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A study of spar buoy floating breakwater

Ocean Engineering 31 (2004) 43–60

A study of spar buoy floating breakwater
Nai-Kuang Liang Ã, Jen-Sheng Huang, Chih-Fei Li
Institute of Oceanography, National Taiwan University, Taipei, P.O. Box No. 23-13, Taipei 106,
Taiwan, ROC
Received 26 February 2002; accepted 23 May 2003

A floating breakwater produces less environmental impact, but is easily destroyed by large
waves. In this paper, the spar buoy floating breakwater is introduced with a study on the
wave reflection and transmission characteristics and mooring line tension induced by the
waves. Mei (The Applied Dynamics of Ocean Surface Waves, Wiley, New York (1983) 740
p) proposed a theoretical solution for the reflection and transmission coefficients as the wave
propagates through a one-layer slotted barrier. For a multiple-layer fence system, the analytical solution is proposed linearly. The results show that the theoretical computations agree
well with the experimental trends. For a multiple-layer fence system, the transmission coefficients become maximal as the layer spacing to wavelength ratio moves to 1/2. Conversely,
the coefficients become minimal, as the ratio moves to 0.3. To estimate the maximum tension of the mooring line, both numerical calculations and laboratory experiments were executed. The numerical calculation results were similar to the experimental results.
# 2003 Elsevier Ltd. All rights reserved.
Keywords: Floating breakwater; Spar buoy; Semi-closed pipe; Vena-contracta; Wave transmission; Slant
wire tension

1. Introduction
Breakwaters are used in near shore sea areas to produce wave amplitude
reduction in areas such as harbors, fishing ports, marinas, power plant in and outtakes and offshore cage aquaculture support bases. The traditional breakwater is
composed of caissons, rubble mounts or a hybrid. This breakwater type could
change the original near shore current system and destroy littoral sand balance and

Corresponding author. Tel.: +886-2-236-92-034; fax: +886-2-239-25-294.
E-mail address: liangnk@ccms.ntu.edu.tw (N.-K. Liang).

0029-8018/$ - see front matter # 2003 Elsevier Ltd. All rights reserved.


N.-K. Liang et al. / Ocean Engineering 31 (2004) 43–60

Fig. 1. Schematic diagram of single spar buoy.

ecological system. The breakwater construction is expensive and time-consuming.
Breakwaters are also difficult to remove. The traditional breakwater is required for
highly stable harbor. A floating breakwater can be employed for shore facilities
that require a lower level of stability. Many studies have been produced on floating
breakwaters (Twu and Lee, 1983; Guo et al., 1996; Murali and Mani, 1997; etc.).
The floating breakwater has low sheltering efficiency and maintenance difficulties.
The floating breakwater has therefore been seldom used.
The first author proposed a spar buoy floating breakwater design, i.e. the Semiclosed Pipe Floating Breakwater (SPFB), registered as a new type patent in
Taiwan, ROC (Liang, 2000). A pipe made of polyethylene is closed at one end.
Holes are drilled for anchoring at the other end. The semi-closed pipe is aerated
from the open end. This pipe becomes a tautly moored spar buoy if the water is
deep enough. To suppress spar buoy pitching, two slant wires are anchored at the
top of the buoy (Fig. 1). There is pretension in the slant wire. Successive spar
buoys are installed on a line like a slotted vertical column fence (Fig. 2). More fences can be added to increase the sheltering effect. A rod is used to pierce the lower
end of the pipe with used tires piled on it to enlarge the cross section and protect
the pipe (Fig. 3). Several application possibilities are suggested in Section 2 of this
There are two questions that should be answered, i.e. the wave sheltering effect
(or wave transmission) and the maximum tension of the slant wire during huge

waves. Theoretical and experimental studies are presented in Sections 3 and 4
(Huang, 2002; Li, 2002).

Fig. 2. Schematic diagram of spar buoy fences.

N.-K. Liang et al. / Ocean Engineering 31 (2004) 43–60


Fig. 3. Schematic diagram of practical spar buoy.

2. Practical design concept and possible applications
For a small island with tourism value, such as the Tung-Sa corral reef island in
the northern South China Sea, a multiple-layered semi-closed pipe fence system
could be used to build a breakwater and established a simple harbor (Fig. 4). The
environmental impact of such a breakwater is minimal, the cost is the lowest and
the breakwater fence can be easily removed. There are many islands in the South
Pacific where the sea is rather calm year round. A floating breakwater is to provide
effective shelter in these areas.
A beach for swimming is an important recreation area across the world. However, many beaches are open only part of the year due to high waves. An offshore
floating breakwater could increase the beach utilization rate.
Traditional breakwaters are commonly old and dangerous in large waves. Often
the harbor basin or entrance is not stable enough due to poor breakwater design.
A spar buoy floating breakwater can be installed outside of the weak part of the
old breakwater in the former case. In the latter case, such a breakwater could be
installed at a proper location that the entrance becomes calm and ships can easily

Fig. 4. Schematic diagram of a simple harbor with floating breakwater.


N.-K. Liang et al. / Ocean Engineering 31 (2004) 43–60

come into and out of the harbor. Ships will be unharmed even if they collide with
the floating breakwater.

3. Theoretical approach
As regards to the wave sheltering of the spar buoy floating breakwater, an
assumption is made for simplicity that fixed vertical pipes are assumed to simulate
the aerated semi-closed pipes in studying the wave reflection and transmission
characteristics. There is much published literatures on vertical slotted barrier wave
shelters. Wiegel (1960, 1961) proposed the power transmission theory which states
that if the energy dissipation and reflection of waves transmitted through the
porous portion of the barrier is neglected, the wave transmission coefficient is
pffiffiffiffiffiffiffiffiffi pffiffiffiffi
Ht =Hi ¼ b=B ¼ P. P is the porosity and is equal to b/B, where B is equal to
D þ b (Fig. 5). Hi is the incident wave height, and Ht is the transmitted wave
height. Hayashi et al. (Hayashi et al., 1966; Hayashi et al., 1968) proposed a transmission coefficient Kt and a reflection coefficient Kr for a closely spaced pile breakwater. The long wave assumption considers that only the horizontal water particle
current exists. A jet flow in the slot and a vena-contracta could take place (Fig. 5).
Mei (Mei et al., 1974, Mei, 1983) proposed a solution for the transmission coefficient under the long wave assumption (shallow water wave). Their study pointed
out that the velocity variation in the jet flow could result in energy losses and the
wave steepness, porosity and relative depth are the main factors. Referring to Mei’s
theory (1983), Kriebel (1992) integrated the momentum equation in the water
depth direction and obtained a transmission coefficient solution for any water
depth. The solution can approach Mei’s result for a shallow water wave. Several
researchers (Williams et al., 2000; Suh et al., 2001; Zhu and Chwang, 2001) executed serial studies on the reflection of an absorbing-type caisson breakwater. This
type of breakwater is a caisson with permeable thin structures that are installed
at equal spacing. As the S=L ¼ ð2n þ 1Þ=4, in which n ¼ 0,1,2,3, . . .and L is the

Fig. 5. Schematic diagram of vena-contracta through slotted pile barriers.

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wavelength, the reflection wave height is minimal. Conversely, as S=L ¼ n=2, the
reflection becomes maximal. There is little literature on the slant wire tension.
3.1. Wave sheltering
The reflection coefficient is K r ¼ H r =H i and the transmission coefficient is K t ¼
H t =H i where Hr is the wave height of the reflected wave. The energy loss coefficient is
ELOSS ¼ 1 À Kr2 À Kt2 :
For a single-layer structure or fence, Mei (1983) proposed the theoretical result as:
À1 þ 1 þ 2ð4=3Þðf =khÞðHi =LÞ
Kt ¼
ð4=3Þðf =khÞðHi =LÞ
1 À Kr ¼ Kt


where f is the dissipation coefficient and is equal to ðð1=CPÞ À 1Þ and C is the
vena-contracta coefficient.
For multiple-layer fences, it is assumed that the successive incident, transmitted
and reflected waves are linearly superimposed (Huang, 2002). A two-layer fence
case is used as an example (Fig. 6). As the incident wave g0 passes the 1st fence,
the 1st reflected wave gr1 and the 1st transmitted wave gt1 are generated. As the 1st
transmitted wave passes the 2nd fence, the 2nd reflected wave gr2 and the 2nd transmitted wave gt2 take place. As the 2nd reflected wave propagates to the 1st fence,

Fig. 6. Schematic diagram of the linear superimposition of wave components in two-layer fence system.


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the 3rd reflected wave gr3 and the 3rd transmitted wave gt3 come out, and so on.
There will be theoretically infinite number of reflected and transmitted waves. They
gr1 ¼

cosðkx þ rtÞ

H1r ¼ H0 Á RðH0 Þ

cosðkx À rtÞ H1t ¼ H0 Á TðH0 Þ
gr2 ¼ 2 cosðkð2S À xÞ þ rtÞ H2r ¼ H1t Á RðH1t Þ
gt1 ¼

cosðkx À rtÞ H2t ¼ H1t Á TðH1t Þ
gr3 ¼ 3 cosðkðx þ 2SÞ À rtÞ H3r ¼ H2r Á RðH2r Þ
gt2 ¼

cosðkðx þ 2SÞ þ rtÞ
gr4 ¼ 4 cosðkð4S À xÞ þ rtÞ

gt3 ¼

gt4 ¼

cosðkðx þ 2SÞ À rtÞ


H3t ¼ H2r Á TðH2r Þ


H4r ¼ H3r Á RðH3r Þ


H4t ¼ H3r Á TðH3r Þ


TðHi Þ ¼

À1 þ

1 þ 2ð4=3Þðf =khÞðHi =LÞ
ð4=3Þðf =khÞðHi =LÞ

RðHi Þ ¼ 1 À TðHi Þ


The total number of reflected and transmitted waves are determined as follows:
grTotal ¼ gr1 þ

gt2iþ1 ;



x ! 2S




gtTotal ¼ gt2 þ

gt2i ;

This principle can be applied to any layered fence system.
3.2. Tension of slant wire
A two-dimensional rectangular coordinate system is assumed (Li, 2002). As
shown in Fig. 7, x is the horizontal axis and z the vertical axis. The origin is at
point a, which is the anchor point of the slant wire.
For simplicity, the assumptions are as follows:

N.-K. Liang et al. / Ocean Engineering 31 (2004) 43–60


Fig. 7. Sketch definition for wave propagation on an anchored spar buoy.

1. The wire elongation and buoy deformation are very small and can be neglected.
2. The diameter of the wire is small. The drag, inertial, buoyancy and gravity
forces are all neglected.
3. Only waves are considered and there is no current.
4. The entire system is in a static state.
5. The entire buoy is submersed in the water.
The environmental forces acting at the buoy or pipe are as shown in Fig. 8. They
are gravity, buoyancy, tension and wave forces. Because the wire cannot sustain
compressive force, the right slant wire is idle, as the wave force directs to the right,
and vice versa. The force balance equations for the positive wave force are as

Fig. 8. Sketch definition for environmental forces on a spar buoy.


N.-K. Liang et al. / Ocean Engineering 31 (2004) 43–60

For the x direction

Fxi ¼ 0



For the z direction

Fzi ¼ 0



For the moment


Mi ¼ 0



where Fxi is the force in the x direction, Fzi the force in the z direction and Mi the
moment referring to the lower end of the buoy. The sub-index i indicates the various environmental forces, introduced as follows:
Gravity (i ¼ 1):
Fx1 ¼ 0


Fz1 ¼ ÀW


M1 ¼ 0


˜ is the mass of the buoy and g the gravitational acceleration.
in which W
Buoyancy force (i ¼ 2):
Fx2 ¼ 0


Fz2 ¼ qVg


M2 ¼ 0


where q is the water density and V the volume of the buoy.
Drag force (i ¼ 3): according to the Morison equation, we have
ð r2
qCDX DðUÞjUjdz
Fx3 ¼
r1 2
qCDZ AðW ÞjW j
1 r2

W dz
L0 r1
ð r2
qCDX DðUÞjUjðz À r1 Þdz
M3 ¼
r1 2
Fz3 ¼


where r1 is the z-coordinate of the buoy lower end, r2 the z-coordinate of the buoy
upper end, D the spar buoy diameter, A is the cross-sectional area, CDX is the horizontal drag coefficient, CDZ is the vertical drag coefficient, U is the horizontal velo the average vertical velocity of the water particles
city of the water particles, W
and L0 the spar buoy length.

N.-K. Liang et al. / Ocean Engineering 31 (2004) 43–60


Inertial force (i ¼ 4): according to the Morison equation, the inertial forces are
as follows:
ð r2
Fx4 ¼ qCMX AU_ dz

Fz4 ¼ qCMZ V W
ð r2
_ ¼ 1
_ dz
L0 r1
ð r2
M4 ¼ qCMX AU_ ðz À r1 Þdz



CMX ¼ 1 þ kMX


CMZ ¼ 1 þ kMZ


_ is the average vertical acceleration of the water particles, K
where W
MX the horizontal added mass coefficient and KMZ the vertical added mass coefficient.
Left slant wire tension (i ¼ 5): the slant tension TE is decomposed into x and z
FX5 ¼ ÀTE cosh


FZ5 ¼ ÀTE sinh


M5 ¼ ÀTEL0 cosh


Buoy bottom wire tension (i ¼ 6): this tension is divided into x and z components:
FX6 ¼ ÀT2X


FZ6 ¼ ÀT2Z


M6 ¼ 0


After rearrangement, we have the following equations: the force balance equation in the x direction:
ÀTE cosh À T2X ¼ ÀWFX


The force balance equation in the z direction:
ÀTE sinh À T2Z ¼ ÀWFZ þ Wg À qVg


The moment balance equation:
ÀTEL0 cosh ¼ ÀWFM



N.-K. Liang et al. / Ocean Engineering 31 (2004) 43–60


ð r2

qCDX DðUÞjUjdz þ

ð r2

qCMX AU_ dz



qCDZ AðW ÞjW j þ qCMZ V W
ð r2
ð r2
qCDX DðUÞjUjðz À r1 Þdz þ qCMX AU_ ðz À r1 Þdz
r1 2





Eqs. (40), (41) and (42) are the governing equations for numerically calculating
the slant wire tension TE.

4. Laboratory experiments and comparison with theories
These experiments were carried out at the wave flume at the Institute of Oceanography, National Taiwan University. This flume has the following dimensions: 17
m in length, 0.8 m in height and 0.6 m in width. The wave maker is piston type
with a 1:6 slope at the end of the flume to eliminate the reflection waves. Capacitance wave meters and tension meters were used to measure the wave and tension.
The data acquisition was accomplished using a personal computer.
4.1. Wave sheltering
The layout of the wave sheltering experiment is shown in Fig. 9. The fixed vertical cylinders used to simulate the spar buoy floating breakwater were made of PVC
pipe, 3.5 cm in diameter. The pipes were fixed in a steel framework mounted on the
flume. The pipe spacing was 0.5 cm. The porosity P was equal to 0.125 (0.5/4). In
this experiment, the water depth h was a constant, i.e. 45 cm. The model wave

Fig. 9. Schematic diagram of wave sheltering experiment setup.

N.-K. Liang et al. / Ocean Engineering 31 (2004) 43–60


period was between 0.8 and 1.2 s, of which the corresponding wavelength was
between 0.99 and 2 m. The wave height ranged from 5 to 15 cm. The Goda and
Suzuki (1976) method was employed to separate the incident and reflected wave
components in front of the wave barrier (Huang, 2002). As mentioned in Section
3.1, the vena-contracta coefficient C was an empirical constant. From the literature,
the C constant is a function of the slot shape and varied between 0.5 and 1.0. Mei
(1983) suggested that for a sharp-edge orifice C ¼ 0:6 þ 0:4P2 . Hayashi et al.
(1966) compared the experimental result with the theoretical calculation by substituting C ¼ 0:9 or 1:0. According to Fig. 10, C ¼ 1:0 is a better choice. From Fig.
10, as the wave steepness Hi/L increases, Kr increases, Kt decreases and ELOSS
increases. However, Kr, Kt and ELOSS gradually approach constant, as the wave
steepness Hi/L increases. As shown in Fig. 11, the comparisons for the two-layer
fence reveal that Kr, Kt and ELOSS oscillate with the relative spacing S/L in a
sinusoidal wave. As S=L ¼ 1=4, the Kr and Kt values are minimal but ELOSS
becomes maximal. Conversely, as S=L ¼ 1=2, the Kr and Kt values become maximal but ELOSS becomes minimal. However, for the experimental Kt value, the minimum is at S=L ¼ 0:3 instead of 0.25. The results are shown in Fig. 12 for the threelayer fence system. Both for theory and experiment Kr, Kt and ELOSS also oscillate
with the relative spacing. As S=L ¼ 1=2, the Kr and Kt values become maximal but
ELOSS becomes minimal. This is the same as the two-layer fence system. However,
as S=L ¼ 1=4, the Kr, Kt and ELOSS become a little different from that in the twolayer fence system. The Kr and Kt minimums appear at the two sides of the point
S=L ¼ 1=4 for the theoretical calculations. This phenomenon is not clear for the

Fig. 10. Comparisons of theory (solid curve for C ¼ 1:0 and dotted curve for C ¼ 0:6) and experiment
(symbols) in the one-layer fence system.


N.-K. Liang et al. / Ocean Engineering 31 (2004) 43–60

Fig. 11. Comparisons of theory (solid curve) and experiment (symbols) in the two-layer fence system.

Fig. 12. Comparisons of theory (solid curve) and experiment (symbols) in the three-layer fence system.

N.-K. Liang et al. / Ocean Engineering 31 (2004) 43–60


Fig. 13. Schematic diagram of slant wire tension experiment setup.

experiment data. The minimum of the Kt experimental value which is smaller than
that for the calculated value also appears at about S=L ¼ 0:3.
4.2. Maximum tension of slant wire
This experiment was carried out in the same wave flume (Fig. 13). There are
three kinds of models. The 1st model is composed of a spar buoy (40 cm long,
3.5 cm in diameter and 110 g weight) with three nylon wires, of which two are
54 cm length and the other 4.3 cm (Fig. 14). The 2nd model adds a soft pipe to the
spar buoy in the 1st model to simulate used car tires in Fig. 3 (Fig. 15). The dimensions of the soft pipe are 30 cm in length, 6.3 cm in outer diameter and 5 cm in
inner diameter. The 3rd model adds a fixed pipe fence used in the previous wave
sheltering experiment, of which one pipe is substituted by the 1st model buoy
(Fig. 16). The water depth in the experiment was 47.6 cm. There are four wave periods, i.e. 0.8, 1.0, 1.2, and 1.5 s, and five wave heights, i.e. 3.0, 4.0, 5.0, 6.0, and
7.0 cm, in the experiment (Li, 2002).
The slant wire tension variation for the 1st model is shown in Fig. 17. The corresponding theoretical result is shown in Fig. 18. Because only the positive half
cycle of the particle velocity is considered for the left slant wire, only the half cycle
wire tension is calculated. We were interested in the maximum tension TEmax .

Fig. 14. The 1st slant wire tension model.


N.-K. Liang et al. / Ocean Engineering 31 (2004) 43–60

Fig. 15. The 2nd slant wire tension model with a soft pipe.

A non-dimensional comparison between the experimental and numerical data is
shown in Fig. 19, where B0 ¼ qVg À Wg. They coincide with one another well. The
experimental data for the 2nd model are shown in Fig. 20. The maximum tension
is larger than that in the 1st model. This is obvious due to the enlarged diameter.
In the 3rd model, the maximum tension is a little larger than that in the 1st model
(Fig. 21). The gap between adjacent pipes is 0.5 cm.

5. Discussions and conclusions
The reflected waves in the two-layer fence system are calculated as follows:
grTotal ¼ gr1 þ gt3 þ gt5 þ gt7 þ Á Á Á

for x


As S=L ¼ 1=4, the phase lag between gr1 and gt3 is p and the super-position reduces the wave. Although gt5 has a phase lag of 2p with gr1 and strengthens the superposition, it does not have an influence because gt5 is much smaller than gt3 due to its
two more reflections. As regards to the total transmission wave, the superposed
wave is mainly composed of gt2 and gt4 . As the phase lag is p, i.e. S=L ¼ 1=4, the
superposed wave is the minimum. However, gt4 is much smaller than gt2 . Hence, the
oscillation amplitude of Kt is smaller than that for Kr (Fig. 11). Another reason to
explain that Kr and Kt are minimal as S=L ¼ 1=4 is that two adjacent fences are
both the reflection wall and node point for one another. At the node point, the
horizontal velocity of the water particles in a partial standing wave is the greatest.
This results in larger energy loss at the slotted barrier.

Fig. 16. The 3rd slant wire tension model.

N.-K. Liang et al. / Ocean Engineering 31 (2004) 43–60

Fig. 17. Experimental results of 1st slant wire tension model.

Fig. 18. Theoretical results of 1st slant wire tension model.



N.-K. Liang et al. / Ocean Engineering 31 (2004) 43–60

Fig. 19. Comparison between non-dimensional experimental and numerical data for the 1st model.

The theoretical calculation for the maximal slant wire tension was verified
by the laboratory experiment. Using the numerical calculation, the maximal
slant wire tension is influenced mainly by the pipe diameter and is almost not

Fig. 20. Theoretical results of the 2nd slant wire tension model.

N.-K. Liang et al. / Ocean Engineering 31 (2004) 43–60


Fig. 21. Experimental results of the 3rd model slant wire tension.

affected by the net buoyancy for the same wave condition. A prototype estimation
is as follows: water depth ¼ 10 m, wave height ¼ 7:8 m, wave period ¼ 12 s,
wave length ¼ 113 m, pipe diameter ¼ 0:5 m, pipe length ¼ 9 m, middle anchor
wire length ¼ 1 m, slant wire length ¼ 12:5 m, distance between the slant wire
anchor and the middle anchor ¼ 7:5 m, pipe and tire weight ¼ 200 kg, tire diameter
¼ 0:6 m, tire column length ¼ 8 m. The maximum slant wire tension is estimated
to be 3.3 tons. In practical use, the slant wires should be pre-tensioned so that the
buoy will be more stable and the wire connection will grind less. To lower the
demand of derricks, geotubes or geobags made of geotextile and sand can be used
for the anchorage. The following conclusions were made:
1. The proposed ‘Semi-closed Pipe Floating Breakwater’ is feasible for simple harbors for fishing, cage farming, yachts, or as a supplementary breakwater for a
traditional breakwater or a beach for swimming. This breakwater is economical
and environmentally benign.
2. The transmission coefficient Kt is a function of the porosity P, the relative spacing S/L and the number of layers. For a three-layer breakwater Kt can be kept
under 0.3, as P is equal to 0.125 and S=L ¼ 0:3.
3. The maximum slant wire tension is influenced mainly by the pipe diameter and
the wave, not the net buoyancy of the spar buoy. For an 8 m height wave with a
12 s period and 0.6 m pipe diameter and 10 m water depth, the maximum tension is about 4 tons. In the practical use, the wire should be pre-tensioned so
that the wire connection parts grind less. To lower the demand of derricks, geotubes or geobags made of geotextile and sand can be used for the anchorage.

Goda, Y., Suzuki, Y., 1976. Estimation of incident and reflected waves in random wave experiments. In:
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