Waves In Nature: Summary Non-dispersive wave: C constant: not a function of wavelength; original...
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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