ocean water waves i
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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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