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Water Waves
Ekeoma Rowland IjiomaSupervisor : Dr. J.H.M. ten Thije Boonkkamp
16 December 2009
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Outline
IntroductionEquations for water wavesLinear wave theoryClassification of water wavesBehavior near the front of the wavetrainSolution through the dispersion relationConclusion
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Introduction
The general idea of dispersive waves originated from theproblems of water waves. The problems are of great interest inthe Maritime and Offshore settings.
Figure: http://weblogs.sun-sentinel.com
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Equations of water waves
Basic Assumptionswe consider an inviscid incompressible fluidconstant density the spatial domain is given in (x1, x2, y) andthe components of the velocity vector u by (u1,u2, v)F = gj be an external force on the fluid.assume the flow to be irrotational, = u = 0introduce a velocity potential such that u =
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Equations of water waves
Basic Assumptionswe consider an inviscid incompressible fluidconstant density the spatial domain is given in (x1, x2, y) andthe components of the velocity vector u by (u1,u2, v)F = gj be an external force on the fluid.assume the flow to be irrotational, = u = 0introduce a velocity potential such that u =
/centre for analysis, scientific computing and applications
Equations of water waves
Basic Assumptionswe consider an inviscid incompressible fluidconstant density the spatial domain is given in (x1, x2, y) andthe components of the velocity vector u by (u1,u2, v)F = gj be an external force on the fluid.assume the flow to be irrotational, = u = 0introduce a velocity potential such that u =
/centre for analysis, scientific computing and applications
Equations of water waves
Basic Assumptionswe consider an inviscid incompressible fluidconstant density the spatial domain is given in (x1, x2, y) andthe components of the velocity vector u by (u1,u2, v)F = gj be an external force on the fluid.assume the flow to be irrotational, = u = 0introduce a velocity potential such that u =
/centre for analysis, scientific computing and applications
Equations of water waves
Basic Assumptionswe consider an inviscid incompressible fluidconstant density the spatial domain is given in (x1, x2, y) andthe components of the velocity vector u by (u1,u2, v)F = gj be an external force on the fluid.assume the flow to be irrotational, = u = 0introduce a velocity potential such that u =
/centre for analysis, scientific computing and applications
Equations of water waves
Basic Assumptionswe consider an inviscid incompressible fluidconstant density the spatial domain is given in (x1, x2, y) andthe components of the velocity vector u by (u1,u2, v)F = gj be an external force on the fluid.assume the flow to be irrotational, = u = 0introduce a velocity potential such that u =
/centre for analysis, scientific computing and applications
Equations of water waves
Basic Assumptionswe consider an inviscid incompressible fluidconstant density the spatial domain is given in (x1, x2, y) andthe components of the velocity vector u by (u1,u2, v)F = gj be an external force on the fluid.assume the flow to be irrotational, = u = 0introduce a velocity potential such that u =
/centre for analysis, scientific computing and applications
Equations of water waves
Basic Assumptionswe consider an inviscid incompressible fluidconstant density the spatial domain is given in (x1, x2, y) andthe components of the velocity vector u by (u1,u2, v)F = gj be an external force on the fluid.assume the flow to be irrotational, = u = 0introduce a velocity potential such that u =
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Linear wave theory
Plot of velocity potential with the velocity field
(velocity.avi)
velocity.aviMedia File (video/avi)
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Equations of water waves
then, the inviscid incompressible equations are
.u = 0DuDt
=ut
+ (u.)u = 1p gj
so thatut
+(
12u2)
+ u = 1p gj
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Equations of water waves
on integrating foru =
we havep p0
= t 12 ||2 gy
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Equations of water waves
Boundary conditions
Figure: sketch of the flow domain and its boundaries
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Equations of water waves
Define an interfacef (x1, x2, y , t) = 0
or for convenience as
y = (x1, x2, t) such that f (x1, x2, y , t) (x1, x2, t) y
Kinematic free surface condition
DDt
= t + u1x1 + u2x2 = v
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Equations of water waves
and we obtainDynamic free surface condition
p = p0
Boundary conditions at the free surface
t + x1x1 + x2x2 y = 0, on y = ,t +
12||2 + g = 0, on y =
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Equations of water waves
Consider the bottom
y = h0(x1, x2)
we obtain Kinematic bottom boundary condition
x1h0x1 + x2h0x2 + y = 0, on y = h0and for a horizontal flat bottom,
y = 0, on y = h0
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Linear wave theory
the linearized free surface conditions are
t = y , on y = ,t + g = 0, on y = .
tt + gy = 0, on y = 0
the surface elevation is
= 1gt(x1, x2,0, t)
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Linear wave theory
The linearized formulation
x1x1 + x2x2 + yy = 0, on h0 < y < 0,tt + gy = 0, on y = 0,
x1h0x1 + x2h0x2 + y = 0, on y = h0
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Linear wave theory
Analytic solution of the wave problemForm of solution for water waves
= Aexp {i(x t)}
where A is the Amplitude of the wave
= Y (y)exp {i(x t)}
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Linear wave theory
The Laplace equation
xx + yy = 0
Method:Separation of VariablesIn 2-D, we define
(x , y , t) = X (x)Y (y)T (t)
divide by X(x)Y(y)T(t) such that
X
X= Y
Y= 2
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Linear wave theory
The Laplace equation
xx + yy = 0
Method:Separation of VariablesIn 2-D, we define
(x , y , t) = X (x)Y (y)T (t)
divide by X(x)Y(y)T(t) such that
X
X= Y
Y= 2
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Linear wave theory
The Laplace equation
xx + yy = 0
Method:Separation of VariablesIn 2-D, we define
(x , y , t) = X (x)Y (y)T (t)
divide by X(x)Y(y)T(t) such that
X
X= Y
Y= 2
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Linear wave theory
where 2 is the separation constant.We obtain
X + 2X = 0
Y 2Y = 0
solving these equations, we obtain
X = B cosx + D sinxY = Eey +Gey
The solution is given by
(x , y , t) = (B cosx + D sinx)(Eey +Gey )T (t)
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Linear wave theory
We choose for T (t)
cost or sint
where =
2piT
and arrive at 4 possible solutions, periodic in x and t
1 = A1Y (y) cosx cost2 = A2Y (y) sinx sint3 = A3Y (y) sinx cost4 = A4Y (y) cosx sint
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Linear wave theory
The first equation gives the following solution
1 =gA
cosh(h0 + y)coshh0
cosx cost and
1 = A cosx sint
The dispersion relation is
W () = 2 = g tanhh0
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Classification of water waves
Water waves are classified into three main categories:Shallow water or long waves, if
h0
12
Intermediate water waves, if
120
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