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.)

2

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!

3

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.

4

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

5

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

6

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

7

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

8

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