1
Wave Breaking Analysis with Laboratory Tests
André José Figueira Martins
Instituto Superior Técnico – Technical University of Lisbon, Av. Rovisco Pais 1, 1049-001 Lisbon, Portugal
Abstract
Wave breaking is a phenomenon characterized by energy dissipation, turbulence effects and air emulsion.
The importance of studying this phenomenon is due to the effects it may have, as the process of wave breaking is both one of the most visually
dramatic, and one of the most important physically for the wave motion and for the development of near shore currents.
Since the knowledge of the processes involved in this phenomenon is still far from complete, the experimental results play an important role in
their clarifying. This study began with a brief literature review on the current status of the subject and, at the practical level, the presented work
describes a range of wave channel tests performed at the National Laboratory of Civil Engineering (LNEC), with the main objective of
introducing an extensive analysis of the waves, especially the analysis of wave propagation in conditions prone to wave breaking. Therefore,
this paper shows the experimental setup, the incident wave conditions and the measurements of the free surface elevation along the wave
channel. Based upon the time series of the wave data measurements, a statistical time domain analysis, a standard Fourier based spectral
analysis and a wavelet analysis was performed and presented.
This study also aims to compare the results, acquired in these tests with predictions obtained from the application of empirical formulations
relating to geometries similar to the structure under study. Thus, the analysis of similar cases to the case study will contribute to a better
understanding of these empirical formulations, specially their range of application, and represent a move towards the systematization of the
knowledge we can gain using a combination of experimental results and simple theoretical approximations.
Key-Words: Waves, Breaking, Trials, Time series analysis, Spectral analysis.
1. Introduction
treatment was done by using classical analyses in time and
frequency domains.
The determination of the wave breaking zone is essential for
studies referring coastal hydrodynamics and sediment transport
2. Main subjects
issues. Since wave breaking is a nonlinear complex phenomenon
that occurs with different scales, research in this area, specifically
2.1. Time Domain Parameters
the location and extension of wave breaking are two of the main
The wave period
factors for these studies, since they determine the coastal
is the time distance between two
structures location and stability of the subsequent sediment
consecutive down crossings (or up crossings), whereas the wave
transport.
height
is
the
In order to obtain data of the free surface elevation and
vertical distance from
velocity field measured along the channel (Neves et al., 2011)
a trough to the next
conducted a series of tests in LNEC’s waves channel, for 15
crest as it appears on
incident wave condition. The tests were conducted under the
the
Project “BRISA - Breaking waves and Induced Sand transport”,
Another
financed by the Fundação para a Ciência e Tecnologia
commonly used kind
wave
record.
and
more
(PTDC/ECM/67411/2006 contract). The project’s main objective is
Figure 1 – Sample of a wave record.
of wave height is the zero-crossing wave height
, being the
to contribute to the understanding and numerical modeling of the
vertical distance between the highest and the lowest value of the
wave breaking phenomena and sediment transport in coastal
wave record between two zero-down crossings (or up crossings).
areas.
A typical wave record is shown in Figure 1. When the wave record
The bottom profile consisted in a series of ramps, with variable
contains a great variety of wave periods, the number of crests
slopes, in which a depth of 10 cm water column was set on top of
becomes greater than the number of zero-down crossings. In that
the second ramp, as a way of inducing wave breaking in that area.
case, there will be some difference between the crest-to-trough
A great number of experimental data was obtained, and whose
wave height and
neglected and
. In this chapter, however, this difference will be
will be used implicitly.
2
A measured wave record never repeats itself exactly, due to
under the spectral curve therefore has a physical meaning which is
the random appearance of the sea surface. But if the sea state is
used in practical applications for the definition of wave-height
“stationary”, the statistical properties of the distribution of periods
parameters derived from the spectrum. Recalling that for a simple
and heights will be similar from one record to another. The most
wave the wave energy (per unit area),
appropriate parameters to describe the sea state from a measured
height by:
, was related to the wave
wave record are therefore statistical (Laing et al., 1998). The
following are frequently used:
then, if one replaces the actual sea state by a single sinusoidal
̅ – Average wave height;
wave having the same energy, its equivalent height would be given
– Maximum wave height occurring in a record;
̅ – Average zero-crossing wave period; the time obtained by
by:
dividing the record length by the number of down crossings (or up
√
crossings) in the record;
̅̅̅̅̅̅
⁄ – The average height of the 1/n highest waves (i.e. if all wave
the so-called root-mean-square wave height.
heights measured from the record are arranged in descending
total energy (per unit area) of the sea state.
now represents the
order, the one-nth part, containing the highest waves, should be
We would like a parameter derived from the spectrum and
taken and ̅̅̅̅̅̅
is then computed as the average height of this
⁄
corresponding as closely as possible to the significant wave height
part);
̅̅̅̅̅̅
⁄ (as derived directly from the wave record). It has been shown
̅̅̅̅̅̅
⁄ – The average period of the 1/n highest waves. A commonly
that
used value for n is 3;
the required value. Thus, the spectral wave height parameter
̅̅̅̅̅̅
⁄ – Significant wave height (its value roughly approximates to
commonly used can be calculated from the measured area,
visually observed wave height);
under the spectral curve as follows:
should be multiplied by the factor √ in order to arrive at
̅̅̅̅̅
⁄ – Significant wave period (approximately equal to the wave
√ √
period associated with the spectral maximum).
state (
A wave spectrum is the distribution of wave energy (or
) as the total energy, but we must be mindful here that the
total energy
is really
between
frequency and direction, etc.). Thus, as a statistical distribution,
do not occur often in
many of the parameters derived from the spectrum parallel similar
nature. However, the
parameters from any statistical distribution. Hence, the form of a
difference is relatively
wave spectrum is usually expressed in terms of the moments of
small in most cases,
the distribution (spectrum). The nth-order moment,
with
, of the
=1.05. ̅̅̅̅̅̅
⁄ on
average. The significant
spectrum is defined by:
∫
( )
(1.1)
wave
height
frequently
( ) denotes the variance density at
, as in Figure 2, so that
( )
represents the
⁄ contained in the ith interval between
variance
and
.
In practice, the integration in Equation 1.1 is approximated by a
finite sum, with =
. In theory, the correspondence
and ̅̅̅̅̅̅
⁄ is valid only for very narrow spectra which
variance of the sea surface) over frequency (or wavelength or
frequency,
√
Note that we sometimes refer to the total variance of the sea
2.2. Spectral Parameters
In this formula,
,
:
is
also
denoted
by
. In that case, it must
be
indicated
quantity (4√
which
or ̅̅̅̅̅̅
⁄ )
Figure 2 - Typical wave-variance
spectrum for a single system of wind
waves. By transformation of the vertical
axis into units of
( ), a wave
energy ( ) spectrum is obtained.
is being used (Laing et al., 1998). Wave spectra systems typically
have a form like that shown in Figure 2.
∑
The derivation of parameters for wave period is a more
complicated matter, owing to the great variety of spectral shapes
it follows that the moment of zero-
related to various combinations of sea and swell. There is some
, represents the area under the spectral curve. In finite
similarity with the problem about defining a wave period from
From the definition of
order,
statistical analysis. Spectral wave frequency and wave period
form this is:
parameters commonly used are:
∑
which is the total variance of the wave record obtained by the sum
of the variances of the individual spectral components. The area
– Wave frequency corresponding to the peak of the spectrum
(modal or peak frequency);
– Wave period corresponding to , i.e., =
;
3
– Wave period corresponding to the mean frequency of the
Spilling breakers are characterized by the forward slope of the
wave top becoming unstable. A plume of water and air bubbles
spectrum:
slides down the slope from the crest. The volume of the plume
increases, and it travels with the wave as a surface roller. Spilling
– Wave period theoretically equivalent with mean zero-down
crossing period ̅ :
breakers are also found among waves in deep water, where their
energy dissipation is an important part of the energy budget for
wind generated waves.
√
For a plunging breaker the crest of the wave moves forward
and falls down at the trough in front of it as a single structured
Note that the wave period
is sensitive to the high
frequency cut-off in the integration (Equation 1.1) which is used in
practice. Therefore this cut-off should be noted when presenting
and, in particular, when comparing different data sets. For
buoy data, the cut-off frequency is typically 0.5 Hz as most buoys
do not accurately measure the wave spectrum above this
frequency. (Goda, 1978) has shown that, for a variety of cases,
average wave periods of the higher waves in a record, e.g. ̅̅̅̅̅
⁄ ,
remain within a range of 0.87.
to 0.98.
.
mass of water or a jet. The impact of the jet generates a splash-up
of water which continues the breaking process and creates large
coherent vortices, which can reach the bottom and stir up
considerable amounts of
sediment. The flow caused
by entrained air further
spreads the sediment over
the vertical, and clouds of
suspended sediment are
often
2.3. Wave breaking
The surf zone is the name of shallow water areas where
waves break, for example on a beach. The wave-breaking is
associated with a conversion of the energy from ordered wave
energy to turbulence and to heat. The surf zone is the area with
the most intense sediment transport because of the high intensity
of the turbulence and the shallow water which makes agitation of
sediment from the bottom easy.
As waves propagate into shallower water, the process of
shoaling leads to increasing wave heights. This process cannot
continue, and at a certain location the wave breaks. The wavebreaking will typically take place when the wave height is about 0.8
times the local water depth. The waves break because their
steepness becomes very large as the depth becomes shallower.
The forward wave orbital velocity at the crest becomes large, and
the crest topples because it is unstable.
While the shoaling process is characterized by a very small
energy loss, the wave breaking is associated with a very large loss
of wave energy. The surf zone along the beach is where the wave
energy flux from offshore is dissipated to turbulence and heat. Due
to the strong energy dissipation, the wave height decreases
towards the shore in the surf zone. The breaking waves can be
divided into several different types, the three most important are
observed
location
of
at
a
plunging
breakers.
In surging breakers it
is not the crest of the wave
that becomes unstable. It
is the foot of the steep front
that
rushes
forward,
causing the wave crest to
decrease and disappear.
Figure 3 – Comparison between the
main types of wave breaking and
four distinct moments in their
evolution (Derived from Dally et al.,
1985).
The occurrence of the different types of breakers depends on
the character of the incoming waxes and of the beach. The most
important factor is the slope of the beach and the steepness of the
incoming waves. Spilling breakers occur at very gentle beach
slopes and relatively steep incoming waves, while plunging
breakers are found for steeper bed slopes and less steep waves.
Surging breakers are found on very steep beaches. (Galvin, 1968)
found a relationship between the wave geometry and the breaker
type. The waves can be characterized by the surf similarity
parameter
(Battjes, 1974), which is the ratio between the beach
slope and the square root of the wave steepness. The wave
steepness can be calculated from the deep water wave height
or the wave height at breaking
wave length
. In both cases the deep water
is used in the expressions for :
(Fredsøe & Deigaard, 1992):
√

Spilling breakers;

Plunging breakers;

Surging breakers.
√
√
√
where Figure 3 presents the comparison of the 3 breaking waves
where
types.
Galvin’s experimental data.
√
√
is the beach slope. Table 1 shows the domains from
4
Table 1 – Breaker type criteria.
Breaker type
Spilling
Plunging
Surging
Zone
Deep water
Breaking point
< 0.5
0.5 < < 3.3
> 3.3
< 0.4
0.4 < < 2.0
> 2.0
3. Experimental settings
wave maker by the National Instruments acquisition system. For
the data acquisition, the sensors signals elapsed through the
Wave Probe Monitor in order to convert the analogical signal in a
digital format. Therefore the signal was recorded at the CPU1
computer.
3.1. Incident wave characteristics and
experimental tests
The wave tank experiments were conducted in a wave channel
A piston-type wave maker generated a combination of regular
with 32.4 m length and 0.6 m width. A beach profile, with different
waves combining four wave periods ( =1.1, 1.5, 2.0 and 2.5 s)
bottom slopes, was constructed as shown in Figure 5. The slope
with four wave heights ( =12, 14, 16, and 18 cm). The wave with
angle of the front face of the bar and the beach section was fixed
=1.1 s and
=18 cm presented a very steep wave that broke in
with 1:20 and the slope of the lee side of the bar was inexistent.
front of the wave maker, therefore, this incident wave condition
Water depth was measured to be 0.1 m at the crest of the 1:20
was excluded from the experimental tests. Thus, in total, for each
bar. Figure 4 presents a plant of the wave channel.
position along the canal, fifteen combinations of waves were
tested, as indicated in Table 2.
Table 2 – Incident wave conditions.
[s]
[cm]
12
14
16
18
Figure 4 – Wave channel plant and positions along the longitudinal (x)
axis (derived from Conde, 2012).
1.5
x
x
x
x
2.0
x
x
x
x
2.5
x
x
x
x
The total set of experiments was divided in two phases:
I.
Phase I – the objective of the first phase was to measure
free surface elevations along the channel with an 8 wave
Along the wave channel, several equipments were installed:
gauges mobile structure;
- The measurement equipment (wave resistive gauges and an
Acoustic Doppler Velocimeter [ADV]) to measure free surface
1.1
x
x
x
-
II.
Phase II – the objective of the second phase was to
elevations and particle velocities;
measure the particle velocity along the channel in the
- The National Instruments acquisition system to forward the wave
middle of the water column, using the ADV sensor. At the
generation signal to the wave maker (‘‘piston’’ type);
same time, a resistive gauge was located near the ADV
- A computer named CPU2 to connect and customize the ADV
sensor.
sensor to measure the free surface elevation.
This work focused on the Phase I experiments results.
3.2. Equipment and experimental procedures
In Phase I, a mobile structure with eight wave gauges was
placed along the channel to measure the free surface elevations.
The mobile structure provided an easy transport and allocation of
the wave gauges along the channel. The covered length of the
wave channel was from the beginning of the first ramp (x=-1000
cm) till x=560 cm when the wave breaking is shown to be
Figure 5 – Wave channel profile and positions along the longitudinal (x)
axis (Derived from Conde, 2012).
At the monitoring office, the computer named CPU1 was
installed to generate the incident wave signal and to acquire and
record the data from both the wave resistive gauges and the ADV
sensor. The CPU1 computer received the sensor signals through
intermediate devices, like the Wave Probe Monitor and the
National Instruments acquisition system. The connection between
the computer CPU1 and the wave maker was made through the
Signal Express software generating the signal to be sent to the
completely over for every incident wave condition.
To calibrate the input wave height, and since the bottom profile
was not flat, a wave gauge, named “AØ”, was installed at the toe
of the front face of the first slope (x=-1080 cm). Each gauge in the
mobile structure was separated by a fixed distance (20 cm) and
measurements separated by 10 cm were taken along the covered
area. The sampling frequency of “AØ” was 25 Hz, whereas the
other eight gauges were 100 Hz.
It is important to note that some positions (only two) are
repeated due to limitations of the channel, which contain a set of
5
transverse metal bars that sometimes prevent the placement of the
(i.a) Fourier, with Sam Mod 7 software (Capitão, 2002);
eight gauges structure. Each experimental test (incident wave) had
(i.b) Wavelet, with Matlab™ software (Mori, 2009).
the duration of 490 s.
- Breaking analysis:
4. Analysis of results
(i) Relative wave height;
(ii) Wave breaking type;
In this chapter, we will analyze the results obtained in the
incident wave tests. The wave breaking section will be the basis
(iii) Limiting breaker height;
(iv) Wave transformation on the internal part of surf zone.
for comparisons with some previous studies made by other
authors.
4.2. Wave breaking zone
The main parameter to define the wave height in the breaking
, was to consider the maximum wave height registered for
The wave breaking section was defined for each one of the 15
each wave incident condition. Using a similar reasoning, the water
incident waves. Table 3 presents the values of the initial and final
depth at breaking point,
location of the wave breaking section for all the incident waves in
point,
, was considered to be the minimum
depth in which the maximum wave height was measured.
For each of the 15 incident wave conditions, the time series of
the free surface elevation was obtained in the various positions,
the tests described above.
Table 3 – Breaking waves locations, wave height and water depth at
breaking point for the 15 incident waves.
from x=-1000 cm to x=560 cm. Based on those data, different
Incident
wave
types of data analysis were considered, but as in this study only
the Phase I experiments were relevant, the results obtained were
T11H12
T11H14
T11H16
T15H12
T15H14
T15H16
T15H18
T20H12
T20H14
T20H16
T20H18
T25H12
T25H14
T25H16
T25H18
mainly focused on time, spectral, and statistical analysis of the free
surface elevations. Also the relative wave height ( ⁄ ) along the
surf zone was calculated for each incident wave condition.
In the next sections, the experimental location values for the
beginning and ending of the wave breaking zone are presented.
Then some samples are presented of the obtained data and the
performed analysis, with varying incident wave conditions for each
presented sample.
4.1. Time, breaking, statistical and spectral
analysis
Breaking positions (x)
Initial
Final
[cm]
[cm]
-270
210
-530
200
-530
200
-270
270
-330
250
-390
250
-470
250
-270
330
-410
330
-410
310
-470
300
-270
450
-330
450
-350
430
-410
420
[cm]
[cm]
18.021
19.552
23.208
19.946
22.002
24.145
27.408
23.577
29.537
30.242
31.414
27.330
27.594
31.339
32.335
23.350
35.020
35.020
23.350
25.960
28.680
32.200
23.350
29.580
29.580
32.200
23.350
25.960
26.840
29.580
4.3. Time domain analysis
Based on the free surface records along the channel, the
35.0
following parameters were calculated:
Significant Wave Height (Constant Period T=2.0 s)
30.0
(i)
(ii)
(iii)
(iv)
(v)
(Maximum wave height);
(Significant wave height);
(Average wave height);
(Significant wave period);
(Average wave period), based on the up-crossing method;
Hs (cm)
25.0
- Time analysis of the free surface elevation values:
H12
20.0
H14
15.0
H16
H18
10.0
5.0
0.0
-1000
-800
-600
-400
-200
x (cm)
0
200
400
600
Figure 6 – Significant wave height throughout the channel, for incident
waves with 2.0 s period.
- Statistical analysis of the free surface elevation values:
(i) Average;
(ii) Standard deviation;
(iii) Skewness;
(iv) Kurtosis.
Figure 6 shows the significant wave heights, for incident waves
with =2.0 s. We can see an initial decrease of the
followed by
a significant increase throughout the section close to x=-700 cm
due to the shoaling effect. Some oscillations on the wave height
values are also observed. Subsequently, the wave breaking effect
- Spectral analysis of the free surface elevation values:
produces a significant reduction in the wave height preceded by a
(i) Spectral density variance was calculated in order to define the
section with a constant value for the wave heights at the lee side of
energy distribution in the frequency spectrum, with help of two
the channel due to the ending of the wave breaking.
different sets of analysis:
6
0.40 and 0.60, which can be considered a bit high for the values
Average Period (Constant Wave Height H=16 cm)
we were comparing.
3.0
2.5
Another comparison was made with data from another study
T1.1
Tm (s)
2.0
1.5
1.0
T1.5
(Fredsøe & Deigaard, 1992). After a wave has broken as a spilling
T2.0
or plunging breaker, a transition occurs. For the spilling breaker
T2.5
0.5
For the plunging breaker the jet of water that plunges down pushes
0.0
-1000
the surface roller grows and the wave height decreases rapidly.
-800
-600
-400
-200
0
200
400
600
up a very turbulent mass of water which continues the wave-
x (cm)
Figure 7 – Average period throughout the channel, for incident waves
with 16 cm wave height.
breaking process.
In both cases the wave is transformed into a bore-like broken
wave, and this inner part of the surf zone can be described as a
Figure 7 shows the average wave periods along the channel,
for incident waves with
=16 cm. Tests show that the wave period
remains approximately constant throughout the channel, only to be
disturbed at the wave breaking zone. In this last section we
observe major changes in the wave period, particularly the ones
with bigger periods.
series of periodic bores (Svendsen et al., 1978). The ratio between
the local wave height and mean water depth decreases from the
value of about 0.8 at the point of wave-breaking to become almost
constant at about 0.5 in the inner zone. Figure 9 shows the
experimental results together with the empirical relation by
(Andersen & Fredsøe, 1983):
Relative Wave Height (T15H16) after breaking, with
constant bottom slope
4.4. Relative wave height
1.0
The relative wave height ( ⁄ ) is often used as an index for
(1985) considers this index as reference for a stabilized wave
H/h (-)
the wave breaking section in shallow water conditions. Dally et al.
0.9
0.8
0.7
0.6
condition. Unlike the beginning of the wave breaking, there is no
0.5
standard or commonly used value for the end of the wave breaking
0.4
0
2
4
6
Δx/hb (-)
section. (Dally et al., 1985), indicates multiple values ( ⁄ =0.35 to
0.47) for different bottom slopes in order to get a curve that fits
8
10
12
14
best the experimental results. It is expected that at the end of the
Figure 9 – Comparison between empirical expression and experimental
data for incident wave with 1.5 s period and 16 cm wave height.
surf section the relative wave height reveals values close to
The obtained results show that, in spite of a certain degree of
⁄ =0.35 to 0.40, which is the value for an horizontal bottom case.
dispersion of the acquired data, it clearly shows a similar tendency
Relative Wave Height (H=12 cm)
to the theoretical expression, represented by the orange line. Also,
1.2
it is important to note that the blue data dots are comprehended
1.0
Hs/h (-)
0.8
T11H12
T15H12
0.6
T20H12
0.4
T25H12
between the breaking zone point and the end of the 1:20 bottom
slope. The subsequent points, in the horizontal slope area, were
not used in this analysis.
0.2
0.0
-1000
4.5. Breaking analysis
-800
-600
-400
-200
x (cm)
0
200
400
600
Figure 8 – Relative have height throughout the channel, for incident
waves with 12 cm wave height.
Figure 8 shows, for the 4 incident wave conditions with
=12
cm, the relative wave height evolution from the top of the bar until
the end of the surf section. It can be concluded that the behavior of
⁄ has two phases. Before the wave breaking zone the curve is
steeper, increasing rapidly, while after the wave breaking, there is
a decrease of
⁄ and the slope presents a smoother curve. After
breaking, this value steadily decreases, with the exception of the
x=0 cm area (bottom slope changing zone), where it momentarily
increases. After that, it decreases until it remains almost constant
in the end of the breaking zone. For these tested incident waves,
the rate of
⁄ at the end of the wave breaking section is between
Table 4 – Wave breaking type for the fifteen different incident waves.
Incident wave
[-]
T11H12
T11H14
T11H16
T15H12
T15H14
T15H16
T15H18
T20H12
T20H14
T20H16
T20H18
T25H12
T25H14
T25H16
T25H18
[m]
1.888
3.510
6.240
9.750
[m]
0.180
0.196
0.232
0.199
0.220
0.241
0.274
0.236
0.295
0.302
0.314
0.273
0.276
0.313
0.323
[-]
0.095
0.104
0.123
0.057
0.063
0.069
0.078
0.038
0.047
0.048
0.050
0.028
0.028
0.032
0.033
[-]
0.162
0.155
0.143
0.210
0.200
0.191
0.179
0.257
0.230
0.227
0.223
0.299
0.297
0.279
0.275
Breaking
type
Spilling
Spilling
Spilling
Spilling
Spilling
Spilling
Spilling
Spilling
Spilling
Spilling
Spilling
Spilling
Spilling
Spilling
Spilling
7
As expected, all the incident waves had a surf similarity
parameter inside the range of spilling wave breaking, essentially
base. This data was compared to our own values and Figure 11
shows the comparison.
characterized by very low slopes, as was the case of this particular
Relative breking height vs Hb/g.T2
wave channel (1:20).
1.6
According to an author (Goda, 1985), we can easily observe
1.4
shoaling over a sloping bottom and breaks at a certain depth. The
location at which waves break is almost fixed for regular waves,
Hb/hb (-)
that a train of regular waves in a laboratory flume undergoes
1.2
T1.1
T1.5
1.0
and there is a distinct difference between the oscillatory wave
T2.0
0.8
motion before breaking and the turbulent wakes with air
entrainment after breaking. The terminology wave breaking point,
T2.5
0.6
0.000
depth and height is employed do denote the location, water depth,
0.004
0.008
and height of wave breaking, respectively. The expression “limiting
breaker height” is sometimes also used, in the sense of the upper
limit of progressive waves physically possible at a certain water
0.012
0.016
0.020
Hb/(g.T2) (-)
Figure 11 – Relative breaking height (Derived from Corps of Engineers,
2003).
Again we have a tendency pretty much the same as the wave
depth for a given wave period. The ratio of limiting breaker height
to water depth depends on the bottom slope and the relative water
channel slope (1:20, i.e.,
=0.05). However, to a certain extent,
depth. Compilation of a number of laboratory results has yielded
we still have a discrepancy in the larger periods, in this case,
the design diagram of Figure 10 as an average relation, although a
bigger or equal to 2.0 s.
scatter in the data of more than 10% must be mentioned. Using
Wanting to make an analysis of the wave transformations
our experimental results and putting them together with Goda’s
occurring after the breaking point, (Horikawa & Kuo, 1966) made a
data, we reached the results presented in Figure 10.
stud with the objective of presenting an approach to this problem,
having analytical and experimental treatments as a backup.
Relative breking height vs hb/L0
Having some results for the same slope was we had, 1:20, we
1.4
tried to see the differences and similarities, as illustrated in Figure
1.2
T1.1
Hb/hb (-)
1.0
12.
T1.5
0.8
0.6
T2.0
0.4
T2.5
Normalized wave heigh vs normalized depth
1.2
0.2
1.1
0.0
0.00
0.01
0.10
1.0
1.00
hb/L0 (-)
0.9
0.8
Figure 10 – Relative breaking height (Derived from Goda, 1985).
similar arrangement to the author’s results, in the sense that, for
bigger periods and respective wave heights, it exist a curve with an
H/Hb (-)
The obtained results say that, in relative terms, our data had a
T2.0H12
0.7
T2.0H14
0.6
T2.0H16
0.5
T2.0H18
0.4
0.3
approximate tendency to his values. However, only for the incident
0.2
waves with the smallest period, 1.1 s, subsists a great
0.1
synchronism with the existing slope (1:20). The other incident
0.0
waves, with bigger periods, had the tendency to move away.
Weggel (1972) presented a reevaluation of some breaking
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
h/hb (-)
Figure 12 – Normalized wave height and depth, for a bottom slope of
1:20 (Derived from Horikawa & Kuo, 1966).
waves studies, in a way to establish some kind of norms for
monochromatic waves, with gentle slopes. His studies implied a
great number of theoretical and experimental data, therefore
having a great range of results which, by not being exactly the
same, enabled a broader vision of the phenomenon and of the
parameters he wished to find. Later on, a revision of is results was
made (Corps of Engineers, 2003), having Weggel’s studies as a
For this analysis, we had a slight difference in the wave
periods, 2.2 versus 2.0 s, which had to be count on in the
beginning. Despite this, it is visible that the data had a very good
correlation with the author’s data. Being a graphic that evolves
form the right to the left, we can see in the first phase a concavity
leaning down, to a concavity leaning up, with an inflexion point in
the middle of them, a kind of transition area.
8
Relative wave heigth vs normalized depth
Standard deviation (Incident wave T=1.5 s e H=18 cm)
Standard deviation (cm)
5.0
1.5
H/h (-)
T2.0H
12
T2.0H
14
T2.0H
16
1.0
4.0
3.0
2.0
1.0
0.0
-1000
-800
-600
-400
-200
x (cm)
0
200
400
600
Figure 15 - Standard deviation for an incident wave with 1.5 s period
and 18 cm wave height.
The standard deviation (Figure 15) essentially reproduces the
same as the wave significant heights: there is slightly increase
0.5
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
related with the depth decreasing (shoaling effect) until x=-470 cm,
h/hb (-)
Figure 13 – Relative wave height, for a bottom slope of 1:20 (Derived
from Horikawa & Kuo, 1966).
In the case of the analysis in Figure 13, we had contradictory
more or less. During the wave breaking section, the standard
deviation decreases significantly until right around x=100 cm,
where the standard deviation is almost constant.
Skewness (Incident wave T=1.5 s e H=18 cm)
data against the author’s results. It is clear our data has a
1.8
tendency to descend, instead of going up. The great difference lies
1.4
normalized depth, which necessarily must come down as we move
forward in the channel, with lower depths, , in the vertical axis our
Skewness (-)
is in the vertical axis. While in the horizontal axis we have the
values have a tendency to decrease. This is because the wave
1.0
0.6
0.2
-0.2
height decreases at a bigger rate than the local depth.
-1000
4.6. Statistical analysis
-800
-600
-400
-200
x (cm)
0
200
400
600
Figure 16 – Skewness for an incident wave with 1.5 s period and 18 cm
wave height.
The performed statistical analysis of the free surface elevation
Kurtosis (Incident wave T=1.5 s e H=18 cm)
records corresponds to the average, standard deviation, skewness
channel of the above mentioned parameters for an incident regular
wave of =1.5 s and
=18 cm.
Average (Incident wave T=1.5 s e H=18 cm)
Average (cm)
2.0
0.0
Kurtosis (-)
and kurtosis. The next figures present the evolution throughout the
2.5
2.0
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
-2.0
-1000
-800
-600
-400
-200
x (cm)
0
200
400
600
-2.0
-4.0
Figure 17 – Kurtosis for an incident wave with 1.5 s period and 18 cm
wave height.
-6.0
-8.0
-10.0
-1000
In Figures 16 and 17, skewness and kurtosis values behave
-800
-600
-400
-200
x (cm)
0
200
400
600
Figure 14 – Average for an incident wave with 1.5 s period and 18 cm
wave height.
similarly. Initially, both present small values and begin to increase
until the wave breaking point. During the surf section, the
skewness and the kurtosis have great variability. After that, they
begin to rise significantly (approximately at x=0 cm) and fall again
Figure 14 presents the average values of the free surface
returning to values closer to the initial range (at x=400 cm), most
elevation related to the Mean Water Level (MWL). A slight set-
probably due to the end of the generated turbulence of the wave
down is observed between the x=-900 cm and x=-800 cm, before
breaking process.
increasing steadily until the wave breaking begins. After the
breaking point, the average free surface starts to decrease until
x=200 cm, where it starts to stabilize. Around x=400 cm the
average free surface begins to increase.
9
4.7. Spectral analysis
The next set of Figures, ranging from Fig. 18 to Fig. 25, shows
some of the calculated spectra for the incident wave condition of
=2.5 s and
=18 cm, at the positions, x=-1000 cm, x=-500 cm,
x=0 cm, and x=400 cm. They were the selected positions because
they represent 4 distinct moments in the wave evolution: (i)
generated wave with few interferences, (ii) wave before breaking
Figure 21 – Fourier analysis of incident wave with 2.5 s period and 18
cm wave height. Position x=-500 cm.
zone, (iii) wave after breaking zone and (iv) wave outside of
breaking zone, near to the end of its cycle.
It is important to note an important fact: to ensure that the
wavelet transforms at each scale are directly comparable to each
other and to the transforms of other time series, i.e., Fourier, the
wavelet function at each scale is normalized to have unit energy.
To make it easier for the wavelet spectra comparison, it is
desirable to find a common normalization. For white noise time
series, the expected value of the wavelet transform is the variance,
i.e.,
(Torrence & Compo, 1998).
Figure 22 – Wavelet analysis of incident wave with 2.5 s period and 18
cm wave height. Position x=-150 cm.
Figure 23 –Fourier analysis of incident wave with 2.5 s period and 18
cm wave height. Position x=-150 cm.
Figure 18 – Wavelet analysis of incident wave with 2.5 s period and 18
cm wave height. Position x=-1000 cm.
Figure 19 – Fourier analysis of incident wave with 2.5 s period and 18
cm wave height. Position x=-1000 cm.
Figure 204 – Wavelet analysis of incident wave with 2.5 s period and 18
cm wave height. Position x=400 cm.
Figure 25 – Fourier analysis of incident wave with 2.5 s period and 18
cm wave height. Position x=400 cm.
The results for the spectral analysis show that as the wave
Figure 20 – Wavelet analysis of incident wave with 2.5 s period and 18
cm wave height. Position x=-500 cm.
propagates along the channel, there is an increasingly number of
harmonics and a strong reduction in the amplitude of the main
frequency.
Comparatively, it is possible to observe that after the wave
breaking, the spectrum energy of the main frequency decreases to
the point of being more similar to the other generated frequencies.
10
5. Conclusions
References
In this paper, recent physical modeling tests on a wave
channel from the National Laboratory of Civil Engineering in Lisbon
(LNEC), Lisbon, Portugal, were presented. The tests aimed mainly
to introduce a new analysis based on the directional spread of the
wave on a
wave channel
to study the wave breaking
hydrodynamics on complex bathymetries, namely bathymetries
with variable bottom slopes. This work represented a step forward
to understand and better define the wave breaking process
throughout the surf section since the beginning till the very end,
considering incident wave conditions.
The physical modelling was performed on a wave channel,
built for wave propagation studies, with a bottom characterized by
a ramp with 1:20 and then by a zone with horizontal slope,
respectively. The tested waves resulted from a combination of 1.1
s, 1.5 s, 2.0 s and 2.5 s periods with wave heights of 12 cm, 14
cm, 16 cm and 18 cm. The measured data from the resistive
gauges (free surface elevation) enabled a time, spectral and
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From the physical model (wave channel), a wide set of wave
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appear as important parcels, stacked to the results. These
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quality to draw relevant conclusions for the proposed study.
Future work, concerning a more complete analysis of the
breaking criteria here presented and the comparison with other
parameters for the wave breaking definition will be a sure plus. A
further analysis on the wave breaking physical process is also
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