LOCAL CONCENTRATION OF SLOWLY VARYING WAVE AND CURRENT FIELDSAROUND THE ABRUPTLY CHANGING BOTTOM SLOPES ALONG THE SHORE
Yoshimitsu Tajima1 and Shinji Sato1
This study aims to investigate physical mechanisms of locally concentrated nearshore hydrodynamic forces underinteractive fields of waves and currents on abruptly changing coastal bathymetry. Laboratory experiments were first
performed to represent such phenomena and a newly developed monitoring system based on image processing techniques
successfully captured the local concentrations of wave fields as quantitative high-resolution data sets. Numericalexperiments were then performed to investigate the physical mechanisms of the observed features. It was found that,among the other various possible factors, slowly varying wave-induced nearshore currents have one of the most significant
impacts on local concentrations of waves around the abruptly changing bathymetry.
Keywords: wave-current interaction, wave breaking, wave envelope, nearshore current, long wave

INTRODUCTION

The typhoon No.9, T0709, hit the Pacific Coast of Japan in September 2007 and caused severe
damages along the coasts. Seisho coast, located in the middle of Sagami bay, was one of coasts whichsuffered most significant damages due to T0709 (Tajima and Sato, 2009). Figure 1 shows thebathymetry around the most damaged area of the Seisho coast and alongshore distributions ofinundation heights measured after the storm. While Seisho coast has long and straight coast line, as
seen in the figure, damaged area was concentrated behind Oiso-Spur, where the area with relatively
shallow water depth of h < 100m is convexly extended toward the sea. Measured inundation heights
show somewhat periodic undulation along the coast and the locations of the highest peak of theinundation heights surely correspond to that of most damaged area where a part of highway along thecoast was collapsed. According to Sato et al. (1998), the typhoon No.20, in 1997 (T9720) also hit near
the Seisho coast and caused locally concentrated coastal damages while the most damaged area due toT9720 was about 6km west from that of T0709. Similar locally concentrated coastal damages can beobserved in Japan and these damages seem to have certain similarities such as: (i) the coast has
abruptly changing bathymetry along the shore; and (ii) slowly varying long wave components werealso significant during the storm.
The goal of this study is to investigate physical mechanisms of such features of locallyconcentrating hydrodynamic forces that eventually cause severe local damages along the coast. In thisstudy, laboratory experiment is first carried out to represent such locally concentrating wave andcurrent fields near the abruptly changing bathymetry under waves with low frequency wave

Figure 1 Measured inundation heights due to T0709 along the Seisho coast.

1

Department of Civil Engineering, The University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-8656, Japan

1

2

COASTAL ENGINEERING 2010

components. Image-based measuring system is then newly developed to capture the hydrodynamic
characteristics in a wider area with higher resolution both in time and space domains. Finally,
numerical analysis is performed to further investigate the physical mechanisms of the observed
features of locally concentrated hydrodynamic forces.
LABORATORY EXPERIMENT

Laboratory experiment was first performed to explore the physical mechanisms of locallyconcentrating hydrodynamic force due to various factors such as abruptly changing bathymetry andinteractions among wind waves, currents and low-frequency wave components.
Experimental Setups

Figure 2 shows plane view of the experimental setups. As seen in the figure, the original bed of
the basin consists of a plane panel with constant water depth near the wave generator and two
different shore-parallel inclined planes with cross-shore slopes of 1:10 and 1:20, respectively. In thealongshore half of the basin, stainless attachments were placed so as to create gently sloping 1:40
plane on top of the 1:20 sloping plane. In this manner, the basin somewhat represents the bathymetryat Seisho coast where relatively straight coasts with different cross-shore slopes merge each other at
the Oiso Spur (Figure 1). Boundary between panels with different slopes, 1:20 and 1:40, were linearly
connected by plane panels with an alongshore slope of 1:10. As seen in Figure 2, furthermore, whitelight source of rectangular panel was imbedded in the 1:40 sloping panel near the boundary of two
different bottom slopes. The water was colored in blue using artificial coloring, food blue No.2 CI
73015 indigo carmine. Video camera was then installed above the light source and captured the imageof the light penetrating through the colored water. Captured pixel data was then transferred to thebrightness and compared to the local water depth. Near the shoreline, vertical sea wall was placedalong the straight 2cm-deep contour lines of still water depth. The sea wall was colored in yellow andthe other four video cameras, illustrated in Figure 2, were installed to capture the surface waterfluctuations on the sea wall. A part of the seawall, ranging within -40(cm) < y < -20(cm) as shown in
the Figure 2, was replaced by a transparent acrylic panel and, behind the transparent seawall, a plane
mirror plate was placed. Green laser sheet was then introduced from the top of the mirror and, by
adjusting the angle of the mirror plate, the reflected laser sheet was introduced through thetransparent panel to the water and radiated thin layer parallel to the sloping floor and at height of
5mm above the bottom bed. This system enabled us to introduce the thin laser layer from the fixedboundary to the water but not from the fluctuating surface water boundary without disturbing the laser
layer due to refraction at the fluctuating air-water boundary. Aluminum powder was then put into thewater and the illuminated powder in the thin green laser layer was captured by high-speed camera
with frame rate of 100 fps. Particle tracking velocimetry, PTV, was then applied to the obtained
images to yield instantaneous horizontal velocity components along the green laser layer.

unit : cm
camerawave
generator

wave

-340
100

1:10

-240

1:10
80

steeper slope1:20

0yvertical seawall-400

milder slope
1:40

160

Light panel
laser sheet

-200

200

x

line A

Figure 2 Layout of the experimental setup

400

12

COASTAL ENGINEERING 20103

In this basin, this study introduced either monochromatic sinusoidal wave or bichromatic waves,i.e., combinations of two monochromatic sinusoidal waves whose wave frequencies are slightly offset
from each other. In this manner, bicromatic wave develops slowly varying wave envelope which
induces low-frequency band waves as well as free long waves especially after the wave breaking in thesurf zone. Wave period and height of the regular monochromatic wave were set to be T=1s and
H0=2.8cm, respectively. Similarly, wave properties of combining two wave components wererespectively set to be H0=2cm and T=1.111s or 0.909s. These two wave heights were selected so that
averaged energy of the combined waves becomes equivalent to the monochromatic wave and the timeperiod of the envelope becomes 5s. At the wave generator with uniform water depth 25cm, thesewaves were introduced with incident angle of 10 degrees. The wave generator was controlled so as toabsorb the reflected wave components whose properties were estimated from the measured surface
water fluctuations in front of the wave generator.
Image-Based Data Acquisition System

Based on the succeeding still digital images captured by all these six cameras including the highspeed camera, we developed a data acquisition system which yields widely covering and highresolution data sets of current and surface water fluctuations.
Obtained images were first rectified to quantify the distance and directions in the image based on
the actual xyz coordinate system. Based on seven bench-mark points, which determine geometric
relationships between actual xyz coordinates and image pixel coordinates, rectification parameters,such as camera locations, angles, focal length and lens distortion, were estimated so that the obtained
parameters yield the minimum root-mean-square errors of estimated pixel coordinates at seven benchmark points. This study also applied weight function proposed by Tajima et al.(2009) so that therelative importance of the errors at each bench mark points based on the xyz coordinates but notimage pixel coordinates become equivalent to each other. The other detailed procedures, validations,and primary initial results of the measured system are summarized in the following respective threemeasuring items.
Surface Water Fluctuations along the Seawall Based on the succeeding still images recordedby four cameras which captured the seawall from seaside toward the shore, instantaneous surface
water fluctuations along the sea wall were estimated. The horizontal range of the image captured ineach camera was about 1m along the sea wall and frame rate was set to be 30 fps.
Based on RGB-values in each pixel, the surface water boundary on the sea wall was detected by afollowing parameter,
S1 R G B(1)
Since yellow is represented by larger values of both R and G and blue, on the other hand, isexpressed by larger B and smaller R and G, this parameter, S1, gives a larger value on the sea wall andsmaller value on the blue water. Figure 3(a) shows a snap-shot image of the sea wall when the broken
wave was splashing on the wall. In this image, pixel data were replaced by the parameters S1. Figure3(b) shows the downward vertical distributions of S1 along the line specified in the Figure 3(a). As
seen in these figures, the parameter S1 abruptly decreases on the splashing water even though thewater layer is relatively thin and its color is not as dark as the one on the still deeper water. Since S1 isnearly constant on the yellow sea wall, this study determined the surface water boundary condition of
S1 by a single critical value, S1 =290, which is very close to the value on the yellow seawall. Startingfrom the arbitrary points on the seawall, the system search the pixel location in the vertical downward
direction where S1 first goes below this critical value and the detected pixel coordinates were thentransferred to the xyz-coordinates on the sea wall based on the obtained rectification parameters inadvance. Estimated averaged error of the rectified coordinates was less than one pixel and theaveraged pixel size in the xyz coordinate system on the sea wall was around 1.6mm. The possible
error of this surface water detection system should therefore be of the same order.
Figure 4(a) compares the time-varying surface water fluctuations estimated by the present imagebased system and the ones measured by wave gauge. As seen in the figure, the estimated surface water
profiles agree well with the ones measured by the wave gauge.
Surface Water Fluctuations on the Rectangular Light Panel Horizontal distributions of
instantaneous surface water fluctuations was estimated on the rectangular light panel embedded in the

4

COASTAL ENGINEERING 2010

1:40 sloping bottom based on the succeeding still images. Frame rate of the video image was set to be
60 fps and all the light except the embedded light source was turned off so that the recorded images
only capture the brightness of the bottom light source penetrating through the colored water column.
Using the image taken on the still water condition, known water depth increasing in the offshore
direction was compared with the local RGB values at each pixel. Through various comparisons, it was
found that the parameter, S2=R+G showed the best correlations with the water depth. Relationshipbetween this parameter, S2, and the local water depth, h, was then fitted in the following simple
exponential curve,
S2
exp
h
(2)
with and , fitting coefficients which should be determined through the obtained data sets. Even on
the same water depth along the shore-parallel contour line, the parameter S2 was found to vary withthe alongshore locations. This study therefore separately determined the optimum values of andalong each cross-shore line at different alongshore location. Local water depth was then inversely
estimated through equation (2) with local S2 and optimum sets of and on the entire light source.
The maximum error of estimated water depth was 2.5mm.
Figure 4(b) compares the surface water profile at the offshore corner of the light panel estimatedby the present image-based measuring system and the one measured by the wave gauge. As seen in the

(a)

400

S1

0

400

S1=290

S=290

S1
(b)

0
0

pixel number

100

Figure 3 (a) Recorded image with pixel values replaced by S1 and (b) vertical distributions of S1 along the line
specified in (a)

波高計
gauge

image
画像計測

cm

6 (a) seawall
3
0
-3
0

cm

2

1

2

s

3

(b) light panel

0

-2
0

gauge
波
高計
image
画
像計測

1

2

3

4

s

5

Figure 4 Comparisons of surface water fluctuations measured by a wave gauge and by the present imagebased measuring system at (a) seawall and (b) offshore corner of the light panel

COASTAL ENGINEERING 2010

5

figure, the image-based system tends to estimate milder wave crest but is able to capture overallsurface water profile within acceptable errors.
Horizontal Current Field Succeeding still images taken by a high speed camera in front of theseawall were finally applied for estimation of the horizontal velocity fields in the nearshore area.
Aluminum powder illuminated by green laser layer was used as tracer and directly estimate theinstantaneous and local horizontal velocity by taking time derivative of the obtained tracks of the
tracers. By introducing the green laser outside of the seawall, a rigid transparent panel, laser was not
disturbed by fluctuating surface water boundary and successfully illuminate a fixed thin layer about
5mm above the bottom bed. To remove noise in obtained instantaneous velocity, we separated all theobtained data into a number of groups in each of which the data is in the same 1cm-square grid area
and also in the time span of 0.1s. In each group, mean and standard deviation of the velocity
components were estimated and the individual data whose deviation from the mean was 1.8 times
greater than the standard deviation was removed as noise.
Results and Discussions

This section summarizes and analyzes experimental results of surface water fluctuations andhorizontal velocity fields in front of the sea wall, respectively.
Characteristics of the Surface Water Fluctuations Figure 5 compares surface waterfluctuations on the sea wall for both monochromatic and bichromatic incident wave cases, respectively.
In the figure, the vertical axis is the measurement time while the horizontal axis corresponds to thelocations along the seawall. In the figure, the darker color indicates the higher elevation of theinstantaneous surface water level. While the figure shows only for the first 15s of the measurements,
the data was obtained for three minutes and all these data were used for computations ofrepresentative wave properties such as root-mean-square wave heights.
Since waves were obliquely incident on the basin, arrival time of the waves at the seawall has
certain phase lag along the shore. Away from the vicinity of abruptly changing bottom slopes, i.e., in
the area, y < -100 (cm) or 100(cm) < y, for example, the observed crest lines in the Figure 5 are nearlystraight but are sloping in the up-rightward direction. These up-rightward slopes indicate that thebichromatic wave

monochromatic wave
0

steeper slope (1:20)

milder slope (1:40)

milder slope (1:40)

steeper slope (1:20)

elev.
(cm)

time (s)

8

10
0
-2

-200

0

y(cm)

200

-200

0

y(cm)

200

Figure 5 Surface water fluctuations at the seawall in both time and horizontal domains

6

COASTAL ENGINEERING 2010

wave crest line arrived at the seawall earlier at the larger y since the incident wave angle was from theright-offshore side of the basin. In the vicinity of the boundary of two different bottom slopes, i.e., in 100< y(cm) <100, wave refraction and diffraction curved the wave crest lines and, especially in thecase of monochromatic wave, curving crest line profile has clear peak at around y(cm)~50 on themilder 1:40 bottom slope. While the wave refraction should cause wave concentration around thispeak, the observed wave heights did not show such clear evidence of the wave concentration in thecase of monochromatic waves. Besides similar refraction and diffraction characteristics to themonochromatic waves, bichromatic incident wave case showed slowly varying envelopes with periodof 5s. Alongshore locations of the time-peak of the curving wave crest line shown in Figure 5 also has
slowly varying fluctuations with period of 5s.
In order to investigate the characteristics of slowly-varying long wave components, time-series ofthe surface water fluctuation at each alongshore location on the seawall was decomposed into discrete
frequency components and, integrating these components whose frequency is either greater or smaller
than 1/3 Hz, the time series of either “long” or “short” components of surface water fluctuations werereproduced. Based on the obtained long and short components of surface water fluctuations, Hrms,
H1/10, time-averaged water level, mean, was estimated. Figure 6 shows the alongshore distributions ofmean and wave heights of both short and long wave components for either monochromatic orbichromatic wave cases. As seen in the figure, there is clear difference between monochromatic andbichromatic wave cases and, especially in the case of bichromatic incident waves, Hrms of the longwave components, Hrms,L was significantly large. In the case of bichromatic incident wave, moreover,Hrms and H1/10 of the short wave components had local peak around y~100(cm) on the milder 1:40
slope, while monochromatic incident wave case did not have such a local peak. The observed local
peak of the wave heights on the milder slope is somehow consistent with the observed local peak ofsteeper slope (1:20)

(cm)

10

milder slope (1:40)

monochromatic wave

Hrms
H1/10

5
0

(cm)

H

(cm)

(cm)

rms,L
1
0
2
1
0
-200
-100
10 bichromatic wave

mean

0

100
Hrms
H1/10

5

Hrms,L

(cm)

(cm)

0
2
1
0
2
1
0
0.5
0
-0.5
-200

200

/

mean

phase
gap
位相差

-100

0

alongshore distance (cm)

沿岸方向距離(cm)

100

200

Figure 6 Alongshore distributions of measured wave heights, mean water levels and phase gaps.

COASTAL ENGINEERING 20107

the inundation height on the milder slope near the abruptly changing bottom slopes behind the OisoSpur on the Seisho coast.
Surface Water Fluctuations on the Rectangular Light Panel Taking the same procedures as
the surface water fluctuations on the seawall, surface water fluctuations on the rectangular light panel
was also decomposed into each frequency components and, based on the reproduced fluctuations oflong or short wave components, Hrms for both short and long wave components were estimated. Figure
7 compares horizontal distributions of the estimated Hrms of short and long wave components formonochromatic waves and bichromatic waves, respectively. Similar to the one observed in the surface
water fluctuations on the seawall, clear difference between monochromatic and bichromatic waves
were observed in terms of Hrms of long wave components. It is seen from the figure that Hrms,L has
clear dominant undulations in the cross-shore direction and this feature somewhat indicates that longwave response may be dominant in the cross-shore direction.
In order to explore the interactive characteristics between low-frequency surface waterfluctuations and slowly varying envelopes of the bichromatic waves, phase relationships of thesewaves were first investigated under the following procedures. Following List(1992), we first estimatedthe envelope function, A(t), as a low-pass filtered profile of | (t)| with (t), time-varying surface
water fluctuations at arbitrary location. Low-frequency component of the surface water fluctuations,
L(t), was also extracted using the same low-pass filter with maximum frequency of the low-pass filter
set to be 0.5Hz. Obtained low-frequency surface water fluctuations, L(t), was then separated into
incident onshore-ward and reflected offshore-ward components, L,i (t), and L,R(t). Following Janssen
and Battjes (2003), cross-correlation coefficient, R( ), was then estimated by the following formulae,
1R( )A2 t L ,i t
(3)
A2

L

Here A2 and L,i are standard deviations of A2 (t) and L,i , respectively, and the overbar on the lasttwo variables denotes the time-average over the entire time-series of the data. All these procedures
were then applied to the time-series data of surface water fluctuations along the cross-shore line A
shown in Figure 2 and also the alongshore line on the seawall.
Figure 8 shows the distributions of computed cross-correlation coefficients, R( ), as function oftime shift, , and horizontal distance either along the line A or along the seawall, respectively.
Horizontal axis of the figure denotes the location either along the line A or along the seawall and thevertical axis denotes the phase shift, =/TL, defined as a function of the time shift, , and TL=5s,

monochromatic-130

bichromaticbichromatic

monochromatic

x

-50
40

y

80
0

40

y

3
cm
(a) short wave components (T < 3s)

80

40

y

80

40

y

80

unit : cm
10cm(b) long wave components (T > 3s)

Figure 7 Horizontal distributions of Hrms of (a) short and (b) long wave components on the light panel

8

COASTAL ENGINEERING 2010
(a)

1

0.3
0.6

0.3
0.6
0.3

0.6

0

0.5
0

0

R

0

-0.5

-2 0 0y (cm) 1 0 0

0

200
1

(b)
0
0.3
0.3

0.6
0

x (cm)

0.5
0

0

-100

0.6

0.3
0

R

0

-1 0 0-0.5

seawall (x=0) -50

Figure 8 Alongshore (y) and cross-shore (x) distributions of cross-correlation coefficient, R, as a function of thephase gap, , of wave group envelopes and long waves

the period of slowly varying wave envelope. White dashed line in the figure indicates the phase shift
at which the cross-correlation coefficient reached its peak, i.e., profiles of envelope and low-frequency
surface water fluctuation best matches with each other.
As seen in the figure, phase shift decreases as the wave propagate toward the shore along the line
A and this feature surely corresponds to the similar experimental results (e.g., Janssen and Battjes,
2003, and Seki and Mizuguchi, 2008). On the seawall, on the other hand, the phase of envelope andthe low-frequency surface water fluctuations match with each other. This feature suggests that thepeak of the wave heights in the envelope comes with the peak of the surface water fluctuations. Since
the higher water level yields larger water depth and the larger water depth should decrease the broken
wave dissipation rate, i.e., increase the broken wave heights, observed match of the phases between
envelope and slowly varying surface water fluctuations may yield larger wave height in front of theseawall. This feature is further investigated in the following numerical experiments.
Current Velocity Fields Figure 9 shows the time-averaged horizontal current velocity fields forboth monochromatic and bichromatic wave cases estimated from succeeding images recorded by thehigh-speed camera. The figure also shows the cross-shore distributions of the horizontal velocity
components in the alongshore and the cross-shore directions, respectively. These representative
velocity components were obtained by averaging local velocity components along the respective shoreparallel line. The figure also shows the cross-shore distributions of predicted depth-averaged
horizontal velocity components in the alongshore and cross-shore directions, which will be discussed
later in the next section of the numerical experiments. As seen in the figure, the time-averaged
velocity fields in the measured area were nearly uniform in both monochromatic and bichromaticwave cases but magnitude of the current velocity was about two times greater in the case ofbichromatic waves than monochromatic waves while the time-averaged incident wave energy andincident wave angles of theses waves are equivalent to each other. This feature is reasonably
represented by the numerical model in predictions of the depth-averaged longshore currents.
NUMERICAL ANALYSIS

Laboratory experiments successfully represented the locally concentrating wave field in thevicinity of abruptly changing bottom slopes near the shore. Clear difference between monochromaticwaves and bichromatic waves with slowly varying wave envelopes was also observed especially in thatwave concentration on the milder slope appeared only in the case of bichromatic waves. This section
aims to further investigate the physical mechanisms of such wave concentrations observed through theCOASTAL ENGINEERING 2010
horizontal velocity (bichromatic wave)

longshore
沿岸方向流速(b)Case 2(群波)における平均流速9cross-shore
岸沖方向流速

(c)

(a)

10

5

-18

(cm/s)

0

-35

-30

y(cm) -25

-20 -3 (cm/s) 0

Figure 9 Horizontal distributions of the time-averaged current velocity under bichromatic waves (a) andlongshore (b) and cross-shore(c) components of horizontal velocity both under monochromatic andbichromatic waves (solid and blank circles; measured data and solid and dashed lines; predictions
respectively for monochromatic and bichromatic waves)laboratory experiments. Special focus is put on comparisons of relative importance of possible factorsthat could cause such wave concentration near the shore.
In order to explore the hydrodynamic difference between monochromatic and bichromatic
incident wave conditions, the numerical model should be able to represent the propagations of slowlyvarying wave envelopes and resulting slowly fluctuating wave radiation stress that induces lowfrequency waves as well as the near shore currents. Interactions among short waves, long waves, andnearshore mean current should also be accounted for in the model.
Model Descriptions

While time-dependent non-linear dispersive model such as Boussinesq-type model can directly
solve such interactive features of waves and currents, computed horizontal current velocities andsurface water fluctuations contain all of these hydrodynamic properties and it is therefore virtuallydifficult to separate those into each component to investigate their direct impact on the wave
deformations. In this study, therefore, we applied time-dependent mild slope equations based on linear
wave theory for computations of the short wave components and time-dependent non-linear shallow
water equations for the computations of nearsore currents as well as slowly varying low-frequency
waves. These two models were iterated so that the interactive features among short wave, long wave
and time-averaged nearshore curernts and wave setups are reasonably accounted for.
This study applied the following Oonaka and Watanabe’s (1987) time-dependent mild slope
equations for computations of short wave propagations,
Cg
Cg
1
1
U
Q 0
(4)
C
t
C
Q
t

C2

Db

(5)

where
and
are the wave angular frequencies respectively on fixed coordinate system and on
moving coordinate systems with current speed, U. Momentum equation, (5), moreover, introduces
breaking and broken wave dissipation term, Db, proposed by Tajima et al. (2007) which is verified tobe applicable to relatively complex arbitrary wave fields including partially reflected waves. As seenin these momentum and continuity equations, the time-dependent mild slope equations first need to
specify the dispersion relationships of the propagating waves, i.e., to determine both wave phasevelocity and group velocity on all the computation domain. Since the wave dispersion relationship isdetermined as a function of local water depth and a wave period, the model requires special treatments
for computations of waves with multiple frequency components such as bichromatic waves. For
computations of bichromatic waves, this study simultaneously solved two separate sets of momentum
and continuity equations for each wave frequency components and linearly integrated these results to10

COASTAL ENGINEERING 2010

obtain instantaneous surface water fluctuations and wave orbital velocity fields. These combined
instantaneous properties of bichromatic waves were then used to determine time-varying non-linear
breaking and broken wave dissipation term, Db, which should be highly affected by combinations ofthese two wave components.
For computations of slowly varying long wave including time-averaged nearshore current andwave setups, this study simply applied non-linear shallow water equations with additional terms ofbottom shear stress, mixing dispersions, and wave radiation stress. Madsen’s (1994) bottom boundary
layer model was used to evaluate bottom shear stress due to nearshore current under wave-current
interactive field and spatially varying dispersion coefficients for nearshore current fields were
determined by Larson and Kraus’s (1991) model. In order to account for the time-varying waveinduced forces due to slowly varying wave envelopes, this study computed the wave radiation stressthrough the following numerical procedures. Following Longuet-Higgins and Stewart (1962), themodel first computed “instantaneous wave radiation stress,” R’i , defined by
S ' jiR 'i(6)xj
in the computations of short wave field based on the time-dependent mild slope equations. Here
subscripts i and j (=1, 2) denote the components in the horizontal x and y directions and02

uiu j w2 ij dz(7)2
his the instantaneous radiation stress tensor with , instantaneous surface water levels of combined
waves, ij, Kronecker’s delta, ui , wave-associated horizontal velocity components in the i-th direction,
and w, wave-associated vertical velocity component. In the case of monochromatic waves,monochromatically changing R’i were averaged over a wave period to determine the time-averaged
Radiation stress and the estimated constant Radiation stress was then simply introduced in the nonlinear shallow water equations for computations of time-averaged nearshore current. In the case ofbichromatic waves, on the other hand, slowly-varying radiation stress was first computed byextracting the low-frequency components of R’i . Obtained R’i were then introduced in the non-linear
shallow water equations by changing their values as the time proceeds with the actual numericalcomputations.
S 'ij

ij

Experimental Cases and Conditions

Based on the numerical models described in the previous section, this study performed thenumerical experiments. In the experiments, wave fields based on the mild slope equations and thenearshore current fields including the slowly varying components based on the non-linear shollow
water equations were iterated so that each model can feed back their computed results to the other.
During these iterations, this study neglected a part of hydrodynamic features as indicated by “x” inTable-1 for different computation cases. Relative importance of various hydrodynamic factors cantherefore be observed through comparisons of experimental cases in Table-1.
Table 1 Cases of numerical experiments
cases
A
B
C
D
E
F

wave
breaking

x
○
○
○
○
○

time-averaged components
water level
current

x
x
○
○
○
○

x
x
x
○
○
○

long wave components
water level
current

x
x
x
x
○
○

x
x
x
x
x
○

The other computation conditions were set so as to represent the present laboratory experiments.
As shown in Figure 2, the computation was performed on a rectangular area with cross-shore andalongshore distances of 4 and 8 meters, respectively. Incident waves were introduced uniformly along
the straight offshore boundary and perfect reflection boundary conditions were applied on bothalongshore sides of the basin to represent the vertical wall boundaries of the basin. In order to absorb
the short wave components, 30-cm-wide sponge layers were installed on both sides of the basin and toCOASTAL ENGINEERING 201011

represent this feature, the model applied artificial dissipation term which is proportional to local
instantaneous current velocity. Along the on shore seawall, perfect reflection condition was also
applied.
Numerical Results

Figure 10 shows computed wave and current fields when either monochromatic or bichromaticwaves were obliquely incident on the basin. In the figure, horizontal distributions of the root-meansquare wave heights are expressed in gradually changing color and contour lines whereas thehorizontal current velocity is represented in vectors. It is clearly seen in the figure that the wave tends
to concentrate where the seaward current velocity component is dominant in both monochromatic andbichromatic wave cases and the different current patterns somehow appear to determine the horizontal
locations of the wave concentrations. It is also seen in the figure that computed nearshore current
velocity under bichromatic waves is uniformly distributed on the steeper slope near the abruptly
changing bottom slopes as was observed in the experimental results shown in Figure 9. Figure 9 also
compares the predicted and measured cross-shore distributions of the current velocity components
averaged over the shore-parallel lines. As seen in this comparison, the predicted longshore current
velocity agree well with the measured data even in a quantitative manner in both monochromatic(solid line) and bichromatic (dashed line) incident wave cases. It should be noted that the model
underestimated the seaward current velocity components since the model only predicts depth-averaged

steeper slope (1:20)

milder slope (1:40)

200

rms (cm)y (cm)

6

00 monochromatic wave
200

rms (cm)y (cm)

500 bichromatic wave
-200

0x (cm)

200

Figure 10 Predicted root-mean-square wave heights and time-averaged nearshore current velocity for bothmonochromatic wave (case D) and bichromatic wave (case F).

12COASTAL ENGINEERING 2010

current velocity which does not account for the influence of the undertow. On the other hand, theobserved velocity is the near-bottom velocity components measured at 5mm above the bed and shouldbe affected by the undertow.
Finally, Figures 11 compare the alongshore distributions of the predicted and measured wave
heights along the seawall. The figure compares computed results based on different cases outlined inTable 1. Figure 11(a), for example, compares the wave heights along the seawall when the model only
accounted for wave deformations due to shoaling, refraction and diffraction, i.e., the model neglectedvarious deformation factors such as breaking and broken wave dissipations and interactions among
short waves, long waves and time-averaged nearshore currents. As seen in the figure, bothmonochromatic and bichromatic waves had a local peak on the milder 1:40 slope near the boundary
where the bottom slope changes while observed distributions of monochromatic waves had no such
local peak on the milder slope. Moreover, the predicted peaks were milder than the measured one and,compared to the measured data, the alongshore location of the predicted peak was closer to the centerof the bathymetry, i.e., the boundary between steeper and milder bottom slopes. Based on theseobservations, it is clear that the simple wave refraction phenomena of the short wave components do
not reasonably explain the nearshore wave concentrations. It is also interesting to note that thepredicted wave heights were slightly larger on the milder slope while the measured wave heights
tended to be larger on the steeper slope both in monochromatic and bichromatic wave cases. Thisfeature indicates the importance of the broken wave dissipation terms as functions of different bottom

steeper slope (1:20)

milder slope (1:40)

H/Hin

5

(cm)

0
4

(cm)

0
4

(cm)

0
4

(cm)

0
8

(cm)

0
3

規則波

(a)モデル(非砕波)

群波(b)実測値(規則波；Hrms)

B(c)モデル(規則波)D1(d)モデル(規則波)

C

D

D3
D2

(e)実測値(群波；H1/10)(f)モデル(群波)

C

D

-100

0

y(cm) 100

0
-200

E

F

200

Figure 11 Comparisons of predicted and measured wave heights along the seawall for monochromatic wave (a,b, c, d) and bichromatic wave (a, e and f). (a) predicted non-breaking (case A) monochromatic (solidline) and bichromatic (dashed line) waves, (b) measured Hrms, (c) predicted cases B, C and D, (d) case
D with different iteration number for computations of wave-current interactions, (e) measured H1/10,
and (f) predicted cases C, D, E, and F.

COASTAL ENGINEERING 201013

slopes.
Figure 11(c) compares cases B, C and D when monochromatic waves were incident on the basin.
As outlined in Table 1, the cases B, C and D added the following wave-deformation factors one by
one: (i) wave breaking; (ii) increase of the water depth due to wave setup; and (iii) wave-current
interactions due to nearshore mean current. As seen in the case C, breaking wave model successfully
captured the feature that the wave heights on the steeped slope tended to be larger than those on themilder slope except around the local peak. Difference between C and D, i.e., the influence of meancurrent on wave deformation, is more significant compared to the difference between B and C, i.e., theinfluence due to wave setup. By introducing the interactions between nearshore mean current andshort waves, as seen in the case D, predictive skills of the alongshore distributions of the wave heights
are largely improved especially in that the local peak on the milder slope at around y = 40cm observedin computed cases A, B and C, were diminished and instead a local peak was developed on the steeper
bottom slope at around y = -70(cm) while the observed peak, located around y = -50(cm), was slightly
offset.
Figure 11 (d) compares the wave heights in case D but when the number of iteration of wave andcurrent computations was altered. In the figure, the numbers in D1, D2, and D3 denote how manytimes the wave-current interactions were iteratively computed. In the case of D1, for example, wave
field was computed twice so that the second wave computation accounts for the influence of thenearshore current induced by the wave radiation stress based on the first wave computation. As shownin the figure, the computed results changed especially in the vicinity of the boundary of milder andsteeper bottom slopes and the best predictive skills was observed in the case of D3. It was also
confirmed that the computed results nearly converged after three-time iterations.
Figures 11 (e) and (f) compare the measured wave heights and the predicted ones in cases, C, D,
E and F, respectively, when the bichromatic waves were incident on the basin. In order to investigate
the influence of slowly varying components of the surface water fluctuations or current velocity, weapplied the same computed results of the slowly-varying nearshore current field for computations ofwave-current interacting wave fields but extracted either fluctuating or time-averaged components ofcomputed results depending on the cases outlined in Table 1. The case E, for instance, directly
substituted computed U, which contains both time-averaged and slowly fluctuating components, into
the mild slope equations, (4) and (5), whereas the case E applied only the time-averaged componentsof the surface water level, mean, for computations of the instantaneous local water depth used in theequations (4) and (5). In Figures (e) and (f), alongshore distributions of H1/10 were compared in steadof Hrms since this study focus more on the concentrations of larger waves that directly cause severe
coastal damages. As seen in the Figure (f), the model tends to underestimate the wave heights along
the seawall while the model successfully captured the alongshore profiles of wave heights, i.e., locallyconcentrating features of the waves. This underestimation may be because the model is based on thelinear wave theory and thus could not capture the sharp crest of the surface water profile in front of
the seawall. Comparison of cases C, D, E, and F reveals that the influence of the slowly-varying water
depth, represented by the difference between cases D and E, is less significant compared to the
influence of interactions between waves and time-averaged mean current, represented by a difference
between cases C and D. Since, as seen in Figure 8, the phases of wave envelope and slowly-varying
surface water fluctuations match with each other, increase of the instantaneous water depth due to
low-frequency surface water fluctuations may decrease the broken wave dissipation rate and thusincrease the broken wave height. In this numerical experiment, the influence of such surface waterfluctuations was not as significant as the influence of wave-current interactions. Slowly-varying
component of the nearshore current, as seen in comparisons between E and F, affect to change thealongshore locations of the local peaks of the wave heights while its impact was less significant thanthat of time-averaged components of the nearshore current.
CONCLUSIONS

Laboratory experiments were performed to investigate the physical mechanisms of locallyconcentrating wave fields around the abruptly changing bottom slopes. Besides simple monochromaticwaves, this study applied bichromatic waves with slowly varying wave envelopes, which induced thegeneration of low-frequency wave components. Image-based measuring system was newly developed
and applied to the experiments. This measuring system enabled us to capture the wave and current
fields in wider area with higher spatial and time resolutions. The experiments with this measuring

14

COASTAL ENGINEERING 2010

system successfully captured such wave concentration phenomena especially around the abruptly
changing bottom slopes.
Through the experiments, clear difference between monochromatic waves and bichromatic waves
were observed while their time-averaged wave energy and the mean wave periods were equivalent
with each other. While the monochromatic wave had no local peak of the wave height on the milder
slope, bichromatic wave case had a clear local peak of the wave height near the alongshore boundaryof the two different bottom slopes.
Numerical experiment was then performed to further investigate the detailed physicalmechanisms of the observed wave concentrations. In order to decompose various factors of wavedeformations which affect wave concentrations, this study applied linear time-dependent mild slope
equations for computations of short wave fields and non-linear shallow water equations with timevarying wave radiation stress for computations of nearshore current and slowly-varying long wave
components. These two models were iteratively applied so that the entire model can accommodate
interactions among short waves, long waves and time-averaged nearshore currents. During theiteration, different numerical cases selected a different part of various interactive factors so thatcomparisons of computed results in these different cases represent the influence of some specificfactors.
Accounting for all the interactive features of short waves, long waves and time-averagednearshore currents, the present model reasonably explained the difference between monochromaticand bichromatic incident wave particularly in that the local peak of the wave height on the milder
slope was observed only in the case of bichromatic incident waves. Through the numericalexperiments, it was also found that, among various interactive factors, interactions between timeaveraged nearshore current and the short wave components were one of the most dominant factorsthat determine the nearshore wave fields around the abruptly changing bottom slopes.
ACKNOWLEDGMENTS

Authors would like to acknowledge with sincere gratitude that this research was supported by agrant-in-aid of JRTT, Japan Railway Construction, Transport and Technology Agency.
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