Reprintedfrom Coastal Dynamics Proceedings of the

International ConferenceonCoastal Research

in Terms of Large Scale Experiments pp

Gdansk Poland September

Lo oking for Wave Groups in the Surf Zone

Merrick C Haller and Rob ert A Dalrymple

Abstract

This pap er prop oses two new parameters temp oral group steepness

S and spatial group steepness S for evaluating wave grouping The

t x

new parameters provide information on two imp ortant qualities of groups

the amplitude variation of the groups and their length b oth of which

have an impact on coastal problems Computation of the parameters

involves the use of the Hilb ert transform to determine the waveenvelop e

of incidentwaves The Hilb ert transform metho d is reviewed and shown

to b e sup erior to other metho ds involving or j j and the new parameters

are shown to b e resp onsivetochanges in the wave eld due to shoaling

and wave breaking

The new parameters are applied to eld data to study the variabilit y

of wave grouping b oth spatially in the crossshore direction and temp o

rally at xed lo cations Wave grouping is shown to p ersist through the

surf zone indicating the necessity for the inclusion of nonsteady radiation

stress gradients shoreward of the mean break p oint in mo dels of long

wave generation In addition the temp oral change in group steepness

is shown to correlate with the increase in longp erio d surf zone current

mo dulations

Intro duction

The imp ortance of wave groups is well recognized by the engineering community

Wave groups can have a serious eect on the stability of rubble mound structures

Johnson et al and pip elines Dean Wave groups inuence the

resp onse of ships mo ored at sea Hsu and Blenkarn Pinkster and

can excite strong harb or oscillations Bowers Wave groups are also

Graduate student Center for Applied Coastal Research UniversityofDelaware Newark

DE USA email merrickcoastaludeledu

Professor and Director Center for Applied Coastal Research University of Delaware

Newark DE USA email radcoastaludeledu

utilized as a forcing mechanism in mo dels for long waves b oth oshore Longuet

Higgins and Stewart and in the nearshore Tucker Symonds et

al and others More recently lab oratory and eld exp eriments have

identied wave groups as a source of edge waves Bowen and Guza rip

currents Tang and Dalrymple and p ossibly shear waves Hamilton et

al

This pap er compares metho ds of identifying wave groups that are commonly

used to day and identies some of the problems and inadequacies of existing

metho ds Amethodofwave group detection involving the Hilb ert transform

is describ ed and shown to have sup erior characterisics to comp eting metho ds

The metho d has the advantage of b eing utilized strictly in the frequency domain

Current parameters used to describ e the amountofwave grouping are evaluated

and shown to b e of limited usefulness A new parameter group steepness is

intro duced and evaluated for its usefulness in measuring groupiness in real seas

In contrast to most other studies whichhave fo cused on grouping in deep

water this pap er will analyze grouping in the surf zone If the dynamics of the

surf zone are expressed bya waveaveraged equation suchas

u S

xx

g

t x h x

where u is the crossshore comp onentofvelo city is setup and S is the cross

xx

shore comp onent of radiation stress and dep endent on the lo cal waveheight

then it is evidentthatwave groups p ersisting into the surf zone will lead to long

wave generation and longp erio d current mo dulations It will b e shown that

the group steepness in the surf zone has some correlation with observed current

oscillations

Identication of Wave Groups

In order to quantify the imp ortance of wave groups in the eld a wave group

detector must b e employed Wave group detectors taketwo forms One form

involves counting discrete waves via a zero crossing metho d Runs of successive

waves higher than an arbitrary threshold wave heightareidentied and consid

ered a wave group or a run of waves In this manner wave groups are treated

as isolated events

The second general form of wave group detector involves waveenvelop e the

ory Various techniques are used to determine a con tinuous measure of wave

heightH orofwave energy E H In this waywave groups are treated

as oscillations ab out a mean value and they can b e analyzed in the frequency

domain This is the form of wave group detection that will b e utilized herein

SIWEH

One of the well known wave group detectors is the Smo othed Instantaneous

Wave Energy History SI W EH asdenedby Funke and Mansard The

SI W EH is a running time series of waveenergythatisaveraged over the p eak

wave p erio d by means of a smo othing Bartlett lter The SI W EH is found by

the following

Z

E t t Q d

T

p

where E t is the energy envelop e function T is the p eak sp ectral p erio d of the

p

incidentwaves and Q is a Bartlett window smo othing function as dened

by

j jT j j T

p p

Q

j j T

p

A similar metho d the Lo cal Variance Time Series was prop osed by Thomp

son and Seelig It also involves squaring the surface displacement and

smo othing over the p eak p erio d There are several diculties inherent to b oth

metho ds First the metho ds contain no provision for removing low frequency

waves from the time series unrelated to the windswell record In the nearshore

region where low frequency waves b ecome highly energetic the energy time

series will b e unduly inuenced bysuch motions Another diculty arises from

the squaring of the surface displacement record t This can b e easily shown

by rst expressing the wave eld as a Fourier series expansion given as

N

X

t a cos t

i i i

i

where a and are the angular frequency amplitude and random phase of

i i i

the individual waves resp ectively Then asshown by Naess contains

four distinct terms as follows

a mean term prop ortional to a

i

dierence interaction terms with frequencies

i j

sum interaction terms with frequencies j

i

harmonic terms with frequencies

i

It is the rst and second terms which are related to wave grouping Thus a

ltering op eration is required to remove the extraneous terms created in the

squaring pro cess In addition the ltering op eration is based on knowledge

of the p eak p erio d which can b e essentially arbitrary in a widebanded sea

ave group Medina and Hudspeth demonstrate that the determination of w

parameters eg GF using these metho ds b ecomes highly dep endent on the

ltering op eration In addition these metho ds involve p erforming b oth envelop e

detection and ltering in the time domain which requires heavy CPU time

Mo dulus Metho d

Instead of using a time series of wave energy to determine wave group parame

ters the envelop e signal can b e used directlyThewaveenvelop e is essentially

a time series of instantaneous wave amplitude List prop osed using the

mo dulus of the measured surface displacement j tj to nd the waveenvelop e

The metho d can b e describ ed as follows rst t is ltered to remove the low

est frequencies usually Hz asso ciated with infragravitybandwaves

Next the mo dulus of the resulting time series is determined and multipliedby

a normalization factor The resulting series At j tj contains terms

and from the list ab ove Again a lowpass lter must b e employed on At

in order to isolate the dierence interaction terms which are directly related to

amplitude mo dulations of the incidentwave eld

Hilb ert Transform

An alternative to b oth of the previous metho ds involves the Hilb ert transform

and can b e done quickly and easily in the frequency domain The use of the

Hilb ert transform in envelop e theory has its origins in the seminal pap er on

noise in electrical circuits by Rice Melvil le was the rst

to apply the metho d to deep water wave mo dulation and subsequentwork was

done by Medina and Hudspeth among others The Hilb ert transform can

b e applied in either the time or frequency domains here we p erform almost all

op erations in the frequency domain utilizing Fast Fourier Transform routines

requiring very little CPU time

In the time domain an analytic signal z t can b e dened from a given time

series t and its corresp onding Hilb ert transform t

z t t i t

In the frequency domain the relation b etween z tand t is dened in terms

of their Fourier transforms as

F for f

F f F f for f

z

for f

and so

z t F F f

z

Once the analytic signal is computed we can determine time series of wave energy

or wave amplitude from whichwe can quantify wave grouping The denition

are as follows

Atjz tj



E tzz jz tj

The most signicant advantage to the Hilb ert transform metho d is that the

energy time series contains only two terms

a mean term prop ortional to the variance of t

dierence interaction terms

i j

Thus the squared envelop e function as determined by the Hilb ert transform

metho d directly isolates the mean and the dierence interaction terms with

out ltering The waveenvelop e At gives a direct measure of amplitude

mo dulations that are o ccuring at all frequencies Narrowbanded seas will have

narrowbanded wave group sp ectra S f The advantage here is that S f

A A

only contains amplitude mo dulations Hence for widebanded seas with wide

banded group sp ectra if ltering is applied to S f only high frequency wave

A

groups are removed eg T s and only low frequency wave groups remain

Quantifying Groupiness

Run Length

Perhaps the most commonly used parameter to quantify wave groupiness is the

mean run length whichistheaverage number of waves p er wave group with wave

groups b eing dened as a sequence of waves whose heights exceed a threshold

value Goda related wave group statistics to sp ectral information in

an attempt to achieve predictability Kimura extended Go das theory by

assuming that a sequence of wave heights can b e treated as a Markovchain which

allows for correlations b etween successivewaves Signicant improvements have

b een made since eg Battjes and Van Vledder and the metho d do es a

go o d job predicting run lengths in deep water However the run length itself is

basically a measurement of groups in one dimension It do es not give an explicit

description of the amplitude of the wave groups

Groupiness Factor

As opp osed to the run length the Groupiness Factor is a parameter that mea

sures the amplitude mo dulation of the incidentwave eld As originally dened

by Funke and Mansard the GF in terms of the SI W EH function E t

is as follows

p

m

GF

SI W EH

m

where m and m are the zeroth moment of the SI W EH sp ectrum S f

E

and the sp ectrum of the incidentwaves S f resp ectively In this way the

GF relates the variance of the wave energy to the variance of the underlying

pro cess Hudspeth and Medina p oint out that for narrowbanded linear

waves if the dierence frequencies are exactly isolated in the energy time series

then S f can b e approximated by the following

E

S f m f

E

E t m

where f istheenvelop e sp ectral density function as is dened by

Z

S x f S xdx f

m

It can b e easily shown that the GF computed from Eq will b e approximately

unity Any deviations from unity can b e attributed to the ltering op eration

applied to E t or to deviations from linearity

List suggests a similar GF dened in terms of an amplitude time

series as

p

A

GF

At

where and At are the standard deviation and the mean of the amplitude

A

function At resp ectively Again for narrowbanded linear waves the sp ectrum

of At can b e approximated by

m f S f

A

p

meanA m

In a similar fashion the GF determined from Eq will reduce to a constant

For these reasons the GF has limited ability to describ e wave grouping

Group Steepness

In the following we dene a new parameter group steepness for quantifying

wave grouping Group steepness combines the amplitude of wave mo dulation

and the number of waves p er group in a matter similar to the ka parameter

which is a measure of wave steepness The parameter as dened herein is not a

predictive parameter but rather a deterministic quantitywhich can b e applied

to wave data The metho d of computing the group steepness is as follows

removelow frequencies from wave time series Hz

compute At with Hilb ert transform Eq

select wave group frequency band of interest eg f Hz

compute sp ectrum of S f

A

compute group steepness S or S

x t

The group steepness can b e dened spatially by the following

A

S

x

L

g

p

m L is where is the standard deviation of At dened sp ectrally as

A g A

the mean group wavelength as dened by smallamplitude wave theory

L nL T T c

g c g

where n C C is the ratio of phase sp eeds T m m is a sp ectral

g g A A

estimate of the mean group p erio d and T m m is a sp ectral esti

c

mate of the mean wave p erio d L is the mean wavelength found from T and

c c

the disp ersion relationship The spatial steepness parameter can b e applied to

remotely sensed data where the data has a known spatial distribution

The temp oral Group Steepness S is dened as

t

A

S

t

gT

g

This dynamic parameter is a measure of the amplitude of the wave groups and

their mean p erio d A widebanded sea with essentially at groups will have large

mean group p erio d and hence small S Moreover this parameter can evaluate

t

narrowbanded seas with similar mean group p erio ds by distinguishing b etween

highamplitude groups and lowamplitude groups It is exp ected the parameter

can b e applied to mo ored structures and will applied here to analyze the forcing

of surf zone current mo dulations

a b 4 4

2 2

0 0 cm

−2 −2

−4 −4

S  S 

t t

S  S  x

−6 x −6

GF GF

0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350

time sec time sec

Figure Grouping parameters for synthetic waves in a m b m depth

To test the viability of the group steepness parameter Figure a and b com

pares the group steepness and GF in simulated waves Figure a shows a well

mo dulated synthetic wave train in deep water If the waves shoal according to

linear theory their height will increase and visually their grouping will b ecome

more pronounced In Figure b the same synthetic waves are shown in two

meters depth and S has increased by while the Groupiness Factor is essen

t

tially unchanged Since the wavelength of the waves has also shortened

considerably the spatial steepness has increased the most

Field Data Analysis

Two eld data sets were analyzed The rst set was obtained during the SU

PERDUCK exp eriments conducted by the US Army Corps of Engineers in

Octob er of at the Field ResearchFacilityatDuck NC Crowson et al

The b each prole was characterized by a linear bar system with the in

nermost bar lo cated approximately m oshore Pressure data was recorded by

a crossshore array of b ottommounted pressure sensors lo cated in depths rang

ing from to m of water Pressure data was converted to surface displacement

using linear theory In addition current records were obtained by an alongshore

array of MarshMcBirney bidirectional electromagnetic current meters lo cated

approximately m from the shoreline in the trough of the innermost bar Thor

ton and Guza The wave elds considered here generally consist of long

p erio d swell from the south along with highenergy windgenerated waves from

the north Data for all sensors were sampled at Hz during hour measuring

p erio ds centered ab out high and low tides b a 50

0 2.5

−50 2000 2100 2200 2300 2400 2500 2600 2700 2800 2900 2

50

0 1.5 cm

−50

2000 2100 2200 2300 2400 2500 2600 2700 2800 2900 1

S t

50

S 0.5 x

0

o GF −50 0

2000 2100 2200 2300 2400 2500 2600 2700 2800 2900 50 100 150 200 250 300 350 400

time sec oshore dist m

Figure a Wave records at crossshore p osition m top m middle

and m b ottom b Grouping parameters vs crossshore distance param

eters are normalized by their oshore value NSTS Santa Barbara

Signicantwave breaking o ccured at m oshore

The second data set was obtained during the Nearshore SedimentTransp ort

Study conducted at Leadb etter Beach Santa Barbara California The

site consists of near planar b each top ographyThorton and Guza its

orientation is such to encourage incidentwaves with high angles of incidence

Wave records were obtained from a crossshore array of pressure sensors m

in a similar fashion to the previous data set

Figures a and b demonstrate the change in grouping parameters with shoal

ing for real wave data The shoaled waves at xm a middle are mostly

unbroken and their grouping is more pronounced than at xm a top

Corresp ondingly S has increased Since the wavelegths of the waves have

t

shortened during shoaling while the number of waves p er groups has not changed

signicantly S has increased to of its oshore value Groupiness Factor

x

has increased At the most inshore p osition x m a b ot

tom waves are mostly broken and app ear p o orly group ed Here S is

t

of its oshore value S remains twice its oshore value and GF is essentially

x

unchanged from oshore

100 100

a b

50 50

0 0 cmsec

cmsec −50 −50 u u

−100 −100

−150 −150

5 6 7 8 9 10 11 12 13 14 15 200 201 202 203 204 205 206 207 208 209 210

time min time min

Figure Crossshore velo city records at athr bthr current meter LX SUPERDUCK

50 200

a b

40 150

30 100 20

50 10

0 0 cm cm

−10 −50

−20 −100 −30

−150 −40

−50 −200

10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15 199 199.5 200 200.5 201 201.5 202 202.5 203 203.5 204

time min time min

Figure Wave record at athr bthr pressure gage LA

SUPERDUCK Note scale change

Figures and show the crossshore current record measured in the inner surf

zone and the oshore wave record at two dierent times during data collection

on Octob er at SUPERDUCK The current record indicates a spinup

of a longp erio d O s current oscillation that can b e termed a shear wave

or large scale eddy in the nearshore circulation OltmanShay et al The

wave records in Figure a and b demonstrate an increase in waveheight and

wave grouping as a storm blew in from the northeast Figure compares the

variation of grouping parameters with time The values of GF showed signicant

net increase from thr to thr only at the oshore gage while the GF at

all the surfzone gages showed a net decrease Both group steepness parameters

increased at all gages with the oshore gage showing the largest increase It is

suggested that the signicant increase of group steepness in the surfzone over

time which corresp onds to the growing longp erio d mo dulations of the wave

eld indicates that wavegroupsplay a role in the inducement of longp erio d

current oscillations or shear waves measured in the surfzone

S GF S

t x

2 8 8

xm xm

1.8 7 7

1.6 6 6 m

1.4 xm 5 5

1.2 m

4 4 m 1 m

3 3 m

0.8 m 2 2

0.6 m m

1 1 m 0.4

1 2 3 1 2 3 1 2 3

time hr

Figure Grouping parameters vs time All parameters are normalized by the

value at oshore gage LA at thr Distance of sensor from origin is also

given SUPERDUCK

Conclusions

Two parameters are intro duced to quantify wave grouping by utilizing informa

tion regarding mo dulation of the incidentwave eld and its spatial and temp oral

scales The parameters are shown to b e more resp onsivethan GF to changes in

the wave eld due to shoaling and breaking The parameters are based on using

the Hilb ert transform to determine the waveenvelop e from the wave record

This metho d is shown to b e sup erior to other metho ds involving and j j in

b oth accuracy and sp eed of computation

Analysis of eld data demonstrates that signicantwave grouping remains

after waves have broken This gives further evidence that mo dels of long wave

generation should account for nonsteady radiation stress gradients in the surf

zone In addition increased group steepness is shown to correlate well with

increased long p erio d mo dulation of nearshore currents

Acknow ledgements Funding for this study was provided by the Army

Research Oce under pro ject DAALG The authors would also like

to thank Joan OltmanShay for her help in obtaining SUPERDUCK and NSTS

data sets

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