UFL/COEL2002/007
WAVE FIELD MODIFICATION BY BATHYMETRIC ANOMALIES
AND RESULTING SHORELINE CHANGES: A STATE OF THE
ART REVIEW
by
Christopher J. Bender
and
Robert G. Dean
Partial funding by:
Bureau of Beaches and Wetland Resources
Florida Department of Environmental Protection
Tallahassee, Florida
2002
Wave Field Modification by Bathymetric Anomalies and
Resulting Shoreline Changes: A State of the Art Review
by
Christopher J. Bender
and
Robert G. Dean
Partial funding by:
Bureau of Beaches and Wetland Resources
Florida Department of Environmental Protection
Tallahassee, Florida
Executive Summary
This report provides a review of available studies on wave transformation by
bathymetric changes and the resulting shoreline impacts. Changes in the bathymetry
cause four wave transformation processes: wave refraction, wave reflection, wave
diffraction and wave dissipation. These four processes result in a modified wave field on
the leeward and seaward side of bathymetric changes. The modified wave field will
impact the shoreline and results in an altered equilibrium planform. A better
understanding of the effects of removing large quantities of sediment from offshore
borrow areas on the local wave field and associated shoreline changes is needed to allow
utilization of offshore sediment resources with minimal shoreline impacts.
Three case studies of beach nourishment projects with significant nearshore
borrow areas are examined. The beach nourishment projects of Grand Isle, Louisiana in
1984, Anna Maria Key, Florida in 1993, and Martin County, Florida in 1996 all include
segments of shoreline leeward of the project borrow areas with dissimilar behavior from
the adjacent shorelines. While the accretion leeward of the borrow area in the Grand Isle
project can be directly linked to the bathymetric changes, the causes of the increased
erosion leeward of the borrow areas in the other two projects are less certain.
A review is presented of field and laboratory scale studies that have examined the
impact of offshore pits on the local wave field and sediment dynamics. The field studies
have investigated the depth where a pit ceases to have significant impact on the local
wave field and sediment transport. The laboratory studies have examined the shoreline
response to the presence of an offshore pit with results indicating the formation of a
salient leeward of the pit.
Solutions for wave transformation by changes in the bathymetry are outlined
primarily in chronological order following the development from analytical solutions for
long waves in one horizontal dimension (1D) through numerical models for arbitrary
bathymetry that include many waverelated nearshore processes. Complete transmission
past infinitely long trenches or sills is found for certain trench or sill and wavelength
combinations in the 1D domain. Models in two horizontal dimensions (2D) show the
development of a partial standing wave pattern seaward of a pit with a shadow zone of
decreased wave action on the leeward side.
Shoreline responses due to wave field modification from changes in offshore
bathymetry have been studied with models that include both wave field and shoreline
changes and by coupling models that evaluate these processes independently. The
complexity of the shoreline response models has evolved from models that include only
wave refraction to complex numerical models for wave transformation combined with
detailed sediment transport models. Both erosion and accretion have been modeled for
the shoreline response leeward of an offshore pit depending on the model used and the
sediment transport coefficients contained. The wave transformation processes included
in the model appear to be an important factor in obtaining a salient leeward of a pit; the
shoreline response observed in the limited laboratory experiments.
Table of Contents
Executive Sum m ary ....................................................................................................... i
List of Figures .............................................................................................................. iii
List of Tables...................................................................................................................... vi
1. Introduction ........................................................................................................... ......... 1
2. M motivation ................................................................................................................ 1
3. Case Studies and Experim ents ....................................... ............... ......................... 3
3.1. Case Studies ...................................................................................................... 3
3.1.1. Grand Isle, Louisiana (1984).................................. ................................... 3
3.1.2. A nna M aria Key, Florida (1993)................................................ ............... 6
3.1.3. M artin County, Florida (1996).................................................. .............. .. 9
3.2. Field Experim ents ........................................................................................... 10
3.2.1. Price et al. (1978) ..................................................................................... 10
3.2.2. Kojim a et al. (1986) ....................................... .............. .......................... 11
3.3. Laboratory Experim ents................................................. ............................... 12
3.3.1. Horikawa et al. (1977)................................................... .................................. 12
3.3.2. W illiam s (2002) ........................................................................................ 12
4. W ave Transform ation............................................................................................. 15
4.1. Analytic M ethods............................................................................................ 15
4.1.1. 1D M ethods............................................................................................. 16
4.1.2. 2D M ethods............................................................................................. 24
4.2. Num erical M models ........................................................................................... 30
5. Shoreline Response ................................................................................................ 34
5.1. Longshore Transport Considerations............................................. ............... 34
5.2. Refraction M models ........................................................................................... 35
5.2.1. M otyka and W illis (1974) ............................................... ....................... 35
5.2.2. H orikaw a et al. (1977)................................................... .......................... 37
5.3. Refraction and Diffraction M odels ................................................ ....... ..... 39
5.3.1. G ravens and Rosati (1994) ............................................... ....................... 39
5.3.2. Tang (2002) ............................................................................................... 41
5.4. Refraction, Diffraction, and Reflection Models........................................41
5.4.1. Bender (2001)............................................................................................ 41
6. Sum m ary and Conclusions..................................................... ........................... 44
7. Acknow ledgem ents.............................................................................................45
8. References ........................................................................................................... 46
List of Figures
Figure 1: Aerial photograph showing salients shoreward of borrow area looking East
to West along Grand Isle, LA in August, 1985 (Combe and Solieau, 1987).............. 4
Figure 2: Aerial photograph showing salients shoreward of borrow area along Grand
Isle, LA in 1998 (modified from Louisiana Oil Spill Coordinator's Office
(LO SCO ), 1999)................................................................................................... 5
Figure 3: Bathymetry off Anna Maria Key, FL showing location of borrow pit
following beach nourishment project (modified from Dean et al., 1999)...................7
Figure 4: Beach profile through borrow area at R26 in Anna Maria Key, FL
(modified from Wang and Dean, 2001). ............................................................8
Figure 5: Shoreline position for Anna Maria Key Project for different periods
relative to August, 1993 (modified from Dean et al. 1999). .....................................
Figure 6: Project area for Martin County beach nourishment project (Applied
Technology and Management, 1998)............................................... ............. 9
Figure 7: Fouryear shoreline change for Martin County beach nourishment project:
predicted versus survey data (modified from Sumerell, 2000)............................. 10
Figure 8: Setup for laboratory experiment (Horikawa et al., 1977)............................. 13
Figure 9: Results from laboratory experiment showing plan shape after two hours
(H orikaw a et al., 1977) ........................................................................................ 13
Figure 10: Experiment sequence timeline for Williams laboratory experiments
(W illiam s, 2002)....................................................................................................... 14
Figure 11: Volume change per unit length for first experiment (Williams, 2002).......... 14
Figure 12: Shifted even component of shoreline change for first experiment
(W illiam s, 2002)................................................................................................. 16
Figure 13: Reflection and transmission coefficients for linearly varying depth [hi/h3]
and linearly varying breadth [b12/b32] (modified from Dean, 1964) ..................... 18
Figure 14: Approximate reflection and transmission coefficients for the rectangular
parallelepiped of length 8.86h0 in infinitely deep water (Newman, 1965b)............. 19
Figure 15: Reflection cofficient for a submerged obstacle (Mei and Black, 1969).........20
Figure 16: Transmission coefficient as a function of relative wavelength (h=10.1
cm, d=67.3 cm, trench width =161.6 cm) (modified from Lee and Ayer, 1981)......21
Figure 17: Transmission coefficient as a function of relative wavelength for
trapezoidal trench; setup shown in inset diagram (modified from Lee et al.,
1980).......................................................................................................................... 22
Figure 18: Reflection coefficient for asymmetric trench and normally incident waves
as a function of Khl: h2/hl=2, h3/hl=0.5, L/h1=5; L = trench width (Kirby and
D alrym ple, 1983a).....................................................................................................23
Figure 19: Transmission coefficient for symmetric trench, two angles of incidence:
L/hi=10, h2/hl=2; L = trench width (modified from Kirby and Dalrymple,
19 83a)........................................................................................................................ 23
Figure 20: Transmission coefficient as a function of relative trench depth; normal
incidence: klhl= 0.2: (a) L/hi=2; (b) L/h1=8, L = trench width. (Kirby and
D alrym ple, 1983a) ..................................................................................................... 24
Figure 21: Total scattering cross section of vertical circular cylinder on bottom
(modified from Black et al., 1971)................................................... ................. 26
Figure 22: Contour plot of relative amplitude in and around pit for normal
incidence; kl/d = n/10, k2/h=7r/1012, h/d=0.5, b/a =1, a/d=2, a = crossshore pit
length, b = longshore pit length, h = water depth outside pit, d = depth inside
pit, L2 = wavelength outside pit. (Williams, 1990). ..............................................27
Figure 23: Contour plot of diffraction coefficient in and around pit for normal
incidence; a/L=l, b/L=0.5, d/h =3, kh=0.167 (McDougal et al., 1996). ..................28
Figure 24: Contour plot of diffraction coefficient around surfacepiercing
breakwater for normal incidence; a/Ll1, b/L=0.5, kh=0.167 (McDougal et al.,
1996) .................................................................................................. 28
Figure 25: Maximum and minimum relative amplitudes for different koa, for normal
incidence, a/b=6, a/d=x, and d/h=2. (modified from Williams and Vasquez,
199 1)............................................................................................... ............... 29
Figure 26: Comparison of wave height profiles for selected models along transect
parallel to shore located 9 m shoreward of shoal apex[*=experimental data]
(M aa et al., 2000). ..................................................................................................... 33
Figure 27: Comparison of wave height profiles for selected models along transect
perpendicular to shore and through shoal apex [*=experimental data] (Maa et
al., 2000).................................................................................................................... 33
Figure 28: Calculated beach planform due to refraction after two years of prototype
waves for two pit depths (modified from Motyka and Willis, 1974).................... 36
Figure 29: Calculated beach platform due to refraction over dredged hole after two
years of prototype waves (Horikawa et al., 1977)................................................ 38
Figure 30: Comparison of changes in beach plan shape for laboratory experiment
and numerical model after two years of prototype waves (Horikawa et al.,
1977)............................................................................................... ............... 38
Figure 31: Nearshore wave height transformation coefficients near borrow pit from
RCPWAVE study (modified from Gravens and Rosati, 1994)............................... 39
Figure 32: Nearshore wave angles near borrow pit from RCPWAVE study; wave
angles are relative to shore normal and are positive for westerly transport
(modified from Gravens and Rosati, 1994)...................................... .............. 40
Figure 33: Reflection coefficients versus dimensionless pit diameter divided by
wavelength inside and outside the pit; water depth = 2 m, pit depth = 4 m
(B ender, 2001).................................................................................................... 42
Figure 34. Shoreline evolution resulting from each transport term individually for
transect located 80 m shoreward of a pit with a radius = 6 m, last time step
indicated with [+] (modified from Bender and Dean, 2001)...................................43
Figure 35. Shoreline evolution using full transport equation and analytic solution
model for transect located 80 m shoreward of a pit with a radius = 6 m, last time
step indicated with [+] (modified from Bender and Dean, 2001)..........................44
List of Tables
Table 1: Capabilities of selected nearshore wave models..............................................31
1. Introduction
Modifications of offshore bathymetry by removal of large quantities of sediment
alter the local wave field, which in turn modifies the equilibrium planform of the leeward
beach. These effects as well as the impact on the sediment dynamics near the pit or shoal
have become of concern as the removal of offshore sediment for beach nourishment,
construction materials, and other purposes has increased. Unexpected shoreline
planform changes in and adjacent to completed beach nourishment projects have been
attributed to offshore borrow pits. These include shorelines that erode more than adjacent
areas during storms and then recover more rapidly after storms. Thus, a better
understanding of the effects of bathymetric changes on the wave field and the resulting
impacts on shorelines would be beneficial to more appropriate utilization of offshore sand
resources.
Several studies including field and laboratory scales have been conducted to
investigate this issue. These studies examined the wave transformation over a
bathymetric anomaly with the shoreline changes caused by the altered wave field.
Earlier, dating back to the early 1900's, the focus was on the modification of a wave train
encountering a change in bathymetry. This early research included development of
analytical solutions for bathymetric changes in the form of a step, or a pit, first of infinite
length (in one horizontal dimension; 1D models), and, more recently, of finite
dimensions (in two horizontal dimensions; 2D models). The complexity of the 2D
models has advanced from a pit/shoal with vertical sidewalls and uniform depth
surrounded by water of uniform depth to domains with arbitrary bathymetry. Some
models combine the calculation of the wave transformation and resulting shoreline
change, whereas others perform the wave calculations separately and rely on a different
program for shoreline evolution.
This report presents a state of the art review of studies relating to wave
transformation by bathymetric anomalies and the resulting shoreline changes.
2. Motivation
Changes in offshore bathymetry modify the local wave field, thus causing an
equilibrium planform that may be significantly altered from the previous, relatively
straight shoreline. Not only can a bathymetric change cause wave transformation, but
also may change the sediment transport dynamics by drawing sediment into it from the
nearshore, or by intercepting the onshore movement of sediment. Knowledge of wave
field modifications and the resulting effects on sediment transport and shoreline evolution
is essential in the design of beach nourishment projects and other engineering activities
that alter offshore bathymetry.
Beach nourishment has become the preferred technique to address shoreline
erosion. In most beach nourishment projects, the fill placed on the eroded beach is
obtained from borrow areas located offshore of the nourishment site. The removal of
large quantities of fill needed for most projects can result in substantial changes to the
offshore bathymetry through the creation of borrow pits, or by modifying existing shoals.
The effect of the modified bathymetry in the borrow area on the wave field and the
influence of the modified wave field on the shoreline can depend on the incident wave
conditions, the nourishment sediment characteristics and some features of the borrow
area including the location, size, shape and orientation.
The large quantities of sediment used in beach nourishment projects combined
with the increase in the number of projects constructed, and an increased industrial need
for quality sediment has, in many areas, led to a shortage of quality offshore fill material
located relatively near to the shore. This shortage has increased interest in the mining of
sediment deposits located in Federal waters, which fall under the jurisdiction of the
Minerals Management Service (MMS). Questions have been raised by the MMS
regarding the potential effects on the shoreline of removing large quantities of sediment
from borrow pits lying in Federal waters (Minerals Management Service, 2002).
A better understanding of the effects of altering the offshore bathymetry is
currently needed. There are four wave transformation processes that can occur as a wave
train encounters a change in the bathymetry: refraction, diffraction, reflection and
dissipation. The first three of these processes are referred to collectively as "scattering".
These transformation processes modify the wave field in a complex manner dependent on
the local conditions. A more complete understanding and predictive capability of the
effect of bathymetric changes to the wave field and the resulting shoreline modification
leading to less impactive design of dredge pit geometries should be the goal of current
research.
3. Case Studies and Experiments
Several methods have been employed to quantify the impact on the shoreline
caused by changes in the offshore bathymetry including case studies, field experiments,
analytical developments, numerical models, and laboratory studies. The intriguing
behavior of the shoreline following beach nourishment projects at Grand Isle, Louisiana,
Anna Maria Key, Florida, and Martin County, Florida have led to questions and
investigations, regarding the impact of the significant offshore borrow areas present in
each case. Field studies have been used to investigate the impact of offshore dredging in
relatively deeper water to attempt to define a depth at which bathymetric changes will not
induce significant wave transformation. Laboratory experiments have documented wave
transformations caused by changes in the bathymetry and the resulting effects on the
shoreline in controlled settings possible only in the laboratory.
3.1. Case Studies
3.1.1. Grand Isle, Louisiana (1984)
The beach nourishment project at Grand Isle, LA provides one of the most
interesting, and well publicized examples of an irregular planform resulting from the
effects of a large borrow area lying directly offshore. One year after the nourishment
project was completed, two large salients, flanked by areas of increased erosion,
developed immediately shoreward of the offshore borrow area. Combe and Solieau
(1987) provide a detailed account of the shoreline maintenance history at Grand Isle,
specifications of the beach nourishment project that was completed in 1984, and details
of the shoreline evolution in the two years following completion.
The project required 2.1x106 m3 of sediment with approximately twice this
amount dredged from an area lying 800 m from the shore (Combe and Soileau, 1987) in
4.6 m of water (Gravens and Rosati, 1994). The dredging resulted in a borrow pit that
was "dumbbell" shaped in the planform with two outer lobes dredged to a depth of 6.1 m
below the bed, connected by a channel of approximate 1,370 m length dredged to 3.1 m
below the bed (Combe and Soileau, 1987). The salients seen in Figure 1 started to form
during storm events that occurred during the winter and spring of 1984/85 (Combe and
Soileau, 1987).
Figure 1: Aerial photograph showing salients shoreward of borrow area looking East
to West along Grand Isle, LA in August, 1985 (Combe and Solieau, 1987).
By August 1985 the salients and associated areas of increased erosion were prominent
features on the shoreline. An aerial survey of the area that was completed by the New
Orleans District of the Army Corps of Engineers and the Coastal Engineering Research
Center concluded that the size and location of the borrow area were such that its presence
could affect the local wave climate (Combe and Soileau, 1987). Oblique aerial
photography identified the diffraction of the wave field as a result of the borrow area
(Combe and Soileau, 1987). The area of increased erosion near the salients was found to
"affect 25% of the project length and amounted to about 8% of the neat project volume"
(Combe and Soileau, 1987).
Three major hurricanes impacted the project area in the hurricane season
following the project's completion; the first time that three hurricanes struck the
Louisiana coastline in the same season (Combe and Soileau, 1987). While these storms
did tremendous damage to the newly formed berm and caused large sediment losses, the
location and size of the salients remained relatively unchanged. The salients have
remained on the Grand Isle shoreline as shown by an aerial photograph from 1998
(Figure 2). It appears that the eastern salient has decreased in size while the western
salient has remained the same size or even become larger.
Figure 2: Aerial photograph showing salients shoreward of borrow area along Grand
Isle, LA in 1998 (modified from Louisiana Oil Spill Coordinator's Office (LOSCO),
1999).
A series of detached offshore breakwaters was constructed along the eastern part of
Grand Isle in the 1990's, which terminate at the eastern salient and may have affected its
shape.
Bathymetric surveys taken through the borrow area in February 1985 and August
1986 revealed that the outer lobes had filled to about half their original depth and the
channel connecting the lobes had reached the sea bed elevation (Combe and Soileau,
1987). Currently, the borrow area is reported to be completely filled by fine material
(Combe, personal correspondence) which would have required the same approximate
volume of sediment that was dredged for the initial placement. Although the origin of the
sediment that has refilled the borrow pit is not known, it is reported to be finer than the
sediment dredged for the nourishment project, indicating that the material did not
originate from the project. While no longer a bathymetric anomaly, the borrow areas are
reported to continue to modify the wave field as local shrimpers use the waters shoreward
of the pit as a harbor to weather storms. The reason for the sheltering effect of the filled
pit may be due to the energydissipating characteristic of the finer material that has filled
the pit.
3.1.2. Anna Maria Key, Florida (1993)
The 1993 beach nourishment project at Anna Maria Key, Florida is another
example of a project with a large borrow area lying offshore in relatively shallow water.
The project placed 1.6x106 m3 of sediment along a 6.8 km segment (DNR Monuments R
12 to R35*) of the 11.6 km long barrier island (Dean et al., 1999). The borrow area for
the project was approximately 3,050 m long and ranged from 490 to 790 m offshore in
approximately 6 m of water (Dean et al., 1999). A planview of the bathymetry near the
project including the borrow area is shown in Figure 3. A transect through the borrow
area, indicated in the previous figure at Monument R26, is shown in Figure 4 and shows
dredging to a depth of 3.1 m below the local seabed. This figure shows one preproject
transect, a transect immediately following completion, and two postnourishment
transects. The postnourishment transects indicate minimal infilling of the borrow pit.
The shoreline planform was found to show the greatest losses shoreward of the
borrow area. Figure 5 shows the shoreline position relative to the August, 1993 data for
seven different periods. A large area of negative shoreline change indicating erosion is
found from DNR Monument numbers 25 to 34 for the July, 1997 and February, 1998
data. This area lies directly shoreward of the borrow area shown in Figure 3. The
behavior of the shoreline directly leeward of the borrow area is seen be the opposite of
the Grand Isle, LA response where shoreline advancement occurred.
Volume changes determined from profiles in the project area did not show large
negative values near the southern end of the project. The difference between the
shoreline and volume changes at the southern end of the project implies that the
constructed profiles may have been steeper near the southern end of the project as
compared to those near the northern end (Wang and Dean, 2001).
* The "DNR Monuments" are permanent markers spaced at approximately 300 m along the Florida sandy
beaches for surveying purposes
Passage Yp.'r In' er
It '. .. ,R.I O
12
R14
R15
P.18
R17
wR20
R21
23
24
&* *ft 25
1 28
29
*T30
R32
S R33A
34
l R35
.36
S R37
R 38
"R39
40
*R41
Longboat Pass
Figure 3: Bathymetry off Anna Maria Key, FL showing location of borrow pit
following beach nourishment project (modified from Dean et al., 1999).
The proximity of the borrow area to the shoreline is one possible contribution to
the local erosion. Although the reason for the increased erosion in this area is not clear, it
is interesting that the anomalous shoreline recession did not occur until the passage of
Hurricanes Erin and Opal in August and October 1995, respectively. Hurricane Opal was
a category 4 hurricane with sustained winds of 67 m/s when it passed 600 km west of
Manatee County (Liotta, 1999). A reported storm surge of 0.3 to 1.0 m, combined with
the increased wind and wave action resulted in overtopping of the beach berm, flooding
of the back area of the project and transport of sediment to the back beach or offshore
(Liotta, 1999). The average shoreline retreat for the project area was approximately 9.1
to 15.2 m, based on observations (Liotta, 1999).
........ PreProject (Dec. 1992)
o PostProject
February 1995
February 1999
*1 d
*e Y
457.2 609.6 762.0 914.4
from Monument (m)
Figure 4: Beach profile through borrow area at R26 in Anna Maria Key, FL
(modified from Wang and Dean, 2001).
DNR Morun rt No
Figure 5: Shoreline position for Anna Maria Key Project for different periods relative
to August, 1993 (modified from Dean et al. 1999).
3.05 .
3.05 F
6.10 .......
9.15E
0 152.4 304.8
Distance
I" .i ..... .... .... '
...... .... .... .
3.1.3. Martin County, Florida (1996)
The Hutchinson Island beach nourishment project in Martin County, Florida was
constructed in 1996 with the placement of approximately 1.lx106 m3 of sediment along
6.4 km of shoreline, between DNR Monuments R1 and R25 (Sumerell, 2000). The
borrow area for this project was a shoal rising 4.9 m above the adjacent bed and lying 910
m offshore in 12.8 m of water (Sumerell, 2000). Figure 6 shows the borrow area location
offshore of the southern end of the project area. An average of 3 m of sediment was
dredged from the central portion of the shoal.
Figure 6: Project area for Martin County beach nourishment project (Applied
Technology and Management, 1998).
The 3year and 4year postnourishment shoreline surveys show reasonable
agreement with modeling conducted for the project, except at the southern end, near the
borrow area (Sumerell, 2002). Figure 7 shows the predicted shoreline and the survey
data for the 4year shoreline change. This case differs from the previous two as the
borrow area did not create a pit, but reduced the height of an offshore shoal. By lowering
the height of the shoal the shoreline leeward of the borrow area was exposed to more
wave action, which is the opposite of the sheltering (through reflection) effect of an
offshore pit. The borrow area, with its large extent and proximity to the project, is a
possible reason for the higher than expected erosion at the southern end of the project.
30.5
E Predicted
15.2
Survey Data (December 99)
15.2
45.7
Figure 7: Fouryear shoreline change for Martin County beach nourishment project:
predicted versus survey data (modified from Sumerell, 2000).
3.2. Field Experiments
Field studies have been conducted to examine the effects of offshore dredging on
the coastal environment. The purposes of the these studies have varied and include the
tendency of a dredged pit to induce sediment flows into it from the nearshore, the
interception of sediment transport, and wave transformation effects of a newly dredged
pit on the shoreline.
3.2.1. Price et al. (1978)
Price et al. (1978) investigated the effect of offshore dredging on the coastline of
England. The tendency of a dredge pit to cause a drawdown of sediment and to prevent
the onshore movement of sediment was investigated. The study by Inman and Rusnak
(1956) on the onshoreoffshore interchange of sand off La Jolla, California was cited.
This threeyear study found vertical bed elevation changes of only +/ 0.03 m at depths
greater than 9 m. Based on the consideration that the wave conditions off the southern
coast of England would be less energetic than off La Jolla, California, Price et al. (1978)
concluded that beach drawdown at a depth greater than 10 m would not occur.
A radioactive tracer experiment off Worthing, on the south coast of England was
performed to investigate the mobility of sediment at depths of 9, 12, 15, and 18 m. The
20month study found that at the 9 and 12 m contours there was a slight onshore
movement of sediment and it was concluded that the movement of sediment beyond a
depth contour of 18 m on the south coast of England would be negligible. Therefore, at
these locations and in instances when the onshore movement of sediment seaward of the
dredge area is a concern, dredging in water beyond 18 m depth below low water level
was considered acceptable (Price et al., 1978).
A numerical model of the shoreline change due to wave refraction over dredged
holes was also employed in the study, the details of which will be examined later in
Section 5.2.1. The model found that minimal wave refraction occurred for pits in depths
greater than 14 m for wave conditions typical off the coast of England.
3.2.2. Kojima et al. (1986)
The impact of dredging on the coastline of Japan was studied by Kojima et al.
(1986). The wave climate as well as human activities (dredging, construction of
structures) for areas with significant beach erosion and/or accretion were studied, in an
attempt to determine a link between offshore dredging and beach erosion. The study area
was located offshore of the northern part of Kyushu Island. The wave climate study
correlated yearly fluctuations in the beach erosion with the occurrence of both storm
winds and severe waves and found that years with high frequencies of storm winds were
likely to have high erosion rates. A second study component compared annual variations
in offshore dredging with annual beach erosion rates and found strong correlation at some
locations between erosion and the initiation of dredging although no consistent
correlation was identified.
Hydrographic surveys documented profile changes of dredged holes over a four
year period. At depths less than 30 m, significant infilling of the holes was found, mainly
from the shoreward side, indicating a possible interruption in the longshore and offshore
sediment transport. This active zone extends to a much larger depth than found by Price
et al. (1978) and by Inman and Rusnak (1956). The explanation by Kojima et al. is that
although the active onshore/offshore region does not extend to 30 m, sediment from the
ambient bed will fill the pit causing a change in the supply to the upper portion of the
beach and an increase in the beach slope. Changes in the beach profiles at depths of 35
and 40 m were small, and the holes were not filled significantly.
Another component of the study involved tracers and seabed level measurements
to determine the depths at which sediment movement ceases. Underwater photographs
and seabed elevation changes at fixed rods were taken at 5 m depth intervals over a
period of 3 months during the winter season for two sites. The results demonstrated that
sediment movement at depths up to 35 m could be significant. This depth was found to
be slightly less than the average depth (maximum 49 m, minimum 20 m) for five
proposed depth of closure equations using wave inputs with the highest energy
(H = 4.58 m, T = 9.20 s) for the 3month study period.
3.3. Laboratory Experiments
3.3.1. Horikawa et al. (1977)
Laboratory studies have been carried out to quantify wave field and nearshore
modifications due to the presence of offshore pits. Horikawa et al. (1977) performed
wave basin tests with a model of fixed offshore bathymetry and uniform depth containing
a rectangular pit of uniform depth and a nearshore region composed of moveable
lightweight sediments. The experimental arrangement is shown in Figure 8. The
incident wave period and height were 0.41 s and 1.3 cm, respectively. With the pit
covered, waves were run for 5.5 hours to obtain an equilibrium planform followed by
wave exposure for three hours with the pit present. Shoreline measurements were
conducted at 1 hour intervals to determine the pit induced changes. The results of the
experiment are presented in Figure 9. Almost all of the shoreline changes with the pit
present occurred in the first two hours. At the still water level, a salient formed
shoreward of the pit, flanked by two areas of erosion that mostly extend to the sidewalls
of the experiment; however, the depth contour at a water depth, h = 0.85 cm, also shown
in Figure 9, shows only a slightly seaward displacement at the pit centerline.
3.3.2. Williams (2002)
Williams (2002) performed wave basin experiments similar to those of Horikawa
et al. (1977). The experimental setup of a fixed bed model containing a pit with a
moveable sand shoreline was constructed for similar trials by Bender (2001) and was a
larger scale version of the Horikawa et al. (1977) arrangement. The Williams
experimental procedure consisted of shoreline, bathymetric, and profile measurements
after specified time intervals that comprised a complete experiment. Figure 10 shows the
experiment progression sequence that was used. For analysis, the shoreline and volume
measurements were made relative to the last measurements of the previous 6hour phase.
The conditions for the experiments were 6 cm waves with 1.35 s period and a depth of 15
cm in the constant depth region surrounding the pit. The pit was 80 cm long in the cross
shore direction, 60 cm in the longshore direction and 12 cm deep relative to the adjacent
bottom.
100 0
F "!IXED BED
400 300 200 100 0
Offshore distance
Figure 8: Setup for laboratory experiment (Horikawa et al., 1977).
125
o gC
4,T
u
0
Q I
0 C0 40 60 0a 100 10Z
Longshore distance, X (cm)
Figure 9: Results from laboratory experiment showing plan shape after two hours
(Horikawa et al., 1977).
Previous Test
Phases
(open pit)
I I I
6.0 7,5 9.0
Complete Experiment
Current Control
Phases
(covered pit) Current Test
Phases
(open pit)
I I I I I
I I I
0.0 1.5 3.0
(12.0)
I I5
6.0 7.5 9.0
Next Control
Phases
(covered pit)
I I I
12.0 1.5 3.0
(0,0)
Time Step (Hours)
Figure 10: Experiment sequence timeline for Williams laboratory experiments
(Williams, 2002)
Shoreline and volume change results were obtained for three experiments. Figure
11 shows the volume change per unit length versus longshore distance for results with pit
covered (control phase) and uncovered (test phase). The dashed line represents volume
changes for a covered pit relative to the time step zero which concluded 6 hours of waves
with the pit uncovered. The solid line shows the change with the pit uncovered relative
to time step six when six hours of wave exposure with the pit covered ended. The
volume change results show the model beach landward of the pit lost volume at almost
every survey location during the period with the pit covered and experienced a gain in
volume with the pit uncovered.
150
0 120 10080 60 40 20= 20 40 60 ,8 100 120 1,
sh 00Po n
Longshore Position (cm)
I "Oto6 Hours 6 to 12 Hours
(pit covered)
(pit present)
Figure 11: Volume change per unit length for first experiment (Williams, 2002).
E
c
0. C)
p)S
0C
lc1
i5
Em
a
The net volume change for the first complete experiment (control and test phase) was
approximately 2500 cm3. Different net volume changes were found for the three
experiments. However, similar volume change per unit length results were found in all
three experiments indicating a positive volumetric relationship between the presence of
the pit and the landward beach.
The shoreline change results showed shoreline retreat, relative to Time Step 0.0,
in the lee of the borrow pit during the control phase (pit covered) for all three
experiments with the greatest retreat at or near the centerline of the borrow pit. All three
experiments showed shoreline advancement in the lee of the borrow pit with the pit
uncovered (test phase). With the magnitude of the largest advancement being almost
equal to the largest retreat in each experiment, it was concluded that, under the conditions
tested, the presence of the borrow pit resulted in shoreline advancement for the area
shoreward of the borrow pit (Williams, 2002).
An evenodd analysis was applied to the shoreline and volume change results in
an attempt to isolate the effect of the borrow pit. The even function was assumed to
represent the changes due solely to the presence of the borrow pit. The even components
were adjusted to obtain equal positive and negative areas, which were not obtained using
the laboratory data. For each experiment, the shifted even results shoreward of the pit
showed positive values during the test phase for both the shoreline and volume changes
with negative values during the control phase. The shifted even component of shoreline
change for the first experiment is shown in Figure 12. These results further verify the
earlier findings concerning the effect of the pit.
4. Wave Transformation
4.1. Analytic Methods
There is a long history of the application of analytic methods to determine wave
field modifications by bathymetric changes. Early research centered on the effect on
normally incident long waves of an infinite step, trench or shoal of uniform depth in an
otherwise uniform depth domain. More complex models were later developed to remove
the long wave restriction, add oblique incident waves and allow for the presence of a
current. More recently, many different techniques have been developed to obtain
solutions for domains containing pits or shoals of finite extent. Some of these models
focused solely on the wave field modifications, while others of varying complexity
examined both the wave field modifications and the resulting shoreline impact.
0
11
5
c E
c 
co
v,
Longshore Position (cm)
0 to 6 Hours 6 to 12 Hours
(pit covered) (pit present)
Figure 12: Shifted even component of shoreline change for first experiment
(Williams, 2002).
4.1.1. 1D Methods
By matching surface displacement and mass flux normal to the change in
bathymetry Lamb (1932) was one of the first to develop a long wave approximation for
the reflection and transmission of a normally incident wave at a finite step.
Bartholomeauz (1958) performed a more thorough analysis of the finite depth step
problem and found that the Lamb solution gave correct results for the reflection and
transmission coefficients for lowest order (kh) where k is the wave number and h is the
water depth prior to the step. Sretenskii (1950) investigated oblique waves over a step
between finite and infinite water depths assuming the wavelength to be large compared to
the finite depth. An extensive survey of early theoretical work on surface waves
including obstacle problems is found in Wehausen and Laitone (1960).
Jolas (1960) studied the reflection and transmission of water waves of arbitrary
relative depth over a long submerged rectangular parallelepiped (sill) and performed an
15
10
10 120 100 60 4S^ 20 20 60 d 100 120 1
4 0
experiment to document the wave transformation. To solve the case of normal wave
incidence and arbitrary relative depth over a sill or a fixed obstacle at the surface Takano
(1960) used an eigenfunction expansion of the velocity potentials in each constant depth
region and matched them at the region boundaries. The set of linear integral equations
was solved for a truncated series. A laboratory experiment was also conducted in this
study.
Dean (1964) investigated long wave modification by linear transitions. The linear
transitions included both horizontal and vertical changes. The formulation allowed for
many domains including a step, either up or down, and converging or diverging linear
transitions with a sloped wall. A proposed solution was defined with planewaves of
unknown amplitude and phase for the incident and reflected waves with the transmitted
wave specified. Wave forms, both transmitted and reflected, were represented by Bessel
functions in the region of linear variation in depth and/or width. The unknown
coefficients were obtained through matching the values and gradients of the water
surfaces at the ends of the transitions. Analytic expressions were found for the reflection
and transmission coefficients. The results indicate that the reflection and transmission
coefficients depend on the relative depth and/or width and a dimensionless parameter
containing the transition slope, the wavelength and the depth or width (Figure 13). In
Figure 13 the parameter Z, = 4 for the case of linearly varying depth and
LI S,
Z, for linearly varying breadth where I indicates the region upwave of the
LSH
transition, Sv is equal to the depth gradient, and S is equal to onehalf the breadth
gradient. These solutions were shown to converge to those of Lamb (1932) for the case
of an abrupt transition (ZI=0).
Newman (1965a) studied wave transformation due to normally incident waves on
a single step between regions of finite and infinite water depth with an integralequation
approach. This problem was also examined by Miles (1967) who developed a plane
wave solution for unrestricted kh values using a variational approach (Schwinger &
Saxon 1968), which for this case essentially solves a single equation instead of a series of
equations (up to 80 in Newman's solution) as in the integral equation approach. The
difference between the results for the two solution methods was within 5 percent for all
kh values (Miles, 1967).
Value 'of h/ht i, b I /bT
coefficient, Kr, and the transmission coefficient, Kt, versus ho where K is the wave
I I I
L L K I/ I / I I
l I I
I_ . , ... 
001 0.02 0.05 0, 0.2 0.5 1 2 5 10 20 50 10 200 500 1000
Value of h1/h111 b21/b2111
Figure 13: Reflection and transmission coefficients for linearly varying depth [hi/h3]
and linearly varying breadth [b12/b32] (modified from Dean, 1964).
Newman (1965b) examined the propagation of water waves past long obstacles.
The problem was solved by constructing a domain with two steps placed "back to back"
and applying the solutions of Newman (1965a). Complete transmission was found for
certain water depth and pit length combinations. Figure 14 shows the reflection
coefficient, K,, and the transmission coefficient, Kt, versus Koho where Koo is the wave
number in the infinitely deep portion before the obstacle and ho is the depth over the
obstacle. The experimental results of Takano (1960) are included for comparison. It is
evident that the Takano experimental data included energy losses.
 Numerical results of Newman (1965b)
o Takano (1960) experimental results
Kr
0 02 04 06 08 10 1 I2 14 16 18 24
Figure 14: Approximate reflection and transmission coefficients for the rectangular
parallelepiped of length 8.86ho in infinitely deep water (Newman, 1965b).
The variational approach was applied by Mei and Black (1969) to investigate the
scattering of surface waves by rectangular obstacles. For a submerged obstacle, complete
transmission was found for certain kho values where ho is the depth over the obstacle. A
comparison of the results of Mei and Black (1969) and those of Newman (1965b) is
shown in Figure 15, which presents the reflection coefficient versus kho for a submerged
obstacle. Data from the Jolas (1960) experiment are also included on the plot and
compared to the results of Mei and Black (1969) for a specific / ho, where is the half
length of the obstacle.
[Mei and Black (1969)], o [data from Jolas (1960) experiment]
S(e h =4.43, h/ho= 2.78)
 [Mei and Black (1969)],  [Newman (1965b)]
I ( / ho =4.43, h/ho = infinity)
1Ri I
V, i
0 02 04 V 6 0 1.0 1 14 16
kho
Figure 15: Reflection cofficient for a submerged obstacle (Mei and Black, 1969).
Black and Mei (1970) applied the variational approach to examine the radiation
caused by oscillating bodies and the disturbance caused by an object in a wave field.
Two domains were used for both submerged and semiimmersed (surface) bodies: the
first domain was in cartesian coordinates, with one vertical and one horizontal dimension,
for horizontal cylinders of rectangular cross section and the second domain was in
cylindrical coordinates, for vertical cylinders of circular section. The second domain
allowed for objects with two horizontal dimensions to be studied for the first time (see
Section 4.1.2.). Black et al. (1971) used the variational formulation to study the radiation
due to the oscillation of small bodies and the scattering induced by fixed bodies. Black et
al. demonstrated the scattering caused by a fixed object in a single figure; see Black and
Mei (1970) for further results.
Lassiter (1972) used complementary variational integrals to solve the problem of
normally incident waves on an infinite trench where the depth on the two sides of the
trench may be different (the asymmetric case). The symmetric infinite trench problem
was studied by Lee and Ayer (1981), who employed a transform method. The fluid
domain was divided into two regions, one an infinite uniform depth domain and the other
a rectangular region representing the trench below the uniform seabed level. The
transmission coefficient for the trench is shown in Figure 16 with the theoretical results
plotted along with data from a laboratory experiment conducted as part of the study.
Results from a boundary integral method used to compare with the theoretical results are
also plotted.
090
S085 [Numerical Solution]
S \ [Experimental Results]
o= \ x [Boundary Integral Method]
0 75 
070 ____ j i
0 005 0 10 0 15 020 025
Depth to wavelength rati (h,')
Figure 16: Transmission coefficient as a function of relative wavelength (h=10.1 cm,
d=67.3 cm, trench width =161.6 cm) (modified from Lee and Ayer, 1981).
The results show six of an infinite number of relative wavelengths where complete
transmission (Kt=1) will occur, a result that had been found in prior studies (Newman
(1965b), Mei and Black (1969)). The laboratory data show the general trend of the
theoretical results, with some variation due to energy losses and reflections from the tank
walls and ends.
Lee et al. (1980) proposed a boundary integral method for the propagation of
waves over a prismatic trench of arbitrary shape, which was used for comparison to
selected results in Lee and Ayer (1981). The solution was found by matching the
unknown normal derivative of the potential at the boundary of the two regions. A
comparison to previous results for trenches with vertical sidewalls was conducted with
good agreement. A case with "irregular" bathymetry was demonstrated in a plot of the
transmission coefficient for a trapezoidal trench (Figure 17).
S*1.00f .*.. *.*.*. ...
0 .* .
,
44
0. o.os 0. 1S 0.20 C.l
Depth to Wave Len:gth Ratio th/.k
Figure 17: Transmission coefficient as a function of relative wavelength for
trapezoidal trench; setup shown in inset diagram (modified from Lee et al., 1980).
Miles (1982) solved for the diffraction by an infinite trench for obliquely incident
long waves. The solution method for normally incident waves used a procedure
developed by Kreisel (1949) that conformally mapped a domain containing certain
obstacles of finite dimensions into a rectangular strip. Kreisel (1949) presented this
method without derivation and with no consideration of the phase. To add the capability
of solving for obliquely incident waves, Miles used the variational formulation of Mei
and Black (1969).
The problem of obliquely incident waves over a symmetric trench was solved by
Kirby and Dalrymple (1983a) using a modified form of Takano's (1960) method. Figure
18 compares the reflection coefficient for the numerical solution for normally incident
waves and the results of Lassiter (1972), along with results from a boundary integral
method used to provide verification. Differences in the results of Kirby and Dalrymple
and those of Lassiter are evident. Lee and Ayer (1981, see their Figure 2) also
demonstrated differences in their results and those of Lassiter (1972). The effect of
oblique incidence is shown in Figure 19 where the reflection and transmission
coefficients for two angles of incidence are plotted.
a "'it
('*W>* ^H' 161.6 cm *H*
coefficients for two angles of incidence are plotted.
0.5 [Kirby and Dalrymple (1983a)]
, * [Lassiter (1972, Fig. 7)]
S'. [Boundary Integral Method]
0.2 '\
. .. iJ' I I
0 U. 1 0 0 3 5' 0.5 0 ( ,
Kh,
Figure 18: Reflection coefficient for asymmetric trench and normally incident waves
as a function of Khl: h2/hl=2, h3/h1=0.5, L/h1=5; L = trench width (Kirby and
Dalrymple, 1983a).
K 0.  [Numerical solution, 01 = 0 deg]
[Numerical solution, 01 = 45 deg]
02 04 06 0O 1.0 1 1 4
k1h,
Figure 19: Transmission coefficient for symmetric trench, two angles of incidence:
L/hl=10, h2/h1=2; L = trench width (modified from Kirby and Dalrymple, 1983a).
This study also investigated the planewave approximation and the longwave
limit, which allowed for comparison to Miles (1982). Figure 20 shows transmission
coefficients with the results of the numerical solution, the long wave solution, and values
from the Miles (1982) solution, which is only valid for small kh values in each region.
For the first case, with a small relative trench width, the numerical results from Kirby and
Dalrymple compare well with the results using the Miles (1982) method and the plane
wave solution is seen to deviate from these. For the case of a relative trench length equal
to eight, the numerical results differ from the planewave solution, which diverge from
the values using Miles (1982) for this case where the assumptions are violated. The
difference in scales between the two plots is noted. An extension of this study is found in
Kirby and Dalrymple (1987) where the effects of currents flowing along the trench are
included. The presence of an ambient current was found to significantly alter the
reflection and transmission coefficients for waves over a trench compared to the no
current case. Adverse currents and following currents made a trench less reflective and
more reflective, respectively (Kirby and Dalrymple, 1987).
I .o000
0.9981
0.994'
l.00o
 [Long Wave Solution]
S[Numerical S
 [Miles (1982)
N.
S"NI
solution]
Solution]
0.941
h/hi
Figure 20: Transmission coefficient as a function of relative trench depth; normal
incidence: klhl= 0.2: (a) L/h1=2; (b) L/h1=8, L = trench width. (Kirby and Dalrymple,
1983a).
4.1.2. 2D Methods
Extending the infinite trench and step solutions (one horizontal dimension) to a
twodimensional domain is a natural progression allowing for the more realistic case of
wave transformation by a finite object or depth anomaly to be studied. Changes in
bathymetry can cause changes in wave height and direction through the four wave
transformation processes noted earlier. Some of the twodimensional models study only
the wave transformation, while others use the modified wave field to determine the
"
r
0.96
impact of a pit or shoal on the shoreline. Several models use only a few equations or
matching conditions on the boundary of the pit or shoal to determine the wave field and
in some cases the impact on the shoreline in a simple domain containing a pit or shoal.
Other, much more complex and complete models and program packages have been
developed to numerically solve for the wave field over a complex bathymetry, which may
contain pits and/or shoals. Both types of models can provide insight into the effect of a
pit or shoal on the local wave field and the resulting impact on the shoreline.
The first study on the wave transformation in a twodimensional domain was
Black and Mei (1970), which solved for the radially symmetric case of a submerged or
floating circular cylinder in cylindrical coordinates. A series of Bessel functions was
used for the incident and reflected waves, as well as for the solution over the shoal with
modified Bessel functions representing the evanescent modes. As mentioned previously
in the 1D section, a variational approach was used and both the radiation by oscillating
bodies and the disturbance caused by a fixed body were studied. The focus of the fixed
body component of the study was the total scattering cross section, Q, which is equal to
the width between two wave rays within which the normally incident wave energy flux
would be equal to that scattered by the obstacle and the differential scattering cross
section, which shows the angular distribution of the scattered energy (Black and Mei,
1970). Figure 21 shows the total scattering cross section for a circular cylinder at the
seabed for three ratios of cylinder radius (a) to depth over the cylinder (h).
Williams (1990) developed a numerical solution for the modification of long
waves by a rectangular pit using Green's second identity and appropriate Green's
functions in each region that comprise the domain. This formulation accounts for the
diffraction, refraction and reflection caused by the pit. The domain for this method
consists of a uniform depth region containing a rectangular pit of uniform depth with
vertical sides. The solution requires discretizing the pit boundary into a finite number of
points at which the velocity potential and the derivative of the velocity potential normal
to the boundary must be determined. Applying matching conditions for the pressure and
mass flux across the boundary results in a system of equations amenable to matrix
solution techniques. Knowledge of the potential and derivative of the potential at each
point on the pit boundary allows determination of the velocity potential solution
anywhere in the fluid domain. The effect of a pit on the wave field is shown in a contour
plot of the relative amplitude in Figure 22. A partial standing wave pattern of increased
and decreased relative amplitude is seen seaward of the pit with a shadow zone of
decreased wave amplitude landward of the pit flanked by two areas of increased relative
amplitude.
20
16 a/h=3
12
08 a/h=2
04 a/h=l
0 1 2 3 4 5 6
ka
Figure 21: Total scattering cross section of vertical circular cylinder on bottom
(modified from Black et al., 1971).
McDougal et al. (1996) applied the method of Williams (1990) to the case of a
domain with multiple pits. The first part of the study reinvestigated the influence of a
single pit on the wave field for various pit geometries. A comparison of the wave field in
the presence of a pit versus a surface piercing structure is presented in Figures 23 and 24,
which present contour plots of the transformation coefficient, K, (equal to relative
amplitude) that contain the characteristics discussed in the last paragraph. For this case
with the pit depth equal to 3 times the water depth a greater sheltering effect is found (K
= 0.4) landward of the pit than for the case of the full depth breakwater.
TT 1 T7 T r 7r TTI1FC7 
iI
... ............
D.. .....
X7
. .. .. .
.:: :: : ... .. ...
__ *11
1 107
I.O,.l
1.021
0 970C
0 9357
0 8929
Figure 22: Contour plot of relative amplitude in and around pit for normal incidence;
kl/d = n/10, k2/h=n/1042, h/d=0.5, b/a =1, a/d=2, a = crossshore pit length, b =
longshore pit length, h = water depth outside pit, d = depth inside pit, L2 =
wavelength outside pit. (Williams, 1990).
An analysis of the effect of various pit characteristics on the minimum value of K
found in the domain was also performed. The dimensionless pit width, a/L, (a = cross
shore dimension, L = wave length outside pit) was found to increase the distance to the
region where K < 0.5 behind the pit and the value of K was found to decrease and then
become approximately constant as a/L increases. The minimum values of the
transformation coefficient for a wide pit are much lower than those values found in Lee
and Ayer (1981) and Kirby and Dalrymple (1983), which may be explained by the
refraction divergence that occurs behind the pit in the 2D case (McDougal et al., 1996).
Figure 23: Contour plot of diffraction coefficient in and around pit for normal
incidence; a/L=l, b/L=0.5, d/h =3, kh=0.167 (McDougal et al., 1996).
Figure 24: Contour plot of diffraction coefficient around surfacepiercing breakwater
for normal incidence; a/L=l, b/L=0.5, kh=0.167 (McDougal et al., 1996).
The effect of the dimensionless pit length, b/L, indicates that K decreases as b/L increases
to 1 with a change in the trend, and an increase in K, from b/L = 0.55 to b/L = 0.65.
Increasing the dimensionless pit depth, d/L, was found to decrease the minimum value of
K with a decreasing rate. The incident wave angle was not found to significantly alter the
magnitude or the location of the minimum K value, although the width of the shadow
zone changes with incident angle. For the case of multiple pits, it was found that
placement of one pit in the shadow zone of a more seaward pit was most effective in
reducing the wave height. However, adding a third pit did not produce significant wave
height reduction as compared to the twopit results.
Williams and Vazquez (1991) removed the long wave restriction of Williams
(1990) and applied the Green's function solution method outside of the pit. This solution
was matched to a Fourier expansion solution inside the pit with matching conditions at
the pit boundary. Once again the pit boundary must be discretized into a finite number of
points and a matrix solution for the resulting series of equations was used. Removing the
shallow water restriction allowed for many new cases to be studied, as the wave
conditions approach deep water, the influence of the pit diminishes. A plot of the
minimum and maximum relative amplitude versus the dimensionless pit length (the wave
number outside of the pit times the crossshore pit dimension, koa) is shown in Figure 25.
1 "4[I
S. . .. B
ka
Figure 25: Maximum and minimum relative amplitudes for different koa, for normal
incidence, a/b=6, a/d=7r, and d/h=2. (modified from Williams and Vasquez, 1991).
The maximum and minimum relative amplitudes are seen to occur near koa = 2n or when
L = a and then approach unity as the dimensionless pit length increases. The reason that
the extreme values do not occur exactly at koa = 27r are explained by Williams and
Vazquez (1991) as due to diffraction effects near the pit modifying the wave
characteristics.
4.2. Numerical Methods
The previous twodimensional solutions, while accounting for most of the wave
transformation processes caused by a pit, are simple in their representation of the
bathymetry and their neglect of many waverelated processes including energy
dissipation. Berkhoff (1972) developed a formulation for the 3dimensional propagation
of waves over an arbitrary bottom in a vertically integrated form that reduced the problem
to twodimensions. This solution is known as the mild slope equation and different forms
of the solution have been developed into parabolic (Radder, 1979), hyperbolic, and
elliptic (Berkhoff et al., 1982) models of wave propagation, which vary in their
approximations and solution techniques. Numerical methods allow solution for wave
propagation over an arbitrary bathymetry. Some examples of the parabolic and elliptic
models are RCPWAVE (Ebersole et al., 1986), REF/DIF1 (Kirby and Dalrymple, 1994),
and MIKE 21's EMS Module (Danish Hydraulics Institute, 1998). Other models such as
SWAN (Holthuijsen et al., 2000) and STWAVE (Smith et al., 2001) model wave
transformation in the nearshore zone using the waveaction balance equation. These
models provide the capability to model wave transformation over complicated
bathymetries and may include processes such as bottom friction, nonlinear interaction,
breaking, wavecurrent interaction, windwave growth, and white capping to better
simulate the nearshore zone. An extensive review of any of the models is beyond the
scope of this paper; however, a brief outline of the capabilities of some of the models is
presented in Table 1.
Maa et al. (2000) provides a comparison of six numerical models. Two parabolic
models are examined: RCPWAVE and REF/DIF1. RCPWAVE employs a parabolic
approximation of the elliptic mild slope equation and assumes irrotationality of the wave
phase gradient. REF/DIF1 extends the mild slope equation by including nonlinearity
and wavecurrent interaction (Kirby and Dalrymple, 1983b; Kirby, 1986). Of the four
other models included, two are defined by Maa et al. (2000) as based on the transient
mild slope equation (Copeland, 1985; Madsen and Larson, 1987) and two are classified
as elliptic mild slope equation (Berkhoff et al., 1982) models.
RCPWAVE REF/DIF1 Mike 21 (EMS) STWAVE SWAN
3rd Generation
Elliptic Mild Slope
Solution Parabolic Mild Parabolic Mild Equation Conservation of Conservation of
Method Slope Equation Slope Equation (Berkhoff et al. Wave Action Wave Action
(__1982)
Phase Averaged Resolved Resolved Averaged Averaged
Spectral No NoYes Yes
REF/DIFS) (Use NSW unit)
Shoaling Yes Yes Yes Yes Yes
Refraction Yes Yes Yes Yes Yes
Diffraction Yes Yes (Wide Yes No No
(SmallAngle) Angle) (Total) (Smoothing)
Yes Yes Yes
Reflection No Yes No
(Forward only) (Total) (Specular)
Stable Energy Stable Energy Bore Model: Depth limited: Bore Model:
Breaking Flux: Dally et al. Flux: Dally et al. Battjes & Janssen Miche (1951) Battjes & Janssen
(1985) (1985) (1978) criterion (1978)
Komen et al.
White (1984),
capping No No No Resio (1987) Janssen (1991),
capping
Komen et al.
(1994)
Dalrymple et al. Qu c F n Hasselmann et al.
Quadratic Friction
Bottom (1984) both (1973), Collins
No Law, Dingemans No
Friction laminar and Lw (1972), Madsenet
turbulent BBL ( ) al. (1988)
Currents No Yes No Yes Yes
Cavaleri &
MalanotteRizzoli
Wind No No No Resio (1988) (1981),
Snyder et al.,
1981), Janssen et
al. (1989, 1991)
Availability Commercial Free Commercial Free Free
Table 1: Capabilities of selected nearshore wave models.
The transient mild slope equation models presented are Mike 21's EMS Module and the
PMH Model (Hsu and Wen, in review). The elliptic mild slope equation models use
different solution techniques with the RDE Model (Maa and Hwung, 1997; Maa et al.
1998a) applying a special Gaussian elimination method and the PBCG Model employing
a Preconditioned Biconjugate Gradient method (Maa et al., 1998b).
A table in Maa et al. (2000) provides a comparison of the capabilities of the six
models. A second table summarizes the computation time, memory required and, where
required, the number of iterations for a test case of monochromatic waves over a shoal on
an incline; the Berkhoff et al. (1982) shoal. The parabolic approximation solutions of
REF/DIF and RCPWAVE required significantly less memory (up to 10 times less) and
computation time (up to 70 times less) than the elliptic models, which is expected due to
the solution techniques and approximations contained in the parabolic models. The
required computation times and memory requirements for the transient mild slope
equation models were found to be intermediate to the other two methods.
Wave height and direction were calculated in the test case domain for each model.
The models based on the transient mild slope equation and the elliptic mild slope
equation were found to produce almost equivalent values of the wave height and
direction. The parabolic approximation models were found to have different values, with
RCPWAVE showing different wave heights and directions behind the shoal and
REF/DIF showing good wave height agreement with the other methods, but no change in
the wave direction behind the shoal. Plots of the computed wave heights for the six
models and experimental data along one transect taken perpendicular to the shoreline and
one transect parallel to the shoreline are shown in Figures 26 and 27. Only four results
are plotted because the RDE model, the PMH model and PBCG model produced almost
identical results.
The wave directions found with REF/DIF1 in Maa et al. (2000) were found to be
in error by Grassa and Flores (2001), who demonstrated that a second order parabolic
model, equivalent to REF/DIF1 was able reproduce the wave direction field behind a
shoal such as in the Berkhoff et al. (1982) experiment.
,\ RCPWAVE
Longshore (inm)
Figure 26: Comparison of wave height profiles for selected models along transect
parallel to shore located 9 m shoreward of shoal apex[*=experimental data] (Maa et
al., 2000).
2 ,
RDE
S10 a2
Cross Shore ()
Figure 27: Comparison of wave height profiles for selected models along transect
perpendicular to shore and through shoal apex [*=experimental data] (Maa et al.,
2000).
Application of numerical models to the problem of potential impact on the
shoreline caused by changes to the offshore bathymetry was conducted by Maa and
Hobbs (1998) and Maa et al. (2001). In Maa and Hobbs (1998) the impact on the coast
due to the dredging of an offshore shoal near Sandbridge, Virginia was investigated using
RCPWAVE. National Data Buoy Center (NDBC) data from an offshore station and
bathymetric data for the area were used to examine several cases with different wave
events and directions. The resulting wave heights, directions, and sediment transport at
the shoreline were compared. The sediment transport was calculated using the
formulation of Gourlay (1982), which contains two terms, one driven by the breaking
wave angle and one driven by the gradient in the breaking wave height in the longshore
direction. Section 5.1. provides a more detailed examination of the longshore transport
equation with two terms. The study found that the proposed dredging would have little
impact on the shoreline for the cases investigated.
Later, Maa et al. (2001) revisited the problem of dredging at the Sandbridge Shoal
by examining the impact on the shoreline caused by three different dredging
configurations. RCPWAVE was used to model the wave transformation over the shoal
and in the nearshore zone. The focus was on the breaking wave height; wave direction at
breaking was not considered. The changes in the breaking wave height modulation
(BHM) along the shore after three dredging phases were compared to the results found
for the original bathymetry and favorable or unfavorable assessments were provided for
ensuing impact on the shoreline. The study concluded that there could be significant
differences in the wave conditions, revealed by variations in the BHM along the shoreline
depending on the location and extent of the offshore dredging.
Regions outside the inner surf zone have also been studied through application
of nearshore wave models. Jachec and Bosma (2001) used the numerical model
REF/DIFS (a spectral version of REF/DIF1) to study borrow pit recovery time for
seven borrow areas located on the inner continental shelf off New Jersey. The input
wave conditions were obtained from Wave Information Study (WIS) data with
nearshore bathymetry for the existing conditions and also different dredging scenarios.
Changes in the waveinduced bottom velocity were obtained from the wave height and
direction changes determined by REF/DIFS. The waveinduced bottom velocities
were coupled with ambient nearbottom currents to determine the sediment transport
and then recovery times of the borrow areas. The recovery times from the numerical
modeling were the same order of magnitude as recovery times estimated from two
independent data sets of seafloor change rates offshore of New Jersey.
5. Shoreline Response
5.1. Longshore Transport Considerations
The previous discussion on one and twodimensional models focused first on
simple and complex methods of determining the wave transformation caused by changes
in the offshore bathymetry and then applications that determined the changes to the wave
height, direction and even longshore transport at the shoreline. However, none of the
applications were intended to determine the change in shoreline planforms due to an
anomaly or a change in the offshore bathymetry. With wave heights and directions
specified along the shoreline, sediment transport can be calculated and, based on the
gradients in longshore transport, the changes in shoreline position can be determined.
The longshore transport can be driven by two terms as was discussed previously
in the review of Maa and Hobbs (1998). In most situations where the offshore
bathymetry is somewhat uniform, the magnitude and direction of the longshore transport
will depend mostly on the wave height and angle at breaking as the longshore gradient in
the breaking wave height will be small. In areas with irregular bathymetry or in the
presence of structures, the transformation of the wave field can lead to areas of wave
focusing and defocusing resulting in considerable longshore gradients in the wave height.
Longshore transport equations containing a transport term driven by the breaking wave
angle and another driven by the longshore gradient in the wave height can be found in
Bakker (1971), Ozasa and Brampton (1980), who cite the formulation of Bakker (1971),
Gourlay (1982), Kraus and Harikai(1983), and Kraus (1983). While the value of the
coefficient for the transport term driven by the gradient in the wave height is not well
established, the potential contribution of this term is significant. It is shown later that
under steady conditions the diffusive nature of the angledriven transport term is required
to modify the wave height gradient transport term in order to generate an equilibrium
planform when the two terms are both active.
5.2. Refraction Models
5.2.1. Motyka and Willis (1974)
Motyka and Willis (1974) were one of the first to apply a numerical model to
predict shoreline changes due to altered offshore bathymetry. The model only included
the effect of refraction caused by offshore pits for idealized sand beaches representative
of those found on the English Channel or North Sea coast of England. A simplified
version of the Abernethy and Gilbert (1975) wave refraction model was used to
determine the wave transformation of uniform deep water waves over the nearshore
bathymetry. The breaking wave height and direction were calculated and used to
determine the sediment transport and combined with the continuity equation to predict
shoreline change. The longshore transport was calculated using the Scripps Equation as
modified by Komar (1969):
0.045
pgH C, sin(2ab) (1)
rs
where Q is the volume rate of longshore transport, ys is the submerged unit weight of the
beach material, p is the density of the fluid, Hb is the breaking wave height, Cg is the
group velocity at breaking, and OLb is the angle of the breaking wave relative to the
shoreline. This form of the Scripps Equation combines the transport and porosity
coefficients into one term; the values used for either parameter was not stated. This
process was repeated to account for shoreline evolution with time.
The model determined that erosion occurs shoreward of a pit, bordered by areas
of accretion. For the wave conditions used, stability was found after an equivalent period
of two years. During the runs, 'storm' waves (short period and large wave height) were
found to cause larger shoreline changes than the 'normal' waves with longer periods and
smaller wave heights, which actually reduced the erosion caused by the storm waves.
Figure 28 shows a comparison of the predicted shorelines for the equivalent of two years
of waves over 1 m and 4 m deep pits with a longshore extent of 880 m and a crossshore
extent of 305 m.
I30. WATER DEPTH m) DISTANCE OFFSHORE (m)
1708 2740
20 1762 3050
H
410
 Pit depth = 1 m
S. Pit depth = 4 m
0 500 dWO 1500 2000 50ooO 3oo000 d0 0 4500
DISTANCE ALONG SHOREm
PLANSHAPE OF BEACH
DUE TO REFRACTION OVER DREOGED HOLE, 2740m OFFSHORE
Figure 28: Calculated beach planform due to refraction after two years of prototype
waves for two pit depths (modified from Motyka and Willis, 1974).
The detailed pit geometries were not specified. The erosion directly shoreward of the pits
is shown in Figure 28 with more erosion occurring for the deeper pit.
5.2.2. Horikawa et al. (1977)
Horikawa et al. (1977) developed a mathematical model for shoreline changes due
to offshore pits. The model applies a refraction program and the following equation for
the longshore sediment transport:
0.77pg
Q 7 g H Cg sin(2ab) (2)
16(p, p)(A) b)
where X is the porosity of the sediment. Equation 2 is identified as the Scripps Equation
in Horikawa et al. (1977); however to match the Scripps Equation and for a
dimensionally correct expression, the g term in the numerator should be removed. A
model by Sasaki (1975) for diffraction behind breakwaters was modified to account for
refraction only. The model computes successive points along the wave ray paths.
Interpolation for the depth and slope is used along the ray path with an iteration
procedure to calculate each successive point. The wave conditions were selected to be
typical of the Eastern Japan coast facing the Pacific Ocean. Several pit dimensions and
pit locations were used with the longshore dimension of the pit from 2 km to 4 km, a
crossshore length of 2 km, pit depth of 3 m and water depths at the pit from 20 m to
50 m.
For the configurations modeled, accretion was found directly shoreward of the pit,
flanked by areas of erosion. The magnitude of the accretion behind the pit and the
erosion in the adjacent areas were found to increase with increasing longshore pit length
and for pits located closer to shore. The shoreline planform for a model after the
equivalent of 2 years of waves is shown in Figure 29 with a salient directly shoreward of
the pit.
Although Horikawa et al. state that good qualitative agreement was found with
Motyka and Willis (1974), the results were the opposite with Horikawa et al. and Motyka
and Willis (1974) predicting accretion and erosion shoreward of a pit, respectively. The
proposed reason for the accretion given in Horikawa et al. was that sand accumulates
behind the pit due to the quiet water caused by the decrease in wave action behind the pit.
However, a model that considers only refraction caused by a pit and only includes a
transport term dependent on the breaking wave angle would have wave rays that diverge
over the pit and cause sand to be transported away from the area behind the pit, resulting
in erosion. The two models used different refraction programs and basically the same
transport equation with Horikawa et al. having a pit that was 2 or 3 times as large, lying
in deeper water and with longer period incoming waves. The refraction grid was 250 m
square in the Horikawa et al. model and 176 m square in Motyka and Willis. Regardless
of the differing results from Motyka and Willis, the mathematical model results of
Horikawa et al. follow the trend of the lab results contained in that study showing
accretion behind a pit (Figure 30); however the aforementioned anomalous prediction of
accretion considering only wave refraction remains.
ce depth 0
E AFTER 2 YEARS I 2m 
4, 2 accretion
S0.5
5 3 1 0 1 2 3 5S
Longshore distance from center of dredged hole (km)
Figure 29: Calculated beach planform due to refraction over dredged hole after two
years of prototype waves (Horikawa et al., 1977).
HHOLE
E
SObserved
a.
2 Prodicted
a
0 ?0 1. EO
Longshore distance from center
of dredged hole (cm )
Figure 30: Comparison of changes in beach plan shape for laboratory experiment and
numerical model after two years of prototype waves (Horikawa et al., 1977).
5.3. Refraction and Diffraction Models
5.3.1. Gravens and Rosati (1994)
Gravens and Rosati (1994) performed a numerical study of the salients and a set
of offshore breakwaters at Grand Isle, Louisiana (Figures 1 and 2). Of particular interest
is the analysis and interpretation of the impact on the wave field and the resulting
influence on the shoreline, of the 'dumbbell' shaped planform borrow area located close
to shore. The report employs two numerical models to determine the change in the
shoreline caused by the presence of the offshore pits: a wave transformation numerical
model (RCPWAVE) and a shoreline change model (GENESIS (Hanson, 1987, Hanson,
1989)) using the wave heights from the wave transformation model. RCPWAVE was
used to calculate the wave heights and directions from the nominal 12.8 m contour to the
nominal 4.3 m contour along the entire length of the island for 3 different input
conditions. Figures 31 and 32 show the wave height transformation coefficients and wave
angles near the pit (centered about alongshore coordinate 130). Significant changes in
the wave height and direction are found near the offshore borrow area. The shadow zone
centered at Cell 130 suggests the presence of one large offshore pit as opposed to the
'dumbbell' shaped borrow pit for the project described in Combe and Soileau (1987).
Nea ore Wave Highl Tranroarmano Comefu
Anlu Bna 5 Pefi.d BI 2 *
1.4 1 
100 110 120 130 140 150 Irl
ALmkhore Card.nm lDI tpn ms li10 fil
Figure 31: Nearshore wave height transformation coefficients near borrow pit from
RCPWAVE study (modified from Gravens and Rosati, 1994).
120 130 140
Alongsihore Coordinat (cell spcint 100 t)
Figure 32: Nearshore wave angles near borrow pit from RCPWAVE study; wave
angles are relative to shore normal and are positive for westerly transport (modified
from Gravens and Rosati, 1994).
The shoreline changes were calculated using a longshore transport equation with
two terms; one driven by the breaking wave angle, and one driven by the longshore
gradient in the breaking wave height. Each of these terms includes a dimensionless
transport coefficient. In order for GENESIS to produce a salient leeward of the borrow
pit, an unrealistically large value for the transport coefficient associated with the gradient
in the breaking wave height (K2 = 2.4) was needed, whereas 0.77 is the normal upper
limit. While a single salient was modeled after applying the large K2 value, the
development of two salients leeward of the borrow pit, as shown in Figures 1 and 2, did
not occur. The nearshore bathymetry data used in the modeling was from surveys taken
in 1990 and 1992. Significant infilling of the borrow pit occurred prior to the surveys in
1990 and 1992; however details of how the pit filled over this time period are not known.
The authors proposed that the salient was formed by the refractive divergence of
the wave field created by the borrow pit that resulted in a region of low energy directly
shoreward of the borrow area and regions of increased energy bordering the area. The
gradient in the wave energy will result in a circulation pattern where sediment suspended
_I __
in the highenergy zone is carried into the low energy zone. For GENESIS to recreate
this circulation pattern K2 must be large enough to allow the second transport term to
dominate over the first transport term.
5.3.2. Tang (2002)
Tang (2002) employed RCPWAVE and a shoreline modeling program to evaluate
the shoreline evolution leeward of an offshore pit. The modeling was only able to
generate embayments in the lee of the offshore pits using accepted values for the
transport coefficients. This indicates that wave reflection and/or dissipation are
important wave transformation processes that must be included when modeling shoreline
evolution in areas with bathymetric anomalies.
5.4. Refraction, Diffraction, and Reflection Models
5.4.1. Bender (2001)
A study by Bender (2001) extended the numerical solution of Williams (1990) for
the transformation of long waves by a pit to determine the energy reflection and shoreline
changes caused by offshore pits and shoals. An analytic solution was also developed for
the radially symmetric case of a pit following the form of Black and Mei (1970). The
processes of wave refraction, wave diffraction, and wave reflection are included in the
model formulations, however wave dissipation is not. Both the numerical and analytic
solutions provide values of the complex velocity potential at any point, which allows
determination of quantities such as velocity and pressure.
The amount of reflected energy was calculated by comparing the energy flux
through a transect perpendicular to the incident wave field extending to the pit center to
the energy flux through the same transect with no pit present (Figure 33). The amount of
energy reflected was found to be significant and dependent on the dimensionless pit
diameter and other parameters. Subsequently a new method has been developed which
allows the reflected energy to be calculated using a farfield approximation with good
agreement between the two methods.
The shoreline changes caused by the pit were calculated using a simple model that
considers continuity principles and the longshore transport equation with values of the
wave height and direction determined along a transect representing the shoreline. A
nearshore slope and no nearshore refraction were assumed. The impact on the shoreline
was modeled by determining the wave heights and directions along an initially straight
shoreline, then calculating the transport and resulting shoreline changes. After updating
the shoreline positions, the transport, resulting shoreline changes, and updated shoreline
positions were recalculated for a set number of iterations after which the wave
transformation was recalculated with the new bathymetry and values of the wave height
and direction were updated at the modified breaker line.
Pit Diameter/Wavelength(inside pit,d)
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
+ (radius = 6 m)
0.24
o o (radius = 12m)
(radius = 25 m)
0.22
x (radius = 30 m)
<> (radius = 75 m)
0.2
U 0.18 x
0
o o
O 0.16 + *
o +
00.14 +
0.12 +
+
0.1 O
0.08 
0.06
0 0.5 1 1.5 2 2.5
Pit Diameter/Wavelength(outside pit,h)
Figure 33: Reflection coefficients versus dimensionless pit diameter divided by
wavelength inside and outside the pit; water depth = 2 m, pit depth = 4 m (Bender,
2001).
The impact on the shoreline was found to be highly dependent on the transport
coefficients. Considering transport driven only by the breaking wave angle and wave
height, erosion was found to occur directly leeward of the pit flanked by two areas of
accretion as in Motyka and Willis (1974). Following an initial advancement directly
shoreward of the pit, erosion occurs and an equilibrium shape was reached. Examining
only the effect of the second transport term (driven by the longshore gradient in the wave
height) accretion was found directly shoreward of the pit, with no equilibrium planform
achieved. Figure 34 shows the shoreline evolution for each transport term. Including
both transport terms with the same transport coefficients resulted in a shoreline with
accretion directly shoreward of the pit that was able to reach an equilibrium state
(Figure 35). The twoterm transport equation used to determine the shoreline in Figure
35 is:
SK1H sin8(ab)cos(Oab) K2H2.s K cos(Oab)dH
K (3)
8(s 1X1 p) 8(s 1X1 p)tan(y) dy
where H is the wave height, g is gravity, K'is the breaking index, 0is the shoreline
orientation, a is the wave angle at the shoreline, yis the beach slope, s and p are the
specific gravity and porosity of the sediment, respectively, and Ki and K2 are sediment
transport coefficients, which were set equal to 0.77 for the results presented here.
First Transport Term (wave angle) Second Transport Term (dHb/dx)
200 0200
150 150
100 100
0 500
0
50 50
100 >1 100
,
150.  150
2 Erosion
0.5 80 80.5 81 40 60 80 100 120
Shoreline Position (m) Shoreline Position (m)
Figure 34. Shoreline evolution resulting from each transport term individually for
transect located 80 m shoreward of a pit with a radius = 6 m, last time step indicated
with [+] (modified from Bender and Dean, 2001).
In these models the water depth and pit depth are 2 m and 4 m, respectively, the
period was 10 s, the incident wave height is 1 m, and averaging over 5 wave directions
was used to smooth out the longshore variation in the wave height at large distances from
the pit. The time step was 120 s and 10 iterations of shoreline change are calculated
between wave height and direction updates for a total modeling time of 48 hours. The
diffusive nature of the angledriven transport term is seen to modify the much larger
wave height gradient transport term in order to generate an equilibrium planform when
the two terms are used together. Comparison of these results with those described earlier
establishes the significance of wave reflection and the second transport term.
Full Transport Equation (both terms)
200
150
100
50
o 0
100 
150 *Erosion
150
200
78 78.5 79 79.5 80 80.5 81 81.5 82
Shoreline Position (m)
Figure 35. Shoreline evolution using full transport equation and analytic solution
model for transect located 80 m shoreward of a pit with a radius = 6 m, last time step
indicated with [+] (modified from Bender and Dean, 2001).
6. Summary and Conclusions
Recent interest in extracting large volumes of nearshore sediment for beach
nourishment and construction purposes has increased the need for reliable predictions of
the wave transformation and associated shoreline changes caused by such anomalies.
This predictive capacity would assist the designer of such projects in minimizing
undesirable shoreline changes.
The available laboratory and field data suggest that the effect of wave
transformation by an offshore pit can result in substantial shoreward salients. Of the four
wave transformation processes, a significant number of wave models include effects of
wave refraction and diffraction; however, fewer incorporate wave reflection and
dissipation over a soft medium in the pit. Computational results incorporating only
refraction and diffraction and accepted values of sediment transport coefficients appear
incapable of predicting the observed salient landward of borrow pits. Therefore,
improved capabilities to predict wave transformation and shoreline response to
constructed borrow pits will require improvements in both: (1) wave modeling,
particularly in representing wave reflection and dissipation, and (2) longshore sediment
transport particularly by the wave angle and wave height gradient terms.
7. Acknowledgements
An Alumni Fellowship granted by the University of Florida sponsored this study
with partial support from the Bureau of Beaches and Wetland Resources of the State of
Florida.
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