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SOLITARY WAVES INCIDENT ON A SUBMERGED
HORIZONTAL PLATE
Hong-Yueh Lo
1 2
and Philip L.-F. Liu
1 3
ABSTRACT
Wave scattering of a solitary wave traveling over a submerged horizontal plate was stud-
ied. Experiments for normal incidence were conducted in a wave flume with a horizontal
plate suspended at two different depths in the middle of the flume. Gauge pressures above
and underneath the plate, surface elevations on and near the plate, and flow velocities at
three representative fields of view were measured. The flow underneath the plate was found
to behave almost like a plug flow, driven by the time-dependent, spatially uniform pressure
gradient between the two openings. Complex vortices formed near the two edges of the plate
as the plate acted like a flow divider. A numerical model based on 2D Navier-Stokes equations
was used to further confirm the main features captured by the experimental measurements.
Analytical solutions based on the linear long wave theory, which admits a “soliton-like”
impulse wave solution, were also derived. The linear theory was applicable for obliquely
incident impulse waves. When the analytical solutions were applied to the wave conditions
used in the experiments, it was found that the theory described pressure and surface eleva-
tion satisfactorily when the non-linearity is insignificant. Based on the pressure distribution
above and beneath the plate the total vertical force and moment exerted on the plate were
calculated. As the wave passed over the plate, the plate first experienced a lift, followed by
a force in the downward direction, and then a lift again. As a result a non-zero time-varying
moment existed. The analytical solution was also utilized to examine the effects of relative
plate width on the transmitted and reflected waves. The effects of the angle of incidence
1
Department of Civil and Environmental Engineering, Cornell University, NY, USA.
2
Corresponding author. E-mail: hl645@cornell.edu.
3
Institute of Hydrological and Oceanic Sciences, National Central University, Taiwan.
1
Lo and Liu, July 23, 2013
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
Copyright 2013 by the American Society of Civil Engineers
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were also briefly discussed.
Keywords: solitary wave, submerged plate, breakwater, wave energy converter
INTRODUCTION
As has been discussed by Yu (2002), water waves normally incident on a strip of submerged
horizontal plate with a finite width and infinite length have been widely studied for two main
reasons: on one hand, the plate can act as an alternative breakwater design that imposes
less environmental impact; on the other hand, the plate can be considered as a component
of a wave energy converter or some other underwater structure. A blend of the two is also
possible.
Yu (2002) presented an extensive literature review on the subject and summarized the
major findings known at the time. The majority of the work done is analytic based on the
potential flow theory for small amplitude periodic waves. Some such studies include Siew
and Hurley (1977), Patarapanich (1984a and 1984b), and Patarapanich and Cheong (1989).
A major finding is that optimal submersion depth and plate width exist that minimize the
transmitted wave and maximize the reflected wave. Limited laboratory data are available,
which nonetheless support this finding (e.g., Aoyama 1988 and Yu 1990).
Wave scattering by a submerged circular disk was analytically studied by Yu and Chwang
(1993) and Zhang and Williams (1996). It was found that, for periodic waves, wave focusing
is possible on the lee-side of the disk, resulting in a higher wave height compared to the
incident wave height. This discovery opens up the possibility of using submerged disks in
combination with some wave energy converters for higher efficiency.
While for the plate problem the boundary integral equation method was used relatively
frequently to acquire numerical results (e.g., Liu et al. 2009), few studies are available that
adopt direct numerical method. Yu and Dong (2001) developed a numerical model based on
the Navier-Stokes equations, and showed that vortices form near the two edges of the plate
as a result of the wave-plate interaction.
2
Lo and Liu, July 23, 2013
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
Copyright 2013 by the American Society of Civil Engineers
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Although it has been noted that turbulence and vortices may appear close to the plate,
detailed examination of the flow field near the plate and the actual applicability of potential
flow remained under-addressed until the recent work by Poupardin et al. (2012). Using
monochromatic waves, they conducted careful Particle Image Velocimetry (PIV) measure-
ments on the velocity field around the plate and confirmed the existence of intense vortices
near the two edges of the plate. While the results imply local invalidity of potential flow and
possible complication in sediment transport due to the plate, the exact degree of applicability
of potential flow theory to the plate problem requires further examination.
On the other hand, more application-oriented studies are available that focus on the
feasibility of designing a wave energy converter based on the plate concept. Noting the
uniform flow under the plate, Graw (1993a, 1993b, and 1996) advocated a new type of wave
energy converter/breakwater that converts the uniform flow into usable energy via submerged
turbines, while the device also acts as a breakwater that damps the waves. Carter (2005)
furthered the evaluation of this idea and gave a summary of various types of wave energy
converters that are currently being tested.
For periodic waves, additional experimental studies have also been carried out for practi-
cal purposes where alternative breakwater designs based on the plate concept were proposed.
Koraim (2013) summarized the results from several studies with different breakwater setups,
and proposed the use of connected half pipes in place of a smooth plate for higher wave-
damping efficiency.
Instead of a rigid plate, water-wave scattering by submerged porous or elastic plates
has also been studied mathematically by using linear potential wave theory. Mahmood-ul-
Hassan et al. (2009) presented a solution to the water-wave interaction with a submerged
elastic plate of negligible thickness by the eigenfunction-matching method; the solution can
also be extended to the case of oblique incidence and that of circular disk. Williams and
Meylan (2012) studied the same problem for normal incidence and proposed more efficient
solutions based on Wiener-Hopf and residue calculus methods. For a submerged porous
3
Lo and Liu, July 23, 2013
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
Copyright 2013 by the American Society of Civil Engineers
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plate, Wu et al (1998) first investigated wave reflection by a vertical wall connected to a
submerged horizontal porous plate, Liu and Li (2011) suggested an alternative solution for
the water-wave motion due to a submerged horizontal porous plate, and Evans and Peter
(2011) showed an explicit expression for the reflection coefficient for obliquely incident plane
waves upon a semi-infinite porous plate in water of finite depth.
Since existing works on this topic have focused on periodic waves, no information is
available on the interaction between an impulse (transient) wave and a submerged plate in
shallow water environment. An impulse wave is a well defined single wave, which changes
its shape very slowly in constant water depth - a special example is a solitary wave, which is
highly suitable for experimental and theoretical studies. While periodic incident waves may
be more realistic for practical studies, by using an impulse wave the transient effects of the
plate on a single wave can be easily identified, whereas only the quasi-steady state behavior
can be observed if periodic waves are considered. Thus, in this paper we seek to explore the
interaction mechanism involving a solitary wave impinging on a submerged horizontal rigid
plate in constant water depth, by conducting laboratory experiments, computing numerical
results, and developing a simplified small amplitude wave theory.
We remark that while to our best knowledge, the submerged horizontal plate problem has
not been studied for solitary waves, the transformation of a solitary wave over a submerged
step has been examined quite extensively. Two phenomena have been of primary interest:
the solitary wave fission and breaking process above a submerged step (e.g., Losada et al.
1989, Liu and Cheng 2001, and Lara et al. 2011), and the generation and evolution of
vortices that form near the two edges of a step (e.g., Chang et al. 2001, Lin et al. 2006, Ho
et al. 2012, and Wu et al. 2012).
In this paper we shall present and discuss the results from a combined experimental and
theoretical study. We will first introduce the relevant physical parameters of the problem,
directly followed by the derivation of theoretical solutions based on the linear long wave
theory. The theory is developed for obliquely incident impulse waves, which can be ap-
4
Lo and Liu, July 23, 2013
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
Copyright 2013 by the American Society of Civil Engineers
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proximately employed for small-amplitude solitary waves. The experimental setup and the
numerical model, COBRAS (Lin and Liu 1998), based on the Navier-Stokes equations, will
then be discussed. The experiments and numerical calculations were conducted only for
normally incident solitary waves. The analytical, experimental, and numerical results, in
terms of water surface elevation, gauge pressure along the plate, and net force and moment
exerted on the plate, will then be compared. Based on the linear theory, we will quantify the
reflected and transmitted waves, examine the effects of relative plate width, and also discuss
the similarity between the plate problem and the step problem. Numerical results are also
further utilized to check against the PIV measurements. Since the theory is applicable to
oblique incidence, analytical results for obliquely incident solitary waves will be discussed
briefly.
LINEAR LONG WAVE THEORY
In this section, based on linear shallow water wave theory we formulate and obtain
analytical solutions for a wave scattering problem involving an impulse wave and a horizontal
plate with a finite width. Here we shall consider the general situation that allows oblique
incidence. We remark that oblique incidence of periodic waves over a trench has been studied
by Miles (1981), and Kirby and Dalrymple (1983). For long waves King and LeBlond (1981)
considered the possible lateral wave at a depth discontinuity, and Carrier and Noiseux (1983)
considered the reflection of obliquely incident impulse long waves from a plane slope. Finally,
Liu (1984) used a similar approach for studying the diffraction of solitary waves by a thin
breakwater.
Consider an obliquely incident impulse wave with angle of incidence θ, where normal
incidence occurs when θ = 0. The incident wave is scattered by a submerged plate of a
constant width L (in the x-direction), infinite length (in the y-direction), and finite thickness
δ (see Figure 1). In open water, x < 0 and x>L, the water depth is h, and on top of the
plate, the depth is reduced to d. We are interested in impulse waves with long wavelength
5
Lo and Liu, July 23, 2013
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
Copyright 2013 by the American Society of Civil Engineers
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so that the free surface elevation, η, is the solution of wave equations,
∂
2
η
∂t2
− c2
(
∂
2
η
∂x2
+
∂
2
η
∂y2
)
= 0,
(1)
where c denotes the wave celerity; c = c1 =
√
gh in the open water and c = c2 =
√
gd above
the plate. Defining the wave amplitude spectrum, ζ(x, y, ω), as the Fourier transform of free
surface elevation η(x, y, t), i.e.,
ζ(x, y, ω) =
∫ +∞
−∞
η(x, y, t)e
−iωt
dt,
(2)
we can express the free surface elevation in terms of the wave spectrum:
η(x, y, t) =
1
2π
∫ +∞
−∞
ζ(x, y, ω)e
iωt
dω.
(3)
From the linear wave equation (1), the wave amplitude spectrum must satisfy the following
Helmholtz equation
∂
2
ζ
∂x2
+
∂
2
ζ
∂y2
+ k
2
ζ = 0,
(4)
where k represents the wave number in the region of interest; k = k1 = ω/
√
gh in open water
and k = k2 = ω/
√
gd above the plate. The corresponding horizontal velocity components in
the x− and y− directions can be expressed as
u(x, y, ω) =
ig
ω
∂ζ
∂x
, v(x, y, ω) =
ig
ω
∂ζ
∂y
.
(5)
Now we consider the incident wave spectrum that can be written as
ζinc(x, y, ω) = Ainc(ω)e
−i(α1x+βy)
,
(6)
6
Lo and Liu, July 23, 2013
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
Copyright 2013 by the American Society of Civil Engineers
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where
α
2
1
+ β
2
= k
2
1
=
ω
2
gh
and tan θ =
β
α1
.
(7)
Thus, both β and α1 depend on ω since θ, the angle of incidence, is fixed.
On the incident wave side of the plate, x < 0, the wave amplitude spectrum can be found
from (4), the velocity spectrum from (5), and they can be expressed as the superposition of
incident wave and reflected waves:
ζ1 = Aince
−iβy
(
e
−iα1x
+ Re
iα1x
)
,
u1 =
gα1Ainc
ω
e
−iβy
(
e
−iα1x − Reiα1x
)
, v1 =
gβAinc
ω
e
−iβy
(
e
−iα1x
+ Re
iα1x
)
, x< 0,
(8)
in which R is the reflection coefficient for the wave amplitude spectrum. On the downstream
side of the plate, x>L, the solutions, representing the transmitted wave, are
ζ3 = Aince
−iβy
Te
−iα1(x−L)
,
u3 =
gα1Ainc
ω
e
−iβy
Te
−iα1(x−L)
, v3 =
gβAinc
ω
e
−iβy
Te
−iα1(x−L)
, x > L,
(9)
where T is the transmission coefficient. We remark that due to the long wave approximation,
evanescent modes are ignored here.
Above the plate, 0 <x<L and −d<z< 0, the Helmholtz equation, (4), still holds
with k2 replacing k1, and the modified wave numbers are determined by
α
2
2
+ β
2
= k
2
2
=
ω
2
gd
.
(10)
Thus, above the plate the wave amplitude and velocity spectrum can be written as
ζ2 = Aince
−iβy
(
De
−iα2x
+ Ee
iα2x
)
,
7
Lo and Liu, July 23, 2013
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
Copyright 2013 by the American Society of Civil Engineers
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u2 =
gα2Ainc
ω
e
−iβy
(
De
−iα2x − Eeiα2x
)
, v2 =
gβAinc
ω
e
−iβy
(
De
−iα2x
+ Ee
iα2x
)
, 0 <x<L.
(11)
Underneath the plate, 0 < x < L and −h < z < −(d + δ), the flow is forced by
the pressure difference between the entrance (x = 0) and the exit (x = L). Under the
assumption of shallow water and a wide plate (i.e., the plate width has the same order of
magnitude as the wavelength), beneath the plate the momentum equation simplifies into a
Laplace equation for pressure spectrum. The solutions assume the form
p = ρgAince
−iβy
(a1e
βx
+ a2e
−βx
), 0 <x<L
(12)
with the corresponding velocity components
U =
i
ωρ
∂p
∂x
=
igβAinc
ω
e
−iβy
(a1e
βx − a2e
−βx
),
V =
i
ωρ
∂p
∂y
=
gβAinc
ω
e
−iβy
(a1e
βx
+ a2e
−βx
), 0 <x<L.
(13)
The coefficients, R, D, E, T, a1, and a2, in the solution forms are to be determined from
the matching conditions at the two ends of the plate, x = 0 and x = L.
The continuity of free surface elevation and pressure is required at x = 0 and x = L, i.e.,
ζ1 = ζ2 =
p
ρg
at x = 0,
(14)
and
ζ2 = ζ3 =
p
ρg
at x = L.
(15)
Moreover, the volume fluxes in the x−direction must also be continuous at x = 0 and x = L,
i.e.,
u1h = u2d + U(h − d − δ) at x = 0,
(16)
8
Lo and Liu, July 23, 2013
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
Copyright 2013 by the American Society of Civil Engineers
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and
u2d + U(h − d − δ) = u3h at x = L.
(17)
Substituting solution forms, (8) ∼ (13), into the matching conditions, after some lengthy,
but straightforward algebra, we obtain the following solutions:
R =
−2α1h(Δ+e
iα2L
+ Δ−e
−iα2L
)
Δ
2
+eiα2L − Δ2
−e−iα2L
− 1,
(18)
T =
−2α1h(Δ+ + Δ−)
Δ
2
+eiα2L − Δ2
−e−iα2L
,
(19)
D =
−2α1hΔ+e
iα2L
Δ
2
+eiα2L − Δ2
−e−iα2L
,
(20)
E =
−2α1hΔ−e
−iα2L
Δ
2
+eiα2L − Δ2
−e−iα2L
,
(21)
a1 = D
e
−iα2L − eβL
e−βL − eβL
+ E
e
iα2L − eβL
e−βL − eβL
,
(22)
a2 = D
e
−βL − e−iα2L
e−βL − eβL
+ E
e
−βL − eiα2L
e−βL − eβL
,
(23)
with
Δ± = ∓α1h − α2d ± iβ(h − d − δ)
2e
∓iα2L − e−βL − eβL
e−βL − eβL
.
(24)
For normal incidence, β → 0, α1 → k1, α2 → k2, and Δ± can be found to be
Δ0± = ∓k1h − k2d ∓
i
L
(h − d − δ)(e
∓ik2L − 1).
(25)
The pressure solution underneath the plate becomes linear in x:
p = ρgAinc(a3x + a4), 0 <x<L
(26)
with
a3 =
1
L
[D(e
−ik2L − 1) + E(eik2L − 1)],
(27)
9
Lo and Liu, July 23, 2013
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
Copyright 2013 by the American Society of Civil Engineers
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a4 = D + E.
(28)
We remark that the surface elevations on the incident side (η1), on top of the plate (η2),
and past the plate (η3), can be obtained from (3). Similarly the pressure field underneath
the plate can also be obtained by the inverse Fourier transformation
P−(x, y, t) =
ρg
2π
∫ +∞
−∞
Aince
iωt
e
−iβy
(a1e
βx
+ a2e
−βx
)dω.
(29)
On the other hand, pressure acting on the plate from above can be evaluated as
P+(x, y, t) =
ρg
2π
∫ +∞
−∞
Aince
iωt
e
−iβy
(De
−iα2x
+ Ee
iα2x
)dω.
(30)
Thus, the net vertical force per length acting on the plate can be calculated by integrating
the pressure difference (P− − P+), along the plate, with lift force defined as positive:
F(y, t) =
∫ L
0
(P− − P+)dx
=
ρg
2π
∫ +∞
−∞
Aince
iωt
e
−iβy
[
a1
β
(1 − e
−βL
) −
a2
β
(1 − e
βL
) +
iD
α2
(1 − e
−iα2L
) −
iE
α2
(1 − e
iα2L
)
]
dω.
(31)
For normal incidence, β = 0, the total force formula can be simplified:
F0(y, t) =
ρg
2π
∫ +∞
−∞
Aince
iωt
[
a3L
2
2
+ a4L +
iD
k2
(1 − e
−ik2L
) −
iE
k2
(1 − e
ik2L
)
]
dω.
(32)
Likewise, the moment per length about the center of the plate (with counter-clockwise
rotation defined as positive), μ, can be calculated:
μ(y, t) =
ρg
2π
∫ +∞
−∞
Aince
iωt
e
−iβy
[a1(−
L
β
e
−βL −
1
β2
e
−βL
+
1
β2
) + a2(
L
β
e
βL −
1
β2
e
βL
+
1
β2
)
− D(
iL
α2
e
−iα2L
+
1
α2
2
e
−iα2L −
1
α2
2
) − E(−
iL
α2
e
iα2L
+
1
α2
2
e
iα2L −
1
α2
2
)]dω −
L
2
F (33)
10
Lo and Liu, July 23, 2013
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
Copyright 2013 by the American Society of Civil Engineers
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for oblique incidence. In the special case of normal incidence, the formula for calculating
moment becomes
μ0(y, t) =
ρg
2π
∫ +∞
−∞
Aince
iωt
[
a3L
3
3
+
a4L
2
2
− D(
iL
k2
e
−ik2L
+
1
k2
2
e
−ik2L −
1
k2
2) − E(−
iL
k2
e
ik2L
+
1
k2
2
e
ik2L −
1
k2
2)]dω −
L
2
F0. (34)
We note that the step problem, where no opening exists below the plate, is simply a
special case of the solutions shown here. By setting δ = h − d, the flow under the plate
is completely blocked and the present solutions reduce to the corresponding step problem.
The solutions for a step problem can be further converted to those for a trench problem by
setting d>h.
We remark that for the wave conditions and time scales considered, accurate and con-
vergent numerical results can be obtained for the improper integrals in analytical solutions
with discretization consisting of 200 – 500 points between ω = ±5π – ±15π.
Incident impulse waves
The incident impulse waves can be described in many different ways as long as their
Fourier transforms can be expressed in analytical forms. For instance, for a Gaussian impulse
wave of the shape
ηG(ξ,t) = He
−[K(ξ−ct)]2
,
(35)
we find the wave spectrum to be
ζG(ξ,ω) =
H
√
π
Kc
e
−( ω
2Kc )
2
e
−ikξ
.
(36)
in which ξ = xcos θ + y sin θ, H is the wave height, c =
√
gh is the phase speed, and K is
the arbitrary wave number of the impulse wave.
Similarly, we can also find the analytical expression for the wave spectrum of an impulse
11
Lo and Liu, July 23, 2013
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
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wave expressed in terms of a sech
2− distribution, i.e.,
ηsech(ξ,t) = Hsech
2
[K(ξ − ct)].
(37)
The corresponding wave spectrum can be written as
ζsech(ξ,ω) =
Hπω
K2c2
csch
(
πω
2Kc
)
e
−ikξ
.
(38)
In both cases, (35) and (37), the wave height and wave number are independent of each
other; the wave number can be specified arbitrarily.
For a special case where K, the effective wave number of the sech
2− type impulse wave,
is constrained by the wave height, H, and the water depth, h, in the following manner
K =
1
h
√
3H
4h
, and c =
√
g(H + h),
(39)
the impulse wave is a solitary wave. For a solitary wave the effective wavelength and period
can be defined as
λ =
2π
K
, and T =
2π
Kc
.
(40)
The corresponding wave spectrum is also known (Miles 1976):
ζsoli(ξ,ω) = Ainc(ω)e
−ikξ
, with Ainc(ω) =
4πωh
3
3c2
csch
[(
h
3
3H
)1/2 (πω
c
)
]
.
(41)
Clearly, in applying the linear theoretical solution to solitary waves, we must recognize its
limitations. One obvious limitation is the need to use the linearized phase speed c =
√
gh in
(41) for solitary waves. Since true solitary waves are not solutions to the linear wave theory,
we should be careful and refer to these impulse waves as “soliton-like” impulse waves. As is
the case with the use of all linear theory, we will show in the following discussions that the
analytical solutions capture most of important physical features and agree with experimental
12
Lo and Liu, July 23, 2013
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
Copyright 2013 by the American Society of Civil Engineers
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and numerical data reasonably well in the cases of small amplitude incident waves.
EXPERIMENTS
Experiments examining the scattering of normally incident solitary waves were conducted
in the 32 m long and 0.6 m wide wave flume in the DeFrees Hydraulics Laboratory at Cornell
University. The wave flume has a piston type wave maker with a 4 m stroke installed at
one end and a 1 : 20 slope at the other end, which gives 22 m of constant depth region
in the flume. A glass plate of dimension 1.156 m (width; L) × 0.6 m (length) × 0.01 m
(thickness), suspended from the top of the wave flume with two nylon straps, were installed
in the middle of the constant depth region. At each corner of the plate a metal L-bracket
connects the plate to the sidewall, restraining the plate from moving. Waterproof caulking
foams were used to seal the thin gaps between the plate and the flume sidewalls. Under this
configuration, the location and height of the plate can be adjusted relatively easily, and the
plate support produces minimal intrusion to the flow. The height of the plate was adjusted
to yield two different d values: d = 0.1 m and d = 0.05 m. In the experiments the constant
water depth was fixed at h = 0.2 m. We note that it is impractical to change L in the
experiments, thus different L/λ values result due to different wavelengths.
To measure water surface elevations, acoustic wave gauges (Banner Engineering S18U)
with 0.5 mm resolution were used. The wave gauge (WG) locations are illustrated in Figure
2 and listed in Table 1. Note that WG1 and WG4 are a quarter wavelength away from
the plate edge, thus they need to be moved whenever the wavelength changes. Attached
to plastic tubes of 3 mm diameter, pressure transducers (Omega PX26) are connected to
specific locations on the plate surface. A total of six pressure sensors were available for use
during the experiments. The pressure sensors were moved and the experiments repeated so
that pressures at a total of ten different locations were measured. Figure 2 and Table 1 show
the sensor locations. Note that at each of the five horizontal locations there is one pressure
13
Lo and Liu, July 23, 2013
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
Copyright 2013 by the American Society of Civil Engineers
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sensor on top of the plate and one below, so that pressure differences at these five locations
can be determined. The final pressure was obtained by ensemble-averaging measurements
from three repeated experimental runs. The resolution in terms of pressure head is estimated
to be 1 mm of water.
Solitary waves were generated in the wave flume by using Goring’s (1978) method. Grimshaw’s
(1971) second-order solitary wave solution is used to check the accuracy of wave shapes.
Overall, solitary wave shapes generated in the experiments are highly accurate; see Figure
3 for comparisons. However, we remark that an oscillatory tail smaller than 5% of the wave
height exists beyond the 1.2 effective periods shown in the figure. Such oscillatory tail is
common in laboratory experiments and cannot be eliminated completely. Six different wave
heights ranging from H/h = 0.05 to H/h = 0.3, defined in the constant-depth region away
from the plate, were generated; see Table 2 for experimental conditions.
The PIV setup consists of a continuous Argon-ion laser (Coherent Innova 90) paired with
a high speed camera (Phantom v9.1). Near the field of view (FOV) the water was seeded
with 30 μm glass spheres (Scotchlite Glass Bubbles S60). In all FOVs, the distance between
the laser sheet and the flume wall was about 10 cm, and that between the laser sheet and the
camera was approximately 1 m. We remark that these distances were not explicitly measured
or fixed. The camera operates at 300 frames per second with 1 ms exposure time. In the
analysis, image pairs separated by 1/300 s were considered and flow velocity field acquired at
30 Hz, by using a cross-correlation-based in-house PIV code with three sub-window-shifting
passes. The location, dimension, and resolution of the three FOVs considered in the study
are shown in Figure 4 and Table 3. In outputting the results, 50% sub-window overlap was
applied, giving a plotting resolution half the sub-window size. The experimental condition
used in the PIV experiments is H/h = 0.2 and H/d = 0.4.
NUMERICAL SIMULATION
Since the simplified long wave theory presented in the previous section is based on the
14
Lo and Liu, July 23, 2013
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
Copyright 2013 by the American Society of Civil Engineers
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depth-averaged equations, the theoretical results will not accurately describe the velocity
field around the plate. In this study we also use the numerical model, Cornell Breaking
Waves and Structures (COBRAS), to simulate solitary wave and the flows around the plate
(for normal incidence only). COBRAS is a 2D Reynolds-averaged Navier-Stokes equations
solver, which adopts the volume of fluid method for tracking the free surface and the k − ϵ
model for turbulence closure. The version of COBRAS adopted here was first developed
at Cornell University (see Lin 1998, Lin and Liu 1998, Liu et al. 1999, and Hsu et al.
2002), and later on improved at the University of Cantabria (see Losada et al. 2008). Since
turbulence is not significant and not considered in our case, COBRAS is essentially a 2D
viscous Navier-Stokes equations solver. Kinematic viscosity of 10
−6
m
2
/s was used in all
numerical simulations. To simulate the laboratory experiments, a computational domain of
size 11 m × 0.3 m is employed, where the solitary wave is sent in from the left (upstream)
boundary. The right boundary (downstream) is a standard reflective solid wall, and the
left boundary turns into a reflective wall too once the inflow signal ends. A plate of width
L = 1.156 m and thickness δ = 0.01 m is set up so that the incident side of the plate
is 4 m away from the left (upstream) boundary. The plate is treated as a solid obstacle
so that normal velocity is zero along its surface. The domain size and the plate location
are carefully chosen so that within the time scale considered, the reflected waves from the
domain boundaries do not interfere with the flow near the plate. The time step (∼ 0.001s) is
dynamically adjusted at each iteration, while the data output is at 10 Hz. After performing
convergence tests with various grid sizes, uniform cells of size 4 mm × 2 mm, which is
of similar magnitude of the resolution of PIV measurement, are consistently used in this
study. Since boundary layers cannot be adequately resolved given the cell size (and the
PIV resolution), free-slip conditions are applied along the domain boundaries and the plate
surface. Typical computation time for one case is about 40 minutes with Intel Core i7-2600K
(3.40 GHz) CPU.
RESULTS: COMPARISONS BETWEEN EXPERIMENTAL MEASUREMENTS AND
15
Lo and Liu, July 23, 2013
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
Copyright 2013 by the American Society of Civil Engineers
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THEORETICAL/NUMERICAL RESULTS
In this section we will present the results in terms of surface elevation, pressure, total
force and moment, velocity field and the observed vortices. For each quantity, experimental
data will be discussed first and then compared with analytical and numerical results. The
analytical solution will then be utilized to examine the problem further where experimental
data are not available. Unless otherwise specified, the plate thickness δ = 0.01 m is used in
theoretical calculations. In showing the results two representative wave heights are consid-
ered here: H/h = 0.1 where non-linearity is relatively small and H/h = 0.2 where non-linear
effects become more significant. For all experimental cases we have compared analytical
and numerical results with experimental data, and remark that the two representative wave
heights chosen here capture the overall characteristics well.
For a leading-order solitary wave, (39), normalizing η by H and t by T collapses different
solitary waves into one hyperbolic secant squared curve. Thus, we adopt this normalization
in plotting the results so that they can be compared more directly.
To show the wave evolution, experimental surface elevation measurements at four differ-
ent locations are compared in Figure 5. Throughout the discussion, t/T = 0 is consistently
defined at when the wave peak passes WG2. We remark here that surface elevation mea-
surements are unavailable at moments when the wave front became too steep for the wave
gauges to correctly measure the free surface elevation. It can be seen that the greater
the wave non-linearity above the plate (approximately characterized by H/d), the greater
the wave deforms. Wave breaking above the plate was observed in the experiments where
H/d ≥ 0.8. As can be seen in Figure 5, the wave height seems to decrease slightly as the
wave first encounters the plate (WG2), and the wave height increases as the wave travels
on top of the plate (WG3). The transmitted wave height (WG4) is lower than the incident
wave height.
For the four representative cases, (H/h = 0.1 and H/d = 0.2), (H/h = 0.2 and H/d =
0.4), (H/h = 0.1 and H/d = 0.4), and (H/h = 0.2 and H/d = 0.8), experimental, theoretical,
16
Lo and Liu, July 23, 2013
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
Copyright 2013 by the American Society of Civil Engineers
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and numerical surface elevations are compared in Figure 6-9. In general, while experimental
data and numerical results compare very well in all cases, the theoretical solution deviates
from the two as non-linearity increases. Above the plate the linear theory tends to over-
predict wave heights, but under-predict wave speed. We note that due to the use of linear
shallow water wave speed, the phase difference associated with our analytical solution is
evident in all cases. Nonetheless, for weak non-linearity (H/d % 0.2), the linear analytical
solutions capture the overall trend well.
Upon close inspection of Figures 6-9, one may notice small oscillations in the tails of
the waves that show in both numerical results and experimental data, especially for higher
non-linearity. The oscillations likely result from evanescent modes and non-linearity, which
were not accounted for in the linear theory. Further discrepancy in the oscillations also exists
between numerical and experimental results, which is likely due to the imperfection in the
generation of solitary waves in the experiments, as has been discussed in the Experiments
section.
Similar comparisons were done for gauge pressures (hydrostatic pressure due to still water
is subtracted from all pressure data) in terms of hydraulic head at ten different locations.
Again, analytical pressure becomes inaccurate when non-linearity is significant; see Figure
10 for comparison when non-linearity is insignificant, and Figure 11 when non-linearity
is significant. Different from surface elevations, the magnitude of experimental pressure
measurements appears to deviate noticeably from both numerical and analytical results.
Although boundary layer effects, absent in numerical model and theory, certainly contribute
to this discrepancy, we remark that the relatively low resolution and sensitivity of the pressure
sensors are likely the main causes. Despite the discrepancy in magnitude, the overall trend
of experimental measurements is highly consistent with that of numerical results.
We now focus on the pressure distribution underneath the plate. The pressure distribu-
tions underneath the plate at three different representative times are plotted in Figure 12 for
the case where H/h = 0.1 and H/d = 0.2, and Figure 13 for the case where H/h = 0.2 and
17
Lo and Liu, July 23, 2013
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
Copyright 2013 by the American Society of Civil Engineers
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H/d = 0.8. We note that it is the spatial linearity that is of interest here - the highly linear
spatial pressure profile confirms our assumption that the flow below the plate is pressure-
driven. Due to phase differences, at a fixed time the numerical, experimental, and analytical
results do not always match each other nicely in Figure 12 and Figure 13, especially for (b).
The overall picture of the phase difference can be better observed in the bottom panels of
Figure 10 and Figure 11, where at a fixed time, the magnitudes of the three pressures can
differ greatly due to phase differences.
With the pressure distribution along the plate known (and thus the pressure difference
between the top and bottom of the plate), we can calculate the net vertical force and moment
(about the center of the plate) acting on the plate. While analytically the pressure difference
can be directly integrated to obtain force and moment, discretized approximation is needed
to obtain the two quantities experimentally and numerically. Since the numerical model has
a relatively high spatial resolution (pressure was computed in each 4 mm × 2 mm cell), the
trapezoidal rule was used to estimate total force and moment. However, experimental data
suffer greatly from low spatial resolution (only 5 locations along the width of the plate).
To tackle this issue, cubic spline interpolation was applied to a total of seven points - five
measured pressure differences along the plate and two assumed zero pressure differences at
the two edges of the plate. We note that due to the low spatial resolution in experimental
data, the calculated net moments become much less reliable than pressure measurements
and calculated net forces. With uplifting forces and counter-clockwise moments defined as
positive, the results (per length) are presented in Figure 14-17. We normalize net vertical
force per length, F0, by Fs = ρgHL, and moment per length about plate center, μ0, by
μs = FsL/2.
As observed in all cases, when a solitary wave passes over the plate, the plate first experi-
ences an uplifting force, followed by a force in the downward direction, and then an uplifting
force again. This phenomenon can be explained as follows: as a solitary wave approaches
the plate, a linear pressure distribution beneath the plate responds instantaneously, whereas
18
Lo and Liu, July 23, 2013
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
Copyright 2013 by the American Society of Civil Engineers
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the pressure along the top side of the plate remains mostly zero, and thus a positive net force
results; as the wave starts riding on top of the plate, the weight of the water wave is exerted
on the top side of the plate, while the bottom side experiences a smaller force due to low
pressure gradient, and thus a negative net force results; as the wave travels away from the
plate, a process similar to that when the wave approaches the plate occurs, and a positive
net force again results. A time-dependent non-zero moment that changes sign results from
the changing force distribution. As expected, linear theory performs well when non-linearity
is insignificant. For the force, interestingly, the experimental data seem to compare well with
analytical and numerical results, even though noticeable deviation in pressure measurement
exists (Figure 10 and Figure 11). For the moment, experimental data become less reliable
due to low spatial resolution - numerical results are the most reliable in this case.
We will now examine the velocity field from PIV measurements and COBRAS numerical
results, at three representative locations: L, M, and R (for information on the FOVs, see
Figure 4 and Table 3). To better describe the representative phases shown in the discussion,
the free surface elevation corresponding to each of the three FOVs, M, L, and R, is shown in
Figure 18(right), Figure 19(left), and Figure 19(right), respectively. We remark that the x−y
coordinates are used here for convenience simply to denote the length scales of the FOVs.
Not surprisingly, away from the edges, the flows both on top and below the plate remain
more or less spatially uniform within the FOV and velocity profiles at a fixed x−location are
plotted in Figure 18 at five different times. While the PIV and COBRAS results compare
well, slight boundary layer effects can be seen in the PIV data. The uniform flow below
the plate again validates the assumption made in the theoretical derivation. Flows near the
two edges, however, are more complicated. Similar to what has been observed in studies
on solitary waves incident on a submerged step (instead of a plate), strong vortices form
near the two edges. The overall characteristics of the vortices due to a plate are not very
different from those due to a step - on the incident side of the plate, as the wave passes a
strong vortex first forms on top of the plate very close to the edge (Figure 20 and Figure
19
Lo and Liu, July 23, 2013
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
Copyright 2013 by the American Society of Civil Engineers
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21), and then it evolves into another vortex underneath the plate (Figure 22), which then
moves to the left (upstream) of the plate and lingers in the FOV. The phases associated with
Figures 20-22 are marked in Figure 19(left) in terms of the free surface elevation at x = 0 in
the FOV. On the lee side of the plate, as the wave passes a strong vortex first forms below
the plate just downstream of the plate edge (Figure 23), and then it lingers and induces a
coupled vortex (Figure 24 and Figure 25). Again, the phases associated with Figures 23-25
are marked in Figure 19(right) in terms of the free surface elevation at x = 0 in the FOV.
We remark that although both experiments and COBRAS consistently show the formation
of the vortices, the locations of the vortices differ slightly by a maximum of 2 cm (see Figure
25 for example). Compared to studies on solitary waves incident on a submerged step, it
appears that in terms of the vortices, the main difference between a plate setup and a step
setup is that in the plate setup the vortices can extend to underneath the plate (e.g. Figure
22 and Figure 24), and thus the locations of the vortices can be different. Given the intense
locally rotational flow, whether potential flow theory can still be applied to approximate the
flow fields in such problems, as has been done in many studies for periodic waves, remains
questionable. Nonetheless, as has been shown earlier, for small amplitude waves a simple
linear long wave theory, which completely omits depth-wise variations and rotationality, still
suffices to yield satisfactory results in terms of water surface elevation, pressure, and net
force acting on the plate.
FURTHER DISCUSSIONS
Having established the validity of our analytical solution, at least for small non-linearity,
we will now use the analytical solution to explore effects of different physical parameters on
the wave scattering process.
Effects of plate dimensions on reflection and transmission
Since the plate thickness δ can be adjusted in the solutions, we can examine the effect of
δ on the wave evolution process. Specifically, we will investigate three representative cases:
δ = 0 (plate of infinitesimal thickness), δ = h − d (step with no opening), and δ = (h − d)/2
20
Lo and Liu, July 23, 2013
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
Copyright 2013 by the American Society of Civil Engineers
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(intermediate case with half opening). An example where H/h = 0.1 and H/d = 0.2 is shown
in Figure 26. In all cases, the transmitted waves are no longer solitary waves. Although the
shapes of the transmitted waves are slightly different in the three cases, the wave heights
are highly similar. On the other hand, the reflected waves show differences and similarities
at the same time. It can be seen that the intermediate case systematically lies between the
plate case and the step case, allowing us to better visualize the transition from the plate
case to the step case. In the step case, the reflected wave consistently consists of one main
peak, C in Figure 26, whereas that in the plate case show two peaks, A and B in Figure
26. The first peak, A, is a result of direct reflection from the incident side of the plate,
and the second peak, B, corresponds to the reverse flow from underneath the plate due to
the pressure difference between the two opening ends after the wave passes the plate (a
similar process in the opposite direction occurs before the wave reaches the top of the plate,
resulting in the observed early rise of surface elevation in the transmitted wave, see Figure
26b). Depending on the wave condition and plate width, the heights of A and B vary.
As an attempt to quantify the transmitted and reflected waves, we examine the wave
heights at the two edges of the plate; specifically, the wave heights of the transmitted waves
at x/L = 1, and the heights of the peaks A, B, and C, at x/L = 0. Normalized by the incident
wave height H, we can then use these wave heights to define a set of physical reflection and
transmission coefficients. It is important to keep in mind that only wave heights at the two
edges are considered here; since the waves assume different shapes, these wave heights do not
straightforwardly represent wave energy or mass flux. With the incident wave height, water
depth, and plate submersion depth fixed, we can vary the plate width (with respect to the
incident wavelength) and examine its effects on the newly defined reflection and transmission
coefficients. The result when H/h = 0.1 and H/d = 0.2 is shown in Figure 27.
The step case can be seen as a limiting case of the plate problem where the plate width
approaches infinity, since the flow beneath the plate is driven by the pressure gradient that
tends to zero in the limiting case. As shown in Figure 27 an optimal plate width that
21
Lo and Liu, July 23, 2013
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
Copyright 2013 by the American Society of Civil Engineers
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minimizes transmission and maximizes reflection exists in the plate case. Physically, this
can be interpreted as the situation where the pressure-driven flow under the plate has the
strongest effect (in terms of wave heights) on the incident wave - via the flow under the
plate more water is taken away from the transmitted wave and propelled backwards as part
of the reflected wave. In Figure 27, the optimal plate width is about L/λ ∼ 0.33. Having
numerically varied the values of the parameters in the theoretical solutions, we find that the
optimal width increases as the plate submersion depth d increases, decreases as d decreases,
and appears to be independent of incident wave height, which isn’t a surprising result coming
from a linear theory.
Effects of oblique incidence
Analytical solutions for oblique incidence will now be discussed. It should be kept in mind
that no experimental nor numerical data are available to validate the results. An example
of free surface elevation contours with θ = π/4 is shown in Figure 28. Four snapshots of
surface contours at different times are shown. While the overall physics remains similar to
the normal incidence case, the propagation directions of the incident wave, reflected wave,
and the transmitted wave, are now allowed to vary due to oblique incidence. The wave crest
propagates as expected - wave crest in region x/L < 0 is parallel to that in region x/L > 1,
since water depths are the same in the two regions. Nonetheless, the crest in region x/L < 0
is slightly ahead, due to the fact that the wave is slowed down on the plate, 0 < x/L < 1.
Because water depth is reduced on top of the plate, the wave crest orientation in this region
is different (the slope of the orientation becomes more negative in Figure 28). It can also
be seen that both the incident wave, A in Figure 28b, and the reflected wave, B, travel into
the positive y−direction, and while the incident wave travels into the positive x−direction,
the reflected wave travels into the negative x−direction. The propagation directions of the
incident wave and the reflected wave form an angle π − 2θ. The reflected waves from the
plate are characterized by the recognizable depression in surface elevation, as can be seen
in both Figure 26 and Figure 28. As θ increases (0 ≤ θ ≤ π/2), the depression in the
22
Lo and Liu, July 23, 2013
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
Copyright 2013 by the American Society of Civil Engineers
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reflected wave becomes larger, and the duration of overlap between the incident wave and
the reflected wave lengthens since the angle between wave propagation directions (π − 2θ)
decreases. In the limiting case where tangential incidence occurs, θ = π/2, the incident
wave is completely cancelled out by the reflected wave, and a trivial solution with η = 0
everywhere is obtained, as can be seen by plugging α1 = 0 into (18)-(21) and obtaining
R = −1 and T = D = E = 0. We remark that similar behavior also shows in Kirby and
Dalrymple’s (1983) solution to the trench problem for periodic waves. One explanation for
this seemingly striking phenomenon lies in the theoretical formulation where quasi-steady-
state periodic waves are assumed. When θ = π/2 and α1 = 0, the incident wave specified for
x < 0 is essentially confined within the region, and a non-trivial quasi-steady state simply
does not exist.
CONCLUDING REMARKS
In this paper we have presented experimental, numerical, and analytical results for
soliton-like waves incident on a submerged horizontal plate. Water surface elevation, gauge
pressure along the plate, and the flow velocity field at three representative locations, were
measured in the experiments. The flow underneath the plate was found to be nearly uni-
formly driven by the pressure gradient between the two openings. It was also found that the
plate is subject to oscillating vertical forces as the wave passes - the plate first experiences
an uplifting net force, followed by a net force in the downward direction, and then an uplift-
ing net force again; a time-dependent non-zero moment results. Similar to what has been
observed in the step problem, complex vortices formed near the two edges of the plate.
A 2D Navier-Stokes equation solver was used to simulate the problem. Although there
are discrepancies in the locations of the vortices, in general the water surface elevation,
pressure distribution, and the velocity field all compare well with experimental data.
Analytically, linear shallow water wave theory was adopted and analytical solutions were
obtained for obliquely incident impulse waves. The linear theory admitted “soliton-like”
impulse wave solutions, and hence the soliton-like impulse waves were used to approximate
23
Lo and Liu, July 23, 2013
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
Copyright 2013 by the American Society of Civil Engineers
J. Waterway, Port, Coastal, Ocean Eng.
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solitary waves in the linear theory. Despite the locally intense vortices observed near the
plate edges in the experiments and numerical data, for small incident wave heights and
thus small non-linearity, the theoretical solutions compare well with both experimental and
numerical data for solitary waves, in terms of surface elevation, pressure along the plate, and
net force acting on the plate. For larger wave heights, however, large discrepancies start to
show, which is an expected limit of linear theory. With its validity verified by experimental
and numerical results, the analytical solution was further utilized to study the reflected and
transmitted waves. It was found that an optimal relative plate width exists that minimizes
the transmitted wave height and maximizes the reflected wave height. The optimal plate
width appears to be independent of wave height, decreases as the plate submersion depth
decreases, and increases as the plate submersion depth increases.
The analytical solutions can be used to examine a broad range of impulse waves inter-
acting with a submerged plate, a step, or a trench.
ACKNOWLEDGEMENTS
The research reported here has been supported by NSF grants to Cornell University.
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Lo and Liu, July 23, 2013
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
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List of Tables
1
The locations of wave gauges and pressure sensors. Note that x = 0 is at the
left edge of the plate. Plate width L = 1.156 m. . . . . . . . . . . . . . . . . 30
2
A complete list of experimental conditions. . . . . . . . . . . . . . . . . . . . 31
3
Dimensions of FOVs and sub-windows. . . . . . . . . . . . . . . . . . . . . . 32
29
Lo and Liu, July 23, 2013
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posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
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Gauge
WG1 WG2 WG3 WG4
P1t
P2t
P3t
P4t
P5t
P1b
P2b
P3b
P4b
P5b
Location (m) -λ/4
0.1
L/2 L+λ/4 0.1 0.35 L/2 L-0.35 L-0.1
TABLE 1. The locations of wave gauges and pressure sensors. Note that x = 0
is at
the left edge of the plate. Plate width L = 1.156
m.
30
Lo and Liu, July 23, 2013
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Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
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Case h (m) d (m) L (m) H/h λ (m) L/λ
T (s)
1
0.2
0.1
1.156 0.05 6.490 0.178 4.521
2
0.2
0.1
1.156 0.10 4.589 0.252 3.123
3
0.2
0.1
1.156 0.15 3.747 0.309 2.494
4
0.2
0.1
1.156 0.20 3.245 0.356 2.115
5
0.2
0.1
1.156 0.25 2.902 0.398 1.853
6
0.2
0.1
1.156 0.30 2.649 0.436 1.659
7
0.2
0.05
1.156 0.05 6.490 0.178 4.521
8
0.2
0.05
1.156 0.10 4.589 0.252 3.123
9
0.2
0.05
1.156 0.15 3.747 0.309 2.494
10
0.2
0.05
1.156 0.20 3.245 0.356 2.115
11
0.2
0.05
1.156 0.25 2.902 0.398 1.853
12
0.2
0.05
1.156 0.30 2.649 0.436 1.659
TABLE 2. A complete list of experimental conditions.
31
Lo and Liu, July 23, 2013
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FOV
dimension
sub-window size
L
1200 × 1200 (pixel)
36 × 24 (pixel)
20.31 × 20.31 (cm) 6.09 × 4.06 (mm)
M
1200 × 1632 (pixel)
48 × 24 (pixel)
15.03 × 20.44 (cm) 6.00 × 3.00 (mm)
R
1200 × 1244 (pixel)
32 × 32 (pixel)
19.90 × 20.76 (cm) 5.34 × 5.34 (mm)
TABLE 3. Dimensions of FOVs and sub-windows.
32
Lo and Liu, July 23, 2013
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List of Figures
1
Plan view definition sketch of the plate problem. . . . . . . . . . . . . . . . . 37
2
Illustration of the locations of wave gauges and pressure sensors. The wave
travels from left to right. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3
Comparisons between experimental measurements in the wave flume (◦ื)
and Grimshaw’s theoretical solutions of solitary waves with different H/h
ratios (- - -). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4
Illustration of the locations of the three representative FOVs, L, M, and R.
Note that the axes do not share the same scales. . . . . . . . . . . . . . . . . 40
5
Comparison of experimental water surface elevation data. (a) H/h = 0.1,
H/d = 0.2; (b) H/h = 0.2, H/d = 0.4; (c) H/h = 0.1, H/d = 0.4; (d)
H/h = 0.2, H/d = 0.8. (—): WG1; (-.-): WG2; (- - -): WG3; (ททท): WG4. . 41
6
Comparison of water surface elevations. H/h = 0.1 and H/d = 0.2. (a)
WG1; (b) WG2; (c) WG3; (d) WG4. (—): theory; (- - -): experiment; (ททท):
COBRAS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
7
Comparison of water surface elevations. H/h = 0.2 and H/d = 0.4. (a)
WG1; (b) WG2; (c) WG3; (d) WG4. (—): theory; (- - -): experiment; (ททท):
COBRAS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
8
Comparison of water surface elevations. H/h = 0.1 and H/d = 0.4. (a)
WG1; (b) WG2; (c) WG3; (d) WG4. (—): theory; (- - -): experiment; (ททท):
COBRAS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
9
Comparison of water surface elevations. H/h = 0.2 and H/d = 0.8. (a)
WG1; (b) WG2; (c) WG3; (d) WG4. (—): theory; (- - -): experiment; (ททท):
COBRAS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
10 Comparison of pressure measurements in terms of hydraulic heads. Pt is on
top of the plate, and Pb beneath; H/h = 0.1 and H/d = 0.2. (a) P1; (b) P2;
(c) P3; (d) P4; (e) P5. (—): theory; (- - -): experiment; (ททท): COBRAS. . . 46
33
Lo and Liu, July 23, 2013
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
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11 Comparison of pressure measurements in terms of hydraulic heads. Pt is on
top of the plate, and Pb beneath; H/h = 0.2 and H/d = 0.8. (a) P1; (b) P2;
(c) P3; (d) P4; (e) P5. (—): theory; (- - -): experiment; (ททท): COBRAS. . . 47
12 Pressure distribution under the plate. Note that it is the spatial linearity
that is of interest here. H/h = 0.1 and H/d = 0.2. (a) t/T = −0.06; (b)
t/T = 0.12; (c) t/T = 0.30. (—): theory; (×): experiment; (ททท): COBRAS.
48
13 Pressure distribution under the plate. Note that it is the spatial linearity
that is of interest here. H/h = 0.2 and H/d = 0.8. (a) t/T = −0.09; (b)
t/T = 0.12; (c) t/T = 0.49. (—): theory; (×): experiment; (ททท): COBRAS.
49
14 Comparison of net vertical force (upper panel) on the plate and the moment
(lower panel) about the center. H/h = 0.1 and H/d = 0.2. (—): theory; (- -
-): experiment; (ททท): COBRAS. . . . . . . . . . . . . . . . . . . . . . . . . . 50
15 Comparison of net vertical force (upper panel) on the plate and the moment
(lower panel) about the center. H/h = 0.2 and H/d = 0.4. (—): theory; (- -
-): experiment; (ททท): COBRAS. . . . . . . . . . . . . . . . . . . . . . . . . . 51
16 Comparison of net vertical force (upper panel) on the plate and the moment
(lower panel) about the center. H/h = 0.1 and H/d = 0.4. (—): theory; (- -
-): experiment; (ททท): COBRAS. . . . . . . . . . . . . . . . . . . . . . . . . . 52
17 Comparison of net vertical force (upper panel) on the plate and the moment
(lower panel) about the center. H/h = 0.2 and H/d = 0.8. (—): theory; (- -
-): experiment; (ททท): COBRAS. . . . . . . . . . . . . . . . . . . . . . . . . . 53
18 Left: snapshots of the horizontal velocity profile in the middle of field of view
M. H/h = 0.2 and H/d = 0.4. y = 0 corresponds to the flume bottom and
y = 20 cm the still water level. (a), (b), (c), (d), and (e) indicate the sequence
of evolution. Each case is separated by 0.3 s. Right: the corresponding water
surface elevation at this location. . . . . . . . . . . . . . . . . . . . . . . . . 54
34
Lo and Liu, July 23, 2013
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19 The corresponding surface elevation at x = 0 in Figures 20-25. Left: (a)
Figure 20; (b) Figure 21; (c) Figure 22. Right: (a) Figure 23; (b) Figure 24;
(c) Figure 25. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
20 A snapshot showing the vortex generation in the field of view L. t/T = −0.21,
H/h = 0.2 and H/d = 0.4. x = 0 corresponds to the upstream edge of the
plate in this figure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
21 A snapshot showing the vortex generation in the field of view L. t/T = 0.07,
H/h = 0.2 and H/d = 0.4. x = 0 corresponds to the upstream edge of the
plate in this figure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
22 A snapshot showing the vortex generation in the field of view L. t/T = 0.36,
H/h = 0.2 and H/d = 0.4. x = 0 corresponds to the upstream edge of the
plate in this figure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
23 A snapshot showing the vortex generation in the field of view R. t/T = 0.45,
H/h = 0.2 and H/d = 0.4. x = 0 corresponds to the lee edge of the plate in
this figure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
24 A snapshot showing the vortex generation in the field of view R. t/T = 1.16,
H/h = 0.2 and H/d = 0.4. x = 0 corresponds to the lee edge of the plate in
this figure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
25 A snapshot showing the vortex generation in the field of view R. t/T = 1.45,
H/h = 0.2 and H/d = 0.4. x = 0 corresponds to the lee edge of the plate in
this figure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
35
Lo and Liu, July 23, 2013
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
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Copyright 2013 by the American Society of Civil Engineers
J. Waterway, Port, Coastal, Ocean Eng.
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Accepted Manuscript
Not Copyedited
26 Theoretical wave evolution due to different plate thickness δ. H/h = 0.1 and
H/d = 0.2. The plate spans from x/L = 0 to x/L = 1. (a) t/T = −0.32; (b)
t/T = 0; (c) t/T = 0.32; (d) t/T = 0.64; (e) t/T = 0.96; (f) t/T = 1.28. (—):
plate of infinitesimal thickness, δ = 0; (- - -): intermediate, δ = (h − d)/2;
(ททท): step, δ = h − d. A: the first peak of the reflected wave due to a plate;
B: the second peak of the reflected wave due to a plate; C: the single peak of
the reflected wave due to a step. . . . . . . . . . . . . . . . . . . . . . . . . . 62
27 The reflected and transmitted wave heights, defined at the two edges of the
plate and normalized by the incident wave height. H/h = 0.1 and H/d = 0.2.
Tp: transmitted wave height due to a plate; Ts: transmitted wave height due to
a step; R1p reflected wave height of the first peak due to a plate; R2p reflected
wave height of the second peak due to a plate; Rs: reflected wave height due
to a step. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
28 Contour plots of η/H at different times with an obliquely incident solitary
wave. H/h = 0.1, H/d = 0.2, L = 1.156 m, h = 0.2 m, δ = 0, and θ = π/4.
(a) t = −1 s in the analytical solution (3); (b) t = 0 s; (c) t = 1 s; (d) t = 2
s. A: incident wave; B: reflected wave. . . . . . . . . . . . . . . . . . . . . . 64
36
Lo and Liu, July 23, 2013
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
Copyright 2013 by the American Society of Civil Engineers
J. Waterway, Port, Coastal, Ocean Eng.
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θ
incident wave
depth h
(open water)
depth d
(submerged plate
of thickness δ)
depth h
(open water)
L
x
y
x=0
FIG. 1. Plan view definition sketch of the plate problem.
37
Lo and Liu, July 23, 2013
Accepted Manuscript
Not Copyedited
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
Copyright 2013 by the American Society of Civil Engineers
J. Waterway, Port, Coastal, Ocean Eng.
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WG1
WG2
WG3
WG4
P1
t
P2
t
P3
t
P4
t
P5
t
P1
b
P2
b
P3
b
P4
b
P5
b
λ/4
λ/4
FIG. 2. Illustration of the locations of wave gauges and pressure sensors. The wave
travels from left to right.
38
Lo and Liu, July 23, 2013
Accepted Manuscript
Not Copyedited
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
Copyright 2013 by the American Society of Civil Engineers
J. Waterway, Port, Coastal, Ocean Eng.
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−0.4
−0.2
0
0.2
0.4
0.6
0
0.05
0.1
0.15
0.2
0.25
0.3
t/T
η/h
FIG. 3. Comparisons between experimental measurements in the wave flume (◦ื
)
and Grimshaw’s theoretical solutions of solitary waves with different H/h
ratios (- - -).
39
Lo and Liu, July 23, 2013
Accepted Manuscript
Not Copyedited
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
Copyright 2013 by the American Society of Civil Engineers
J. Waterway, Port, Coastal, Ocean Eng.
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L
M
R
FIG. 4. Illustration of the locations of the three representative FOVs, L, M, and R.
Note that the axes do not share the same scales.
40
Lo and Liu, July 23, 2013
Accepted Manuscript
Not Copyedited
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
Copyright 2013 by the American Society of Civil Engineers
J. Waterway, Port, Coastal, Ocean Eng.
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−1
0
1
2
−0.5
0
0.5
1
η/H
(a)
−1
0
1
2
−0.5
0
0.5
1
(b)
−1
0
1
2
−0.5
0
0.5
1
t/T
η/H
(c)
−1
0
1
2
−0.5
0
0.5
1
t/T
(d)
FIG. 5. Comparison of experimental water surface elevation data. (a) H/h = 0.1
,
H/d = 0.2
; (b) H/h = 0.2
, H/d = 0.4
; (c) H/h = 0.1
, H/d = 0.4
; (d) H/h = 0.2
,
H/d = 0.8
. (—): WG1; (-.-): WG2; (- - -): WG3; (ททท
): WG4.
41
Lo and Liu, July 23, 2013
Accepted Manuscript
Not Copyedited
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
Copyright 2013 by the American Society of Civil Engineers
J. Waterway, Port, Coastal, Ocean Eng.
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−1
−0.5
0
0.5
1
1.5
−0.5
0
0.5
1
η/H
(a)
−1
−0.5
0
0.5
1
1.5
−0.5
0
0.5
1
(b)
−1
−0.5
0
0.5
1
1.5
−0.5
0
0.5
1
t/T
η/H
(c)
−1
−0.5
0
0.5
1
1.5
−0.5
0
0.5
1
t/T
(d)
FIG. 6. Comparison of water surface elevations. H/h = 0.1
and H/d = 0.2
. (a) WG1;
(b) WG2; (c) WG3; (d) WG4. (—): theory; (- - -): experiment; (ททท
): COBRAS.
42
Lo and Liu, July 23, 2013
Accepted Manuscript
Not Copyedited
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
Copyright 2013 by the American Society of Civil Engineers
J. Waterway, Port, Coastal, Ocean Eng.
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−1
0
1
2
−0.5
0
0.5
1
η/H
(a)
−1
0
1
2
−0.5
0
0.5
1
(b)
−1
0
1
2
−0.5
0
0.5
1
t/T
η/H
(c)
−1
0
1
2
−0.5
0
0.5
1
t/T
(d)
FIG. 7. Comparison of water surface elevations. H/h = 0.2
and H/d = 0.4
. (a) WG1;
(b) WG2; (c) WG3; (d) WG4. (—): theory; (- - -): experiment; (ททท
): COBRAS.
43
Lo and Liu, July 23, 2013
Accepted Manuscript
Not Copyedited
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
Copyright 2013 by the American Society of Civil Engineers
J. Waterway, Port, Coastal, Ocean Eng.
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−1
−0.5
0
0.5
1
1.5
−0.5
0
0.5
1
η/H
(a)
−1
−0.5
0
0.5
1
1.5
−0.5
0
0.5
1
(b)
−1
−0.5
0
0.5
1
1.5
−0.5
0
0.5
1
t/T
η/H
(c)
−1
−0.5
0
0.5
1
1.5
−0.5
0
0.5
1
t/T
(d)
FIG. 8. Comparison of water surface elevations. H/h = 0.1
and H/d = 0.4
. (a) WG1;
(b) WG2; (c) WG3; (d) WG4. (—): theory; (- - -): experiment; (ททท
): COBRAS.
44
Lo and Liu, July 23, 2013
Accepted Manuscript
Not Copyedited
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
Copyright 2013 by the American Society of Civil Engineers
J. Waterway, Port, Coastal, Ocean Eng.
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−1
0
1
2
−0.5
0
0.5
1
η/H
(a)
−1
0
1
2
−0.5
0
0.5
1
(b)
−1
0
1
2
−0.5
0
0.5
1
t/T
η/H
(c)
−1
0
1
2
−0.5
0
0.5
1
t/T
(d)
FIG. 9. Comparison of water surface elevations. H/h = 0.2
and H/d = 0.8
. (a) WG1;
(b) WG2; (c) WG3; (d) WG4. (—): theory; (- - -): experiment; (ททท
): COBRAS.
45
Lo and Liu, July 23, 2013
Accepted Manuscript
Not Copyedited
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
Copyright 2013 by the American Society of Civil Engineers
J. Waterway, Port, Coastal, Ocean Eng.
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−0.5 0 0.5 1
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
P
t/H
(a)
−0.5 0 0.5 1
−0.2
0
0.2
0.4
0.6
0.8
t/T
P
b
/H
−0.5 0 0.5 1
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
(b)
−0.5 0 0.5 1
−0.2
0
0.2
0.4
0.6
0.8
t/T
−0.5 0 0.5 1
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
(c)
−0.5 0 0.5 1
−0.2
0
0.2
0.4
0.6
0.8
t/T
−0.5 0 0.5 1
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
(d)
−0.5 0 0.5 1
−0.2
0
0.2
0.4
0.6
0.8
t/T
−0.5 0 0.5 1
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
(e)
−0.5 0 0.5 1
−0.2
0
0.2
0.4
0.6
0.8
t/T
FIG. 10. Comparison of pressure measurements in terms of hydraulic heads. Pt
is on
top of the plate, and Pb
beneath; H/h = 0.1
and H/d = 0.2
. (a) P1; (b) P2; (c) P3;
(d) P4; (e) P5. (—): theory; (- - -): experiment; (ททท
): COBRAS.
46
Lo and Liu, July 23, 2013
Accepted Manuscript
Not Copyedited
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
Copyright 2013 by the American Society of Civil Engineers
J. Waterway, Port, Coastal, Ocean Eng.
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−0.5 0 0.5 1
−0.5
0
0.5
1
1.5
P
t/H
(a)
−0.5 0 0.5 1
−0.2
0
0.2
0.4
0.6
0.8
1
t/T
P
b
/H
−0.5 0 0.5 1
−0.5
0
0.5
1
1.5
(b)
−0.5 0 0.5 1
−0.2
0
0.2
0.4
0.6
0.8
1
t/T
−0.5 0 0.5 1
−0.5
0
0.5
1
1.5
(c)
−0.5 0 0.5 1
−0.2
0
0.2
0.4
0.6
0.8
1
t/T
−0.5 0 0.5 1
−0.5
0
0.5
1
1.5
(d)
−0.5 0 0.5 1
−0.2
0
0.2
0.4
0.6
0.8
1
t/T
−0.5 0 0.5 1
−0.5
0
0.5
1
1.5
(e)
−0.5 0 0.5 1
−0.2
0
0.2
0.4
0.6
0.8
1
t/T
FIG. 11. Comparison of pressure measurements in terms of hydraulic heads. Pt
is on
top of the plate, and Pb
beneath; H/h = 0.2
and H/d = 0.8
. (a) P1; (b) P2; (c) P3;
(d) P4; (e) P5. (—): theory; (- - -): experiment; (ททท
): COBRAS.
47
Lo and Liu, July 23, 2013
Accepted Manuscript
Not Copyedited
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
Copyright 2013 by the American Society of Civil Engineers
J. Waterway, Port, Coastal, Ocean Eng.
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0
0.5
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x/L
P
b
/H
(a)
0
0.5
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x/L
(b)
0
0.5
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x/L
(c)
FIG. 12. Pressure distribution under the plate. Note that it is the spatial linearity that
is of interest here. H/h = 0.1
and H/d = 0.2
. (a) t/T = −0.06
; (b) t/T = 0.12
; (c)
t/T = 0.30
. (—): theory; (×
): experiment; (ททท
): COBRAS.
48
Lo and Liu, July 23, 2013
Accepted Manuscript
Not Copyedited
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
Copyright 2013 by the American Society of Civil Engineers
J. Waterway, Port, Coastal, Ocean Eng.
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0
0.5
1
−0.2
0
0.2
0.4
0.6
0.8
1
x/L
P
b
/H
(a)
0
0.5
1
−0.2
0
0.2
0.4
0.6
0.8
1
x/L
(b)
0
0.5
1
−0.2
0
0.2
0.4
0.6
0.8
1
x/L
(c)
FIG. 13. Pressure distribution under the plate. Note that it is the spatial linearity that
is of interest here. H/h = 0.2
and H/d = 0.8
. (a) t/T = −0.09
; (b) t/T = 0.12
; (c)
t/T = 0.49
. (—): theory; (×
): experiment; (ททท
): COBRAS.
49
Lo and Liu, July 23, 2013
Accepted Manuscript
Not Copyedited
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
Copyright 2013 by the American Society of Civil Engineers
J. Waterway, Port, Coastal, Ocean Eng.
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−0.5
0
0.5
1
1.5
−0.6
−0.4
−0.2
0
0.2
0.4
t/T
F
0
/F
s
−0.5
0
0.5
1
1.5
−0.1
0
0.1
t/T
μ 0
/μ
s
FIG. 14. Comparison of net vertical force (upper panel) on the plate and the moment
(lower panel) about the center. H/h = 0.1
and H/d = 0.2
. (—): theory; (- - -):
experiment; (ททท
): COBRAS.
50
Lo and Liu, July 23, 2013
Accepted Manuscript
Not Copyedited
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
Copyright 2013 by the American Society of Civil Engineers
J. Waterway, Port, Coastal, Ocean Eng.
Downloaded from ascelibrary.org by NATIONAL TAIPEI UNIVERSITY OF TECHNOLOGY on 11/05/13. Copyright ASCE. For personal use only; all rights reserved.
−0.5
0
0.5
1
1.5
−0.6
−0.4
−0.2
0
0.2
0.4
t/T
F
0
/F
s
−0.5
0
0.5
1
1.5
−0.1
0
0.1
t/T
μ 0
/μ
s
FIG. 15. Comparison of net vertical force (upper panel) on the plate and the moment
(lower panel) about the center. H/h = 0.2
and H/d = 0.4
. (—): theory; (- - -):
experiment; (ททท
): COBRAS.
51
Lo and Liu, July 23, 2013
Accepted Manuscript
Not Copyedited
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
Copyright 2013 by the American Society of Civil Engineers
J. Waterway, Port, Coastal, Ocean Eng.
Downloaded from ascelibrary.org by NATIONAL TAIPEI UNIVERSITY OF TECHNOLOGY on 11/05/13. Copyright ASCE. For personal use only; all rights reserved.
−0.5
0
0.5
1
1.5
−0.6
−0.4
−0.2
0
0.2
0.4
t/T
F
0
/F
s
−0.5
0
0.5
1
1.5
−0.1
0
0.1
t/T
μ 0
/μ
s
FIG. 16. Comparison of net vertical force (upper panel) on the plate and the moment
(lower panel) about the center. H/h = 0.1
and H/d = 0.4
. (—): theory; (- - -):
experiment; (ททท
): COBRAS.
52
Lo and Liu, July 23, 2013
Accepted Manuscript
Not Copyedited
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
Copyright 2013 by the American Society of Civil Engineers
J. Waterway, Port, Coastal, Ocean Eng.
Downloaded from ascelibrary.org by NATIONAL TAIPEI UNIVERSITY OF TECHNOLOGY on 11/05/13. Copyright ASCE. For personal use only; all rights reserved.
−0.5
0
0.5
1
1.5
−0.6
−0.4
−0.2
0
0.2
0.4
t/T
F
0
/F
s
−0.5
0
0.5
1
1.5
−0.1
0
0.1
t/T
μ 0
/μ
s
FIG. 17. Comparison of net vertical force (upper panel) on the plate and the moment
(lower panel) about the center. H/h = 0.2
and H/d = 0.8
. (—): theory; (- - -):
experiment; (ททท
): COBRAS.
53
Lo and Liu, July 23, 2013
Accepted Manuscript
Not Copyedited
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
Copyright 2013 by the American Society of Civil Engineers
J. Waterway, Port, Coastal, Ocean Eng.
Downloaded from ascelibrary.org by NATIONAL TAIPEI UNIVERSITY OF TECHNOLOGY on 11/05/13. Copyright ASCE. For personal use only; all rights reserved.
−5
0
5
10
15
20
25
30
35
0
2
4
6
8
10
12
14
16
18
20
Horizontal velocity (cm/s)
y (cm)
Plate
PIV
COBRAS
−1
−0.5
0
0.5
1
1.5
−0.2
0
0.2
0.4
0.6
0.8
1
t/T
η/H
(a)
(b)
(c)
(d)
(e)
(c)
(b)
(a)
(d)
(e)
(d)
(b)
(c)
(a)
(e)
FIG. 18. Left: snapshots of the horizontal velocity profile in the middle of field of
view M. H/h = 0.2
and H/d = 0.4
. y = 0
corresponds to the flume bottom and y = 20
cm the still water level. (a), (b), (c), (d), and (e) indicate the sequence of evolution.
Each case is separated by 0.3
s. Right: the corresponding water surface elevation at
this location.
54
Lo and Liu, July 23, 2013
Accepted Manuscript
Not Copyedited
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
Copyright 2013 by the American Society of Civil Engineers
J. Waterway, Port, Coastal, Ocean Eng.
Downloaded from ascelibrary.org by NATIONAL TAIPEI UNIVERSITY OF TECHNOLOGY on 11/05/13. Copyright ASCE. For personal use only; all rights reserved.
−0.5
0
0.5
1
1.5
−0.2
0
0.2
0.4
0.6
0.8
1
t/T
η
/H
−0.5
0
0.5
1
1.5
2
−0.2
0
0.2
0.4
0.6
0.8
1
t/T
η
/H
(b)
(a)
(c)
(a)
(b)
(c)
FIG. 19. The corresponding surface elevation at x = 0
in Figures 20-25. Left: (a)
Figure 20; (b) Figure 21; (c) Figure 22. Right: (a) Figure 23; (b) Figure 24; (c) Figure
25.
55
Lo and Liu, July 23, 2013
Accepted Manuscript
Not Copyedited
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
Copyright 2013 by the American Society of Civil Engineers
J. Waterway, Port, Coastal, Ocean Eng.
Downloaded from ascelibrary.org by NATIONAL TAIPEI UNIVERSITY OF TECHNOLOGY on 11/05/13. Copyright ASCE. For personal use only; all rights reserved.
FIG. 20. A snapshot showing the vortex generation in the field of view L. t/T = −0.21
,
H/h = 0.2
and H/d = 0.4
. x = 0
corresponds to the upstream edge of the plate in this
figure.
56
Lo and Liu, July 23, 2013
Accepted Manuscript
Not Copyedited
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
Copyright 2013 by the American Society of Civil Engineers
J. Waterway, Port, Coastal, Ocean Eng.
Downloaded from ascelibrary.org by NATIONAL TAIPEI UNIVERSITY OF TECHNOLOGY on 11/05/13. Copyright ASCE. For personal use only; all rights reserved.
FIG. 21. A snapshot showing the vortex generation in the field of view L. t/T = 0.07
,
H/h = 0.2
and H/d = 0.4
. x = 0
corresponds to the upstream edge of the plate in this
figure.
57
Lo and Liu, July 23, 2013
Accepted Manuscript
Not Copyedited
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
Copyright 2013 by the American Society of Civil Engineers
J. Waterway, Port, Coastal, Ocean Eng.
Downloaded from ascelibrary.org by NATIONAL TAIPEI UNIVERSITY OF TECHNOLOGY on 11/05/13. Copyright ASCE. For personal use only; all rights reserved.
FIG. 22. A snapshot showing the vortex generation in the field of view L. t/T = 0.36
,
H/h = 0.2
and H/d = 0.4
. x = 0
corresponds to the upstream edge of the plate in this
figure.
58
Lo and Liu, July 23, 2013
Accepted Manuscript
Not Copyedited
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
Copyright 2013 by the American Society of Civil Engineers
J. Waterway, Port, Coastal, Ocean Eng.
Downloaded from ascelibrary.org by NATIONAL TAIPEI UNIVERSITY OF TECHNOLOGY on 11/05/13. Copyright ASCE. For personal use only; all rights reserved.
FIG. 23. A snapshot showing the vortex generation in the field of view R. t/T = 0.45
,
H/h = 0.2
and H/d = 0.4
. x = 0
corresponds to the lee edge of the plate in this figure.
59
Lo and Liu, July 23, 2013
Accepted Manuscript
Not Copyedited
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
Copyright 2013 by the American Society of Civil Engineers
J. Waterway, Port, Coastal, Ocean Eng.
Downloaded from ascelibrary.org by NATIONAL TAIPEI UNIVERSITY OF TECHNOLOGY on 11/05/13. Copyright ASCE. For personal use only; all rights reserved.
FIG. 24. A snapshot showing the vortex generation in the field of view R. t/T = 1.16
,
H/h = 0.2
and H/d = 0.4
. x = 0
corresponds to the lee edge of the plate in this figure.
60
Lo and Liu, July 23, 2013
Accepted Manuscript
Not Copyedited
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
Copyright 2013 by the American Society of Civil Engineers
J. Waterway, Port, Coastal, Ocean Eng.
Downloaded from ascelibrary.org by NATIONAL TAIPEI UNIVERSITY OF TECHNOLOGY on 11/05/13. Copyright ASCE. For personal use only; all rights reserved.
FIG. 25. A snapshot showing the vortex generation in the field of view R. t/T = 1.45
,
H/h = 0.2
and H/d = 0.4
. x = 0
corresponds to the lee edge of the plate in this figure.
61
Lo and Liu, July 23, 2013
Accepted Manuscript
Not Copyedited
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
Copyright 2013 by the American Society of Civil Engineers
J. Waterway, Port, Coastal, Ocean Eng.
Downloaded from ascelibrary.org by NATIONAL TAIPEI UNIVERSITY OF TECHNOLOGY on 11/05/13. Copyright ASCE. For personal use only; all rights reserved.
−5
0
5
−0.2
0
0.2
0.4
0.6
0.8
1
η
/H
(a)
−5
0
5
−0.2
0
0.2
0.4
0.6
0.8
1
(b)
−5
0
5
−0.2
0
0.2
0.4
0.6
0.8
1
η
/H
(c)
−5
0
5
−0.2
0
0.2
0.4
0.6
0.8
1
(d)
−5
0
5
−0.2
0
0.2
0.4
0.6
0.8
1
η
/H
x/L
(e)
−5
0
5
−0.2
0
0.2
0.4
0.6
0.8
1
x/L
(f)
C B
A
FIG. 26. Theoretical wave evolution due to different plate thickness δ
. H/h = 0.1
and
H/d = 0.2
. The plate spans from x/L = 0
to x/L = 1
. (a) t/T = −0.32
; (b) t/T = 0
; (c)
t/T = 0.32
; (d) t/T = 0.64
; (e) t/T = 0.96
; (f) t/T = 1.28
. (—): plate of infinitesimal
thickness, δ = 0
; (- - -): intermediate, δ = (h −d)/2
; (ททท
): step, δ = h− d
. A: the first
peak of the reflected wave due to a plate; B: the second peak of the reflected wave
due to a plate; C: the single peak of the reflected wave due to a step.
62
Lo and Liu, July 23, 2013
Accepted Manuscript Not Copyedited
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
Copyright 2013 by the American Society of Civil Engineers
J. Waterway, Port, Coastal, Ocean Eng.
Downloaded from ascelibrary.org by NATIONAL TAIPEI UNIVERSITY OF TECHNOLOGY on 11/05/13. Copyright ASCE. For personal use only; all rights reserved.
0
0.5
1
1.5
2
2.5
3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
L/λ
Normalized reflected or transmitted wave height
T
p
T
s
R
1p
R
2p
R
s
FIG. 27. The reflected and transmitted wave heights, defined at the two edges of
the plate and normalized by the incident wave height. H/h = 0.1
and H/d = 0.2
. Tp
:
transmitted wave height due to a plate; Ts
: transmitted wave height due to a step;
R1p
reflected wave height of the first peak due to a plate; R2p
reflected wave height
of the second peak due to a plate; Rs
: reflected wave height due to a step.
63
Lo and Liu, July 23, 2013
Accepted Manuscript
Not Copyedited
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
Copyright 2013 by the American Society of Civil Engineers
J. Waterway, Port, Coastal, Ocean Eng.
Downloaded from ascelibrary.org by NATIONAL TAIPEI UNIVERSITY OF TECHNOLOGY on 11/05/13. Copyright ASCE. For personal use only; all rights reserved.
0.1
0.1
0.1
0.5
0.5
0.9
0.9
y/L
(a)
−2
−1
0
1
2
3
−2
−1
0
1
2
−0.1
0
0
0.1
0.1
0.1
0.5
0.5
0.5
0.9
0.9
0.9
0.9
(b)
−2
−1
0
1
2
3
−2
−1
0
1
2
−0.1
−0.1
0
0
0
0
0.1
0.1
0.1
0.1
0.1
0.5
0.5
0.5
0.9
0.9
0.9
0.9
y/L
x/L
(c)
−2
−1
0
1
2
3
−2
−1
0
1
2
−0.1
−0.1
0
0
0
0
0
0.1
0.1
0.1
0.5
0.5
0.9
0.9
x/L
(d)
−2
−1
0
1
2
3
−2
−1
0
1
2
B
A
FIG. 28. Contour plots of η/H
at different times with an obliquely incident solitary
wave. H/h = 0.1
, H/d = 0.2
, L = 1.156
m, h = 0.2
m, δ = 0
, and θ = π/4
. (a) t = −1
s
in the analytical solution (3); (b) t = 0
s; (c) t = 1
s; (d) t = 2
s. A: incident wave; B:
reflected wave.
64
Lo and Liu, July 23, 2013
Accepted Manuscript
Not Copyedited
Journal of Waterway, Port, Coastal, and Ocean Engineering. Submitted May 12, 2013; accepted September 6, 2013;
posted ahead of print September 9, 2013. doi:10.1061/(ASCE)WW.1943-5460.0000236
Copyright 2013 by the American Society of Civil Engineers
J. Waterway, Port, Coastal, Ocean Eng.
Downloaded from ascelibrary.org by NATIONAL TAIPEI UNIVERSITY OF TECHNOLOGY on 11/05/13. Copyright ASCE. For personal use only; all rights reserved.
