Capsizing of Ships
Following sea is the most dangerous!
Q#5: Perfect Standing Wave
(Reflection from vertical wall)
  A cos(kx  t )  A cos(kx  t )
2 gA cosh k ( z  h)
 
cos kx sin t

cosh kh
e
u
Total Pressure
kz
(deep)


, w
x
z

p    gz  
t
Partial Standing Wave
Hi
H
cos(kx  t )  r cos(kx  t   )
2
2
 I ( x) cos t  F ( x)sin t
 
Hi
Hr

I
(
x
)

cos
kx

cos(kx   )

2
2
where 
 F ( x)  H i sin kx  H r sin(kx   )


2
2

Max/Min when
0
t
H 
2
H 
2
    i   r  
 2   2 
F ( x)

 tan t 
I ( x)
Hi H r
cos(2kx   )
2
At quasi-antinode
:
At quasi-node
:
1
max  ( H i  H r )
2
1
min  ( H i  H r )
2
distance between max and min
H i  max  min
H r  max  min
Hr
Reflection coefficient 
Hi
L

4
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, nondispersive 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=20200m 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
from geometry
0
c 0t 0
B0
h0
h1
1
B1
c1 t
1
sin  0 
sin 1 
c1t
Diag .
c0 sin a0


 find new heading
c1 sin a1
cos  0 
< Snell’s law >
c0t
,
Diag .
B0
,
Diag .
cos 1 
B1
Diag .
B0 cos a0


 find new B
B1 cos a1
Combined shoaling & Refraction
 reflection 
If 
 negligible
 diffraction 
Power(Energyflux) Conservation
1
1
2
 gA0 B0C g 0   gA2 B C g
2
2
Cg 0 B0
A

 Ks  Kr
A0
Cg B
 K s  shoaling coefficient
where 
 K r = refraction coefficient
 Normal Incidence 
 no refraction

 refraction occurs!
 Oblique Incidence 
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 kh
Deep: H/L=1/7=0.14

Shallow
McCowan’s criterion: flat bottom
H=0.78h
Goda-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 =2.1m
Ho’/GT²=0.00214
From fig.2-72(m=0.05): Hb=3.15m; plunging
Hb/Ho’=1.5 & Hb/GT =0.0032
From fig.2-73
hb/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
Linear Wave Kinematics
u
w
H cosh k (d  z )
T
sinh kd
H sinhk ( d  z )
T
sinhkd
cos( kx  t )
sin(kx  t )
w
az 
t
u
ax 
t
Stokes 2nd-order Wave Kinematics
H gk cosh k ( d  z )
3 H 2k cosh 2 k ( d  z )
u
cos(kx t ) 
cos 2 ( kx t )
4
2  cosh kd
16
sinh ( kd )
H gk sinhk ( d  z )
3 H 2k sinh2 k ( d  z ) sin 2 ( kx t )
w
sin(kx  t ) 
2 
cosh kd
16
sinh4 ( kd )
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= A cos(kx  t ) + 1 kA2 cos(2kx  2t )
2
Valid when
Ursell #: L H/h < 26.3

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 Current
H smaller, L longer: wave steepness
decreased, C faster

Wave in Adverse Current
H 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-18.44
M=7, m=0(Hmax=1m): small damage
M=8, m=2.4(Hmax=10m)
M>8.6, m>4(Hmax=30m): considerable damage
After December 26 Tsunami
Before 2004 Dec. 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/mb
b=0.048(baylength(km)/ bay
meandepth(m))
V=max wind velocity (m/s)
P=(spatial mean – lowest) atmospheric
pressure
D=wind direction
Empirical Storm-surge Forecasting EX

Find the maximum sea-rise when
Lowest atm pressure=0.85 bar
Spatial mean atm pressure=1 bar
Bay length=5km
Bay mean depth=5m
Max wind velocity=50m/s
Normal 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/p
ublic_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 installed
H & L at h=4cm? C=? Cg=? Will it break? What type?
Length of surf-zone? Which wave theory?
