, and the wave-damping ratio
can be estimated as follows, respectively,
D.-X Shen
JCD Co. Ltd., Higasi-Ueno Center Bldg, Higasi-Ueno 2-1-13, Taito-ku, Tokyo, Japan
M. Isobe
Dept., Tokyo Univ., Hongo 7-3-1, Bunkyo-ku, Tokyo, Japan
S. Katori
Coastal Eng. Consultants Co. Ltd., Tamura Bldg, Hongo 5-23-13, Bunkyo-ku,
Tokyo, Japan
Abstract: This paper presents a simple new method to separate damped incident and reflected waves on the muddy bottom and to estimate their dissipation ratios from a simultaneous recordings of the wave profiles. The reflection coefficient of the sea wall is approximately evaluated based on the resolution results. On the other hand, the method of the estimation of the wave-damping ratio by using the mud motion theory is also described. Both the numerical and laboratory experiments are carried out to verify the resolution technique. Finally, the damping ratios evaluated by the proposed technique are compared with those obtained from the theoretical analysis of mud motion.
Keywords: incident and reflected waves, dissipation ratios, resolution techniques, mud motion
Many attempts have been made to separate the incident and reflected waves in regular and irregular wave fields (Goda and Suzuki, 1976; Isobe and Kondo, 1984; Katori et.al, 1992). All of them pay attention only on the waves without damping. However, muddy coasts represent a type of coastline where some ports such as Tianjin Port of China, Calcutta Port of India, and Northern Ports of Mexico, have been built in the world. Waves at muddy coasts always experience high attenuations because of the interaction between waves and muddy beds. Therefore, resolution of damped incident and reflected waves and estimation of wave dissipation ratios are important to understand the damping tendency of the incident and reflected waves, to estimate accurately the reflection coefficient of the reflective structure, and to evaluate qualitatively mud flux under composite waves.
This paper presents a new method to separate damped incident and reflected waves and at the same time give an estimation of their dissipation ratios from a simultaneous recordings of wave profiles. On the basis of the resolution results, the reflection coefficient of the sea wall is evaluated. Then, various numerical experiments are carried out to verify the method. Finally, the wave recordings of the laboratory experiments are resolved using the proposed technique, and the damping ratios are compared with those obtained from the theoretical analysis of mud motion.
Wave dissipation on muddy bottom is studied
generally through proposed rheological models (Gade, 1958; Dalrymple and Liu,
1978; Maa and Meta, 1990; Shibayama et al, 1993). In this paper, we consider
only a simple vertical two-dimensional mud motion based on the proposed
rheological relationship (shen et. al 1993). Generally, the energy dissipation
rate
, and the wave-damping ratio
can be estimated as follows, respectively,
(1)
where
is the velocity of mud particles,
the shear stress,
is the
fluidized mud thickness and the symbol - denotes the average over a wave period;
the water density,
the
gravity acceleration,
the wave height, and
the group
velocity.
Suppose we have an attenuating regular
incident-reflected-wave system in a wave flume shown in Fig. 1 (multi-reflected
waves is assumed to disappear or to be neglected). A group of wave gauges are
set up in a test area with an equivalent space
at each adjacent station. The measured wave surface elevation
at time
is regarded as a superposition of incident wave
and reflected wave
. Then, the surface elevation
by the
th wave gauge is expressed as,
(2)
Fig.
1 Definition sketch
where
and
denote the incident and reflected wave components at the
station of the
th wave gauge, respectively.
represents the deviation component. In order to improve the
resolution accuracy, we define a standard point at the center location of the
gauge group (see Fig.1), and express the incident and reflected waves at the
standard point as
and
, respectively. Hence, the incident component and the
reflected one can be described as,
(3)
where
denotes the damping ratio, c the wave velocity,
and
are the time intervals in which the wave propagates the
distances
and
respectively, where
is the total number of the gauges. Substituting Eqns. (3) and (4) into (2), we have,
(4)
Now we consider the sum
of the squared deviation components, and define it as
. By taking
, we yield,
(5)
In the same way, we obtain
(6)
Furthermore, to
determine the unknown damping ratio
, we redefine
again as
, take an average of
over a wave period and obtain the following equation by taking
,
(7)
where
and
, and the symbol – is described as before. With three or more the wave data
recordings, the unknown variables
,
and
can be solved.
One of the applications of the method is to evaluate reflection coefficients of coastal structures, which could not be estimated in wave attenuation area by the other methods proposed previously.
Assuming that dissipation ratios of the waves
remain a constant between the standard point and the tested structures, we
define a pseudo-reflection-coefficient
at the standard point as if a reflective wall was put at this
point. Furthermore, we define the real reflection coefficient
; where
and
represent the reflected and incident wave components at the
test structure. It is not difficult to verify the following equation,
(8)
where
is the distance between the standard point and the test wall. Actually,
above is so-called the reflection coefficient of the seawall
estimated by the other research.
Several examinations are tried to verify the present resolution method. The conditions of numerical experiments are shown in Fig. 1. The incident wave height is 6.5cm, the period 2.0s, and the sampling time interval of the gauges is 0.05s.
Figure 2 shows the relationships among the
gauge spacing and the accuracies of the resolution and the damping ratio in
which
is the wavelength. The errors of the resolution and the
dissipation ratio
in Fig. 2 are evaluated by
and
, respectively, in which
(9)
The parameters with
superscript primes in the equations represent ideal theory value, and
is the ideal wave height at the standard point. These
figures suggest that if the observed wave phase difference between each gauge at
an arbitrary time is integer times of
, the resolution is not possible theoretically. In
addition, the magnitude of damping ratio is found to have little influence on
the decomposition accuracy except for a certain range nearby the
unable-resolution point.
Fig. 2
Effect of the gauge spacing on the resolution and dissipation ratio (
)
Now we consider the relations between data sampling time intervals and accuracies of the resolution. Figure 3 illustrates the changes of the errors of the resolution and the damping
Fig.
3 Errors of resolution and dissipation ratio and sampling intervals
ratio with the ratio of
the sampling interval
to period
. If the resolution error is required to less than 1%,
should be less than 0.03.
should be less than 0.09 in case of less than 1% of the
damping ratio error.
The method is also applied to nonlinear waves such as damped cnoidal waves with a good accuracy.
Laboratory experiments were carried out to verify the present resolution techniques in case of sinusoidal waves with a high attention. They were performed in a wave flume of 21.00 m long and 0.80 m wide with a wave absorber. A depth of 12 cm muddy bottom was constructed and a depth of 30 cm water was poured. The time histories of the wave profiles at a few stations were recorded.
Figure 4 shows an example of the resolution by RT with three gauge data in a case of a relatively high attenuation. The estimated damping ratio is 0.19 1/m and the reflection coefficient about 0.28.
On the other hand, the damping ratio of the incident wave is also estimated by the theory of mud motion described in Section 2 on the assumption that the reflected wave is too small to be omitted in the above case. Figure 5 compares the calculated damping ratios by the theory of mud motion and by RT based on laboratory data. Agreement is seen between the two kinds of methods.
Fig. 4 Resolution result of the laboratory experimental data by RT
Fig. 5 Comparison of the damping ratios by two methods
(H=4.0-8.0cm, T=1.02-1.20s, h=30cm)
A simple new method has been presented to separate the damped incident and reflected waves on the muddy bottom and to estimate their dissipation ratios from a simultaneous recordings of wave profiles. Both the numerical and laboratory experiments have been carried out, and the damping ratios of the sinusoidal waves estimated by the mud motion theory and the resolution technique proposed show some agreement. As a result, the method has been verified to be applicable for the resolution of the composite waves, and for the estimations of the damping ratios and the reflection coefficients.
References
[1] Dalrymple, R.A. and Liu, P.L.-F. (1978): Waves over soft muds: a two-layer fluid model, J. Physical Oceanogr., Vol. 8, pp. 1121-1131.
[2] Gade, H.G (1958): Effect of nonrigid, impermeable bottom on plane surface waves in shallow water. J. Marine Res., Vol. 16, No. 2, pp. 61-82.
[3] Goda, Y. and S. Suzuki(1976): Estimation of incident and reflected waves in random wave experiments, Proc. 15th Coastal Eng. Conf., ASCE, pp. 828-848.
[4] Isobe, M and K. Kondo(1984): Method for estimating directional wave spectrum in incident and reflected wave field, Proc. 19th Coastal Eng. Conf., ASCE, pp. 467-483.
[5] Katori, S., T. Taira and M. Miziguchi (1992): Resolution of incident and reflected waves by wave trace method, Proc. Japanese Conf. on Coastal Eng., Vol. 39, pp. 16-20 (in Japanese).
[6] Maa, P.Y. and Meta A.J.(1990): Soft mud response to water waves, J. Waterway, Port, Coastal and Ocean Eng., ASCE, Vol. 116, No. 5, pp. 634-650.
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