Focused waves onto a plane beach
Physical modelling in the U.K. Coastal Research Facility
Alison Hunt, Paul Taylor and Alistair Borthwick
Department of Engineering Science, University of Oxford, UK
Peter Stansby and Tong Feng
Centre for Civil and Construction Engineering, UMIST, Manchester, UK
Focused wave groups and the UKCRF
Introduction
This poster demonstrates some preliminary results from a collaborative project between
Oxford University and UMIST carried out in the UK Coastal Research Facility (UKCRF). The
aim of the research was to investigate the interaction of steep focused wave groups with a
plane beach and a sea wall.
Background
Waves change significantly as they propagate inshore. The most extreme event in a
random sea is atypical of the waves in that sea. Thus, the modifications to a large but
localised wave group as it advances inshore, particularly on the set-up and crest elevation is
more relevant for the prediction of sea defence overtopping and lee shore flooding than the
behaviour of regular waves. It is likely that small changes to the incident wave group affect
the surface set-up and wave height and have a significant effect on coastal flooding.
Major advances have been made in the past decade regarding the temporal and spatial
profiles of the most extreme waves in unidirectional and spread deepwater seas. One
contribution has been the so-called NewWave concept of a focused wave group (Tromans
et al. (1991), Jonathan and Taylor (1997), and Taylor and Haagsma (1994)). In the region
of the largest wave crest, the focused wave group has a profile proportional to the autocorrelation function (Lindgren (1970)). By a similar methodology, the shape of the group
about the deepest trough is the inverted auto-correlation function.
UKCRF with sea wall in place
Experimental Set-up
The UKCRF is a 27 m cross-shore by 36 m alongshore multi-element wave basin. The
mean water level at paddles is 0.5 m. Instrumentation includes an array of up to 32 wave
gauges, a 2D laser Doppler anemometer (LDA), several acoustic Doppler velocimeters, and
two video cameras.
Tromans, P.S., Anaturk, A.R., and Hagemeijer, P. (1991) A new model for the kinematics of large ocean
waves – application as a design wave. Proc. 1st Int. Offshore and Polar Eng. Conf., Edinburgh, U.K., 11-16 August,
64-71.
Jonathan, P. and Taylor, P.H. (1997) On Irregular, Nonlinear Waves in a Spread Sea. J. Offshore Mechanics
and Arctic Engineering, 119, 37-41.
Taylor, P.H., and Haagsma, I.J. (1994) Focussing of steep wave groups on deep water. Proc. Int. Symposium:
Waves – physical and numerical modelling, Vancouver, Canada, 21-24 August, 862-870.
Baldock, T.E, Swan, C., and Taylor, P.H. (1996) A laboratory study of nonlinear surface waves on water. Phil.
Trans. R. Soc. Lond. A 354: 649-676.
Lindgren, G. (1970) Some properties of a normal process near a local maximum. Ann. Math. Stat. 41, 1870-1883.
Development of spread sea focused wave
Focused wave groups and the UKCRF
NewWave
A focused wave group has individual wave components that come into phase at a
particular time and place. NewWave theory can be used to generate an extreme wave
group that is representative of a real event in a random sea state by the use of a
realistic energy spectrum, such as Pierson-Moskowitz, as follows:
NewWave = AN
∫ S (ω ) cos(ωt )dω
×
∫ S (ω )dω
where S(ω) is the desired sea spectrum, ω is the frequency and AN is the
linear crest amplitude:
AN = (2m0 (ln N ))
1/ 2
where mo is the first moment of the underlying spectrum and N is the number of
waves.
Experimental results and analysis
Surface elevation results for a single extreme wave group are shown in this poster.
The wave group was the largest event generated in the UKCRF - a Pierson Moskowitz
spectrum with an ωp of 2.91 rad/s and a NewWave crest amplitude of 114mm.
Two sets of results are shown – those where the crests of the individual wave
components coincided – crest focused – and those where the troughs coincided –
trough focused. The surface elevation of a crest focused group is given the symbol η
and the trough focus the symbol η* (after Baldock et al 1996).
The crest focus is composed of free and bound waves:
η = η f + ηb
The free wave is described:
∞
η f = ∑ an cos (k n x − ω nt + φn )
n =1
Where an is the wave component amplitude, k is the wave number, ω is the
frequency and φ is the phase angle.
Assuming a Stokes-like expansion, the bound wave can be represented as:
∞
∞
∞
∞
∞
( 3)
η b = ∑ ∑ a n a m Fnm( 2 ) + ∑ ∑ ∑ a n a m a l Fnml
+ ...
n =1 m =1
n =1 m =1 l =1
where F are the expansion coefficients.
By the addition and subtraction of the crest and trough focused waves it is
possible to separate the free and bound wave components of the
experimental data:
∞
1
2
∞
∞
( 3)
η diff = (η − η * ) = η f + ∑∑∑ an am al Fnml
+ ...
1
2
n =1 m =1 l =1
∞
∞
η sum = (η + η * ) = ∑∑ an am Fnm( 2 ) +...
n =1 m =1
Furthermore, if the data is bandpass filtered, the underlying locally linear structure
of the group can be separated from the various bound harmonics.
Conclusions
Preliminary investigations of sequential wave gauge data shows:
• the evolution of different terms of the Stokes expansion
• the reflection of the ingoing low frequency bound wave as a long free wave
which reverberates across the tank
• the long free wave is dependent on the wave group envelope only i.e. both crest
and trough focused wave groups produce the same free wave.
Crest focus wave groups
Cre s t foc us time s e ries
TimeTime-history (mm)
Amplitude spectra (mm)
6
MEAN SHORE-LINE
100
4
4
0
2
-100
40
3
100
LOCATION 3 – d=0.375m
20
s urface elevation (mm)
LOCATION 1 – d=0.5m
Channel No.
25
0
0
1
2
0
1
2
0
1
2
0
1
fre quency (Hz )
2
4
2
30
-100
BEACH TOE
60
0
i.e. HALF DEPTH
LOCATION 2 – d=0.5m
50
6
35 100
LOCATION 4 – d=0.25m
40
40
50
60
0
6
2
4
0
2
15-100
40
50
60
0
6
10 100
100
0
5
-100
0
40
45
50
55
60
65
time (s ec )
70
75
80
85
90
4
0
-100
35
1
2
40
50
time (s e c)
60
0
Trough focus wave groups
Trough foc us time s e rie s
TimeTime-history (mm)
Amplitude spectra (mm)
6
MEAN SHORE-LINE
100
4
4
0
2
-100
40
40
50
60
0
0
1
2
0
1
2
0
1
2
0
1
frequency (Hz)
2
6
35 100
LOCATION 4 – d=0.25m
i.e. HALF DEPTH
30
3
4
0
2
-100
100
LOCATION 3 – d=0.375m
BEACH TOE
LOCATION 1 – d=0.5m
20
s urface elevation (mm)
LOCATION 2 – d=0.5m
Channel No.
25
15
40
50
60
0
6
2
4
0
2
-100
40
50
60
0
6
10 100
100
0
5
-100
0
40
45
50
55
60
65
time (s ec )
70
75
80
85
90
4
0
-100
35
1
2
40
50
time (s ec)
60
0
‘Sum’ series – even harmonics
Cres t and trough S UM time s e ries
TimeTime-history (mm)
Amplitude spectra (mm)
6
MEAN SHORE-LINE
100
4
4
0
2
-100
40
40
50
60
0
0
1
2
0
1
2
0
1
2
0
1
frequency (Hz)
2
6
35 100
LOCATION 4 – d=0.25m
i.e. HALF DEPTH
30
3
4
0
2
-100
100
LOCATION 3 – d=0.375m
BEACH TOE
LOCATION 1 – d=0.5m
20
s urface elevation (mm)
LOCATION 2 – d=0.5m
Channel No.
25
15
40
50
60
0
6
2
4
0
2
-100
40
50
60
0
6
10 100
100
0
5
-100
0
40
45
50
55
60
65
time (s ec )
70
75
80
85
4
0
-100
35
1
2
40
50
time (s ec)
60
0
90
2nd order difference
(set-up/down)
2nd order
sum
‘Difference’ series – odd harmonics
Note the phase distortion of the 3rd
harmonic compared to linear – due to
shallow water effects
Cres t and trough DIFFERENCE time s eries
TimeTime-history (mm)
Amplitude spectra (mm)
6
MEAN SHORE-LINE
100
4
4
0
2
-100
40
40
50
60
0
0
1
2
0
1
2
0
1
2
0
1
frequency (Hz)
2
6
35 100
LOCATION 4 – d=0.25m
i.e. HALF DEPTH
30
3
4
0
2
-100
100
LOCATION 3 – d=0.375m
BEACH TOE
LOCATION 1 – d=0.5m
20
s urface elevation (mm)
LOCATION 2 – d=0.5m
Channel No.
25
15
40
50
60
0
6
2
4
0
2
-100
40
50
60
0
6
10 100
100
0
5
-100
0
40
45
50
55
60
65
time (s ec )
70
75
80
85
4
0
-100
35
1
2
40
50
time (s ec)
60
0
90
linear group 3rd harmonic
sum
‘Sum’ series – filtered at 0.5Hz
Low pass filtered
High pass filtered
40
40
35
35
30
30
10
0
5
-100
0
35
40
45
50
55
60
time (s )
65
70
75
80
20
s urface elevation (mm)
s urface elevation (mm)
15
100
25
15
100
10
0
5
-100
0
35
40
45
50
2nd order
bound waves
55
60
time (s )
65
70
75
80
Channe l No.
20
Channe l No.
25
‘Difference’ series – filtered at 1Hz
Low pass filtered
High pass filtered
40
40
35
35
30
30
10
0
5
-100
0
35
40
linear
shorewards
group
45
50
55
60
time (s )
65
70
no significant
reflection
75
80
20
s urfac e elevation (mm)
s urface ele vation (mm)
15
100
25
15
100
10
0
5
-100
0
35
40
45
50
55
60
time (s )
local 3rd order
bound waves
65
70
75
80
Channel No.
20
Channel No.
25