Lý thuyết sóng tuyến tính linear wave theory

capillary waves, gravity waves, infra-gravity waves, tidal waves Wave Classification: Generation * impulse free wave - tsunami * constant forcing forced - tide, wind Plane Water Surface

Linear Wave Fundamentals

To describe a wave: H, L, T, C

Wave steepness: H/L

Wave speed: C = L/T

C

a = Wave Am plitude

Linear Wave Theory – Principle of Superposition

capillary waves, gravity waves,

infra-gravity waves, tidal waves

Wave Classification: Generation

* impulse (free wave) - tsunami

* constant forcing (forced) - tide, wind

Plane Water Surface

Wind

friction

High p Low p

Water Surface /w waves at time t 1

Water Surface /w waves at time t 2 Wind Direction

Reduced Static Pressure Stream Lines

Wind Direction

friction

Mechanism: Wind Generating Waves

DEEP WATER Wave Direction

Wave Base

Water Particle

Motion of water particles

Progressive waves

Standing waves

Wave Classification: Motion

• incompressible, homogeneous fluid

• neglect surface tension and Coriolis

• uniform and constant pressure at the surface

• ideal fluid (no viscosity)

• horizontal, impermeable bottom

• small wave height and invariant form

• long-crested waves (2D approach)

George Airy, 1801-1892

Basic Relationships

2

Wave profile:

,

k

L

C



Wave phase speed:

Dispersion relationship

2

tanh

2

C

L



2

tanh

2

C

L



tanh

2

L



Wave phase speed:

Wave length:

t 1 t 2

Waves (short, irreg.) Swell (long, reg.)

Asymptotic Solutions

Deep water: (d/L >1/2)

2 2

o

gT d gT

L



2

o

gT d gT

L



2

L





Shallow Transitional Deep

Shallow water: (d/L < 1/25)

tanh

2

gT d

L



2

tanh

2

gT d

L



L L

2d/L 0.01

0.1 1 10

tanh = 1

ta nh (2d/

d/ L

ta nh (2d/

tanh

o

d

L



2

tanh

o

L



(iterative solution required)

sinh 0.5(x x)

cosh 0.5( x x)

tanh









/ sinh

/ cosh

Hyperbolic Functions

Variation of Wave Parameters with d/L o

1

t

n

T



2

t n T





For a simple harmonic wave train,

the wave period is independent of depth!

Water Particle Velocity

Water Particle Velocity

cosh 2 ( ) / 2 2

cos

2 cosh(2 / )

u

sinh 2 ( ) / 2 2

sin

2 cosh(2 / )

w

 

cosh 2 ( ) / 2 2

sin cosh(2 / )

x

a

 

sinh 2 ( ) / 2 2

cos cosh(2 / )

z

a

 

Water Particle Motion: Equations

cosh 2 ( ) / 2 2

sin

2 sinh(2 / )

sinh 2 ( ) / 2 2

cos

2 sinh(2 / )

2 2

2 2 1

 

cosh 2 ( ) /

2 sinh(2 / )

H A

d L

 





sinh 2 ( ) /

2 sinh(2 / )

H B

d L

 





Water Particle Motion: Trajectory

Shallow water: Deep water:

,





2 /

2

z L H

 

 

Pressure Under Waves

Dynamic component due

to acceleration

Water surface profile

Atmospheric pressure Static component

of pressure

' a

ppp (relative pressure)

Pressure Under Waves

cosh 2 ( ) /

cosh 2 /

z

K

d L

 





Pressure response factor

1 cosh 2 /

z

d L

 



At bottom (z = -d):

z

gK

 

 



Water level from pressure: (N = 1 for linear waves)

Potential energy:

( )

x L

x



( ) 2

PT

d z



 



   

2

1

16

w

Kinetic Energy:

2

1

( )

16

x L







2 2

( )

2

K

Total Energy:

2

1

8

2

1

8

(per wave and unit width) (per wave and unit surface area)

w

Definition (rate of energy transfer):

1x L

D

D

L









  

 

2

1

8 2 sin 2

kd T

Group Velocity

t 1 5 4 3 2 1

Distance from Wavemaker

t 2 6 5 4 3 2

t 3 7 6 5 4 3

t 4 8 7 6 5 4

t 5 9 8 7 6 5

t 6 10 9 8 7 6 5

C

C g

> C g

Group Velocity

1

2 sinh(4 / )

g



o

L

T

g

L

(shallow water)

Two waves:

1 2

cos

env

Envelope:

Envelope of Superimposed Waves

Cnode= Cg

Energy Transport