,
) of a fluid which occupies the domain (
,
) (see Figure 1). Under the assumption that the fluid is inviscid and
incompressible, and the motion is irrotational, there exists a velocity
potential
which satisfies the Laplace
equation
T. Sahoo, T. L. Yip and Allen T. Chwang
Department of Mechanical Engineering, The University of Hong Kong
Pokfulam Road, Hong Kong
Tel: (852) 2859 2634, Fax: (852) 2858 5415, E-mail: atchwang@hkucc.hku.hk
Abstract: The
interaction between waves and a semi-infinite horizontal membrane is analyzed by
means of the linearized wave theory in a fluid domain of constant depth.
Combining the linearized kinematic and dynamic surface conditions with the
dynamic pressure on the membrane, a third-order differential equation is derived
to describe the membrane-covered surface. To deal with the membrane-covered
surface condition and the edge condition, a generalized inner product is
developed. By the eigenfunction expansions method and the orthogonality property
of the eigenfunctions in the membrane-covered region, the complete solution is
obtained. Especially, the influence of different edge conditions on the
wave-induced behaviours of membrane are investigated and compared. The edge
conditions considered in the present study involve (i) a fixed edge, (ii) a free
edge, and (iii) a spring-supported edge. Using Green’s integral theorem, a
relationship between the reflection and transmission coefficients is
established. The free edge condition leads to the maximum membrane deflection
while the fixed edge enhances the wave reflection.
Keywords: membrane, water waves, breakwater, floating barrier
A horizontal barrier would suppress the vertical motion of water oscillation so strongly that it can reduce the wave amplitude effectively. Theoretical investigation on the wave scattering by a horizontal barrier was started by Heins (1950), who studied the scattering of surface waves by a semi-infinite horizontal barrier. Stoker (1957) discussed the problem in terms of shallow water theory. Siew and Hurley (1977) analyzed the effect of long surface waves incident on a submerged horizontal plate. Patarapanich (1984) investigated the scattering of surface waves by a fixed, submerged, horizontal plate using a shallow water approximation developed by Siew and Hurley (1977). McIver (1985) investigated the scattering and radiation of surface waves by a moored horizontal flat plate. Yip and Chwang (1997) studied the control of surface waves by a pitching plate. Yip and Chwang (2000) analyzed the possibility of using a submerged horizontal plate to increase the stability of a perforated wall breakwater against waves.
A membrane can be an attractive alternative for
use in the construction of portable, temporary floating wave barriers. Membranes
have the positive attribute of covering large areas while being light weight,
inexpensive, easily handled and reusable. Since it can be easily removed, it has
a minimum environmental impact on various coastal processes. Due to the
largeness of the structure, many floating breakwater projects are a mere failure
due to the high construction cost. As a result, in the last decade there has
been an increasing interest in the use of
flexible plates or membranes as breakwaters. Oil boom and silt curtain are two
practical examples of membrane structures for use in the sea for specific
purposes (Sawaragi, 1995).
Kim and Kee (1996) investigated the wave interaction with a tensioned vertical membrane which is hinged or elastically-supported at the seabed and at the sea surface. However, the overall performance depends on the magnitude of the tension and the kinds of boundary conditions. Using a buoy-membrane system where the buoy is required to provide the tension, Kee and Kim (1997) investigated the efficiency of the buoy-membrane system as another efficient breakwater. These results were further generalized by Cho et al. (1998) to study the performance of membrane wave barriers in oblique incident waves, and by Lo (1998) to study the efficiency of wave barriers consisting of two vertical membranes spaced at a fixed distance apart. Cho and Kim (1998) investigated the interaction of oblique waves with a submerged horizontal membrane by using the mode expansions method and the boundary integral equation method. Their numerical results are in excellent agreement with the experimental results. Cho and Kim (1999) investigated the interaction of surface waves with a circular membrane submerged horizontally, while Cho and Kim (2000) studied a porous membrane in the presence of gravity waves numerically and experimentally.
In the present paper, the problem of wave interaction with a semi-infinite floating membrane is analyzed in the context of the linearized wave theory in a fluid domain of finite depth. Combining the linearized kinematic and dynamic surface conditions with the dynamic pressure on the membrane, a third-order differential equation is derived which is satisfied on the membrane-covered surface. The hydrodynamic behaviours due to three different types of edge conditions, namely (i) a fixed edge, (ii) a free edge, and (iii) a spring-supported edge are analyzed. To deal with the membrane-covered surface condition and to incorporate the edge conditions at the interaction point, a generalized inner product is developed which makes the eigenfunctions of the membrane-covered region orthogonal. By the matched eigenfunction expansions method and the orthogonality property of eigenfunctions, the full solution is obtained explicitly. Using Green’s integral theorem, a relationship between the reflection and transmission coefficients is derived. In the present analysis, the dispersion relation is obtained for the membrane-covered region from which information regarding the behaviour of wave numbers can be directly obtained.
In the present paper, the problem is studied in
a three-dimensional Cartesian co-ordinate system. A semi-infinite membrane is
kept on the surface (
,
) of a fluid which occupies the domain (
,
) (see Figure 1). Under the assumption that the fluid is inviscid and
incompressible, and the motion is irrotational, there exists a velocity
potential
which satisfies the Laplace
equation
(1)
in the fluid region for all time t. The bottom boundary condition is given by
on
(2)
The fluid domain is divided into two regions:
(1) the upstream open region (
,
) and (2) the membrane-covered region (
,
). The linearized kinematic and dynamic conditions are
(
) on
(3)
(
) on
(4)
where subscript j
denotes the quantity in region j,
is the transverse displacement of
the surface, r the fluid density, Pj the pressure and g the
acceleration due to gravity. It is assumed that the membrane is a thin,
homogeneous and inextensible sheet with uniform mass ms (
, d is the thickness of the membrane,
is the uniform mass density) under
constant tension T. With these assumptions, the displacement
from equilibrium is related to the
differential pressure Ps acting on the membrane by
(5)
If there is no cavitation between the membrane
and the water surfaces, the pressure
in the linearized dynamic condition
(4) is equated to Ps plus the constant atmospheric pressure such that
and
(6)
From (4) to (6), we have
on
,
(7a)
on
,
(7b)
The velocity and pressure are continuous across the boundary between the membrane-covered region and the upstream open region (see Figure 1). The full solution is obtained by matching the velocity and pressure at the boundary which gives
, and
on
,
(8)
A membrane is an elastic plate of negligible flexural rigidity compared to the tension provided at the circumference of the elastic plate. However, the edge conditions for the membrane are very much different from those of an elastic plate physically and mathematically. The membrane edge is assumed to be anchored with zero displacement at the ends or may have a spring-support or a free edge. Furthermore, it is assumed that at the edge of the membrane (x = 0+, y = 0), one of the following edge conditions is satisfied:
for a fixed edge,
(9a)
for a free edge,
(9b)
for a spring-supported edge,
(9c)
where K0 is
the spring stiffness. Spring-supported edge condition (9c) can be used to model
a tension line, a buoy support, and of course a spring support. The
spring-supported edge condition (9c) is derived by considering the membrane with
a spring support deflecting through an angle
and the balance between the spring
force
and the transverse component of
tension
.
We assume that the forcing of the system has a
sinusoidal dependence on time t, and the motion is simple harmonic in time.
Thus, the velocity potential
can be written as
, where
is the angular frequency of
incident waves. The spatial velocity potential
satisfies the Laplace equation
along with bottom condition (2), matching conditions (8) and edge conditions
(9). The surface boundary conditions (7a) and (7b) become
on
,
(10)
on
,
(11)
It may be noted that if ms = 0, then T can be considered as surface tension and surface condition (11) will represent the free surface condition in the presence of surface tension.
The spatial velocity potentials in the corresponding regions are expressed as
for
(12)
+
for
(13)
where
(14a)
(14b)
(14c)
with H0 being the incident wave height. The wave numbers km’s are positive roots of the dispersion relations
(m = 1, 2, 3, …)
(15)
The eigenfunctions
(m = 0, 1, 2, ...) are orthogonal
in the usual sense and complete. On the other hand, the wave numbers pn’s
are positive roots of the dispersion relations
(n =
I, 1, 2, 3, ...)
(16)
where
, and
and pn [(n-1)p/h < pn
< np/h] (n = 1, 2, 3, ...) are positive and real. The additional roots
pI is an additional evanescent mode for the presence of membrane and
can be found around
. It can be expressed explicitly as
only if
, where n is an positive integer. Rm (m = 0, 1, 2, …) and Tn (n
= 0, I, 1 ,2, 3, ...) are unknown constants to be determined.
Define the generalized inner product by
(m, n = 0, I, 1,
2, 3, …)
(17)
Then, it is easy to derive that
for m ¹ n (m, n = 0, I, 1, 2, 3, …)
r
(18)
(19)
(n = I, 1, 2, 3,…)
(20)
which suggests that, the set of functions {fn; n = 0, I, 1, 2, 3, ...} are orthogonal with respect to the generalized inner product (17). The above sequence of functions {fn} is assumed to form a complete set. As a result, in the present study the generalized inner product (17) is similar to the one used for capillary-gravity waves (Rhodes-Robinson, 1979).
Using the generalized inner product (17) and the matching conditions (8), we have
(21)
(22)
Truncating the system of equations (21) and (22) after n = N, we obtain a system of equations for the determination of unknowns Rm (m = 0, 1, 2, …, N+1) and Tn (n = 0, I, 1, 2, …, N). It may be noted that the three prescribed edge conditions are applied based on the relations that
The reflection and transmission coefficients are the ratios of wave heights of reflected and transmitted waves to that of incident waves and can be expressed as
and
(23)
The relationship between the reflection and transmission coefficients can be derived by the application of Green’s integral theorem to equation (1) as
(24)
where S is the region (- x0
< x < x0, -
h < y < 0), C is the boundary described in the
positive sense,
is the line element on C,
is the derivative in the direction
of the outward normal n, and the asterisk * denotes the complex conjugate. The
double integral on the left side vanishes because of (1) and the integral (24)
reduces to an energy flux integral around curve C as
(25)
where Im(.) is the imaginary part. For large x0, it can be shown that
(26)
in which
(27)
Similar derivations were obtained earlier by Keller and Weitz (1953) for waves entering or leaving an ice field and by Fox and Squire (1990) who analyzed the scattering of surface waves by a floating ice.
The variation of reflection coefficient Kr and transmission coefficient Kt versus the wave number k0h is plotted in Figure 2 for various values of T/rgh2 for a fixed edge membrane. For long waves which correspond to a small wave number k0h, the flow is almost uniform along the horizontal direction. As a result, the wave reflection by the horizontal membrane is minimum. The membrane appears to be transparent to the incident wave and the wave reflection disappears. In such a case, Kr tends to zero and Kt approaches one as k0h vanishes, which means that a major part of the wave energy for long waves is transmitted into the covered region. On the other hand, for large wave numbers, Kr increases and Kt decreases to zero. This is due to the fact that for short waves, which correspond to large wave numbers, a major part of the wave energy is concentrated near the free surface. Thus, a large proportion of the wave energy is reflected back by the membrane and a small amount of wave energy is transmitted. The rapid change in the reflection coefficient Kr and the transmission coefficient Kt is observed for moderate k0h. Generally Kr increases and Kt decreases as k0h increases. For different values of T/rgh2, each curve of Kr and Kt has the same basic structure as T/rgh2 varies. As T/rgh2 increases, the membrane behaves like a rigid sheet and a major part of the wave energy which concentrates near the free surface is reflected back and in the process less wave energy is transmitted below the membrane.
The variation of reflection coefficient Kr and transmission coefficient Kt is plotted in Figure 3 versus the wave number k0h for different edge conditions on the scattering of surface waves. It is observed that a free-edge membrane exhibits lower Kr and higher Kt, which indicates that a free-edge membrane allows more incident wave energy to transmit below the membrane and in turn the reflection becomes less. In such a case, reflection is less compared to the membrane having a fixed edge or a spring-supported edge. The edge conditions are not only localised effects but also affect the wave scattering. It may be noted from (9c) that when the spring stiffness K0 is negligible, the spring-supported edge becomes a free edge, and when the spring stiffness K0 becomes very large, the spring supported edge behaves as a fixed edge. In general, an increase in the spring stiffness leads to a larger reflection coefficient and a smaller transmission coefficient. However, a moderate value of K0h/T is sufficient to represent a fixed edge condition.
The interaction of surface waves with a semi-infinite horizontal membrane floating on the free surface is analyzed using the eigenfunction expansions method and the generalized inner product in the membrane-covered region. A relationship between the reflection and transmission coefficients is established based on Green’s integral theorem. The membrane deflection is larger for lower tension on the membrane. The hydrodynamic characteristics are compared for membranes having (i) a fixed edge, (ii) a free edge, and (iii) a spring-supported edge. It was found that the edge condition is not a local effect. The fixed edge condition induces maximum wave reflection and minimum wave transmission. The deflection is maximum for a free edge membrane and minimum for a fixed edge membrane. Spring-supported edge condition is a general case of a free edge and a fixed edge. A large value of the spring stiffness enhances the wave reflection for a spring-supported membrane. It is seen that the membrane can be used as a wave barrier under appropriate tension.
Acknowledgements
This research was sponsored by the Hong Kong Research Grants Council under Grants HKU 7068/00E and NSFC/HKU 8. The second author was supported financially by a Post-Doctoral Fellowship for the Area of Excellence in Harbour and Coastal Environment Studies.
References
Cho, I. H., Kee, S. T. and Kim, M. H. (1998) “Performance of dual flexible membrane wave barriers in oblique waves,” J. Wtrwy. Port Coast. and Ocean Engrg., ASCE, 124(1), 21-30.
Cho, I. H. and Kim, M. H. (1998) “Interaction of a horizontal flexible membrane with oblique incident waves,” J. Fluid Mech., 367, 139-161.
Cho, I. H. and Kim, M. H. (1999) “Wave deformation by a submerged circular disk,” Appl. Ocean Research, 21, 263-280.
Cho, I. H. and Kim, M. H. (2000) "Interaction of horizontal porous flexible membrane with waves," J. Wtrwy. Port Coast. and Ocean Engrg., ASCE, 126(5), 245-253.
Fox, C. and Squire, V. A. (1990) “Reflection and transmission characteristics at the edge of shore fast sea ice,” J. Geophys. Research, 95(C7), 11629-11639.
Heins, A. E. (1950) “Water waves over a channel of finite depth with a submerged plane barrier,” Canadian J. Math., 2, 210-222.
Kee, S. T. and Kim, M. H. (1997) “Flexible membrane wave barrier. II: floating/submerged buoy-membrane system,” J. Wtrwy. Port Coast. and Ocean Engrg., ASCE, 123(2), 82-90.
Keller, J. B. and Weitz, M. (1953) “Reflection and transmission coefficients for waves entering or leaving an ice field,” Commun. Pure Appl. Math., 6, 415-417.
Kim, M. H. and Kee, S. T. (1996) “Flexible-membrane wave barrier. I: Analytic and numerical solutions,” J. Wtrwy. Port Coast. and Ocean Engrg., ASCE, 122(1), 46-53.
Lo, E. Y. M. (1998) “Flexible dual membrane wave barrier,” J. Wtrwy. Port Coast. and Ocean Engrg., ASCE, 124(5), 264-271.
McIver, M. (1985) “Diffraction of water waves by a moored, horizontal, flat plate,” J. Engrg. Math., 19, 297-319.
Patarapanich, M. (1984) “Maximum and zero reflection from a submerged plate,” J. Wtrwy. Port Coast. and Ocean Engrg., ASCE, 110(2), 171-181.
Rhodes-Robinson, P. F. (1979) “On surface waves in the presence of immersed vertical boundaries II,” Quart. J. Mech. Appl. Math., 32, 125-133.
Sawaragi, T. (1995) Coastal Engineering - Waves, Beaches, Wave-Structures Interactions, Elsevier, Tokyo.
Siew, P. F. and Hurley, D. G. (1977) “Long surface waves incident on a submerged horizontal plate,” J. Fluid Mech., 83, 141-151.
Stoker, J. J. (1957) Water Waves, The mathematical theory and applications, Interscience Publishers, Inc., New York.
Williams, A. N., Geiger, P. T., and McDougal, W. G. (1991) “Flexible floating breakwater,” J. Wtrwy., Port, Coast., Ocean Engrg., ASCE, 1229(6), 288-296.
Yip, T. L. and Chwang, A. T. (1997) “Water wave control by a pitching plate,” J. Engrg. Mech., ASCE, 123(8), 800-807.
Yip, T. L. and Chwang, A. T. (2000) “Perforated wall breakwater with internal horizontal plate,” J. Engrg. Mech., ASCE, 126(5), 533-538.
Fig. 1 Schematic diagram of a floating membrane
Fig. 2 Variation of Kr
and Kt versus wave number k0h
for different values
of tension
for a fixed edge membrane with
d/h=0.001 and K0h/T=2.5
Fig. 3 Variation of Kr and Kt
versus wave number k0h for different edge conditions with
d/h=0.001,
=1 and K0h/T=2.5