Lectures 11-12: Gravity waves

• Linear equations

• Plane waves on deep water

• Waves at an interface

• Waves on shallower water

Water waves

The free surface of a liquid in equilibrium in a gravitational field is a plane. If the surface is disturbed, motion will occur in the liquid. This motion will be propagated over the whole surface in the forms of waves, called gravity waves.

Let us consider waves on the surface of deep water. We neglect viscosity, as there are no solid boundaries, at which it could cause marked effects; and we also neglect compressibility and surface tension.

z

x

air

water0

The governing equations are

div 0v

v pv v g

t

Small-amplitude wavesThe physical parameters characterising the wave-motion are the amplitude of oscillations of fluid particles, a, the wavelength, λ, and the period of oscillations, T.The velocity of a fluid particle, . The significant change of the velocity occurs at a distance λ.

The unsteady term can be estimated as .

The non-linear term can be estimated as .

This means,

Ta

v

2Ta

tv

2

2

Ta

vv

a

tvvv

For the small-amplitude waves, when , the non-linear term is negligibly small.

1a

Irrotational motionThe linearised Navier-Stokes equation is

constant

0t

gp

tv

For the oscillatory motion, the average position of a fluid particle is z=0, the average velocity is 0. The average vorticity must be also 0. As the vorticity is time-independent, it must be 0 at every moment (to make the average value being 0).

0 -- small-amplitude wave motion is an irrotational flow

Hence, the velocity field can be represented as . The velocity potential φ is determined by the Laplace equation:

v

0

Boundary conditions: 0pp

z

on the surface (ζ is the surface elevation above the flat position), pressure is atmospheric.

zv waves on deep water, the fluid velocity is

bounded everywhere.

1. We use Bernoulli’s equation to rewrite the first boundary condition in terms of φ.On the surface,

constant0

gp

t z

Here, we neglect the term containing v2, as it originates from the non-linear term, which is small for the small-amplitude waves.

The velocity potential φ is a technical variable that does not have the physical meaning and is needed only to find the velocity ( ).

The velocity does not change if the potential is redefined as follows

tp

tc

constant

0

v

This allows us to rewrite the Bernoulli’s equation as an equation for the elevation ζ :

ztg1

2. On the surface,

zz zt

v

3. This results in the following boundary condition on the surface

01

2

2

ztgz

4. Using the Taylor’s series over small ζ, the leading term of the above equation is

01

02

2

ztgz

Finally, we have got the following mathematical problem

01

02

2

ztgz

Equation:

Boundary conditions:

zv

0

Plane wave solution

Let us seek the solution in the form of a single plane wave,

kxtzf cosT 2

2k

Here, -- the circular frequency (1/T would be the regular frequency)

-- the wavenumber

kc

-- the wave speed

Substitution into the Laplace equation gives

kzkz ececffkf 212 0 As the velocity is

bounded,02 c

Illustration:http://www.youtube.com/watch?v=aKGgsLHN1dc

kxtec kz cos1

Hence,

kxtekcz

v

kxtekcx

v

kzz

kzx

cos

,sin

1

1

Or, in terms of velocity,

Wave dispersion

Applying the boundary condition on the liquid’s surface, we obtain the following dispersion relation

kg

kckg

gk

02 The waves of

different lengths travel at different speeds (see the video).

Video: http://www.youtube.com/watch?v=lWi_KpBy8kU (note that long waves travel faster)

Phase and group velocities

is called the phase velocity, the velocity of travelling of any given phase of a wave.

is called the group velocity, this is the velocity of the motion of the wave packet.

The red dot moves with the phase velocity, while the green dot moves with the group velocity.

kg

dkd

U21

kg

kc

Waves at an interface

z

x0

ρ2

ρ1

Consider gravity waves at an interface between two very deep liquid layers. The density of the lower liquid is ρ1 and the density of the upper liquid is ρ2. ρ2 >ρ1

Motion is irrotational. The governing equations are

0111 ,v

0222 ,v(lower liquid)

(upper liquid)

Boundary conditions:

at infinity, , fluids’ velocities are boundedz

At interface, , z21 ,, zz vv 21 pp

We need to re-write the boundary conditions on the interface in terms velocity potential.

Applying Bernoulli’s equation at an interface, we have

constant

gp

t z 1

11constant

gp

t z 2

22

or, redefining φ1 and φ2

0

gp

t z 1

11 0

gp

t z 2

22

The pressure is continuous at interface, i.e.

gt

gt zz

22

11

The z-component of the velocity is continuous as well, i.e.

tzz zz

21

As a result, we have two following boundary conditions at interface

zz zg

tzg

t2

22

2

21

21

2

1

zz zz21

Expanding over powers of small ζ and leaving only the linear terms, we finally obtain

0

222

2

2

0

121

2

1

zz zg

tzg

t

0

2

0

1

zz zz

Now, seek solution in the form of a plane wave

tkxzf cos11

tkxzf cos22

Solving the Laplace equations, we get

kzkz eBeAfffk 111112 0

kzkz eBeAfffk 222222 0

The boundedness of the velocities leads to

kzeAf 11 kzeBf 22and

tkxeA kz cos11

tkxeB kz cos22

That is,

Let us now use the boundary conditions at the interface

gkgk

BA

kBkA

BgkAgk

22

21

21

21

22

212

1

The resultant dispersion relationgk

21

212

The phase and group speeds are

kg

kU

kg

kc

21

21

21

21

21

dd

;

The fluid velocity (A1 cannot be determined from the linear equations):

tkxekAz

vtkxekAz

v

tkxekAx

vtkxekAx

v

kzz

kzz

kzx

kzx

cos;cos

sin;sin

,,

,,

12

211

1

12

211

1

Water/air interface

ρ2<< ρ1

dispersion relation and the expressions for the phase and group velocities become as those for the gravity waves on a free surface of deep water

kkU

kkcgk

21

dd

,,

Waves on shallow waterz

-h

x0The liquid is now bounded by a rigid wall.

The fluid velocity is determined by the Laplace equation:

Boundary conditions:

-- no fluid penetration through the wall

-- pressure is atmospheric

The boundary condition at a free surface can be rewritten as

0

0z

hz

:

0ppz :

01

02

2

ztgz

Using the boundary condition at the wall gives B=0.

That is,

Solving the Laplace equation, we obtain

Seek the velocity potential in the form of a plane wave,

Using the boundary condition at an interface produces the dispersion relation

Consequently, the phase and group velocities are

khkh

khkhk

gk

U

khkg

kc

coshtanh

tanhdd

tanh

2

tkxzf cos

tkxhzkBhzkA cossinhcosh

tkxhzkA coscosh

khkgkhg

khk tanhcoshsinh 22

0

Deep water and long waves

Let us find the dispersion relation and the phase and group velocities for the cases (a) of very deep water (h/λ>>1) and (b) of long waves (h/λ<<1).

(a) Deep water, h/λ>>1, or kh>>1

(b) Long waves, h/λ<<1, or kh<<1

kg

kU

kg

kc

kgkh

211

dd

,tanh

ghk

Ughk

c

ghkkhkh

dd

,tanh

There are two linearly independent solutions of equation (1), and

Since equation (1) is linear, any linear combination of solutions (2) is a solution of (1). The general solution can be written as

Seeking solution in the exponential form, , gives the auxiliary equation,

Appendix: solution of the amplitude equation

02 fkf

mzef

022 km km

kzkz BeAef

The roots of the auxiliary equation:

(1) This is the linear ordinary differential equation with constant coefficients.

kze kze(2)

Here, A and B are unknown arbitrary constants (to be determined from the boundary conditions).

(3)

kzkzkzkzkzkz

eBA

eBAee

Bee

Af

22221111

11

kzBkzAf sinhcosh 11

Let us show that the general solution of (1) can be also written in the following form

Using the definitions of hyperbolic functions,

and re-defining the constants

we can prove that (4) is equivalent to (1). Similarly, it can be easily proved that the general solution of (1) can be also written in the forms

A1 and B1 are the arbitrary constants, different from A and B.

,,22

1111 BAB

BAA

(4)

)(sinh)(cosh hzkBhzkAf 22

)(sinh)(cosh hzkBhzkAf 33

or

(5)

(6)

Conclusion:

- (3), (4), (5), and (6) are the equivalent forms of the general solution of equation (1); all these forms are the linear combinations of basis solutions (2). All these expressions include two unknown arbitrary constants to be determined from the boundary equations, but the final solution (with determined constants) is unique.

- It is recommended to chose such a form of the general solution that can simplify derivations. For instance, (5) should be convenient when one of the boundary conditions is imposed at z=-h, where , which immediately gives one constant, A3. (3), (4), and (6) can be also used and should produce the same final expression, but intermediate derivations can be lengthier.

3Ahf )(