Water Waves - Faculteit Wiskunde en Informatica · PDF fileOutline Introduction Equations for water waves Linear wave theory Classiﬁcation of water waves Behavior near the front

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Water Waves

Ekeoma Rowland IjiomaSupervisor : Dr. J.H.M. ten Thije Boonkkamp

16 December 2009

Outline

IntroductionEquations for water wavesLinear wave theoryClassification of water wavesBehavior near the front of the wavetrainSolution through the dispersion relationConclusion

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

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 =

Linear wave theory

Plot of velocity potential with the velocity field

(velocity.avi)

velocity.aviMedia File (video/avi)

then, the inviscid incompressible equations are

.u = 0

DuDt

=ut

+ (u.)u = 1p gj

so thatut

+(

12

u2)

+ u = 1p gj

on integrating foru =

we havep p0

= t 12||2 gy

Boundary conditions

Figure: sketch of the flow domain and its boundaries

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

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 =

Consider the bottom

y = h0(x1, x2)

we obtain Kinematic bottom boundary condition

x1h0x1 + x2h0x2 + y = 0, on y = h0

and for a horizontal flat bottom,

y = 0, on y = h0

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)

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

Linear wave theory

Analytic solution of the wave problemForm of solution for water waves

= A exp {i(x t)}

where A is the Amplitude of the wave

= Y (y) exp {i(x t)}

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

Linear wave theory

xx + yy = 0

(x , y , t) = X (x)Y (y)T (t)

X

X= Y

Y= 2

Linear wave theory

xx + yy = 0

(x , y , t) = X (x)Y (y)T (t)

X

X= Y

Y= 2

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)

Linear wave theory

We choose for T (t)

cost or sint

where =

2T

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

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

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

Documents

Water Waves - Faculteit Wiskunde en Informatica · PDF fileOutline Introduction Equations for water waves Linear wave theory Classiﬁcation of water waves Behavior near the front