
m1=m
2=1
m1=m
2=2
m1=m
2=
(a)
0 1 2 3 4 5 6 7 8
2b/1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Kt
K
r
m1=m
2=1
m1=m
2=2
m1=m
2=
(b)
0 0.5 1 1.5 2 2.5 3 3.5 4
2b/1
0
0.5
1
1.5
2
2.5
E=
Kr2 +
Kt2 -1
10-3
m1=m
2=1
m1=m
2=2
m1=m
2=
(c)
Figure 5 Variation in reflection and transmission coefficients versus (a) dock length 2a/λ1 with b/h1 = 4 and(b) trench width 2b/λ1 with a/h1 = 3 for different values of m1 and m2 and (c) variation of error associatedwith energy relation.
In Figs. 5(a) and 5(b), the reflection and transmission coefficients are plotted against (a) docklength 2a/λ1 with b/h1 = 4 and (b) trench width 2b/λ1 with a/h1 = 3 for different values of m1
and m2 with h2/h1 = 1.5 and h3/h1 = 1 whilst in Fig. 5(c), variation of error associated with energyrelation. Figure 5(a) exhibits that wave reflection increases with an increase in dock length irrespectiveof trench configurations and the opposite trend is found in case of wave transmission. However, minorchanges in wave reflection and transmission are observed due to change in trench configurations inthe case of a floating dock of smaller length. On the other hand, Fig. 5(b) reveals certain periodicoscillatory trends in wave reflection due to variations in the trench width 2b/λ1 which is because of the
12
resonating interaction of wave propagation within the trenches. Further, it is noticed that amplitudeof oscillation in wave reflection is maximum in case of rectangular trenches, whilst amplitude ofoscillation is minimum for triangular trenches which may be due to the reduction in trench surfacearea where there is a remarkable reduction in the oscillatory trend of wave motion within the trenches.Figure 5(c) reveals that the wave energy relation concerning trench width 2b/λ1 is satisfied up to threedecimal places for different trench configurations.
4.3.2. Single breakwater
0 0.5 1 1.5 2 2.5 3 3.5 4
k1h
1
0
0.2
0.4
0.6
0.8
1
1.2
Kr
K
t h3/h
1=0.75
h3/h
1=1
h3/h
1=1.5
(a)
0 0.5 1 1.5 2 2.5 3 3.5 4
k1h
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Kr
K
t
m1=m
2=1
m1=m
2=2
m1=m
2=
(b)
0 0.5 1 1.5 2 2.5
k1h
1
0
0.005
0.01
0.015
0.02
0.025
E=
Kr2 +
Kt2 -1
m1=m
2=1
m1=m
2=2
m1=m
2=
(c)
Figure 6 Variations of reflection and transmission coefficients against wave number k1h1 with (a) water depthh3/h1 in case of rectangular type bottom and (b) shapes of breakwater m1,m2 and (c) variation of error associ-ated with energy relation.
Figure 6 exhibits the changes on in the reflection and transmission coefficients Kr and Kt againstwave number k1h1 with change in (a) water depth ratio h3/h1 in case of rectangular breakwater and(b) configurations of breakwater profile with a/h1 = 3, b/h1 = 4 and h2/h1 = 0.5 and (c) variationof error associated with energy relation. Figure 6(a) reveals that the transmission coefficient Kt ismore than one in the shallow water zone in case of the asymmetric trench, which is similar to theobservation made in Fig. 4(a). Figure 6(b) reveals that minor changes in wave reflection and trans-mission due to change in breakwater configurations in the shallow water region, whilst the changesare invariant for a triangular and parabolic breakwater in case of deep water. A comparison of Figs.4 and 6 demonstrates the occurrence of certain variations in the monotonically increasing trend inwave reflection and a decreasing trend in transmission in case of rectangular breakwater comparedto that of the corresponding trench which is because of the resonating effect of wave propagation inthe horizontal direction within the submerged breakwater and the dock. Figure 6(c) reveals that thewave energy relation concerning wave number k1h1 is satisfied up to three decimal places for differentconfiguration of the breakwater.
Figures 7(a) and 7(b) demonstrate the variations of reflection and transmission coefficients against(a) dock length 2a/λ1 and (b) breakwater width 2b/λ1 with water depth h3/h1 = 1 and h2/h1 = 0.5,whilst Figure 7(c) exhibits the error associated with energy relation for different kind of sea bedconfigurations. Figure 7(a) reveals that in case of triangular and parabolic breakwaters in the presenceof floating dock, variation in wave reflection and transmission are negligible compared to that of therectangular breakwater which is due to the shoaling of wave motion by the rectangular breakwater.On the other hand, Fig. 7(b) reveals that both reflection and transmission coefficients follow the
13
0 0.5 1 1.5 2 2.5
2a/1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Kr
K
t
m1=m
2=1
m1=m
2=2
m1=m
2=
(a)
0 1 2 3 4 5 6 7 8
2b/1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Kt
Kr
m1=m
2=1
m1=m
2=2
m1=m
2=
(b)
h2=0.5 h
1
h3=h
1
0 0.5 1 1.5 2 2.5 3 3.5 4
2b/1
0
0.5
1
1.5
2
2.5
E=
Kr2 +
Kt2 -1
10-3
m1=m
2=1
m1=m
2=2
m1=m
2=
(c)
Figure 7 The reflection and transmission coefficients against (a) dock length 2a/λ1 with b/h1 = 4 and (b)breakwater width 2b/λ1 with a/h1 = 3 for different shapes of the seabed configurations and (c) variation oferror associated with energy relation.
certain periodic oscillatory trend with an increase in the width of the breakwater 2b/λ1 which isalmost similar to that of Fig. 5(b). A comparison of Fig. 7(b) with that of Fig. 5(b) reveals thatwave reflection is more in case of the floating dock in the presence of a breakwater compared to thatof the trench whereas the reverse trend is observed in case of wave transmission. This is because theconfined fluid region below the floating dock in the presence of a submerged breakwater reflects amajor part of the wave energy compared to that of the trench. Further, the frequency of occurrencesof optima in wave reflection and transmission is more in case of a breakwater compared to that ofthe trench. The increase in the frequency of oscillation in case of a breakwater is due to an increasein the back and forth motion of gravity waves confined within the floating dock and the submergedbreakwater. Figure 7(c) reveals that the wave energy relation concerning the width of the breakwater2b/λ1 is satisfied up to three decimal places for different breakwater configurations.
4.3.3. Multiple trenches and breakwaters
0 0.5 1 1.5 2 2.5 3 3.5 4
k1h
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Kr
K
t
N=1N=2N=3
(a)
0 0.5 1 1.5 2 2.5 3 3.5 4
k1h
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Kr
Kt
N=1N=2N=3
(b)
Figure 8 Variation in reflection and transmission coefficients versus k1h1 for different number of triangular (a)trenches and (b) breakwaters with a = 60 m, h4/h1 = 1.
Figure 8 exhibits the variation of the reflection Kr and transmission Kt coefficients versus k1h1
for different number N of triangular (a) trenches and (b) breakwaters below the floating dock with14
a = 60m and h4/h1 = 1.0. Figure 8 reveals that wave reflection is least affected due to an increase inthe number of triangular trenches below the floating dock, whilst minor variations in wave reflectionoccur in case of multiple triangular breakwaters for 0 < k1h1 < π corresponding to waves in shallowand intermediate depths. The variation in wave reflection is due to the fact that a part of the incidentwave energy is reflected by multiple breakwaters unlike that of multiple trenches where higher waterdepth does not contribute to wave transformation.
4.4. Bottom undulation at a finite distance from the floating dock
In this subsection, effects of trenches, breakwaters and their combination on wave scattering bythe floating dock are studied in different cases. Here, for the sake of comparison, results for mul-tiple trenches and breakwaters and their combinations are presented in the case of three differentconfigurations. Moreover, although reflection and transmission coefficients are analyzed in specificcases for a check of the energy identity, major results are presented for the reflection coefficients forunderstanding the pattern of wave transformation.
4.4.1. Submerged trenches
0 0.5 1 1.5 2 2.5 3 3.5 4
k1h
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Kr
a/Lg=0
a/Lg=0.25
a/Lg=0.5
(a)
0 0.5 1 1.5 2 2.5 3 3.5 4
k1h
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Kr
a/Lg=0
a/Lg=0.25
a/Lg=0.5
(b)
0 0.5 1 1.5 2 2.5 3 3.5 4
k1h
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1K
r
a/Lg=0
a/Lg=0.25
a/Lg=0.5
(c)
Figure 9 Reflection coefficient Kr versus k1h1 for (a) double, (b) triple and (c) four triangular trenches withvariation in dock length a/Lg.
In Fig. 9, variation in reflection coefficient Kr versus k1h1 is plotted for (a) double, (b) triple and(c) four triangular trenches with variation in dock length a/Lg. It is observed that in the case of nsubmerged trenches, n− 2 sub-harmonic peaks occur in the absence of floating dock as demonstratedin [35] whereas n − 1 sub-harmonic depression occurs in the presence of floating dock. The increasein the sub-harmonic depression is because of the formation of a standing wave between the floatingdock and the trench adjacent to the dock and the standing waves disappear in the absence of thefloating dock. Further, Fig. 9 reveals that with an increase in the length of the floating dock, wavereflection increases and the amplitude of the harmonic and sub-harmonic depressions decreases whichis due to the dominant role of wave reflection by the floating dock of larger length. However, with theintroduction of the floating dock, the reflection coefficient increases and approaches to one with anincrease in wave number k1h1, which is because of the reflection of the gravity waves by the floatingdock concentrating near the free surface in case of deep water corresponding to higher values ofwavenumber k1h1. A similar phenomenon was observed in case of wave scattering by the floatingdock in the presence of submerged trench located below the dock in FIg. 4. The change in pattern in
15
wave reflection is due to the combined effect of wave transformation by the submerged trenches asobserved in Fig. 9(a) along with the wave reflected by the floating dock and the resonating interactionbetween the dock and the adjacent trench. Further, the occurrence of harmonic and sub-harmonicdepressions in wave reflection is observed in the presence of floating dock which is analogous to theoccurrence of harmonic and sub-harmonic peaks in wave reflection in the absence of floating dock.
0 0.5 1 1.5 2 2.5 3 3.5 4
k1h
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Kr
a/Lg=0
a/Lg=0.25
a/Lg=0.5
(a)
0 0.5 1 1.5 2 2.5 3 3.5 4
k1h
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Kr
a/Lg=0
a/Lg=0.25
a/Lg=0.5
(b)
0 0.5 1 1.5 2 2.5 3 3.5 4
k1h
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Kr
a/Lg=0
a/Lg=0.25
a/Lg=0.5
(c)
Figure 10 Variation in reflection coefficient versus k1h1 for (a) double, (b) triple and (c) four parabolic trenchesfor different values of dock length a/Lg with trench width b1/h1 = 2.64 and h4/h1 = 1.
Figure 10 demonstrates the variation of the reflection coefficient Kr versus wave number k1h1 forvarious values of dock length a/Lg for (a) double, (b) triple and (c) four parabolic trenches. In general,the reflection coefficient pattern is similar to that of Fig. 9. However, the amplitude of Bragg reflectionis less in case of multiple parabolic trenches in contrast to that of triangular trenches in the absenceof a floating dock. On the other hand, the reflection coefficient is comparatively smaller in case ofparabolic trenches relative to that of triangular trenches in the presence of a floating dock.
0 0.5 1 1.5 2 2.5 3 3.5 4
k1h
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Kr
a/Lg=0
a/Lg=0.25
a/Lg=0.5
(a)
0 0.5 1 1.5 2 2.5 3 3.5 4
k1h
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Kr
a/Lg=0
a/Lg=0.25
a/Lg=0.5
(b)
0 0.5 1 1.5 2 2.5 3 3.5 4
k1h
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Kr
a/Lg=0
a/Lg=0.25
a/Lg=0.5
(c)
Figure 11 The change in the reflection coefficient Kr against wave number k1h1 in the presence of floating dockto understand the effect of (a) double, (b) triple and (c) four rectangular trenches for different dock length a/Lg
with h2/h1 = 1.5, b1/h1 = 2.64, h3/h1 = 1, h4/h1 = 1.
Figure 11 exhibits the changes in the reflection coefficient Kr against wave number k1h1 to un-derstand the effect of (a) double, (b) triple and (c) four rectangular trenches for various values docklength a/Lg. Figure 11 depicts that the amplitude of wave reflection increases in an oscillatory mannerwith an increase in wave number k1h1 in the presence of a floating dock. Further, sub-harmonic andharmonic peaks/depressions occur with an increase in the number of trenches as discussed in Figs. 9and 10. The number of sub-harmonic depressions between two consecutive harmonic depressions is
16
one less than the number of trenches in the presence of the dock which is analogous to trenches ofother configurations as in Figs 9 and 10. The occurrence of harmonic depressions is due to the back,and forth wave motion between the trench and the floating dock, whilst the sub-harmonic depressionsare because of the formation of the standing wave within the trenches as well as the gap between thetrenches. In the presence of multiple trenches, the combined effect of the resonating interactions dueto wave motion within consecutive trenches increases the peak amplitude of the oscillatory pattern ofwave reflection. Moreover, the amplitude of the harmonic and sub-harmonic depression reduces asthe wave number increases. Figure 11 depicts that the occurrence of Bragg resonance is restricted towaves in shallow and intermediate depths irrespective of the number of trenches and their configura-tions which is consonant to Figs. 9 and 10. However, the amplitude Bragg resonance is negligiblysmall in water of intermediate depth. The vanishing role of Bragg resonance for higher values of thewave number is due to the negligible role of both the progressive and evanescent wave modes in thepresence of trenches.
4.4.2. Submerged breakwaters
0 0.5 1 1.5 2 2.5 3 3.5 4
k1h
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Kr
h2/h
1=0.2
h2/h
1=0.4
h2/h
1=0.8
(a)
0 0.5 1 1.5 2 2.5 3 3.5 4
k1h
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Kr
h2/h
1=0.2
h2/h
1=0.4
h2/h
1=0.8
(b)
0 0.5 1 1.5 2 2.5 3 3.5 4
k1h
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Kr
h2/h
1=0.2
h2/h
1=0.4
h2/h
1=0.8
(c)
Figure 12 Variation of the reflection coefficient Kr versus wave number k1h1 in the case of double (a) triangular,(b) parabolic and (c) rectangular breakwaters for various values of breakwater depth h2/h1 with a/Lg = 0,b1/h1 = 2.64, h3/h1 = 1 and h4/h1 = 1.
In Fig. 12, variation of reflection coefficient Kr is demonstrated versus k1h1 in case of double(a) triangular, (b) parabolic and (c) rectangular breakwaters for various values of breakwater depthh2/h1 in the absence of a floating dock. Figure 12 reveals that Bragg reflection amplitude increasesas breakwater height increases which is irrespective of the breakwater configuration. However, max-imum Bragg reflection occurs for rectangular trenches whilst minimum reflection occurs in case ofa triangular breakwaters which is due to the reflection of a major part of the wave energy by imper-meable breakwaters of larger surface area. The basic difference in Bragg reflection pattern due tochange in undulated bed configurations is the frequency of occurrence of Bragg reflection with anincrease in wave number k1h1.Within the range 0 < k1h1 < 4, Bragg reflection occurs (a) four timesfor rectangular breakwaters, (b) twice for parabolic breakwaters and (c) once for triangular breakwa-ters. Further, it may be noted that wave energy reflection in case parabolic trenches in deep waterregion is the least compared to other configurations which is due to the diffraction of gravity wavesby parabolic trenches compared to other breakwater configurations. Further, the non-occurrence ofBragg reflection for circular/triangular breakwaters compared to that of rectangular breakwaters in
17
deep water region is due to the negligible role of the evanescent modes in the formation of Braggreflection as discussed in Davies and Heathershaw [17].
0 0.5 1 1.5 2 2.5 3 3.5 4
k1h
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Kr
a/Lg=0
a/Lg=0.25
a/Lg=0.5
(a)
0 0.5 1 1.5 2 2.5 3 3.5 4
k1h
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Kr
a/Lg=0
a/Lg=0.25
a/Lg=0.5
(b)
0 0.5 1 1.5 2 2.5 3 3.5 4
k1h
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Kr
a/Lg=0
a/Lg=0.25
a/Lg=0.5
(c)
Figure 13 Reflection coefficient Kr versus k1h1 in the case of (a) double, (b) triple and (c) four triangularbreakwaters for different values of dock length a/Lg with h2/h1 = 0.5 and h4/h1 = 1.
Figure 13 demonstrates the changes in the reflection coefficients Kr against k1h1 in case of (a) apair of, (b) triple and (c) four triangular breakwaters for various values of dock length a/Lg. Fig-ure 13 reveals that amplitude of Bragg reflection increases as the number of breakwaters increasesin the absence of floating dock. However, an increase in the number of breakwaters beyond a pairis invariant of the number of Bragg resonance in the case of triangular breakwaters. Figure 13 re-veals the occurrence of n − 1 sub-harmonic peaks within two consecutive harmonic peaks in thecase of n breakwaters with floating dock unlike the occurrence of n − 2 sub-harmonic peaks withintwo consecutive harmonic peaks without floating dock. A comparison with Fig. 9-11 reveals thatthe occurrence of sub-harmonic peaks within consecutive harmonic peaks for an array of trencheswhereas sub-harmonic depressions occurs for an array of trenches in the presence of a floating dock.However, wave reflection increases at a faster rate to attain unity value with respect to wavenumberk1h1 for multiple breakwaters than that of trenches. Further, unlike the case of multiple trenches ofvaried configurations in which sub-harmonic peaks occur within the bandwidth of Bragg resonance,the occurrence of sub-harmonic peaks is invariant of non-dimensional wave number k1h1 in case oftriangular multiple breakwaters.
0 0.5 1 1.5 2 2.5 3 3.5 4
k1h
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Kr
a/Lg=0
a/Lg=0.25
a/Lg=0.5
(a)
0 0.5 1 1.5 2 2.5 3 3.5 4
k1h
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Kr
a/Lg=0
a/Lg=0.25
a/Lg=0.5
(b)
0 0.5 1 1.5 2 2.5 3 3.5 4
k1h
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Kr
a/Lg=-0
a/Lg=0.25
a/Lg=0.5
(c)
Figure 14 Reflection coefficient versus k1h1 for different values of dock length a/Lg for (a) double, (b) tripleand (c) four parabolic breakwaters with h2/h1 = 0.5 and h4/h1 = 1.
In Fig. 14, reflection coefficients Kr are plotted against k1h1 for (a) double, (b) triple and (c)four parabolic breakwaters for various values of dock length a/Lg. With an increase in the number
18
of breakwaters, Bragg reflection amplitude increases (as number of breakwaters increases) which issimilar to that of Fig. 10 in case of parabolic trenches. Figure 14 depicts that the Bragg resonanceoccurs twice in case of parabolic breakwaters unlike the case of triangular breakwaters in whichBragg resonance occurs once as observed in Fig. 13. Moreover, the amplitude of Bragg reflectionis higher in the case of parabolic breakwaters compared to that of triangular breakwaters as in Fig.13, whilst the amplitude of wave reflection in the case of triangular breakwaters is more than that ofparabolic breakwaters in deep water. Further, unlike the occurrence of sub-harmonic peaks in the caseof triangular breakwaters, the same does not occur in the case of parabolic breakwaters in higher waterdepth corresponding to k1h1 > 2.5. The occurrence of the number of sub-harmonic peaks/depressionsbetween two consecutive harmonic peaks/depressions is analogous to the observations made in Figs.9 and 13 with and without floating dock.
0 0.5 1 1.5 2 2.5 3 3.5 4
k1h
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Kr
a/Lg=0
a/Lg=0.25
a/Lg=0.5
(a)
0 0.5 1 1.5 2 2.5 3 3.5 4
k1h
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Kr
a/Lg=0
a/Lg=0.25
a/Lg=0.5
(b)
0 0.5 1 1.5 2 2.5 3 3.5 4
k1h
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Kr
a/Lg=0
a/Lg=0.25
a/Lg=0.5
(c)
Figure 15 The reflection coefficient Kr against wave number k1h1 for (a) double, (b) triple and (c) four rectan-gular breakwaters for change in dock length a/Lg with h2/h1 = 0.5, b1/h1 = 2.64, h3/h1 = 1, h4/h1 = 1.
Figure 15 exhibits the reflection coefficient Kr versus k1h1 to understand the effect of (a) double,(b) triple and (c) four rectangular breakwaters for different values of dock length a/Lg. Figure 15reveals that the Bragg reflection amplitude increases as the number of breakwaters increases in theabsence of a floating dock. Moreover, a number of resonant peaks (which has been referred here as theharmonic peaks) are found as the wave number k1h1 increases whilst each resonant peak is coupledwith a number of sub-harmonic peaks as observed in Guazzelli et al. [44] for in the presence of multi-ple rectangular breakwaters. However, the amplitude of resonant peaks decreases as the wave numberk1h1 increases. This phenomenon is quite different from that of triangular breakwaters in which Braggreflection occurs involving a single resonating peak coupled with a number of sub-harmonic peaks.Further, it may be noted that the number of resonating peaks and the bandwidth of Bragg reflectionis invariant of the number of breakwaters which demonstrates that the bandwidth depends upon therelative water depth, whilst the number of sub-harmonic peaks depends on the number of breakwaterswhich is similar to the observation made by Zeng et al. [45]. A comparison of Fig. 15 with that ofFigs. 9 - 11 reveals that unlike the occurrences of multiple resonating peaks in the case of multiplerectangular breakwaters, a single resonating peak occurs in case of multiple trenches in the absence ofa floating dock. The occurrence of multiple resonating peaks is due to the dominating role of evanes-cent wave modes in case of breakwaters compared to that of trenches in water of relatively higherdepth corresponding to higher values of k1h1. Moreover, Bragg reflection amplitude is more in case of
19
an array of breakwaters than that of trenches of similar configurations. The pattern of wave reflectionin the presence of floating dock increases with the occurrence of harmonic and sub-harmonic peaksin the case of multiple breakwaters whereas sub-harmonic and harmonic depressions are observed incase of trenches of similar configurations. A comparison with Fig. 9(a) reveals the occurrence of n−1sub-harmonic peaks/depressions within two consecutive harmonic peaks/depressions in the case of ntrenches/breakwater with floating dock unlike the occurrence of n− 2 sub-harmonic peaks within twoconsecutive harmonic peaks without floating dock which is similar to that of multiple trenches andbreakwaters as discussed earlier.
4.4.3. Combined effect of trenches and breakwaters
Here, the combined effects of trenches and breakwaters of the three different configurations asmentioned above are analyzed by varying the position of trenches and breakwaters in different cases.Moreover, the results are compared with those of multiple trenches and breakwaters for efficiency asa wave barrier. Moreover, the role of gap length between the floating dock and the adjacent bottomundulation on wave scattering by a floating dock is studied.
0 0.5 1 1.5 2 2.5 3 3.5 4
k1h
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Kr
a/Lg=0
(a)
0 0.5 1 1.5 2 2.5 3 3.5 4
k1h
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Kr
a/Lg=0
(b)
0 0.5 1 1.5 2 2.5 3 3.5 4
k1h
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Kr
a/Lg=0
(c)
Figure 16 Variation of reflection coefficient Kr versus wave number k1h1 for a pair of trenches and a pair ofbreakwaters placed alternately for (a) rectangular, (b) parabolic and (c) triangular configurations with h2/h1 =
0.5, h3/h1 = 1.5 for various values of dock length a/Lg.
Figure 16 exhibits the variation of the reflection coefficients Kr versus wave number k1h1 due tothe surface wave scattering by a pair of trenches as well as breakwaters of different configurationsplaced alternately. Figure 16 reveals that the frequency of occurrence of Bragg resonance dependson the configurations of the seabed profiles as discussed in the case of multiple breakwaters in Figs.13-15. However, the Bragg reflection amplitude is comparatively less in the case of alternately placedtrenches and breakwaters compared to that of multiple breakwaters. Moreover, the sub-harmonicpeaks vanishes and the frequency of occurrence of harmonic peaks within each band increases in thecase of alternately placed trenches and breakwaters compared to that of multiple breakwaters in caseof deep water. The vanishing of sub-harmonic peaks may be due to the increase in water depth in thepresence of trenches whilst the increase in the frequency of occurrence of harmonic peaks is due tothe back and forth motion between the consecutive undulations of the sea bed.
Figure 17 illustrates the change in the reflection coefficient Kr as a function of wave number k1h1
to understand the combined effect of (a) a single, (b) double and (c) triple trenches with single break-water for different values of dock length a/Lg. Figure 17 reveals that amplitude in the oscillatory
20
0 0.5 1 1.5 2 2.5 3 3.5 4
k1h
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Kr
a/Lg=0
a/Lg=0.25
a/Lg=1
(a)
0 0.5 1 1.5 2 2.5 3 3.5 4
k1h
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Kr
a/Lg=0
a/Lg=0.25
a/Lg=1
(b)
0 0.5 1 1.5 2 2.5 3 3.5 4
k1h
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Kr
a/Lg=0
a/Lg=0.25
a/Lg=1
(c)
Figure 17 The reflection coefficient Kr versus wave number k1h1 for understanding the combined effect of (a) asingle trench, (b) a pair of trenches and (c) triple trenches with a breakwater in case of rectangular type bottomfor different values of dock length a/Lg with b2/b1 = 1, b1/h1 = 2.64, h2/h1 = 1.5 and h3/h1 = 0.5.
trend of wave reflection increases as the number of trenches increases. Further, it is observed thatthe amplitude of resonating peak and corresponding sub-harmonic peaks in each bandwidth of Braggresonance decreases as wave number k1h1 decreases and diminishes for larger wave number in the ab-sence of floating dock. However, the number of Bragg resonance and the corresponding band widthsare invariant on the combined effect of trenches and breakwater which is similar to that of multiplebreakwaters as in Fig. 14. Unlike the occurrence of Bragg reflection for an array of trenches withoutbreakwater as observed in [35], the amplitude of Bragg reflection increases. This is because of the in-crease in wave reflection in the presence of the submerged breakwater. However, with the introductionof a floating dock, amplitude of wave reflection increases in an oscillatory manner and approaches toone for higher values of wavenumber k1h1 corresponding to waves in deep water. Further, with anincrease in dock length, the amplitudes of harmonic and sub-harmonic peaks diminish. Moreover,a comparison of the sub-figures reveals the decrease in sub-harmonic depression magnitude and anincrease in the amplitude of harmonic depression as the number of trenches increases.
0 0.5 1 1.5 2 2.5 3 3.5 4
k1h
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Kr
a/Lg=0
a/Lg=0.25
a/Lg=1
(a)
0 0.5 1 1.5 2 2.5 3 3.5 4
k1h
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Kr
a/Lg=0
a/Lg=0.25
a/Lg=1
(b)
0 0.5 1 1.5 2 2.5 3 3.5 4
k1h
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Kr
a/Lg=0
a/Lg=0.25
a/Lg=1
(c)
Figure 18 The reflection coefficient Kr versus wave number k1h1 for understanding the combined effect of (a)a single, (b) double and (c) triple rectangular breakwaters along with a rectangular trench for different valuesof dock length a/Lg.
Figure 18 exhibits the change in the reflection coefficient Kr against k1h1 for understanding thecombined effect of (a) a single breakwater, (b) double and (c) triple breakwaters along with a singletrench for different values of dock length a/Lg with h2/h1 = 0.5, h3/h1 = 1.5, h4/h1 = 1, b1/h1 = 2.64and b2/b1 = 1. Figure 18 reveals that the amplitude of wave reflection increases as the number
21
of breakwater increases which is similar to that of Fig. 15 in the absence of the floating dock. Acomparison of Figs. 17 and 18 depicts that in the absence of floating dock, the reflection pattern isinvariant of the position of the trench and the breakwater in case of a single trench and a breakwater,whilst in case of multiple breakwaters, wave reflection is more compared to that of the trenches. Also,the frequency of occurrence of zero reflection increases in case of multiple breakwaters as the wavenumber k1h1 increases which is because of the dominating role of resonating interaction of wavemotion within the breakwaters. The decrease in water depth in the presence of breakwater playsa significant role in the formation of clapotis which diminishes in the case of multiple submergedtrenches. However, in the presence of floating dock, although the pattern in wave reflection is similarin nature, both wave reflection is more, whilst the associated depression is less when the breakwateris located near the dock compared to that of the trench. Further, the number of the sub-harmonic peakwithin two consecutive harmonic peaks is two less than the number of breakwaters which is differentthan that of multiple trenches in the presence of a single breakwater as discussed in Fig. 17.
0 0.5 1 1.5 2 2.5
L1/
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Kr
a/Lg=0
a/Lg=0.5
a/Lg=1
Lg
L1
(a)
0 0.5 1 1.5 2 2.5
L1/
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Kr
a/Lg=0
a/Lg=0.5
a/Lg=1
(b)
0 0.5 1 1.5 2 2.5
L1/
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Kr
a/Lg=0
a/Lg=0.5
a/Lg=1.0
(c)
0 0.5 1 1.5 2 2.5
L1/
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Kr
a/Lg=0
a/Lg=0.5
a/Lg=1
(d)
Figure 19 Variation in reflection coefficient Kr versus gap between dock and adjacent bottom undulation L1/λ1
with (a) trench and breakwater, (b) breakwater and trench, (c) a pair of breakwaters and (d) a pair of rectangulartrenches for different values of dock length a/Lg.
In Fig. 19, reflection coefficients Kr are plotted against the gap between the floating dock and theadjacent bottom undulation L1/λ1 in case of (a) trench and breakwater, (b) breakwater and trench, (c) apair of breakwaters and (d) a pair of rectangular trenches placed in order, with different values of docklength a/Lg. Figure 19 reveals that wave reflection is invariant of the position of the breakwater andtrenches, whilst amplitude of wave reflection attains maximum for a pair of breakwaters and least fora trench and a breakwater irrespective of the presence of the floating dock. However, in the presenceof a floating dock, irrespective of the position of the trench and breakwater, the reflection coefficientvaries in a periodic oscillatory manner as the gap length L1/λ1 increases. Further, as the dock lengtha/Lg increases, reflection coefficient increases, whilst amplitude of oscillation in the trend of thereflection coefficient decreases. However, altering the position of the trench and breakwater changesthe phase of the oscillatory trend in wave reflection which is due to the change in phase of the bedprofile.
Figure 20 demonstrates the variation in the reflection coefficient Kr versus the non-dimensional gapbetween the dock and the adjacent bottom undulation L1/λ1 for various combination of rectangulartrenches and breakwaters with dock length a/Lg = 0.5 and h4/h1 = 1.0. All the sub-figures revealthat wave reflection changes in a periodic oscillatory manner with an increase in the gap length L1/λ1.However, minor changes in the amplitude of oscillation as well as phase-shift in the oscillatory trend
22
0 0.5 1 1.5 2 2.5
L1/
1
0.9
0.91
0.92
0.93
0.94
0.95
0.96
0.97
0.98
0.99
1
Kr
m1=m
2=1
m1=m
2=2
m1=m
2=
(a)
0 0.5 1 1.5 2 2.5
L1/
1
0.9
0.91
0.92
0.93
0.94
0.95
0.96
0.97
0.98
0.99
1
Kr
m1=m
2=1
m1=m
2=2
m1=m
2=
(b)
0 0.5 1 1.5 2 2.5
L1/
1
0.9
0.91
0.92
0.93
0.94
0.95
0.96
0.97
0.98
0.99
1
Kr
m1=m
2=1
m1=m
2=2
m1=m
2=
(c)
0 0.5 1 1.5 2 2.5
L1/
1
0.7
0.75
0.8
0.85
0.9
0.95
1
Kr
m1=m
2=1
m1=m
2=2
m1=m
2=
(d)
Figure 20 Variation in reflection coefficient Kr versus L1/λ1 for (a) a pair of trenches and a pair of breakwatersplaced alternately (b) a pair of breakwaters and a pair of trenches placed alternatively; (c) four trenches and (d)four breakwaters for different bottom configurations with dock length a/Lg = 0.5 and h4/h1 = 1.0
of the reflection coefficient occur as the gap length increases. Moreover, the amplitude of wavereflection becomes maximum for multiple breakwaters and minimum in the case of multiple trenchesin the presence of a floating dock. Moreover, the phase shift in wave reflection occurs in the caseof a combination of a pair of trenches and a pair of breakwaters placed alternately and in the caseof multiple breakwaters. A comparison of Figs. 19 and 20 depicts that the number of peaks in thereflection coefficient is invariant of the number of breakwaters, trenches or a combination of trenchesand breakwaters in the presence of a floating dock.
5. Conclusion
A numerical model is developed using the boundary element method to study surface gravitywave interaction with a floating dock in the presence of bottom undulation such as multiple trenches,breakwaters or a combination of trenches and breakwaters. The numerical solutions in the boundaryelement method are compared using two different methods to account for the condition in far field.The bottom undulation is assumed to be either below the floating dock or at a finite distance from thefloating dock. The numerical results are validated with results available in the literature in the caseof bottom variation below the floating dock. The computed results for reflection and transmissioncoefficients are further validated with the energy identity for checking the accuracy of the compu-tational results. The reflection and transmission coefficients are invariant of dock length in the caseof single trench located below the dock, whilst the reflection coefficient increases as the dock lengthincrease when the submerged breakwater is placed below the dock. On the other hand, wave reflec-tion follows a periodic oscillatory trend in case of the rectangular trench as the trench width increaseswhich is similar to that of single breakwater except in the case of shallow water region. In case ofasymmetric trench and breakwater in the presence of floating dock, the reflection coefficient is morethan unity in case of small wave number corresponding to waves in shallow water which is similarto that of wave propagation over trench without the floating dock. Further, with an increase in thenumber of trenches below the dock, variation in wave reflection and transmission is negligible. Fur-ther, the reflection and transmission coefficients are almost invariant due to the change in the numberof trenches below the floating dock. However, when the trenches and breakwaters are located at afinite distance from the floating dock, amplitude of wave reflection increases in an oscillatory mannerwith an increase in wave number and almost full reflection occurs in case of deep water. Moreover,
23
harmonic and sub-harmonic depressions are observed in the presence of an array of trenches whereasharmonic and sub-harmonic peaks are observed in case of multiple breakwaters in the presence of afloating dock. Also, the number of sub-harmonic peaks/depressions between consecutive harmonicpeaks/depressions is one less than the numbers of breakwaters/trenches respectively, whilst in case ofmultiple trenches/breakwaters without the floating dock, the number of sub-harmonic peaks betweenconsecutive harmonic peaks is two less than the number of trenches/breakwaters. Compared to mul-tiple trenches, wave reflection increases at a faster rate with an increase in wave number in case ofmultiple breakwaters. In the presence of floating dock, interchanging the position of the trench andbreakwater does not affect the oscillatory trend of the reflection coefficient in case of multiple trenchesand breakwaters, whilst variation of phase change is observed in the oscillatory trend of the reflectioncoefficient. In the case of multiple breakwaters or a combination of trenches and breakwaters, Braggresonance occurs four times in the case of rectangular bed profiles, twice in case of parabolic bedprofiles and once in the case of triangular bed profiles, whilst amplitude of Bragg resonance dimin-ishes for higher water depth in the case of multiple trenches. However, Bragg resonance amplitudedecreases as the wave number increases irrespective of configurations of the breakwater and a com-bination of trenches and breakwaters. The study can be generalized to deal with wave scattering bythe floating dock in the presence of bottom undulation of many other configurations arising in therealistic physical seabed.
Acknowledgments
The financial support received from Department of Science and Technology, Government of Indiathrough Award No.DST/CCP/CoE/79/2017(G)) is gratefully acknowledged.
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Research Highlights
A boundary element model is developed to study Bragg scattering of gravity waves by a floating dock in the presence of bottom undulation
Bottom undulation on the sea side of the floating dock has a significant role on wave scattering compared to that of bottom undulation below the dock.
The frequency of occurrence of Bragg resonance depends on the configuration of the breakwaters or a combination of trenches and breakwaters.
N ‐1 sub‐harmonic peak/depressions occur between two harmonic peaks in the case of N trenches/breakwaters with floating dock.
N‐2 sub‐harmonic peaks within two consecutive harmonic peaks in the case of N trenches/breakwaters without floating dock.
