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
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.

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.
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
Set-up with sandbar field
39 35 41 42
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
2
1
0
-1
0
-2
5 10 15 20
Probe 35
Probe 39
-3 time (s)
Figure 2: Wave elevation vs.time for probes 35 and 39 with T = 1.04 s and h = 4 cm.
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 2 x
1 x
1
+ ∆ 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.

η
1
=
=
= c
1
η
I cos(
φ
1
( x
1
)
+
+ η a
I cos( kx
1
ω t )
R
−
( x
ω
1 t
)
+ ε
I
)
+ a
R cos(
=
=
Φ
Φ
A
1
I
=
R
=
= a
B
1
= a a
I cos(
Φ
A
1 cos kx
+ ε
ω kx
1
1
+ ε
I t
I
I cos sin
Φ
R
I
Φ
I
I
+
− ω
B
1
+
− a a
R
R t ) sin sin
+ cos
ω a
Φ t
Φ
R
R
R cos(
Φ
R
− kx
1
ω
− t )
ω t
+ ε
R
)
(2)
A
B
2
2
=
=
=
A
B
2
2
=
=
η
2
=
=
= c
2
η
I a
I
A
2 a a a a
I
I
I
I
( cos(
φ
2 x
1
+
+
∆ l )
ω
+ cos( k ( x
1
+ t )
η
R
∆ l )
( x
1
−
+
ω t
∆ l )
+ ε
I
)
+ a
R cos( k ( x
1
+ ∆ l )
− ω t
+ ε cos k
ω k k
∆ t
∆
∆ l l
+ cos( k
∆ l sin( cos sin l
B
2
+ Φ sin
)
+ cos
I
Φ
I
Φ
)
I cos
Φ
I
−
ω
+
−
+ t a a a a
I
R
I
R cos( k
∆ l sin( sin cos k
∆ l k k
∆
∆ l l
+
+ sin
Φ
Φ
Φ
R
R
I
)
) sin
Φ
I
+
− a a
R
R cos sin k
∆ k
∆ l l cos cos
Φ
Φ
R
R
R
)
−
− a a
R
R sin cos k
∆ l sin k
∆ l sin
Φ
Φ
R
R
(1)
(3)
(4)
Next, equations (1)-(4) are combined to find the amplitudes of incident and reflected waves. a
I a
R
=
=
2 sin
1 k
∆ l
2 sin
1 k
∆ l
[
(
A
2
[
(
A
2
−
A
1
− cos k
∆ l −
B
1 sin
A
1 cos k
∆ l +
B
1 sin k
∆ l B
2 k
∆ l
) (
B
2
+
A
1 sin
−
A
1 sin k
∆ l −
B
1 k
∆ l −
B
1 cos k
∆ l
)
2
1
]
2 cos k
∆ l
)
2
1
]
2
(5)
(6)
Finally, Kr can be simply calculated:
Kr
= a a
R
I
(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.
A
1
=
2
N i
N ∑
=
1
η
1 i
N t cos

2
π
T i
∆ t
(8)

B
1
=
2
N i
N ∑
=
1
η
1 i
N t sin

2
π
T i
∆ t
(9)
A
2
=
2
N i
N ∑
=
1
η
2 i
N t cos

2
π
T i
∆ t
(10)
B
2
=
2
N i
N ∑
=
1
η
2 i
N t sin

2
π
T i
∆ t
(11)
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
0.2
0.15
0.1
0.05
0
0.70
0.80
0.90
1.00
period (s)
1.10
Figure 3: Kr vs. T, horizontal bed, h = 4 cm.
1.20
1.30
horizontal bed

Reflection Coefficient, h = 8 cm
0.25
0.20
0.15
0.10
horizontal bed
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
0.15
0.1
0.05
0
0.70
0.80
0.90
1.00
period (s)
1.10
1.20
Figure 5: Kr vs. T, sandbar field set-up, h=4 cm.
Reflection Coefficient, h= 8 cm
1.30
run 1 run 2 run 3 horizontal bed
0.3
0.25
0.2
0.15
run 1 run 2 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.
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.
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
3
2
1
0
-1
0 1 2 3 4 5 6 7 8 9 10 11
Boussinesq
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
3
2
1
-2
-3
0
-1
0
-4
1 2 3 4 5 6 7 8 9 10 11
Boussinesq
Experimental 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
3
2
1
0
-1
0
-2
-3
-4
1 2 3 4 5 6 7 8 9 10 11
Boussinesq
Experiment 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
0
-1
0
-2
-3
-4
4
3
2
1
1 2 3 4 5 6 7 8 9 10 11
Boussinesq
Experimental 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.

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.
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.
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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