Waves In Nature: Summary
Non-dispersive wave: C constant: not a
function of wavelength; original signal
conserved; communication possible
Ex. E-M wave, sound wave, shallow-water wave

Dispersive wave: C depends on wavelength
Communication impossible
Ex. Water waves in deep and transitional depth

Elastic wave on tensioned string




Distance of up-down harmonic
movement=0.5cm
5 times per second
Applied Tension=90N
String mass per unit length=0.25
(kg/m³)
Q: find celerity and wavelength
Water Waves : Generated by
Wind: Wind Waves
Submarine earthquake: tsunami
Large-scale atm-pressure: storm surge
Sun-moon-earth gravity: tidal waves
Water waves

Water particle does not propagate with
wave: only energy is propagating
Similarity bet. water waves and
simple pendulum

Water wave (deep)
T=√2π √L/g (when wave slope is small)

Simple pendulum
T=2π √L/g (when angle is small)
Dispersion Relation: L & T (or ω & k)

Lo=gT
/2π (deep water)
L=Lo tanh kh (general h)
ω ² = kg tanh kh

Eckart formula (<5% error)
L=Lo √tanh koh (where ko= ω²/g)

Asymptotic behavior of ω ² = kg tanh
kh
Deep-water limit:
tanh kh ≈ 1 (when kh>π or h>L/2):
ω² = kg
Celerity C= ω/k = g/ω = √2π/g √L

Shallow-water limit:
tanh kh ≈ kh (when h<L/20) :
ω² = k gh
Celerity C= ω/k =√gh


Q1: Is 500-m waterdepth deep
water?
Q2: Compare 10-s and 15-s water
waves in deep water
Which is longer? Faster?

Dispersion relation for general h

Numerical (e.g. iteration)
L=Lo tanh(2πh/L) or ω ² = kg tanh kh


Hunt formula (p72)
Pre-calculated Table (e.g. SPM
TC1)

As a wave propagates to shallowwater region, wave length becomes
shorter (dispersion) and wave
amplitude becomes larger
(shoaling) to make it steeper and
steeper so that it eventually breaks.
Example: 12-s wave

Wave period remains the same
during propagation, while wave
length (dispersion) & amplitude
(shoaling) change.

At h=400m

At h=3m
Wave theory
Governing Equation = Continuity
Eq.
Physically, it means mass
conservation of continum

Assumptions for Airy’s Linear Wave Theory




Ideal Fluid: inviscid, incompressible
Small amplitude: A/L & A/h small
Flat impermeable bottom (bottom
slope<1/10): reflection negligible
Surface tension, Coriolis force neglected
Elastic wave & harmonic oscillator


http://www.youtube.com/watch?v=
InrY9gnwMrA&feature=related
http://www.youtube.com/watch?v=
SZ541Luq4nE&feature=related
Boundary Conditions

Bottom z=-h: zero normal velocity

Free Surface z=0:
Dynamic: P = Pa (atmospheric pressure)
Kinematic:
free-surface velocity=particle velocity
in normal direction
Linear: z=0



t
z
Alternative expression
u
w
H cosh k (d  z )
T
sinh kd
H sinhk ( d  z )
T
sinhkd
cos( kx  t )
sin(kx  t )
(3.13)

A tsunami is detected at 12:00 on
the edge (h=200m) of the
continental shelf (constant
slope=1:0.005) by a warning
system. At what time can the
tsunami be expected to reach the
shoreline?
WOW (Waves On Web)

http://cavity.ce.utexas.edu/kinnas
/wow/public_html/waveroom/
http://ceprofs.tamu.edu/mhkim/wow
http://ceprofs.tamu.edu/jzhang/
Linear Wave Kinematics
Nonlinear Wave Kinematics
Surface Stokes Drift Velocity over
one period
Vs= A ωk
Ex) L=100m, A=3m, deepwater FS
Stokes drift vel.=0.45m/s
(Cf. Hor. Particle vel. u=2.36m/s)
Pressure

Hydrostatic = -ρgz

t

Hydrodynamic = -ρ

t
= ρgA K(z) cos(kx-ωt)
Where K(z)= depth attenuation factor
Wave height can be measured from

Wave-riding Buoy

Resistance type probe

Pressure gage
SYNERGY

http://www.youtube.com/watch?v=
rQtMPdLZ2L4&feature=PlayList&p=
E3BC2CF6376E9715&index=17
CETO


http://www.youtube.com/watch?v=
V27ZBODcv0c
http://www.youtube.com/watch?v=
LkuNr5OMlts&feature=related
WAVE ENERGY
(1) Potential Energy / unit width

dx
d ( PE )   dxg


2

g
 2 dx
2
where   A cos(kx  t )

Potential Energy within 1 wavelength

L
0
A2  g L 2
1
2
d ( PE ) 
cos
(
kx


t
)
dx


gA
L

0
2
4
(2) Kinetic Energy / unit width
dz
dx
d ( KE ) 
1
 (dx)(dz )(u 2  w2 )
2
Deepwater case
u 
kz  cos 

A

e
 

 (kx  t )
 w
 sin 
u 2  w2   2 A2 e2 kz
Kinetic Energy within 1 wavelength

L
0
L
0
1
1
2 2
2 kz
d ( KE )   A   dx  dz e   gA2 L
0

2
4
1
2
Total Wave Energy in 1 wavelength   gA L
2
(valid for any waterdepth)
Wave Energy Density = E =energy per unit area
1
E   gA2
2
Ex) Find wave energy in 1 wavelength along
1-km crestline when T=8s, H=4m in deepwater
gT 2
L
 100m
2
A  2m,
B  breadth  1km
1
E   gA2 LB  2 109 ( J )
2
Difference?

Celerity (Phase velocity)

Particle velocity

Group velocity = speed of energy
transfer
Group Velocity = speed of energy transfer
Celerity>Group Velocity
Consider 2 waves with
 difference
  A cos(k1 x  1t )  A cos(k2 x  2t )
 k1 
 1 
   k
   
,    
 k 
 2
 2
 k2 
 2 
 k 
  
  2 A cos   x 
t   cos(kx  t )
k  
 2 
amplitude modulation
carrier wave


Cg 
d
dk
C

k
Group Velocity
d
Cg 
dk
General Depth:  2  kg tanh kh
2d  ( g tanh kh  kghsech 2 kh)dk
Cg  nC
(eq.4.82)
1
2kh 
where, n  1 

2  sinh 2kh 
Deep :
Shallow :
1
Cg  C
2
Cg  C
WAVE POWER ≡ ENERGY FLUX
1
  gA2 BC g  EBC g
2
Where B=width of crest
Ex) How much power can be extracted when
H=4m, B=1km, T=9s
(a) h=2m : shallow
(b) h=deep
Cg  C  gh  5.4m / s
P  EBCg  10.8 107 ( J / s  Watt )  108MW
(1971 World Energy Consumption  5 109 MW )
Ocean: Future Energy Source
Unlimited, No Pollution
 Wave Energy Conversion
 Wind Energy
 Tidal Energy
 Current Energy
 OTEC (Ocean Thermal Energy
Conversion)
Power Intensity

Solar energy (at 15-N
latitude)=0.17kw/m²

Wind energy = 0.58 kw/m

Wave energy= 8.42 kw/m
Wave & tidal power distribution
Wave Enegy Resource
2TW of energy, the equivalent of twice the world’s electricity production, could
be harvested from the world’s oceans
Floating OWC Wave Power Absorber
Land-based OWC
Oscillating Water Column
Islay, Scotland
Overtopping Wave-E Plant
Wave Energy Conversion
Fixed Oscillating Water Column
Floating Overtopping
Floating Attenuator
Floating
Point
Absorber




http://www.youtube.com/watch?v=
4VplKt-vzK8
http://www.youtube.com/watch?v=
SFD4vgHGEj4&NR=1&feature=fvwp
http://www.youtube.com/watch?v=
F0mzrbfzUpM&NR=1
http://www.youtube.com/watch?v=
uHTsjqC_rmE&feature=response_w
atch
OWC




http://www.youtube.com/watch?v=
gcStpg3i5V8
http://www.youtube.com/watch?v=
Y6AZjeduI0g&feature=fvw
http://www.youtube.com/watch?v=
tt_lqFyc6Co
http://www.youtube.com/watch?v=
XyNQS9sg5l8&NR=1&feature=fvwp
Energy island

http://www.youtube.com/watch?v=
OrzY6cs9Jic&feature=related
Salter’s Nodding Duck



http://www.youtube.com/watch?v=
_bdeNuRF-yE
http://www.youtube.com/watch?v=
P1wZb_RKRQg&feature=related
http://www.youtube.com/watch?v=
QTVwDmMljVs&feature=related
TIDE

http://www.youtube.com/watch?v=
TrhfFLNah24&feature=PlayList&p=E
3BC2CF6376E9715&index=0
Conference

http://www.youtube.com/watch?v=
ovw-pHqyP7E

HW#3 due 2/24 (Thur.)

EX.#1 : 3/1 (Tue.)

Slides will be placed at
http://ceprofs.tamu.edu/mhkim
Difference?

Celerity (Phase velocity)

Particle velocity

Group velocity = speed of energy
transfer
Shoaling: Variation of H with depth
Conservation of wave power (energy flux)
(mild slope)  (negligible reflection)
1
Power  EBCg   gA2 BCg  constant
2
Ex.) SPM2-27 Shoaling: neglect reflection (unit width)
Given:
Lo  156m, H o  2m ( Deep)
Required:
Find L, H at h = 3m
Power?
Total Energy delivered in 1hr?
2
gT
Lo 
 T  10s
2
Lo
1
Co   15.6 m / s  Cgo  Co  7.8 m / s
T
2
Power 
1
 gAo 2 BCgo  39 kw (deep)
2
Accumulated Energy in 1 hr =
39000(J/s) x 3600 (s) = 140 MJ
1
Power  39000(W )   gA2 BCg
2
Assume Shallow
C  C g  gh  5.42 m / s
L  C T  54.2 m
(check h/L=0.055) accurate 53.6 m
C=5.36 m/s
n=0.96
A=1.24 m
(24% increase)

As a wave propagates to shallowwater region, wave length becomes
shorter and wave amplitude
becomes larger to make it steeper
and steeper so that it eventually
breaks.
Reminder!


Wavelength change: dispersion
relation
Wave-height change: conservation
of power; shoaling equation
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
Particle Velocity
Total Pressure
kz
(deep)


u
, w
x
z

p    gz  
t
Perfect Standing Wave
  A cos(kx  t )  A cos(kx  t )
 2 A cos kx cos t
 2 gAk kz

 u  x   e sin kx sin t

 w     2 gAk ekz cos kx sin t

y


At node
  0  cos kx  0  w  0
u  max
At anti-node
  max  cos kx  1  u  0
w  max
WOW (Waves On Web)

http://cavity.ce.utexas.edu/kinnas
/wow/public_html/waveroom/
http://ceprofs.tamu.edu/mhkim/wow
http://ceprofs.tamu.edu/jzhang/
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
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 
Homework #4
Textbook problems
4.1
4.6
4.10
4.12
4.13
Due: 3/9 (Tue.)

Typical Size of LNG Tank
Seiching

Long-period oscillation of harbors
due to resonance sloshing
EXAMPLE 3: Multi-body & Sloshing: SIMULATION RESULTS (FT + LNGC)
Hs=2m, Tp=12s
90º
6m
18%
98
SINGLE BODY CASE … Coupling in Time Domain
Comparison of Roll RAO (LNG-FPSO)
Experiments
Freq. domain
Time domain
2.5
2.0
0%
1.5
1.0
0.5
3.0
Roll RAO (deg/m)
Roll RAO (deg/m)
3.0
0.0
0.4
0.6
0.8
(rad/sec)
Experiments
Freq. domain
Time domain
2.5
2.0
90º
18%
1.5
1.0
0.5
0.0
1.0
0.4
3.0
0.6
0.8
(rad/sec)
1.0
3.0
Experiments
Freq. domain
Time domain
2.5
2.0
37%
1.5
1.0
0.5
0.0
Roll RAO (deg/m)
Roll RAO (deg/m)

Experiments
Freq. domain
Time domain
2.5
2.0
56%
1.5
1.0
0.5
0.0
0.4
0.6
0.8
(rad/sec)
1.0
0.4
0.6
0.8
(rad/sec)
1.0
99
PNU-MPS More Violent Sloshing:
Experiment vs MPS

Time : 3.03 Sec
Pres(N/m2)
10000
5000
0
Time : 3.16 Sec
10000
5000
0
2
Time : 3.29 Sec
4
6
Time(sec)
8
10



[1] (3pt) 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.
[2] (2pt) When the potential energy of a
regular wave for certain area is 20000J,
what is the corresponding kinetic energy?
[3] (3pt) The group velocity of a shallow
water wave is 3m/s. What is the
corresponding water depth?



[4] Consider a deep-water wave with 8-s
period and 4-m height?
(a) (4pt) What is the power of this wave
along the crest width of 500m?
(b) (5pt) 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))
Wave Breaking
Deep & transitional depth:
General: H/L=(1/7)tanh kh
Deep: H/L=1/7

Shallow
McCowan’s criterion: flat bottom
H=0.78h
Goda-Weggel chart:
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
Geometric Comparison
Nonlinear waves



higher and sharper
crests
Shallower and flatter
troughs
Nonlinear theory
enables one to
analyze large
amplitude (large H/L)
waves (Linear theory
assumes small
amplitude)
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 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 )
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
OCEN300 WAVE MECHANICS (Mini
TERM PROJECT)


Form a 5-person team
Use WOW Java applets to research one of the given
topics.
http://cavity.ce.utexas.edu/kinnas/wow/public_ht
ml/waveroom/
http://ceprofs.tamu.edu/mhkim/wow
http://ceprofs.tamu.edu/jzhang/
(OCEN671)


Prepare 3-page report (1. Intro, 2. Results & Major
Finding, 3. Conclusion) (plus relevant graphics and
pictures in Appendix). (due 4/29)
Prepare 10 minute presentation by POWER POINT
(presentation 4/24,4/29)
List of Topics


(1) WAVE KINEMATICS
The change of magnitudes and shapes of
particle orbits of linear waves for various
depths and wave parameters. The comparison
of wave kinematics and particle orbits between
linear and second-order Stokes wave theory for
various water depths and wave parameters.
(2) NUMERICAL WAVE TANK
Generate several partial standing waves and a
perfect standing wave by controlling the
artificial damping coefficient inside the damping
zone. (If damping coefficient=0, then perfect
reflection) Determine the reflection coefficient
for each case by measuring the semi node and
anti-node. (This kind of work has to be done to
assess the performance of wave absorbers in a
physical wave tank.)
LIST of TOPICS

(3) WAVE FORCE ON VERTICAL
PILES
Study the variation of inertia and
drag wave forces on a vertical
cylinder as function of cylinder size,
wave parameters, and water depth.
Compare the relative magnitudes of
inertia vs. drag forces
Team
A: Arms, Baldwin, Bandas, Beck, Blake
B: Blaylock, Clark, Cotton, Dearing, Debowski
C: Drake, Dunbar, Fernandez, Finn, Garcia
D: Garza, Goertz, Gonzalez, Grimes, Hale
E: Hartsfield, Keller, Kidwell, Leon, Lightsey
F: Lister, Little, McCollum, Meader, Mendez
G: Neat, Noble, Parliament, Phillips, Reimer
H: Reitblatt, Sanchez, Schlosser, Scholeman, Sears
I: Smeeton, Sonnenberg, Stone, Tran, Trumble
J: Vallejo, Wacasey, Winkelmann, Zwerneman