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1
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 2
i 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
Homework #4
Textbook problems4.14.64.74.124.13Due: 3/10 (Fr.)
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Refraction : change of wave directiondue 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
0h
1h
0
1
10
1c t0c t 0
B
1B
< 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
onOblique Incidence refraction occurs!
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WAVES
https://www.youtube.com/watch?v=w2s2fZr8sqQ
https://www.youtube.com/watch?v=JppViHtLNlc
https://www.youtube.com/watch?v=RVyHkV3wIyk&list=PL‐QzaGk0yxEuWrFyKFvOhhpuM4kd0IJD0
Plunging Breaking Waves
Waves break when the crest particle velocity exceeds its celerity.
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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 for sloped bottomHo’=KrHo (deepwater unrefracted wave height)
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
Breaking Wave example
Deepwater T=8s, H=2m (normal incidence), beach slope=1/20
Find breaker height, breaker depth, and breaker type using the GW chart
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Wave theory selection example
Water depth=1mWave period=7sWave height=0.3mFind the best wave theory
Stokes’ 2nd-order Wave Theory
η= +
Valid when Ursell #: L²H/h³< 26.3
cos( )A kx t 21 cos(2 2 )2
kA kx t
Geometric Comparison
Nonlinear waves
higher and sharper crests
Shallower and flatter troughs
Nonlinear theory enables one to analyze large amplitude (large H/L) waves more accurately
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Wave Kinematics
)sin(sinh)(sinh
tkxkdzdk
TH
w
)(2cos)(4sinh
)(2cosh2
163)cos(
cosh)(cosh
2tkx
kd
zdkkHtkxkd
zdkgkHu
)cos(sinh
)(cosh tkxkd
zdkTHu
)(4sinh
)(2sin)(2sinh2
163
)sin(cosh)(sinh
2 kd
tkxzdkkHtkxkd
zdkgkHw
Linear Wave Kinematics
Stokes Wave Kinematics
tuax
twaz
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
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
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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/mb)b=0.048*baylength(km)/bay meandepth(m)V=max wind velocity (m/s)P=(spatial mean – lowest) atmospheric
pressure (mb)D=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
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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)
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TsunamiAttackTsunamiAttack
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