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CHAPTER0
FIFTH
ORDER
GRAVITY
WAVE
THEORY
Lars
Skjelbreia,
h.D
Associate
Director
James
Hendrickson,
h.D
Technical
Staff
NationalEngineeringScienceCompany
Pasadena,
California
INTRODUCTION
In
dealing
with
problems
connected
with
gravity
waves,
scien-
tists
and
engineers
frequentlyfinditnecessarytomake
engthy
theoretical
calculationsnvolvingsuchwavecharacteristicsas
waveheight,avelength,period,
andwater
depth. Several
approxi-
mate
theoretical
expressionshave
been
derivedrelatingthe
above
parameters.
Airy,
or
nstance,
contributeda
very
valuableand
complete
heory
for
waves
raveling
over
a
horizontal
bottom
in
any
depth
of
water.
Due
to
the
simplicity
of
theAiry
theory,itis
fre-
quentlyusedbyengineers. This
heory,owever,wasdeveloped
forwavesof
very
small
heightsand
isnaccurate
forwavesoffinite
height.
Stokespresentedasimilarsolution
forwaves
of
finite
heightby
useoftrigonometric
series.
Using ive
terms
n
the
series,
his
solutionwillextendtherangecovered
by
theAirytheory
to
waves
of
greater
steepness.
Noattempthasbeen
made
n
this
papertospecify
the
range
where
the
theory
is
applicable. The
co-
efficientsn
these
seriesare
very
complicatedandforanumerical
problem,he
calculations
becomeverytedious. Becauseofthis
difficulty,
his
heory
would
be
very
little
used
by
engineers
unless
the
value
of
thecoefficient
is
presentedintabularform
Thepur-
poseof
this
paper
s
o
present
theresultsofthe
ifthorder
heory
andvaluesof
the
various
coefficientsas
a
function
of
the
parameter
d/L.
The
co-ordinate
system
and
description
of
the
wave
o
be
consideredinthispaperisshowninFig.
1.
Coordinate
system
Fig.
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FIFTHORDER
GRAVITY
WAVE
THEORY
The
waves
o
beconsidered
in
thisanalysisareoscillatory,
non-viscous,water
waves
of
constant
depth
and
of
infinite
extent
in
a
directionnormaltotheirpropagation.
Hence,
the
particle
velo-
cities
may
be
obtained
from
a
potential
function
as
ollows:
provided
V*
*
*
O
.
and
the
necessary
boundary
conditions
on
p
re
satisfied.
he
boundarycondition
at
he
mud
line
s
hat
thenormalvelocitybe
zero.
*i
-o
Oneofthe
free
surfaceS d+y)boundaryconditionsarisesfrom
theact
that
thewaterparticles
stayonthe
surface.
The
other
free
surface
boundary
condition
is
hatthe
pressures
zero.
Thesecon-
ditions
may
be
written
as
ollows:
r~
Therelationshipbetweenthepressureand
theparticlevelo-
cities
s
expressed
by
the
Bernoulli
equation,
where
K=
a
constant.
Since
we
are
dealing
with
anoscillatory
wave
of
length
L,
the
wave
profileandthepotential
function
maybeexpressed
n
terms
of
a
phaseangle
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where
COASTALENGINEERING
e
*
2L
x-
ct)'
3Cx-ci)
C =
the
wave
celerity.
Thus,combiningequations ,
4,
5 ,and6,
he
freesurface
boundary
conditionsmaybe
written
inthe
form
ix C
-u,
and
-u.c
+
Cu*+v
z
)
=^
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FIFTHORDER
GRAVITY
WAVE
THEORY
4*
- ****_AA.,+
A
3
A
43
X
5
A
4S
)cosK/3Ssin
6
C
C
+A*A^
^
4
A
K4
)cosh
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COASTALENGINEERING
values
are
given
in
tables
,
I
and
III.
The
constants
nvolve
he
ratioof
water
depth
towave
length
(d/L)asaparameter.
Forbrevity
in
listinghecoefficientshe
notation
ismadethat sinh(2TTd/L)
and
cosh(2
TTd/L).
C
Q
2
= g(tanh/3d)
A
1/s
2
?
A
-c
5 _ c
1)
.
-(1184c
10
-1440c
8
-1992c
6
+2641c
4
-249c
2
+18)
A
15
c
,
11
A
22
1536s
J
3
Is*
A
(192c
8
-424c
6
-312c
4
+480c
2
-17)
24
=
*
33
~^7^
A
512cl2
+
4224cl0
-
6800c8
-
12
.808c
6
+l6,
704c
4
-3154c
2
+107)
35
096s
13
(6c'
:
-l)
A
80c
6
-8l6c
4
+l338c
2
-197)
44
536s
10
(6c
Z
-l)
A-(2880c
10
-72.480c
8
+324
s
000c
6
-432,000c
4
+l63,470c
2
-l6,245)
55
l
>
440s
11
(6c
2
-lH8c
4
-llc
2
+3)
_ (2c
2
+l)
c
B
22
4s
3
c(272c
8
-504c
6
-192c
4
+322c
2
+21)
24
=
sl?
n -
3
(
8c6+1
)
B33
~~~
B(8
8
l
2
8c
14
-208,224c
12
+70,848c
10
+54
>
000c
8
-2l,8l6c
6
+6264c
4
-54c
2
-
35
2.288s
12
(6c
2
-l)
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FIFTH
ORDERGRAVITYWAVETHEORY
B
c(768c
10
-448c
8
-48c
6
+48c
4
+106c
2
-21)
44
84s
9
(6c
2
-l)
192,000c
l6
-262,720c
14
+83,680c
12
+20,I60c
i
-7280c
8
)
,
a
^
00
2,288s
lu
6c-l) 8c-lie
+
3 )
7160c
6
-1800c
4
-1050c
+225)
T 5
12,288s
lu
(6c
-l)(8c-lie3)
8c
4
-8c
2
+9)
8s*
C
(3840c
12
-4096c
10
+2592c
8
-1008c
6
+5944c
4
-1830c
2
+147)
2
12s
10
(6c
2
-l)
C
C
3 Ic
~
12c
8
+36c
6
-162c
4
+141c
2
-27)
C
192cs
7
Therestillremainshe
problem
of
determining
hecoefficients
fe
and
X
We
will
assume
hatthe
wave
sdescribed
by
the
nde-
pendent
parameters
H,
d,
L
crest
to
trought
height,
water
depth
and
wave
engthrespectively).
It
is
easily
seen
thatHisrelated
tothe
profile
expression
y
by
the
relation
13.
Thus,
usingequation
0
andrearranging
2
Also,
usingequation
2
and
the
expression
for
C
it
is
easily
shown
that
fc
-(f:)Tanh
3cl(,
+
'
2
C C,)5 .
where
_
I
t
-
9-
O
j~T
T waveperiodand accelerationdueogravity.
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COASTALENGINEERING
Since
theparametersH,dandLareassumed
to
be
known
for
the
wave
andsinceB,_,
B,,..
B,.,.,
C,
nd
C,areunctionsof
only
d/L,
hesimultaneous
solutionofequations
4
and
5
yield
boththe
values
of
d/L
andthe
coefficient
X
..Knowinghevalue
of
d,
hevalue
of
/&
iseasilyobtainedandhencethe
wave
may
be
completely
de-
scribed
in
allitsproperties.
Unfortunately,he
solutionofequations4
and
5
s
rather
complex
andtedious. It
would
be
advantageous
o
perform
acom-
putersolutiontoequations
4
and
5
andlist
theresults
n
the
manner
used
by
Skjelbreia
Ref.
1)
n
asimilar
analysis
of
the
third-
orderapproximation.
Theresults
of
the ifthordertheorypresentedin
this
analysis
will
be
compared
with
both
the
third-order
approximation
and
the
first-
order
Airytheoryfor
the
ollowingwave.
Given:
waterdepth
wave
height
wave
period
d
0
ft.
H
8
2/3ft.
T
7.72
sec.
Determine:
1.aveength
and
wavevelocity.
2.
quation
for
wave
profile
and
horizontal
particle
velocity.
Aftersubstitutingthe
given
values
for
d,
,and
T
nto
equations
4
and
5 ,
thesimultaneoussolutions
ofthese
equations
yieldthecorrect
valuesord/L
and
X
Theollowingresultsareobtained
d/L
.12 0
X= .
1885
(Noteforthis
examplea
digitalcomputer
was
used
tosolvethese
simultaneousequations.)
tained.
Waveprofile:
B
22
=
From
Tables
,
I,
and
III
the
ollowing
coefficients
are
ob-
2.5024,
B
24
=-3.7216
'33
35
= 5.7317
=
-4.8893
B
44
'55
= 14.034
=
37.200
Potentialunction:
11
13
=
1.2085
=-5.1153
A
15
=40.6530
A
22
= 0.7998
24
33
=
-4.9710
= 0.3683
190
35
44
L
55
= 1.5042
=
.0587
=-0.0750
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COASTAL
ENGINEERING
o
l_ UI
IO
O
UI
t
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FIFTH
ORDERGRAVITYWAVETHEORY
Other
constants:
C,
=
4.8600
C
3
=
0.2328
sinh/3
d
= 0.8275
C
2
= 88.6250
C
4
=
0.4314 cosh/3d
=
1.2980
Substitutingthese
constants
ntoequations
2 ,
10, 9,
and
theollowingresultsareobtained.
Wavevelocity:
C
= 32 -
3
9
ft/sec
Wave
ength:
=
CT =
250
ft.
Waveprofile:
y
.
50cos0 +
3.
35
cos 0
+ .48
cos
3
+
.
7 0
cos4
+ .
35
cos
5
Horizontalparticlevelocity:
u 6.186cosh
/3
cos
+
.434cosh ft
s
cos2
+
0.205
cosh
s
cos
.
02 1
cosh
4
/&
a
cos
4
- .
003
cosh
5/3a
cos
The
wave
profile
s
plottedin
Fig.2ogether
with
profileob-
tained
by
the
Airytheory-points
obtained
from
theThird
Order
Approxi-
mation
as
also
shown
on
the
same
graph.
Thehorizontalparticle
velocity
distribution
at
thecreststation
=
)
splottedinFig. 3together
with
thevelocity
distribution
ob-
tained
by
the
Airytheory.
NOMENCLATURE
Cave
celerity
&
S (X Cr) =
phase
angle
/3
T
L
L
ave
ength
d
eanwaterdepth
Hroughtocrest
wave
height
T
ave
period
= L/C
P
bsolute
pressure
X
orizontal
coordinate
distance,
measured
from
crest
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COASTAL
ENGINEERING
t
ime
S
ertical
coordinate
distance,
measured
positivelyup-
wards
from
mud
line
y
rofilecoordinate,
measured
positivelyupwardfrom
mean
water
ine
< J
elocity
potentialunction
V
aplacianoperator = \\ * / M
cosh/3
s
inh3d
v
ertical
particle
velocity
uorizontalparticle
velocity
X
3
a
constanttobedetermined
for
eachwave
Lo
waveengthfor deep-water wave
'ZTC
REFERENCES
1.kjelbreia,Lars,
ravity
Waves
Stokes'
Third
Order
Ap-
proximation
Tables
of
Functions
,
(June
1 ,
1958).
2.
ilton,
.R.,n'DeepWaterWaves,
Phil.
Mag.S.6
Vol.27,
No.
158,(Feb.
1$H).
196