HtUralicsResearctt
Walurgford
A DIRECT METHODOF CALCULATING
BOTTOM ORBITAL VELOCITY UNDER WAVES
R L Soulsby
and J V Smallnan
Report No SR 76
February 1986
Registered Office: Hydraulics Research Limited,
Wallingford, Oxfordshire OXIO 8BA.
Telephone:O49l 35381. Telex: 848552
This report describes work carrled out by Hydraulics Research lnto the
movement of sand by conbined waves and currents.
It has been funded by the
Ministry of Agriculture,
Fi.sheries and Food under contract number CSA 992,
the nominated officer
belng Mr A Allison.
At the time of reporting the
Hydraulics Research nominated project officer was Dr s I.I Iluntington.
The report is published on behalf of the Ministry of Agrlculture,
Fisherles
and Food, but any opinions expressed within it are those of the authors
only, and are not necessarily those of the minlstry who sponsored the
research,
Crown Copyrlght
Published
by permission
1986
of
the
Controller
of
Her Majesty's
Stationery
Office
ABSTRACT
A nethod is presented for calcul.atlng the botton orbLtal velocLty under a
wave slmply and dlrectly
frou lte known helght and perLod, and the water
depth.
Iilhen suitably nondimenelonallsed the results aLl fall on a single
curve, with separate curves for monochromatlc and randon (JONSWAPspectrum)
wave6. The t.rl.s velocity
under random wavea nay be emaller or larger than
that produced by a nonochromatlc wave of helght E. and period T-, dependlng
on the water-depth.
Dlrect methods of obtalniug Ehe effectlve
frerlod of the
bottom veloclty
under random wavea are also preeented; these perlods can be
appreciably longer than Tr.
CONTENTS
Page
INTRODUCTION
t
MONOCIIROMATIC
WAVES
2
A RANDOMSEA
5
PERIODOF BOTTOMORBITALVELOCITY
9
EXAMPLES
II
SUMMARY
L2
ACKNOWLEDGEMENTS
13
REFERENCES
L4
TABLE
Values of non-dimensional bott,om orbital
velocity
for
monochromatic and random (JONS!'IAPspectrum) rdaves, and
peak-period of botton orbital - velocity spectrum.
FIGURES
l.
Dlmensionless
frequency.
transfer
as functions of d i m e n s i o n l e s s
=lJ2n/a4
waves : F
versus * = ,4/g.
mm
funct,ions
Monochromatlc
lFYr
9U
Random waves
(JONSI^IAPspectrun)
^
=
rhr
I' g
- l'v e r s u s
x
z
=
D
2 . Botton velocity
for monochromatic waves (unTn/2H versus
randon \raves (U.r"
APPENDD( -
fltrS 12
l -
rtH/
rUv\.
trr'
g
3 . Period T
|l -
pu
trr/""
versus T'L/T),
lrr/T)
where tr, = (h/S)'.
of the peak of the botton orbital
Relating
monochromatic
velocity
."4
spectrum.
and random wave parameters
IMRODUCTION
In
many aspect,s
oceanography
veloclty
it
at
forces
ls
necessary
include
orbital
measurements
For monochromatic
the
waves
maximun bottom
cycle.
However,
have a broad
veloclty
a naturally
spectrum
of
the peak period
in
E.erms of
T ).
It
ls
H" and period
However,
be a good approxLmation
attenuation
strongly
the
of
orbital
on wave period,
bottom will
with
velocity
so that
have a period
iu
by the
standard
instantaneous
important
to
know the
velocity,
as well
In
straightforward
because it
relation
done graphically,
laborious.
elevation
which
frequency,
spectrum
The usual
spectrum
Lnvolves
is
to
solvLng
and then
elther
T,
to
descrlbe
time-series
of
the
of
it
ls
orbit.al
ltself.
waves is
necessary
to
the wave-number,
Calculation
elevation
the
monochromatic
not
solve
the
which must be
or as a series
iteratively,
approxination.
surface
for
usual
perlod
as Ehe velocity
of U for
waves at
some applications
effective
The calculation
disperslon
is
depends
now be
cannot
Urr"of
deviatior
veloclties.
to
TO).
as the
dominant
different
or T-.
The near-bottom
velocity
p
described
by a single Ur, and it
(or
T"
depth
the
to
by a
height
nay not
the
tempting
monochromaEic wave of
this
wave
zero-crossing
sea can be represented
the
the
Generally,
wave-height
H" and the
is
random sea will
frequencies.
signlflcant
assume that
quantity
occurring
on the waves w111 be given
T_ (or
the
U, during
informatlon
p
' ze-rpi o d
sea bed,
Frequently
appropriate
the
orbital
the
and perlod.
wave height
of
at
be deduced frorn surface
has to
velocity
rraves.
problems,
transport
wave energy.
of
orbltal
by surface
and structures
dlssipation
and
know the
to
sediment
on pipe-llnes
and the
bottom
coastal
sea bed produced
the
Appllcations
engineerlng
of
from a given
*"
rnore
considerably
of U
is
procedure
is
to
a botCon-velocity
the dispersion
integrating
the
convert
Lhe
spectrum
relation
resulting
at
,
each
spectrurn
over the frequency range to yield
U:-^.
Calculation
rms
period ls equally laborlous.
of the effectlve
The purpose of thls
calculaEirg
as single
alt.ernative
in each case are
curves which are given in three
forms : graphieally;
and as explicit
tl" and T, toget,her
The results
the water depth.
presented
period,
and the effective
Ur, Urr",
from the known quantities
directly
with
is to present a method of
report
as tabulated
values;
expressions whlch
algebraic
approximate the curves closely.
uoNocuRouATIc
r{AvEs
Consider
a rdave of
u = 2n/T,
frequency
and period
orbital
just
the
bed).
whieh
U, at
the
outsl-de
the
Then U, is
srnall-anplltude
a = H/2,
and radian
where H and T are
respectively,
velocity
correctly,
near
anplitude
linear
gives
sea-bed
fhin
the wave height
rise
(or,
to
a maximum
nore
wave boundary
obtalned
wave theory
layer
using
from
I
l'
;.rr
\,"
U
m
a
'il
sfrE_(ErhI
The wavenumber k is
dispersion
the water
urh
o
o
Y=kh
related
to the frequency
t'l by che
relation
t,2 = gk tanh
where g is
(1)
(2)
(kh),
the acceleration
depth.
Define
due to gravity
dimensionless
and h is
variables:
(3)
(4)
Uh
(5)
F=B
m"\
Then Equatlon (1) becomes, after
2y
l'=
m
(6)
sffih'(2y-t
and the disperslon
x
use of Equation (2),
=
written
(7)
ln
cannot be written
(7)
gives
between x and yr
parametric
only
quantity
a one-to-one
x alone.
U, to
known quantities
(Fig
waves)
be obtained
1).
the value
very
> 4 (deep-water
The quantities
for
pracEical
squares of
F,
F,
with
the quantit,les
constants.
readily
usable
quantities
natural
scaling
period
versus
x
(6)
tends
of
x
to
one,
it
becomes
waves).
contain
lnceresE
T- defined
the
and also
We therefore
by first
cornplicated
contaLn
define
introducing
nore
rhe
by
r *n = (' g}'l *
Then the
(8)
required
dimensionless
a
frorn the
x until
as they
of
is
and the
Equations
and x are unnecessarily
calculatlons,
some unnecessary
in
F,
and x contain
direcEly
of
monotonically
x
of F,
that
For srnall values
and F* decreases
smal1 for
However,
(6)
h and g,
y as a parameter
and (7) ) a1lows
terms
correspondence
Both F,
H, T,
be
Equation
as y(x).
Uxn. Thus a plot
by using
(shallow-water
relation,
explieitly
of
cannot
and hence in
w€ see from Equation
function
(obtained
x,
F,
dispersion
the known quantlties
required
of
terms
because the
as Equatlon
function
transfer
explicitly
iI and T,
(7),
becomes
ytanhy.
The dlmensionless
of
Equation (2),
relation,
quantities
are
I
UT
- 'mFn- =
F2
m
_T
(e)
and
T
I
n
T-
x2
=
- 2- x
(10)
of U To/2Il versus trr/T (Fie 2) can be used
A plot
dlrectly
for
U, from lI, T, g and h.
obtaining
For
values of UrTrr/2H are tabulated
computer appllcatlon,
against Trr/T in Table 1.
A 3-part
explicit
whlch flts
algebraic
expression can be found
the curve in Figure 2 closely,
as follows:
4
2u- ^*l\
,
I
F _ i = -tl r ,..'g'
f l ) i l = ( 1 - 0 . 6 7 0 x+ 0 . 1 1 0* 2 ) t , 0 s ( < l
m
' - u / ./
( 11)
'9529x
= t.72 *l .-o
, I < x< 3 . 2
= 2*i .-*
, 3.2 (x<-
with
x\2 h
x = 2
tf-J
;'
-o
Equation
(11)
accuracy
of
fits
bett,er
0(x(-.
The first
based
respectively
approximations
and (7),
used
ln
to det.ernine
the
exact
the
entire
and large
some optilnlsation
the
curve.
in
by Equation
is
(11)
Equations
of
Optiuisation
coefficients
because it
range
are
arguoent
with
part.
2 to an
of Equation
y and tanh y in
given
2,
Figure
in
sinh
the first
The approximatl-on
from
parts
on sraall
to
shown on Figure
curve
chan tl% over
and third
together
coefficients
the exaet
the
was also
the niddle
(11)
(6)
has not
indistinguishable
part.
been
A nANDOI{ SEA
Under natural
represented
is
by a spectrum of waves of different
frequencies,
the only
the wave climate
eonditlons
In many cases
anplLtudes and directions.
paramet.ers which are known about the sea-
condlti.ons
wave height
are the significant
zero-crossing
period Tr.
done is to fit
Hs and the
The best that can then be
surface elevatlon
a reallstic
spectruu
SJut) to these triro parameters. One of the nost widely
naccept,ed two-parameter spectra ls the JONSWAP
spectrum
(Hasselnan et al,
L973), glven by
s n ( u r )= 2 x a g 2 t r 5 e * p
{-; q)-4} Y(.,)
where
(12)
4{u,) = exp{- t*bl}.
'p
llere
r0 is che radian
p
y and B are
spectrum,
which
depends
the standard
frequency
at
constants,
of
peak of
and c is
on the wind-speed
values
the
a variable
and duration.
the constanEsr
the
We use
3.3 and F =
y=
o > o_, I = 0.09 for o < o,.
The variables
pp
and op can be related
to Hs and T" respectively,
so
0.07
for
that
a partLcular
T,
corresponds
An additlonal
sea-state
to a particular
complication
Ehere is
an appreciable
which
generally
(12)
is
function
207" (Braapton
bottom
surface
et
orbital
veloeity
elevation
r;
by multiplying
the
lt
calculatlons
form of
Ilowever,
ls
related
is
seen that
Equation
the
rate
of
the
spreading
linearly
function.
of
the
the
spreading
by up to
because the
to
the
relationship
beLweenthe quanrit,les Urft and u"2 {=t6 o.1
independent
that
the wave directions,
the dissipation
1984).
al,
in
For
rate
can influence
by II" and
J O N S W A Ps p e c t r u m .
spread
functlon.
dissipation
only
of a random sea is
expressed
by a spreading
Idave energy
descrLbed
a
is
The bottom velocity
spectrum S,r( t^r) is
the dimenslonal transfer
applytng
obtained
by
given
function
by
Equation (1) to each frequency in the elevation
specErum:
..2
( o) = -+s,,
_
ro)
s".,(
.'
sinh z(kh)
The variance
of
by integratlng
(13)
Ls then
veloclty
bottom
the
obtained
frequency
Su( o) over
@
=
TJ2
rms
We now define
those
for
waves (Equatlons
that
the
elevation
velocity
of
standard
deviation
a random sea is
of
sEandard deviatlons
(1s)
u'
""
of
the
to
3 and 5):
rns l'A)
r. = f__
t
'
r
We not,e here
bottom
analogous
4u^
h
Tc',
surface
variables
dimensionless
rnonochromatic
Qnf
x=
z
(14)
su r- o ) d u r
'I
o
surface
of
the
H"/4,
and the
elevation
and
wave are H/ /2
a monochrornatic
and
U,/ n respectively,
Thus, to make analogous
m
quantities
for monochronatic
and random lraves
correspond
into
the
quantity
If
in
meaning,
definltion
of
we have lntroduced
the
Fr,
2 into
U^T^/Zn deflned
mn
we now further
= funcl(x)
via
defirre
and the
EquaEion (7),
4
the
(see Appendix).
earlLer
xO =
factor
factor
uf;rrtt, and write sinh- 2y
(14)
Ehen Equatlon
becomes
.rX
@
g2
= agh
rms
= agh
I
Jx
o
func
-3/2
\
ot-
( 5rx)exP r 4 \ x )
2tII
x
I
P
tunc ,(x) dx*
P
z (*p)
(16)
Also,
from the definition
of
moment of the spectrum
the zeroth
H" Ln terms of
(12), we have
Equation
Ht
z
Sr-
r
\4)r
S (o)
do
on
+trt"
@
=
clt2
I ; 5 / 2 exp {o
=
ah2
0( :)
,2
ox
i
P
( r 7)
func, (xn)
From Equations (16) and (17) we obtain an expression
the dimensionless transfer
for
F_ given by
r-
function
Equation (15):
func, (xo)
=
( 1 8)
func u (xn)
By expressing Tr2
is a funcLion of x- alone.
rp
as the ratio of the second and zeroth moments of Sn(f)
Thus F
can be related
it
t,o the peak period
spectrum given by Equation (12).
Tn of the JONSI^IAP
With the values of
and y given earlier,
I==
_Zn
(re)
I .281 T
(t)
p
p
2
Thus x, is proportional
of
calculated
given
F - fot a range
r-z
by performing
by Equation
general
existlng
BrampEon eL al
step
of 0.1s
belween
are
curve
(Fig
that
(f984)
have been
the
lntegration
An adaptation
Fr
of
the
the
integration
taken
The resolution
good accuracy.
curve
by
wiEh an integration
and 5To.
give
a more
of
program described
was used,
of O.ls
follows
x
of
values
numerically
computer
ample to
1)
of
(14).
and linits
periods
Iinlts
follows
and it
only of x, fo
is a function
Values
*n,
for
and
The resutting
monochromatic
p
waves for
larger
small
than
For sinpler
of
values
xr,
becomes increasingly
but
as xz increases.
lt
use we have plotted
graphical
Urr"Trr/t"
versus Tn/Tz (Ffg 2) where T' is defined by Equation
(8).
Values of Urr"Trr/H" are tabulated
For the random lrave case iE is not
in Table 1.
straightforward
F, for
expressions for
to obLain asynptotic
srnall and lar8e xr,
monochromatic waves.
fittlng
as was done for
Instead
we have ernployed curve-
techniques to obtain an expliclt
=
S
algebralc
the curve ln Figure 2 closely:
expression whlch fits
UT
rus n
against T /T
o.25
e. Arr;
where
A=
15.54t)a1t/6
fosoo+ ( 0 . s 6 +
(20)
and
n=
l
r-. =
T T-
T
rh ti
t 6; /
zz
(20)
Equation
accuracy of
t h e J O N S W A Pc u r v e
fits,
becter
than 1% in
Again we have not plotted
because it
is
Figure
2 to
an
the range 0< t< 0.55.
Equation
indistinguishable
ln
(20)
fron
on Figure
the exact
2
curve.
F, only up to xz = 1 1 . 8 . i e .
We have calculated
= 0.55, because for larger values of x
Z
is very srnall. For rn/T,
g
v essn sU - _ T / H = 0 . 0 0 3 8 , s o t h a t
- ri m
velocity
T /T
oz
the bottora
= 0.55, t'igure 2
H
=
0 . 0 0 6 9 ;I
S
at
z
T /"1 = 0.55
nz
This provides an upper b o u n d t o v e l o c i t i e s
0,55.
For example, lf
il" = 4m, T,
( 21)
for
'I
,_L/Tz
= 4s, h = 47 .5m,
g = 9.81ns-2, then Tn/T, = o.55 and
Urr" = 0.0069ns- I.
PERIOD OT BOTTOU
ORBITAL VELOCITY
Although
the
it
is
function
transfer
boEt.om velocity
elevations,
period"
ls
should
orbital
to
less
to
This
veloclty.
the
wave-effecEs
often
transport
to
or
in
appear
forces
ls
the
applicat,ions
which
to
near
on structures
bott,om
contribution
in
form
period
the
of
largest
makes the
the
"effective
spectrum
the
of
the wave
of
We examine here
U 2 , which
variance
perlod
that
how the
clear
peak in
the
than
of
dependence
the effective
thaL
be defined.
correspondlng
frequency
be larger
will
lt
the
from
clear
sedlment
the
sea-bed.
We therefore
peak of
=2rl
wLsh to
the
veloclty
in
the elevation
S,r( u:) is
with
the
to
through
the
perlod
Tn
spectrum Sn(o).
Equation
(1)
and (5),
tr'L After
and
settlng
by S,, we obtain:
dS
,dF
lmt
rtr
TO' at
by expressing
Equations
respect
dsu/du: = 0 and dividing
F
found
terms of Fro fron
differentiating
period
Su( ur) with
spectrun
op aE the peak of
The maximum of
(f3)
compare the
=-_
do
Making
the
,l
S
rl
(22)
ao)
0 = rr/op, EquaEion (22)
substitutLon
becomes
I
F*
dtr
dy
q.
dx
SubstitutLon
f ron Equation
dx = -1
Sn
dtt
of
F,
(23)
d0
( 2 3t
du:
from Equation
and dFr/dl
( 7 ) , dx/d rrlf rorn Equation
dSn/dS from Equation
Equation
dQ
5
(L2),
to be wrltten
and d0/d,
in
the
(3 ),
(6)'
S
rl
dYldx
and
= L/ wr, enables
forn
P(x) =
a(0)'
(24)
where
(z++y) e-4Y + 4y -z
P(x) =
L-e-4Y * 4y s-zY
and
r1
O( t - 0 ) e x p ' 2 9 , ( 1 - 0 )2 )
Q ( 0 ) = s q - + - s + 5p '
Q( 0) vs 0 and P(x) vs x (using y as
The functions
parameter) can be plotted.
Then if
we denote by xt
and 01 the values of x and Q which correspond to equal
values of P(x) and Q( $), thereby satisfying Equation
(21+), we obtain,
Txl
Pu =
using the def inition
of
0,
= -1r-g'\i
tx'
TP
0t
(2s)
and
x
x_t
P;"2
Y
I
By picklng
of
off
Equatlon
(Fig
3).
values
(19),
For
waves) we find
of
P(x)
= Q(0),
and naking
a plot
of. T_,-/T can be constructed
puz
values of Tn/T, (shal1ow-\rater
snall
that
TOu tends to Tp =L.281
Tz.
T-/T- increases, T-../T_ increases first
slowly,
nz'Pu-z
rather
raptdly
close to Tn/T, = 0.4, and finally
slowly
again
excess of
Because
for
use
Tn/T,
> 0.5,
at
As
then
which point
TO., is
is
a simple
1.7 T .
z
t.he curve
shape we have not
of T
vsT/T
lt
puzlz
attempted to fit
approximat,ion
to
it.
ae
- na'i zn s t
in
Table
T /T
l0
Values of T
1.
not
an algebraic
pu
lf
z
are
tabulated
in
EXAUPIJS
As illustrations
II = 5m, period
height
a monochrouatic wave of
consider
T = 8s, for
two wat,er-depths
h = 10n and 50n.
For h = 10m, Eq (8) gives Tr, = 1.02s, and hence
Trr/T = 0.L27.
From the "monochronatic" curve tn Fig
w e o b t a l n u - r ^ / z H = 0 . 1 9 6 ', a n d t h u s U - = 1 . 9 2 m s - 1 .
mn'
m
For h = 50n the corresponding
Trr/T = 0.282, u#n/2H
= 0.038, and u
H_ = 5m and T_ = 8s,
sz
with
in
m
a JONSWAPspectrum
a random sea having
Now consider
= 2.26s,
n
= 0.168ns- I.
values a r e T
same ltater-depths.
the
*i
I
Then for h = l0m, Eg (8) gives Tr, = 1.02s, and
''
.'i.rr:*
l
..,.i
i
i',n. ' z ' r :. -l l t s T ^ / T - = 0 . 2 0 4 , a n d . t h u s U - ^
I
i
= 1.00ns-1.
I
I
I
i,.:.
l,'t tt''
For h = 50m the corresponding
T /T = 0.282, IJ
nz-rmsns
= O . 1 9 2 n s -I .
U
rms
T lH
In
the
order
to
compare
wave of
rnonochromatic
firsc
the
necessary
to
uonochromatic
h = 10m the
wlth
random sea with
the
convert
the
random sea value
By contrast,
Tz it
anplitude
is
of
corresponding
= lJ^/ /2.
Ua*"
wave value
h = 50m the
for
velocity
by Ur*"
the monochromatic
a
11" and period
height
value
= 2.26s,
n
= 0.087, and
wave to
root-mean-square
values are T
Then for
= l.OOns-l
Urr"
cotp"r""
= 1.36ns-1.
random sea value
U--- = 0.192ns-l
with the rnonochromatic wave
rms
"orp.r"s
= 0. 119ns-1.
value U
Thus in shallow rdater an
rms
velociEy based on a
estimate of boEtom orbital
monochromatic
a serious
serious
the
wave of
overestimate,
underestLmate.
monochromatic
occurs
height
at
a depth
h = 0 . 0 4 9 e T- 2
2.
11
but
il"
in
and perlod
deep water
The cross-over
(using
Flg
it
point,
and random \{aves give
glven
T,
2) by
the
will
be
will
be a
at
which
""t"
Ur*",
The peak-period
velocity
of the bottom orbital
the random sea with h = 10m, ls
fron Fig 3 with ,n/r, = O.L27, for which
obtained
spectrum for
The corresponding
T pu'
- / T _ = L . 2 9 6 s i v i n g- T - - - = I 0 . 4 s .
Pu
Neither value is very
value for h = 50m is 10.6s.
of the surface
different
from the peak-perlod
elevation
spectrum glven by Eq (19) as Tn = 10.2s.
For a JONSWAPspectrum, which is
strongly
relatively
from T- only
peaked, T
will be appreclably different
pup
for rather short period waves or rather deep water.
SI'}'IMAR]T
Methods have been presented for
and effective
the boLtom orbital-velocity
period
visual
(for
use), as tables
accurate
to
(for
by
computer application
algebraic
and as explicit
fl"7" (fot
in water of
are presented graphlcally
The results
look-up table),
bottom
and period
of waves of known height
depth h.
directly
calculating
expressions
use on pocket calculators
and
The methods can be summarised as
micro-computers).
follows:
1.
b o t h f o r monochromatic
Results,
are
by the natural
scaled
and randon
Tr, =
period
scale
ltaves,
1
(h/il..
2.
the
TrrlT given
in
fron
laboratory
the plot
of UrTrr/2H versus
a s prototype
For a random sea characterised
wave height
bottom
be obtained
frou
given in
T-/fn'
(20 ).
Equation
for
the
Figure
elevation
L2
the
This
orbital
plot
significant
period
veloclty
Tr,
the
U.r"
""t
U----T-/II- versus
rmsn
s
from Table I or
of
2 or
is
rdaves.
by the
Hs ar.d zero-crossing
root-mean-square
or
can be used for
T h e s e results
as well
velocity
2, or from Table I
Figure
(f1 ).
Equation
H and period
the botton orbital
U, of
anplitude
can be obtained
3.
wave of hetght
For a monochronatic
based on a JONS!,IAPf orn
spectrum.
Results
are
presented
for 0 < In/Tz< 0.55 whlch covers the entire
of practical
range
but for values outside
lnterest,
range an upper bound ao Urr"
this
is given by Equation
(21).
4.
velocity
The rns botton orbltal
calculated
spectrum (iten
assuming a JONSWAP
larger
or smaller
than that
(iten
Il" and Period T,
larger
The
resPectively.
12
nay exceed 407" Ln either case.
The perlod tn'
spectrum ls
elevatlon
of the peak of the veloclty
than the period Tp of the
larger
spectrum, by an amount whlch can be
obtained from the plot
of Tru/T,
versus ,n/r"
gLven in Flgure 3, or fron Table 1.
purposes the assumptlorr that
at the sea-bed is T
P
For many
the effective
period
is adequate.
Where necessary, use the relation
Tn = I.281 T,
throughout.
7
or
than 0 .O49g T
difference
6.
by assuming
2 above) depending on whether h is
smaller
5.
3 above) is
calculated
a monochro'matic wave of height
by
ACKNOI{LEDGBUENTS
We are grateful
Eo Mr M W Owen, l1r G Gtlbert
Dr K R Dyer for
rhelr
13
useful
commeots.
and
8
REFERENCES
1.
G and Southgate tI N (f984).
Brampton A lI, Gllbert
A finlte
dlfference
wave refraction
model.
HR
Report EX 1163.
2.
Hasselman K et al
(1973).
wave growth and swell
Sea Wave Project
decay during
L4
the Joint
North
(JONSWAP). Deutsche
Itydrographlsche Zeitschrlft,
12.
l'leasuremenEsof wind-
Supplenent A8, No
Table
Table 1:
Values of non-dlneoeioual
monocbronatlc
of botton
and randon
orbltal
bottom orbltal
velocity
(JONSWAPapectrun)
- vcl,oclty
apectrur.
;-
1 , r' ' i
Il
n - - \\ / (
r,-
MONOCHROMATIC
for
saye8,
and pealc-perlod
i
\
I /
.t /
JONSWAP SPECTRUM
T lT
n
UT /2H
mn
TIT
n z
U TlH
rmsn s
T lT
pu z
0.00
0.250
0.00
0.250
1.281
0.02
0.248
0.02
0.248
0.04
0.244
0.04
0.245
r.281
r.282
0.06
o.237
0.06
0.238
I .283
0.08
0.228
0.08
0.230
L.284
0.10
o.2L6
0.10
0.2L9
L.285
o.L2
0.202
o.L2
0.208
r.286
0.14
0.185
0.14
0.196
I .288
0. 16
0.165
0.16
o.L72
L.290
0.18
0. 143
0.18
0.L67
L.292
0.20
0.120
0.20
0.150
L.296
o.22
0.097
0.22
0.134
t.299
o.24
0 . 0 75
o.24
0. 118
1.304
0.26
0.055
0.26
0.103
1. 3 1 1
0.28
0 . 03 9
0.28
0.088
1 .3 1 9
0.30
0.o27
0.30
0.075
1. 3 2 8
0.32
0 .0 1 8
0.32
0.063
1 .3 3 9
0.34
0.012
0.34
0.052
1. 3 5 3
0.36
0.007
0.36
0.o42
L . 3 72
0.38
0.005
0.38
0.033
1. 3 9 8
0.40
0.003
0.40
o.027
L.448
o.42
0.002
0.42
o.022
I .570
0.44
0.001
o.44
0.017
r.620
o.46
0.000
0.46
0.013
1.653
0.48
0.000
0.48
0.010
t.682
0.50
0.000
0.50
0.008
1. 7 0 8
o.s2
0.000
0.52
0.007
1.731
0.54
0.000
0.54
0.006
I .753
Figures
N
H
H
(L
=
(n
z
o
-t
I
@(o
o-6
l.l.
.t Li
Flgure
1
DlmensLonless transfer
as functions of dlmenslonless
frequency.
Monochromatic waves , F, = U, h/a fu versus x =
4U.Random waves (JoNSWAPspectrun) , Fr= (-ES- )' qO)
whr/e.
2n.'2h
versus xz = tf-.,
g
z
functlons
fL
=
(n
z
o
-)
N
F
c
F
F
c
F
|r)
GI
o
Figure 2
;c{Ftl
fl*
P
o El,E o
=l
Bottom veloclty
f o r monochromatlc
and random naves (U
r
r m s n s/g
n
waves
(umTn/2n versus Tn/T)
versus T /T
,),
where tr, = (h/s)'
N
F
F
H
: ;ls :
Flgure
3
Perlod
TO,, of
t,he peak of
the
bottom
orbltal
velocity
Si^rt5o**r
spectrum.
p*-. D <-tt,-t't<-rt>{
T^
k
V-
tt
V\qz44ez
Appendix
APPENDIX
Rel.atlng
Confusion can easlly
about the
arise
of the varLous ltave Parameters in
Lnterrelatlonships
report.
the quantLt,ies used ln thls
For a slnusoldal
between
here the relationships
We clarify
commonuse.
Yave Paraneters
aad landon
Uonochrolatle
11 and
monochromatic wave of height
period T, the angular frequency is:
w = ZnfT,
(A1)
the anplltude
is
(A2)
a=H/2
of the surface elevation
and the variatLon
tLme t at a particular
n(t) with
ls given by:
point
n = a sin ut.
(A3)
Thus the standard deviation
related
to II by:
H=2{2
6
m
(A4)
is given by
(As)
sintrt
wrltt,en
6 =U
urmsm
as U
.
rms-
is
of
the
the
velocity.
velocity,
as:
The
more commonly
thus:
=lJ /y'2
a random sea the
defined
U(t)
velocity
where U_ is Ehe anplitude
m
standard
devlation
o,, of
For
is
n
The bottorn orbital
U=U
on of the surface
(A6 )
slgnificant
wave height
H" is
H
=46
(A7)
sn
where on ls now the standard devlation
surface
elevation
(The rrns wave height
r(t).
also sometl,mes used, and ls
= 2/2
H
rttrs
dlstinguish
of the random
related
to
is
on by
o . ) The term "random" ls used to
n'
a naturally occurrlng nulti-directional
spectrum of waves from a unidirectlonal
sinusoidal
Hrr"
wave, rather
monochrouatlc
than ln the usual statist,lcal
sense.
The perlod
can be characterised
perlod
zero-crosslng
either
by the
T" or by the angular
at the peak of the surface elevation
frequetrcy op
spectrum, leading
Eo the peak period TO given by:
rl|
l-
2rl u
p
p
(AB)
For any standard
shape of spectrun
(JONSWAP,
Pierson-Moskowitz, etc) Tn is proportional
with
the constant of proportlonality
chosen spectral
to Ehe variance
vector Ut)
lF*"!,tu'
depending on the
shape.
The root-mean-square bott,om orbital
related
to Tr,
veloclty
U.r"
lu
o--2 of Lhe random veloeicy
by:.
ffi
urr"' = ou'=
ltrl
{rI r,t.,t._t{ub"(o}tp ^ | (Ag)
,.)it".t6' '+*r rc.W'-dt','1,5l1.
_
,i
L'J
If
a monochromatie
of
signlflcant
the
they
surface
are
wave of
height
H" have the
elevation,
related
hetght
then,
using
II and a random sea
same variance
o 2 of
n
eqs ( A 4 ) a n d ( A 7 ) ,
by:
H
H=;
If,
(A10)
instead,
bottom orbital
they have the same variance
velocity,
62
of
the
u
then, using eqs ( A 6 ) a n d
(A9), they are relaced by:
U ={2
m
U
rms
Thus, using eqs (A2),
quantity
(A11)
(A10) and (A11), the random wave
deflned in eq (15):
F, = ({Itl)r,*,
corresponds
( A 1 2)
to the monochromat,ic quantity
defined
ln
eq (5):
U4
rm =3-.
a'E
Sinllarly
the monochromatic quantlty
corresponds
UnTn/(2H)
to the random quant.Lty U.r"Trr/H".
DDB 8/86