Experimental and Numerical Modeling of Wave
Propagation and Reflection Over Sinusoidally-Varying
Sandbar Field
Gayle Willis
Department of Civil and Environmental Engineering
Georgia Institute of Technology
Permanent Address:
8 Candlewood Drive, Amherst, NH 03031
Mentor: Yongxue Wang, Professor, Ph.D.
The State Key Laboratory of Coastal and Offshore Engineering
Dalian University of Technology
2 Linggong Road, Dalian 116024, China
Abstract
Longshore sandbars are naturally occurring structures that have the ability to
reflect incident wave energy, reducing wave impact on coasts. If longshore sandbars
occur periodically on a mildly sloping seabed, a Bragg resonance phenomenon is
possible. The Bragg resonance phenomenon, originally seen in crystallography, predicts
a significant increase in reflected wave energy when the undulating seabed has a
wavelength ½ that of incident waves. Physical experiments were carried out to
investigate Bragg resonance using a 21.0 X 0.45 X 0.6 meter wave tank at the State Key
Laboratory of Coastal and Offshore Engineering, Dalian University of Technology. The
purpose of this study was to reproduce Bragg resonance in regular waves traveling over a
sinusoidally- varying seabed. The experiment set-up had a horizontal seabed, 5
sinusoidally- varying sandbars, and a device at the end of the wave tank to completely
absorb propagated wave energy. Non-breaking regular waves were studied, varying
parameters of wave period and wave height. Wave periods measured ranged from 0.80
seconds to 1.20 seconds. Wave heights used were 4 cm and 8 cm, representing the
lower-amplitude weakly nonlinear waves and the higher-amplitude strongly nonlinear
waves, respectively. Experimental results were analyzed with a two-point method to
calculate reflection coefficients as an indication of wave energy reflection by the
sandbars. Wave time series were also used as a basis for analysis. A numerical model
based on Boussinesq theory was used to compare experimental data vis-à-vis theoretical
findings. Experimental results for the 4 cm and 8 cm wave tank runs demonstrated
expected Bragg resonance, with more variability in the 8 cm results. A comparison study
of reflected energy on a horizontal bed yielded higher and more variable reflection
coefficients than expected, indicating a need to reassess experimental techniques. The
Boussinesq numerical model, studied for weakly nonlinear waves, works accurately for
long period waves but displayed increasing inaccuracies with waves of shorter periods.
A broad range of further research is suggested, both in analysis methods and in physical
experimentation.
Introduction
Sandbars offer natural protection against coastal erosion. Occurring naturally in
the marine environment, it is commonly accepted that these undulations of the seabed
cause incident wave energy to partially reflect, lessening the impact of waves on beaches.
Over the past twenty years, interest in sandbar-wave interaction has significantly grown;
both in seeking to understand the physical interaction and in the anticipated use of manmade bar structures in future engineering works. In the research environment, sandbarwave interaction offers a complex puzzle, both in experimentation and in numerical
modeling. In the engineering community, the use of sandbar configurations to design is
an attractive alternative to the current practice of using massive structures such as jetties
and breakwaters. Sandbars are hidden underwater, provide for easier ship navigation,
and have the potential ability to induce further sandbar formation, giving an even higher
level of incident wave reflection.
About 20 years of research in the field of sandbar-wave interaction has been
fueled by a phenomenon known as Bragg resonance. The Bragg resonance phenomenon,
originally observed about 70 years ago in crystallography, predicts a significant increase
in reflected wave energy when a periodic series of long-shore sandbars have a
wavelength ½ that of incident waves. Past experimental work by Heathershaw (1982)
demonstrated that, at Bragg resonance, a partially standing wave forms on the upwave
side of a series of longshore sandbars, linearly changing into a propagating wave
downwave of the sandbars. Other research has been done to further explore Bragg
resonance, with emphasis on providing theoretical models and in understanding the
physical interaction, looking at sediment-transport and future sandbar growth.
This present study focuses on physical experiments modeling Bragg resonance in
regular surface waves passing over a sinusoidally- varying seabed, looking to pinpoint any
inaccuracies produced from experimental methods. The physical experiments were
conducted using a 21 meter long wave tank located at the State Key Laboratory of
Coastal and Offshore Engineering, at the Dalian University of Technology. A second
focus of this study is to examine a proposed numerical model based on Boussinesq
equations (Zou, 1999). This program has been modified to apply to the physical
experiments conducted.
Experiment Set-up
A series of physical experiments were conducted at the State Key Laboratory of
Coastal and Offshore Engineering in June-July, 2001. A 21 X 0.45 X 0.6 meter wave
tank was set- up as shown in Figure 1.
39 35
41 42
39 35
41 42
Set-up with sandbar field
Set-up without sandbar field
Figure 1: Schematic of Experiment Set-up.
Using the wave paddle (on left) as a reference point, model seabed is initially
horizontal for 2 m, has a 1:20 slope for the next 3 m, then is again horizontal for 3.275 m,
resulting in a pre-sandbar water depth of 30 cm. The model sandbar field contains five
sinusoidal undulations, with wavelengths of 0.69 m and amplitudes of 0.03 m.
Following the finite sandbar field, the bottom is again horizontal for 9.275 m. The end of
the wave tank has an energy-dissipating device.
Five wave gauges were set to measure wave elevation, two wave gauge pairs
positioned to measure wave reflection, one gauge located over the sandbar field to
provide the instantaneous wave elevation over the reflective structure. The wave gauge
pair upwave of the sandbar were numbered 39 and 35, the pair downwave of the sandbar
field numbered 41 and 42. From the head of the wave tank, the gauges were located at
6.5 m, 6.7 m, 10 m, 11.725 m, and 11.925 m, with wave gauge pairs located 20 cm apart.
An alternative set- up, without the sandbar field, was used as a basis for comparison with
the experiments done with the sandbar field set-up. The only modification made to the
set-up was the seabed, a horizontal plane in place of the longshore sandbars.
Waves were generated by computer-controlled wave paddle. Data collection was
collected via computer-connection to wave gauges, measurements begun once incident
waves were at equilibrium. Once data collection began, the gauges recorded wave
elevation every 20 ms. Identical series of experiments were run on both the horizontal
bed set-up and the sandbar set- up. Wave parameters varied were wave height and wave
period. Two wave heights were used, 4 cm and 8 cm. Higher nonlinear effects were
expected for the wave height of 8 cm, compared with the wave height of 4 cm. Wave
periods used were .80 s, .85 s, .90 s, .92 s, .94 s, .96 s, .98 s, 1.00 s, 1.02 s, 1.04 s, 1.06 s,
1.08 s, 1.10 s, 1.15 s, and 1.20 s. Bragg resonance was expected at T = 1.00 s. For each
time period and wave height, three trials were run, each run taking approximately 20
seconds, producing 1024 measurements of wave elevation. Figure 2 shows a sample of
data collected concurrently from paired probes 35 and 39, with a time period of 1.04 s
and wave height of 4 cm.
Wave Elevation vs. Time
h = 4 cm, T = 1.04 s
3
elevation (cm)
2
1
Probe 35
0
-1
Probe 39
0
5
10
15
20
-2
-3
time (s)
Figure 2: Wave elevation vs.time for probes 35 and 39 with T = 1.04 s and h = 4 cm.
Calculation of Kr
The commonly accepted indicator of reflected incident wave energy is the
reflection coefficient, Kr, which is simply the amplitude of the reflected wave divided by
the amplitude of the incident wave. The physical set-up of the experiment used a 2-point
measurement method. The pair of gauges (39 and 35) located upwave of the sandbar
field provide simultaneous measurements of wave elevation. The series of wave
elevation measurements can then be used calculate Kr. The reflection coefficient given
from the upwave gauges describes the amount of incident wave energy reflected by the
sandbar field. The pair of gauges located after the sandbar field (41 and 42) will yield a
Kr describing reflection of the propagating wave downwave of the sandbars.
The calculation of Kr is derived below.
1
x1
2
x1 + ∆l
For gauges 1 and 2, η refers to wave elevation, k is wave number, ω is angular
frequency, t is time, and c, A, B, and ε are constants. Subscripts I and R represent
incident and reflected waves, respectively.
η1 = c1 cos(φ1 + ωt)
= ηI ( x1 ) + η R ( x1 )
= a I cos( kx1 − ω t + ε I ) + a R cos( kx1 − ω t + ε R )
= a I cos(Φ I − ω t ) + a R cos( Φ R − ω t)
= A1 cos ω t + B1 sin ω t
Φ I = kx1 + ε I
Φ R = kx1 + ε R
A1 = a I cos Φ I + a R cos Φ R
B1 = a I sin Φ I − a R sin Φ R
(1)
(2)
η2 = c 2 cos(φ 2 + ω t)
= η I ( x1 + ∆l) + η R ( x1 + ∆l )
= a I cos( k ( x1 + ∆l) − ω t + ε I ) + a R cos( k ( x1 + ∆l) − ω t + ε R )
= A2 cos ω t + B2 sin ω t
A2 = a I cos( k∆l + Φ I ) + a R cos( k∆l + Φ R )
B2 = a I sin( k∆l + Φ I ) − a R sin( k∆l + Φ R )
A2 = a I cos k∆l cos Φ I − a I sin k∆l sin Φ I + a R cos k∆l cos Φ R − a R sin k∆l sin Φ R
B2 = a I sin k∆l cos Φ I + a I cos k∆l sin Φ I − a R sin k∆l cos Φ R − a R cos k∆l sin Φ R
(3)
(4)
Next, equations (1)-(4) are combined to find the amplitudes of incident and reflected
waves.
[
]
[
]
1
( A2 − A1 cos k∆l − B1 sin k∆l )2 + ( B2 + A1 sin k∆l − B1 cos k∆l )2
2 sin k∆l
1
( A2 − A1 cos k∆l + B1 sin k∆l )2 + (B2 − A1 sin k∆l − B1 cos k∆l) 2
aR =
2 sin k∆l
aI =
1
2
1
2
(5)
(6)
Finally, Kr can be simply calculated:
Kr =
aR
aI
(7)
In a program written at the Dalian University of Technology, the coefficients are found
by doing a summation of the finite set of wave elevation data generated by the wave
gauges, given by (8)-(11). N is the number of measurements, T is the wave period, and t
is time.
A1 =
2
N
 i 
N
 2π i ∆t 

T 
∑η  N t  cos
1
i =1
(8)
B1 =
A2 =
B2 =
2
N
2
N
2
N
 i 
N
 2π i ∆t 

T 
∑η  N t  sin 
1
i =1
N
∑η
2
i =1
N
∑η
i =1
2
(9)
 2π i∆t 
 i 
 t  cos 

N 
 T 
(10)
 i   2π i ∆t 
 t  sin 

N   T 
(11)
Experimental Results
In order to understand the impact of the sandbar field on incident wave energy, it
is important to first look at the case of horizontal bed, using it as a basis for comparison.
Also, the horizontal bed case reveals inadequacies of the model. In the ideal case, a
horizontal bed would be expected to have no reflected wave energy, all incident waves
passing unaffected over the flat seabed and then being completely absorbed by the
energy-dissipating device at the end of the wave tank.
For the horizontal bed, reflection coefficients were calculated for wave heights of
4 cm and 8 cm, wave periods of .80s, .85 s, .90 s, .92 s, .94 s, .96 s, .98 s, 1.00 s, 1.02 s,
1.04 s, 1.06 s, 1.08 s, 1.10 s, 1.15 s, and 1.20 s. Refer to Figures 3 and 4 for a graphical
representation of reflection coefficient vs. period.
Reflection Coefficient, h = 4 cm
0.25
KR
0.2
0.15
horizontal bed
0.1
0.05
0
0.70
0.80
0.90
1.00
1.10
period (s)
Figure 3: Kr vs. T, horizontal bed, h = 4 cm.
1.20
1.30
Reflection Coefficient, h = 8 cm
0.25
KR
0.20
0.15
horizontal bed
0.10
0.05
0.00
0.70
0.80
0.90
1.00
1.10
1.20
1.30
period (s)
Figure 4: Kr vs. T, horizontal bed, h = 8 cm
Both Figures 3 and 4 demonstrate the impact of the model set- up, showing at least
minor reflection for every given period. Due to the inability of wave tank models to ever
perfectly fit the theoretical ideal, some reflection was anticipated. Also, due to the
nonlinearity of the 8 cm waves, higher reflected energy was expected in the 8 cm case.
The results shown in Figures 3 and 4 meet both of these expectations. However, the
results also demonstrate some significant imperfections with the given model. For both
the 4 cm and the 8 cm case, the reflection coefficients were significantly higher than
expected, ranging up to a peak value of .1076 for the 8 cm case. This horizontal bed
reflection calls for some readjustments to the experimental set-up. Another aspect of
concern is that, for different time periods, the horizontal reflection coefficient varies. If
the horizontal set-up has varying reflection coefficients, comparison with the sandbar setup will have unexpected complications. The impact of this variability has yet to be fully
investigated.
The next set of experiments involved a finite sandbar field set-up, propaga ting
regular non-breaking waves over the sandbar field. Reflection coefficients were
calculated for wave heights of 4 cm and 8 cm, wave periods of .80s, .85 s, .90 s, .92 s, .94
s, .96 s, .98 s, 1.00 s, 1.02 s, 1.04 s, 1.06 s, 1.08 s, 1.10 s, 1.15 s, and 1.20 s. Three wave
tank runs were done for each set of parameters. Figures 5 and 6 show the experimental
Kr values combined with the results from the horizontal bed.
Reflection Coefficient, h = 4 cm
0.25
0.2
KR
run 1
0.15
run 2
run 3
0.1
horizontal bed
0.05
0
0.70
0.80
0.90
1.00
1.10
1.20
1.30
period (s)
Figure 5: Kr vs. T, sandbar field set-up, h=4 cm.
Reflection Coefficient, h= 8 cm
0.3
0.25
run 1
KR
0.2
run 2
0.15
run 3
horizontal bed
0.1
0.05
0
0.70
0.80
0.90
1.00
1.10
1.20
1.30
period (s)
Figure 6: Kr vs. T, sandbar field set-up, h=8 cm.
The Bragg resonance phenomenon predicted maximum reflection at T = 1.00s for
both the 4 cm and the 8 cm cases, with a significantly higher Kr value in comparison to
other given time periods. The 4 cm case strongly confirms Bragg resonance, each run
rising to a nearly identical maximum at T = 1.00s. The 8 cm case confirms Bragg
resonance to a lesser degree of accuracy, with two of the runs maximizing at T = 1.02 s,
one trial maximizing at T = 1.00s.
Comparing the sandbar bed results with the horizontal bed Kr values, the Bragg
resonance in the sandbar trials lines up with lower and more stabilized reflection Kr
values for the horizontal bed. This allows more confidence in the results indicating
Bragg resonance, with the significant rise in reflection attributed solely to the periodic
sandbars. An interesting phenomenon seen in the graphs is the nearly identical shape of
Kr vs. T for each trial, particularly the second peak seen around T = 1.10 s, lining up with
a low Kr for the horizontal bed. The reason for this second peak is currently unknown.
This unexplained local maximum at a 1.10 s merits further investigation.
Numerical Model
Boussinesq equations, first developed by Boussinesq in 1872, describe
propagating random and no nlinear waves, highly accurate in shallow water. Since 1872,
the Boussinesq equations have been modified many times to achieve higher accuracy in
predicting nonlinear water waves, accounting for turbulence, breaking waves, distribution
of velocities, etc. For our present study, the Boussinesq equations are expected to
accurately predict the effect of the longshore sandbars on incident waves, as the modeled
physical set-up involves shallow water and slightly nonlinear waves.
Zou has published a new modification, providing accuracy in the equations to the
third order (Zou, 1999). He developed a numerical model applying the new equations to
the case of incident waves passing over a sill. In the present research, Zou’s numerical
model was modified to simulate the physical experiments, changing the sill seabed
topography to the 1:20 slope followed by 5 periodic sandbars and including four probes
(modeling experimental probes 39, 35, 31, 32) to record wave elevation. Please refer to
Figure 7 for the graphical display of the seabed topography.
Figure 7: Seabed topography modification made to Dr. Zou’s Boussinesq numerical model.
The numerical model provides the capability of selecting wave parameters, time
step, and spatial step. Each run of the model produces a series of instantaneous wave
heights at each probe. At this point, only preliminary comparisons have been made
between the Boussinesq model and the experimental data, focusing solely on 4 cm waves
and comparing wave time series for both the horizontal bed and the sandbar set-up. The
only parameter varied was time period, T, to examine the sensitivity of the numerical
model to parameter change.
Sensitivity Analysis of Boussinesq Model
Before using the Boussinesq numerical model to calculate theoretical reflection
coefficients, it is important to check the model’s accuracy. The most basic comparison is
looking at the model-generated waves, comparing the time series and looking at the effect
of changing time period, T.
For both horizontal and sandbar field set-up, highest
accuracy is expected with a time period of 1.20 s, giving the most linear waves. Figures
8 and 9 show the wave time series for both the horizontal bed set-up and the sandbar field
set-up.
Probe 39, Numerical Model vs. Experiment, horizontal bed,
h = 4 cm, T = 1.20 s
4
elevation (cm)
3
2
1
Boussinesq
0
-1
0
1
2
3
4
5
6
7
8
9
10
11
Experiment
-2
-3
-4
time (s)
Figure 8: Wave elevation recorded at Probe 39 for horizontal bed with wave height of 4 cm and
period of 1.20s, comparing Boussinesq numerical model with experimental data.
Probe 39, Numerical Model vs. Experiment, sandbar bed,
h = 4 cm, T = 1.20 s
4
elevation (cm)
3
2
1
Boussinesq
0
-1
0
1
2
3
4
5
6
7
8
9
10
11
Experimental
-2
-3
-4
time (s)
Figure 9: Wave elevation measured at Probe 39 for both Boussinesq model and experimental
data, for sinusoidally-varying bed with h = 4 cm, T = 1.20 s.
Simply by visual inspection, Figures 8 and 9 show good concordance between the
numerical and experimental wave profiles, only minor deviations in phase and amplitude.
If this level of accuracy continued for all time periods, the Boussinesq model could be
considered accurate enough to take to the next level of comparison, looking at reflection
coefficients. However, when the time period was altered to 1.00 s, significant
discrepancy is seen between the experimental data and the given numerical data, wave
elevations deviating up to over 1 cm. Figure 10 and Figure 11 show the comparison for
the horizontal bed case and for the sandbar case, respectively.
Probe 39: Numerical Model vs. Experiment, horizontal bed
h = 4 cm, T = 1.00s
4
elevation (cm)
3
2
1
Boussinesq
0
-1 0
1
2
3
4
5
6
7
8
9
10
11
Experiment
-2
-3
-4
time (s)
Figure 10: Wave elevation measured at Probe 39 for both Boussinesq model and experimental
data, for horizontal bed with h = 4 cm, T = 1.00 s.
Probe 39, Numerical Model vs. Experiment, sandbar bed,
h = 4 cm, T = 1.00 s
4
elevation (cm)
3
2
1
Boussinesq
0
-1 0
1
2
3
4
5
6
7
8
9
10
11
Experimental
-2
-3
-4
time (s)
Figure 11: Wave elevation measured at Probe 39 for both Boussinesq model and experimental
data, for sinusoidally-varying bed with h = 4 cm, T = 1.00 s.
In comparing Figures 8-11, a high sensitivity to time period change is clear,
affecting both the phase and amplitude of the wave profiles. In the T = 1.00 s cases, the
sensitivity manifests itself over time, resulting in significant phase and amplitude
discrepancies. The reasoning for the program error lies in the intrinsic nature of
Boussinesq equations and in numerical inaccuracies. The Boussinesq equations are
more accurate with increasing wavelength to water depth, most accurate for shallow
water. Therefore, lower time periods result in less accuracy in the equations, resulting in
phase shift errors. Secondly, numerical models created from mathematical theory
produce inaccuracies by truncating solutions, dropping higher order terms. In the given
model, the Boussine sq model is accurate to the third-order. However, at lower time
periods, numerical damping is apparent – pointing to a need for a model accurate to an
even higher-order.
Conclusions
The essential goal of this present research is to further our ability to understand
wave-sandbar interaction, focusing especially on the Bragg resonance phenomenon.
With enough understanding of the ability of sandbars to reflect incident wave energy,
manmade sandbars can be used as an alternative for coastal protection design. In order to
understand this interaction, fine-tuned experimental methods and numerical models are
needed.
The physical experiments conducted in Dalian, China demonstrated a need for
more fine-tuning of the wave tank set-up. In preliminary tests done on a horizontal bed,
unexpected reflection coefficients were calculated – both higher and more variable in
magnitude. This reflection indicates that the energy-dissipating device was not
completely absorbing incident energy, but reflected a portion. Also, some reflection may
be attributed to the 1:20 slope, producing multi-reflections between the slope and the
wave paddle. With the variability in reflection coefficients, more complexity is added to
future analysis of all experiments run in the wave tank. In the given experiments
conducted with the 5 sinusoidal undulations, errors due to wave tank reflection were
considered minimal around the T = 1.00 s point of interest. For both weakly and strongly
nonlinear waves, results indicated strong support of the Bragg resonance phenomenon at
T = 1.00 s. For future research works, however, this researcher suggests modifications
made to the experimental set-up, looking to minimize horizontal bed reflection
coefficient values.
The proposed numerical model (Zou 1999) based on Boussinesq equations was
applied to the case of weakly nonlinear waves propagating over a sandbar field. The
numerical model was compared to the data from the physical experiments, using wave
time series as the basis for comparison. For higher time periods, maximum being 1.20 s,
the Boussinesq model had only minor deviations from experimental results, for both the
sandbar and the horizontal seabeds. For lower time periods, however, errors accumulated
both from the Boussinesq equations and from numerical damping, causing large
discrepancies in amplitude and phase. At this initial stage of analysis, the Boussinesq
model displays high accuracy for larger wave periods, but error-adjustment is needed for
smaller wave periods.
Future Research
There is an extraordinary amount of future research that needs to be done in the
field of wave reflection by periodic sandbars, both this present study and in many other
facets. In this present research, several areas to be further studied include modifying
experimental techniques to minimize horizontal bed reflection, understanding horizontal
bed variability in reflection, investigating the repeated second-peak-phenomena observed
in the periodic sandbar physical experiments, further investigating the effects of
nonlinearity with Bragg resonance, and further developing the Boussinesq numerical
model to provide more accurate wave behavior.
Outside of this present research, many other areas are available to further study
sandbar-wave interaction. The final end product of this field of research has significant
potential both for coastal engineering and for theoretical research. For coastal
engineering, there is a need to understand the exact interaction with incident waves over
sandbars so that sandbars can be used in future coastal protection designs. In order to
understand the wave reflection from a design standpoint, the experiments in this present
study need to be continued, including new investigations in random waves, reflective
shorelines, sediment transport, and multi-reflection between reflective structures. Also,
numerical models need to be further developed, with potential use in design and in longterm prediction of coastal impact by waves. From a scientific point of view, this field
presents numerous unexplored challenges in both experiment design and theoretical
understanding.
Acknowledgements
This researcher would like to offer gratitude to the National Science Foundation
for making this opportunity possible. Also, to Hai Li, Dr. H.T. Shen, and Dr. H.H. Shen
from Clarkson University for their help in coordinating this program. Sincere thanks also
to Dr. P. Dong of University of Dundee and to Dr. Y. Wang, Dr. Z.L.Zou, and Ping Jin of
Dalian University of Technology and for their assistance and guidance in this research.
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