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Lecture20

- Marine HydrodynamicsLecture 20

Chapter 6 - Water Waves6.1 Exact (Nonlinear) Governing Equations for

Surface Gravity Waves, Assuming Potential FlowFree surface definition

,(xB

0),,,(or),,( == tzyxFtzxy

xy

y

z

UnknownvariablesVelocityfield:Positionoffreesurface:Pressurefield:

GoverningequationsContinuity:

BernoulliforP-Flow:Farway,nodisturbance:

y,z, t) = 0

v(x,y,z,t) = (x,y,z,t)y=(x,z,t) or F(x,y,z,t) = 0

p(x, y, z, t)

2

= 0

y <

or

F /2

2 =gkkh=ghk=ghT

(a)Usetablesorgraphs(e.g.JNNfig.6.3)2 =gktanhkh=gk k

k =

=Vp

Vp =tanhkh

2 =gk= g

2T2(inft.)5.12T2 (insec.)

(b)Use

numerical

approximation

(handcalculator,about4decimals)i. CalculateC=2h/gii. IfC > 2: deeperkhC(1+2e2C 12e4C +. . .)

IfC < 2: shallowerkhC(1+0.169C+ 0.031C2 +. . .)

Nofrequency

dispersion

Vp =gh Frequency

dispersion

Vp =

gktanhkh

Frequencydispersion

Vp =

g2

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6.3 Characteristics of a Linear Plane Progressive Wave

2k=

MWL2

=T

H= 2ADefineUA

LinearSolution:=Acos(kxt) ; = Agcoshk(y+h)sin(kxt),where2 =gktanhkh

coshkh6.3.1 Velocity field

A

Vp

h

x

y

(x,t) = y

Velocity on free surfacev(x,y= 0, t)u(x,0, t)Uo =A 1

tanhkhcos(kxt) v(x,0, t)Vo =Asin(kxt) = t

Velocity fieldv(x,y,t)u=

x =Agk

coshk(y+h)

coshkh cos(kxt)= A

Ucoshk(y+h)

sinhkh cos(kxt)

uUo =

coshk(y+h)coshkh

eky deepwater1 shallowwater

v= y =

Agk

sinhk(y+h)coshkh sin(kxt)

= AU

sinhk(y+h)sinhkh sin(kxt)

vVo =

sinhk(y+h)sinhkh

eky deepwater1 + y

h shallowwater

uisinphasewith v isoutofphasewith8

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Velocity fieldv(x,y)Shallow water Intermediate water Deep water

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6.3.2 Pressurefield Totalpressurep=pdgy. Dynamicpressurepd = .t Dynamicpressureonfreesurfacepd(x, y= 0, t)pdo

Pressure fieldShallow water Intermediate water Deep water

pd =g pd =gAcoshk(y+h)coshkh cos(kxt)=gcoshk(y+h)

coshkh pd =geky

pdpdo

samepictureas uUo

pd(h)pdo

=1(nodecay) pd(h)pdo

= 1coshkh

pd(h)pdo

=ekyp= g(

y) hydrostaticapproximation p=g

eky

y

1kh

y

odp )( hp

y

odp

2g

Vp =y

x

)( hp

PressurefieldindeepwaterPressurefield inshallowwater

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6.3.3 Particle Orbits (Lagrangian concept)Letxp(t), yp(t)denotethepositionofparticlePattimet.Let(x; y)denotethemeanpositionofparticleP.The position P can be rewritten as xp(t) = x + x(t), yp(t) = y + y(t), where(x(t), y(t))denotesthedepartureofPfromthemeanposition.Inthesamemanner letv x,v( y, t) denotethevelocity at the mean position andvp v(xp, yp, t)denotethevelocityatP.

),( yx(x )',' y P(x ,y ) vp =v(x+x

, y + y, t) =P P TSE

v v vp = x, x, + (x ,y,t )yv( y, t) + ( y, t)x +. . . x y

ignore - linear theoryvvp =

ToestimatethepositionofP,weneedtoevaluate(x(t), y(t)):x = dtu( y, t) = coshk(y+h) t)x, dtA cos(kx

sinhkhcoshk(y+h)

= A sin(kxt)sinhkh

y = x, dtAsinhk(y+h)dtv( y, t) = sin(kxt)sinhkh

sinhk(y+h)= A cos(kxt)

sinhkhCheck: On y= 0, y =Acos(kxt) = , i.e.,theverticalmotionofafreesurfaceparticle(inlineartheory)coincideswiththeverticalfreesurfacemotion.Itcanbeshownthattheparticlemotionsatisfies

x2 y2 (xp

x)2 (yp

y)2+ = 1 + = 1 a2 b2 a2 b2

coshk(y+h) sinhk(y+h)where a = A and b = A , i.e., the particle orbits form

sinhkh sinhkhclosedellipseswithhorizontalandverticalaxesaandb.

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crest

(a) deep water kh >> 1: a = b = Ae

circular orbits with radii Ae decreasing

exponentially with depth

ky

ky

Vp

A

A

kyAe

trough

A

(b) shallow water kh

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6.3.4 Summary of Plane Progressive Wave Characteristics

f(y) Deepwater/shortwaveskh>(say)

Shallowwater/longwaveskh

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C(x)=cos(kxt)

(inphasewith)

S(x)=sin(kxt)

(outofphasewith)

A =C(x)

uA =C(x)f2(y) vA =S(x)f3(y)pd

gA =C(x)f1(y)

yA =C(x)f3(y) xA =S(x)f2(y)

aA =f2(y) bA =f3(y)

b

a

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