Waves In Nature: Summary

Non-dispersive wave: C constant: not a function of wavelength; original signal conserved; communication possible

Ex. E-M wave, sound wave, shallow-water wave

Dispersive wave: C depends on wavelengthCommunication impossibleEx. Water waves in deep and transitional depth

Elastic wave on tensioned string

Distance of up-down harmonic movement=0.5cm

5 times per second Applied Tension=90N String mass per unit length=0.25

(kg/m³)

Q: find celerity and wavelength

Water Waves : Generated by

Wind: Wind Waves

Submarine earthquake: tsunami

Large-scale atm-pressure: storm surge

Sun-moon-earth gravity: tidal waves

Water waves Water particle does not propagate with

wave: only energy is propagating

Similarity bet. water waves and simple pendulum

Water wave (deep)

T=√2π √L/g (when wave slope is small)

Simple pendulumT=2π √L/g (when angle is small)

Dispersion Relation: L & T (or ω & k)

Lo=gT/2π (deep water)

L=Lo tanh kh (general h)

ω² = kg tanh kh

Eckart formula (<5% error)L=Lo √tanh koh (where ko= ω²/g)

Asymptotic behavior of ω² = kg tanh kh

Deep-water limit:tanh kh ≈ 1 (when kh>π or h>L/2): ω² = kg Celerity C= ω/k = g/ω = √2π/g √L

Shallow-water limit:tanh kh ≈ kh (when h<L/20) : ω² = kgh Celerity C= ω/k =√gh

Q1: Is 500-m waterdepth deep water?

Q2: Compare 10-s and 15-s water waves in deep water

Which is longer? Faster?

Dispersion relation for general h

Numerical (e.g. iteration)

L=Lo tanh(2πh/L) or ω² = kg tanh kh

Hunt formula (p72)

Pre-calculated Table (e.g. SPM TC1)

As a wave propagates to shallow-water region, wave length becomes shorter (dispersion) and wave amplitude becomes larger (shoaling) to make it steeper and steeper so that it eventually breaks.

Example: 12-s wave

Wave period remains the same during propagation, while wave length (dispersion) & amplitude (shoaling) change.

At h=400m

At h=3m

Wave theory

Governing Equation = Continuity Eq.

Physically, it means mass conservation of continum

Assumptions for Airy’s Linear Wave Theory

Ideal Fluid: inviscid, incompressible Small amplitude: A/L & A/h small Flat impermeable bottom (bottom

slope<1/10): reflection negligible Surface tension, Coriolis force neglected

Elastic wave & harmonic oscillator

http://www.youtube.com/watch?v=InrY9gnwMrA&feature=related

http://www.youtube.com/watch?v=SZ541Luq4nE&feature=related

Boundary Conditions

Bottom z=-h: zero normal velocity

Free Surface z=0:

Dynamic: P = Pa (atmospheric pressure)

Kinematic: free-surface velocity=particle velocity in normal direction

Linear: z=0 t z

Alternative expression

)cos(sinh

)(coshtkx

kd

zdk

T

Hu

)sin(sinh

)(sinhtkx

kd

zdk

T

Hw

(3.13)

A tsunami is detected at 12:00 on the edge (h=200m) of the continental shelf (constant slope=1:0.005) by a warning system. At what time can the tsunami be expected to reach the shoreline?

WOW (Waves On Web)

http://cavity.ce.utexas.edu/kinnas/wow/public_html/waveroom/

http://ceprofs.tamu.edu/mhkim/wow

http://ceprofs.tamu.edu/jzhang/

Linear Wave Kinematics

Nonlinear Wave Kinematics

Surface Stokes Drift Velocity over one period

Vs= Aωk

Ex) L=100m, A=3m, deepwater FS

Stokes drift vel.=0.45m/s(Cf. Hor. Particle vel. u=2.36m/s)

Pressure

Hydrostatic = -ρgz

Hydrodynamic = -ρ

= ρgA K(z) cos(kx-ωt)

Where K(z)= depth attenuation factor

t

t

Wave height can be measured from

Wave-riding Buoy

Resistance type probe

Pressure gage

SYNERGY

http://www.youtube.com/watch?v=rQtMPdLZ2L4&feature=PlayList&p=E3BC2CF6376E9715&index=17

CETO

http://www.youtube.com/watch?v=V27ZBODcv0c

http://www.youtube.com/watch?v=LkuNr5OMlts&feature=related

WAVE ENERGY

(1) Potential Energy / unit width

dx

2( )2 2

where cos( )

gd PE dxg dx

A kx t

22 2

0 0

1( ) cos ( )

2 4

L LA gd PE kx t dx gA L

Potential Energy within 1 wavelength

(2) Kinetic Energy / unit width

dz

dx

2 21( ) ( )( )( )

2d KE dx dz u w

2 2 2 2 2

cos( )

sinkz

kz

uA e kx t

w

u w A e

Kinetic Energy within 1 wavelength02 2 2 2

0 0

1 1( )

2 4

L L kzd KE A dx dz e gA L

Deepwater case

Total Wave Energy in 1 wavelength(valid for any waterdepth)

21

2gA L

2

1002

2 , breadth 1

gTL m

A m B km

Wave Energy Density = =energy per unit areaE

Ex) Find wave energy in 1 wavelength along 1-km crestline when T=8s, H=4m in deepwater

2 912 10 ( )

2E gA LB J

21

2E gA

Difference?

Celerity (Phase velocity)

Particle velocity

Group velocity = speed of energy transfer

Group Velocity = speed of energy transfer Celerity>Group Velocity

carrier wave

Ck

Consider 2 waves with difference

amplitude modulation

g

dC

dk

1 1 2 2

1 1

2 2

cos( ) cos( )

,2 2

2 cos cos( )2

A k x t A k x t

k kk

k

kA x t kx t

k

Group Velocity g

dC

dk

2 tanhkg kh

22 ( tanh sech )d g kh kgh kh dk

(eq.4.82)

1 2where, 1

2 sinh 2

gC nC

khn

kh

1:

2:

g

g

Deep C C

Shallow C C

General Depth:

WAVE POWER ≡ ENERGY FLUX

21

2 g ggA BC EBC

Where B=width of crest

Ex) How much power can be extracted when H=4m, B=1km, T=9s (a) h=2m : shallow(b) h=deep

5.4 /gC C gh m s

7

9

10.8 10 ( / ) 108

(1971 World Energy Consumption 5 10 )

gP EBC J s Watt MW

MW

Ocean: Future Energy Source

Unlimited, No Pollution Wave Energy Conversion Wind Energy Tidal Energy Current Energy OTEC (Ocean Thermal Energy

Conversion)

Power Intensity

Solar energy (at 15-N latitude)=0.17kw/m²

Wind energy = 0.58 kw/m

Wave energy= 8.42 kw/m

Wave & tidal power distribution

2TW of energy, the equivalent of twice the world’s electricity production, could be harvested from the world’s oceans2TW of energy, the equivalent of twice the world’s electricity production, could be harvested from the world’s oceans

Wave Enegy ResourceWave Enegy Resource

Floating OWC Wave Power Absorber

Land-based OWCOscillating Water Column

Islay, ScotlandIslay, Scotland

Overtopping Wave-E Plant

Wave Energy Conversion

Fixed Oscillating Water ColumnOscillating Water ColumnFloating AttenuatorAttenuator

Floating OvertoppingOvertopping Floating

Point Point

AbsorberAbsorber

http://www.youtube.com/watch?v=4VplKt-vzK8

http://www.youtube.com/watch?v=SFD4vgHGEj4&NR=1&feature=fvwp

http://www.youtube.com/watch?v=F0mzrbfzUpM&NR=1

http://www.youtube.com/watch?v=uHTsjqC_rmE&feature=response_watch

OWC

http://www.youtube.com/watch?v=gcStpg3i5V8

http://www.youtube.com/watch?v=Y6AZjeduI0g&feature=fvw

http://www.youtube.com/watch?v=tt_lqFyc6Co

http://www.youtube.com/watch?v=XyNQS9sg5l8&NR=1&feature=fvwp

Energy island

http://www.youtube.com/watch?v=OrzY6cs9Jic&feature=related

Salter’s Nodding Duck

http://www.youtube.com/watch?v=_bdeNuRF-yE

http://www.youtube.com/watch?v=P1wZb_RKRQg&feature=related

http://www.youtube.com/watch?v=QTVwDmMljVs&feature=related

TIDE

http://www.youtube.com/watch?v=TrhfFLNah24&feature=PlayList&p=E3BC2CF6376E9715&index=0

Conference

http://www.youtube.com/watch?v=ovw-pHqyP7E

HW#3 due 2/24 (Thur.)

EX.#1 : 3/1 (Tue.)

Slides will be placed at

http://ceprofs.tamu.edu/mhkim

Difference?

Celerity (Phase velocity)

Particle velocity

Group velocity = speed of energy transfer

Shoaling: Variation of H with depthShoaling: Variation of H with depth

Conservation of wave power (energy flux) (mild slope) (negligible reflection)

21

2g gPower EBC gA BC constant

Ex.) SPM2-27 Shoaling: neglect reflection (unit width)

Given:

Required:

156 , 2 ( )o oL m H m Deep

Find L, H at h = 3m

Power?

Total Energy delivered in 1hr?

2

102

115.6 / 7.8 /

2

o

oo go o

gTL T s

LC m s C C m s

T

2139000( )

2 gPower W gA BC

Accumulated Energy in 1 hr =

39000(J/s) x 3600 (s) = 140 MJ

2139 ( )

2 o goPower gA BC kw deep

Assume Shallow

5.42 /

54.2

gC C gh m s

L C T m

(check h/L=0.055) accurate 53.6 m C=5.36 m/s n=0.96A=1.24 m(24% increase)

As a wave propagates to shallow-water region, wave length becomes shorter and wave amplitude becomes larger to make it steeper and steeper so that it eventually breaks.

Reminder!

Wavelength change: dispersion relation

Wave-height change: conservation of power; shoaling equation

Perfect Standing WavePerfect Standing Wave(Reflection from vertical wall)(Reflection from vertical wall)

cos( ) cos( )A kx t A kx t

2 cosh ( )cos sin

cosh

gA k z hkx t

kh

( )kze deep

Particle Velocity

Total Pressure

,u wx z

p gzt

Perfect Standing WavePerfect Standing Wave

cos( ) cos( )

2 cos cos

A kx t A kx t

A kx t

2sin sin

2cos sin

kz

kz

gAku e kx t

xgAk

w e kx ty

At node 0 cos 0 0

max

kx w

u

At anti-node max cos 1 0

max

kx u

w

WOW (Waves On Web)

http://cavity.ce.utexas.edu/kinnas/wow/public_html/waveroom/

http://ceprofs.tamu.edu/mhkim/wow

http://ceprofs.tamu.edu/jzhang/

Partial Standing Wave

cos( ) cos( )2 2( )cos ( )sin

( ) cos cos( )2 2where

( ) sin sin( )2 2

i r

i r

i r

H Hkx t kx t

I x t F x t

H HI x kx kx

H HF x kx kx

2 2

( )Max/Min when 0 tan

( )

cos(2 )2 2 2

i i rr

F xt

t I x

H H HHkx

max

min

max min

max min

max min

1At quasi-antinode : ( )

21

At quasi-node : ( )2

distance between and 4

Reflection coefficient

i r

i r

i

r

r

i

H H

H H

L

H

H

H

H

Refraction : change of wave direction due to bottom topography

0 10 1

0 0

1 1

0 10 1

0 0

1 1

from geometry

sin , sin. .

sinfind new heading

sin

cos , cos. .

cosfind new B

cos

c t c t

Diag Diag

c a

c a

B B

Diag Diag

B a

B a

< Snell’s law >

Combined shoaling & Refraction

2 20 0 0

0 0

0

reflectionIf negligible

diffraction

Power(Energyflux) Conservation

1 1

2 2

shoaling coefficientwhere

= refraction coefficient

Normal Incidence no refracti

g g

gs r

g

s

r

gA B C gA B C

C BAK K

A C B

K

K

on

Oblique Incidence refraction occurs!

Homework #4

Textbook problems4.14.64.104.124.13Due: 3/9 (Tue.)

Typical Size of LNG Tank

Seiching

Long-period oscillation of harbors due to resonance sloshing

98

EXAMPLE 3: Multi-body & Sloshing: SIMULATION RESULTS (FT + LNGC) Hs=2m, Tp=12s

18%

6m90º

Comparison of Roll RAO (LNG-FPSO)

SINGLE BODY CASE …SINGLE BODY CASE … Coupling in Time DomainCoupling in Time Domain

99

90º

PNU-MPS More Violent Sloshing: Experiment vs MPS

Time : 3.03 Sec

Time : 3.16 Sec

Time : 3.29 Sec

Pre

s(N

/m2)

0

5000

10000

Time(sec)2 4 6 8 10

0

5000

10000

[1] (3pt) When a hypothetical sinusoidal wave satisfies the dispersion relation ω²=2k² between circular frequency ω and wave number k, find its celerity and group velocity.

 

[2] (2pt) When the potential energy of a regular wave for certain area is 20000J, what is the corresponding kinetic energy?

[3] (3pt) The group velocity of a shallow water wave is 3m/s. What is the corresponding water depth?

[4] Consider a deep-water wave with 8-s period and 4-m height?

(a) (4pt) What is the power of this wave along the crest width of 500m?

(b) (5pt) If this deepwater wave propagates to the area of 2-m water depth, what is the new wave length and wave height at that location? (Assume 2D wave of normal incidence, shallow-water wave at 2-m depth, and mild bottom slope: use conservation of wave energy flux (power))

Wave Breaking

Deep & transitional depth:General: H/L=(1/7)tanh khDeep: H/L=1/7

Shallow McCowan’s criterion: flat bottomH=0.78hGoda-Weggel chart:

Wave Breaker Type

Spilling: steeper crest-> loose stability: mild beach slope

Plunging: overturning: steeper beach

Surging: bottom part surges over high-sloped beach: very steep beach=high reflection

Geometric Comparison

Nonlinear waves

higher and sharper crests

Shallower and flatter troughs

Nonlinear theory enables one to analyze large amplitude (large H/L) waves (Linear theory assumes small amplitude)

Wave Kinematics

)sin(sinh

)(sinhtkx

kd

zdk

T

Hw

)(2cos)(4sinh

)(2cosh2

16

3)cos(

cosh

)(cosh

2tkx

kd

zdkkHtkx

kd

zdkgkHu

)cos(sinh

)(coshtkx

kd

zdk

T

Hu

)(4sinh

)(2sin)(2sinh2

16

3)sin(

cosh

)(sinh

2 kd

tkxzdkkHtkx

kd

zdkgkHw

Linear Wave KinematicsLinear Wave Kinematics

Stokes Wave KinematicsStokes Wave Kinematics

t

uax

t

waz

WAVE-CURRENT INTERACTION

Wave in Coplanar CurrentH smaller, L longer: wave steepness

decreased, C faster

Wave in Adverse CurrentH larger, L shorter: wave steepness

increased, C slower

If adverse-current velocity > 0.5C: breaking

OCEN300 WAVE MECHANICS (Mini TERM PROJECT)

Form a 5-person team Use WOW Java applets to research one of the given

topics. http://cavity.ce.utexas.edu/kinnas/wow/public_html/waveroom/

http://ceprofs.tamu.edu/mhkim/wow

http://ceprofs.tamu.edu/jzhang/ (OCEN671) Prepare 3-page report (1. Intro, 2. Results & Major

Finding, 3. Conclusion) (plus relevant graphics and pictures in Appendix). (due 4/29)

Prepare 10 minute presentation by POWER POINT (presentation 4/24,4/29)

List of Topics

(1) WAVE KINEMATICS The change of magnitudes and shapes of

particle orbits of linear waves for various depths and wave parameters. The comparison of wave kinematics and particle orbits between linear and second-order Stokes wave theory for various water depths and wave parameters.

(2) NUMERICAL WAVE TANK Generate several partial standing waves and a

perfect standing wave by controlling the artificial damping coefficient inside the damping zone. (If damping coefficient=0, then perfect reflection) Determine the reflection coefficient for each case by measuring the semi node and anti-node. (This kind of work has to be done to assess the performance of wave absorbers in a physical wave tank.)

LIST of TOPICS

(3) WAVE FORCE ON VERTICAL PILES

Study the variation of inertia and drag wave forces on a vertical cylinder as function of cylinder size, wave parameters, and water depth. Compare the relative magnitudes of inertia vs. drag forces

Team

A: Arms, Baldwin, Bandas, Beck, BlakeB: Blaylock, Clark, Cotton, Dearing, DebowskiC: Drake, Dunbar, Fernandez, Finn, GarciaD: Garza, Goertz, Gonzalez, Grimes, HaleE: Hartsfield, Keller, Kidwell, Leon, LightseyF: Lister, Little, McCollum, Meader, MendezG: Neat, Noble, Parliament, Phillips, ReimerH: Reitblatt, Sanchez, Schlosser, Scholeman, SearsI: Smeeton, Sonnenberg, Stone, Tran, TrumbleJ: Vallejo, Wacasey, Winkelmann, Zwerneman