02. partial standing wave
TRANSCRIPT
Capsizing of Ships
Following sea is the most dangerous!
Q#5: Perfect Standing WaveQ#5: Perfect Standing Wave(Reflection from vertical wall)(Reflection from vertical wall)
cos( ) cos( )A kx t A kx t
2 cosh ( ) cos sincosh
gA k z h kx tkh
( )kze deep
Total Pressure
,u wx z
p gzt
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 tH HI x kx kx
H HF x kx kx
2 2
( )Max/Min when 0 tan( )
cos(2 )2 2 2i i rr
F xtt I x
H H HH kx
max
min
max min
max min
max min
1At quasi-antinode : ( )21At quasi-node : ( )2
distance between and 4
Reflection coefficient
i r
i r
i
r
r
i
H H
H H
L
HH
HH
Typical Size of LNG Tank
Seiching
Long-period oscillation of harbors due to resonance sloshing
[1] (24%) Select proper answer When celerity depends on wave length, the wave is
called (dispersive wave, non-dispersive wave). With (dispersive wave, non-dispersive wave),
communication is possible. Acoustic waves are (dispersive waves, non-dispersive
waves). Wave induced dynamic pressure is (linearly,
quadratically) proportional to wave height. When the distance between semi-antinode and semi
node of a partial standing wave is 30m, the wavelength of the incident wave is (120m, 60m)
(Longer, Shorter) water waves travel faster in deep water. The front of water waves in deep water moves with (celerity,
group velocity). The primary restoring force for water waves of wavelength=20-
200m is (gravity, Coriolis, surface tension) force. The wave energy is (linearly, quadratically) proportional to wave
height. Long waves generated by large-scale atmospheric pressure
variation are called (tidal waves, tsunamis, storm surge) The maximum vertical acceleration of water-wave particles occurs
at (crest, crossing point) Water depth=200m is considered to be (deep, transitional) for a
sinusoidal water wave of wavelength=470m.
[2] (6%) 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.
[3] (4%) When the potential energy of a
regular wave for certain area is 20000J, what is the corresponding kinetic energy?
[4] (6%) The group velocity of a shallow
water wave is 3m/s. What is the corresponding water depth?
[5] Consider a deep-water wave with 8-s period and 4-m height?
(a) (10%) What is the power of this wave along the crest width of 500m?
(b) (10%) 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))
[6] (a) (5%) When wave length is 100m at 10-m water depth, what is the corresponding wave period?
(b) (10%) When wave height=2m, what is the major semi-axis of the elliptical particle trajectory at z=-3m?
(c) (10%) What is the amplitude of the horizontal particle velocity at the same location z=-3m?
(d) (15%) If a vertical wall is present at the 10-m depth, a perfect standing wave will be formed in front of the wall. In that case, what is the dynamic pressure amplitude of the standing wave at z=-3m under the anti-node.
Wave Refraction
Change of wave heading due to bottom topography
Cf. reflection, diffraction
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. .
sin find new headingsin
cos , cos. .
cos find new Bcos
c t c tDiag Diag
c ac a
B BDiag Diag
B aB a
< Snell’s law >
Combined shoaling & Refraction
2 20 0 0
0 0
0
reflectionIf negligible
diffractionPower(Energyflux) Conservation
1 12 2
shoaling coefficientwhere
= refraction coefficient
Normal Incidence no refracti
g g
gs r
g
s
r
gA B C gA B C
C BA K KA C B
KK
on
Oblique Incidence refraction occurs!
Wave Breaker Type
Spilling: steeper crest : loose stability at cusp: mild beach slope
Plunging: overturning: steeper beach
Surging: bottom part surges over high-sloped beach: very steep beach=high reflection
Plunging Breaking Waves
Waves break when the crest particle velocity exceeds its celerity.
Wave Breaking
Deep & transitional depth:General: H/L=(1/7)tanh khDeep: H/L=1/7=0.14
Shallow McCowan’s criterion: flat bottomH=0.78hGoda-Weggel chart: with slope
Wave Breaker Ex.) SPM 2-135
Given: Ho=2m, T=10s, beach slope=1/20, Kr=1.05
Find: breaker height Hb, depth hb, type by using Goda-Weggel chart
Unrefracted deepwater height: Ho’=Kr*Ho
Unrefracted deepwater height: Ho’=Kr*Ho =2.1mHo’/GT²=0.00214From fig.2-72(m=0.05): Hb=3.15m; plunging
Hb/Ho’=1.5 & Hb/GT=0.0032From fig.2-73hb/Hb=0.96 therefore hb=3.02m
Surf-zone length=3.02/0.05=60m
Wave breaking (20pt)
Deepwater T=8s, H=2m (Normal Incidence); beach slope=1/20
Find breaker height, breaker depth, and breaker type using the chart
Wave-theory Selection Diagram
Water depth=1m, wave period=7s, wave height=0.3 m
Find the best wave theory
Wave Kinematics
)sin(sinh)(sinh
tkxkdzdk
TH
w
)(2cos)(4sinh
)(2cosh2
163)cos(
cosh)(cosh
2tkx
kd
zdkkHtkxkdzdkgkHu
)cos(sinh
)(cosh tkxkdzdk
THu
)(4sinh
)(2sin)(2sinh2
163
)sin(cosh)(sinh
2 kd
tkxzdkkHtkxkd
zdkgkHw
Linear Wave KinematicsLinear Wave Kinematics
Stokes 2Stokes 2ndnd-order Wave Kinematics-order Wave Kinematics
tuax
tw
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3/27 SNAME Offshore Sym
Rec. Center (Garden Room)9:00 – 3:00
2010 OCEN300 MINI-TERM PROJECT
Research on Ocean Hydro-Power
Team (5-member) selects a research topic related to ocean wave energy and tidal/current energy conversion.
Select a particular concept/system and describe how it works.
Discuss pros and cons compared with other existing concepts.
Discuss its efficiency, survivability, and environmental impacts
Discuss the estimated cost when realized as a proto-type system.
Discuss the ideas how the existing technology and cost-effectiveness can be improved.
Prepare a 5-page report summarizing the study. (Report due on 5/4)
Prepare a team presentation. (Schedule: 4/29 A-E; 5/4)
A: Allahar Jacquelene, Babbitt Charles, Blackburn Megan, Blackmar Philip, Brotzman Duncan
B: Brzezniak Michael, Cantu Felix, Castro Adrian, Dailey David, Demmer Michael
C: Feldman Kyle, Fields Waylon, Finkelshteyn Michael, Fisher Ian, Fluitt Timothy
D: Ford Bryce, Forester Aaron, Freyman Michael, Galatas Joel, Gibson Allison
E: Goebel Kevin, Gonzales Stephanie, Grant Alexander, Holub Chase, Hulsey Jennifer
F: Keel Ryan, Knoll Alex, Lee Sangwook, Lindanger Christopher, McBee Harvey
G: McClung Evan, McNeil Ryan, Medellin Abel, Messina Michael, Mieras Ryan
H: Novasad Nicholas, Outten Kyle, Parker Christopher, Ramsey Paul, Ryan Christopher
I: States William, Stevenson Katy, Tallichet Jules, Thi Andy J: Tipton Craig, Vittone Cynthia, Walsh Andrew, Oyenike
Olaniyi
Wave diffraction: wave deformation by structures
Stokes’ 2nd-order Wave Theory
η= +
Valid when Ursell #: LH/h< 26.3
cos( )A kx t 21 cos(2 2 )2kA kx t
Geometric Comparison
Nonlinear waves
higher and sharper crests
Shallower and flatter troughs
large steepness H/L (Linear theory assumes small amplitude)
Opened Orbit: Stokes’ drift
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
Long waves
Tsunami Storm surge Tide
Tsunami
Long-period (tens of minutes) gravity waves generated by submarine earthquakes, landslides, volcano eruptions, explosion, asteroid impact
Can build up heights in coastal regions as large as 30m
(ex. Hilo, Hawaii: 11m, Wavelength: can be as large as 200km)
Typical speed:Deep: speed of airplane (e.g. 500miles/hr)Coastal: speed of car (e.g.70 miles/hr)
Tsunami
Magnitude of Earthquake Richter Scale M=log(A/Ao)(A: max. amplitude recorded by a seismograph
at 100km from epi-center, Ao=0.001mm)
Tsunami Magnitude m=2.61M-18.44M=7, m=0(Hmax=1m): small damageM=8, m=2.4(Hmax=10m)M>8.6, m>4(Hmax=30m): considerable damage
Before 2004 Dec. 26 Tsunami
After December 26 Tsunami
Storm Surge
Suction effects by large-scale low atmospheric pressure
Wave/water-mass pile up at costal region by strong winds
Max. anomaly=f(max. wind vel., wind direction, lowest atmospheric pressure)
Storm surge
Although the wind shear stress is usually small, its effect, when integrated over a large body of water, can be catastrophic.
Hurricanes, blowing over the shallow continental shelf of GOM, have caused rises in water levels in excess of 6m at the coast.
Empirical storm-surge forecasting
Max anomaly (sea-rise in cm)=a P + b V² cos D
a=0.99 cm/mbb=0.048(baylength(km)/ bay
meandepth(m))V=max wind velocity (m/s)P=(spatial mean – lowest) atmospheric
pressureD=wind direction
Empirical Storm-surge Forecasting EX
Find the maximum sea-rise when
Lowest atm pressure=0.85 barSpatial mean atm pressure=1 barBay length=5kmBay mean depth=5mMax wind velocity=50m/sNormal wind direction
Tidal Wave: sun-moon-earth gravitation
Semi-diurnal tide: 2 highs & 2 lows/day (ex Cape-Cod)
Diurnal tide: 1 high & 1 low/day (ex New Orleans)
Mixed tide: combination 1 semi-high and 1 major high – (ex Los Angeles)
Tidal Current: Ex. 3.1m/s (San Francisco) max=5.2m/s NOS (National Ocean Survey)
Tidal Energy (7)
Promising West Coast Sites
Tidal Energy (3) 21st Century projects
under consideration are based on ‘in stream turbine’ technology at sites with high tidal current velocities
Only a limited number of suitable sites in continental USA with San Francisco the best
Current Energy Conversion
Tidal Energy Conversion
Tidal Energy (2)
La Rance dam and typical turbine/generator configuration
http://www.youtube.com/watch?v=ZcA3e8_j8XA
http://www.youtube.com/watch?v=rQtMPdLZ2L4&NR=1
http://www.youtube.com/watch?v=94iZa96HpUA
Tidal Energy
http://www.youtube.com/watch?v=tSBACzRE3Gw&feature=related
http://www.youtube.com/watch?v=4Iq-h4ShZ8s&feature=related
Have a Good Spring Break!
WOW (Waves On Web)
Ceprofs.tamu.edu/mhkim/wow
cavity.ce.utexas.edu/kinnas/wow/public_html/waveroom
Wavemaker: Review
Flap motion: 1.5 cycles/s, h=80cm, H=3cm Find T=?, L=?, C=?, k=?, w=?, Cg=?, Power(tank
width=90cm)=? Breaking? Speed of wave front=? max horizontal particle velocity? max radius of particle orbit? Total max pressure 10cm below MWL?
Mild-slope (m=0.05) is installedH & L at h=4cm? C=? Cg=? Will it break? What
type? Length of surf-zone? Which wave theory?