Fifth Order Wave Theory

8/9/2019 Fifth Order Wave Theory

1/13

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.

184


2/13

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

185


3/13

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

)

=^


4/13

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


5/13

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)

188


6/13

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.

189


7/13

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


8/13


9/13


10/13


11/13

COASTAL

ENGINEERING

o

l_ UI

IO

O

UI

t


12/13

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

195


13/13

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

Fifth Order Wave Theory

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