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 sin

cosh

gA k z hkx t

kh

( )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 t

H HI x kx kx

H HF x kx kx

2 2

( )Max/Min when 0 tan

( )

cos(2 )2 2 2i 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

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

Homework #4

Textbook problems4.14.64.104.124.13Due: 3/12

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!

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

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:

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 =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 diffraction: wave deformation by structures

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

Stokes’ 2nd-order Wave Theory

η= +

Valid when Ursell # LH/h< 26.3

cos( )A kx t 21cos(2 2 )

2kA kx t

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 2Stokes 2ndnd-order Wave Kinematics-order Wave Kinematics

t

uax

t

waz

3/27 SNAME Offshore Sym

Rec. Center (Garden Room)9:00 – 3:00

Long waves

Tsunamis Storm surge Tide

Tsunami

Long-period ( x tens of minites) gravity waves generated by submarine earthquakes, landslides, volcano eruptions, explosion

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 (500miles/hr)Coastal: speed of car

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(H=1m): small damageM=8, m=2.4(H=10m)M=8.6, m=4(H=30m): considerable damage

Before December 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

http://www.youtube.com/watch?v=tSBACzRE3Gw&feature=related

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

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?