KH4434
KEJ AWAM MARITIM
LECTURE # 3
OCEAN WAVE THEORIES
Dept. of Civil & Structural Engineering
Faculty of Engineering 1
Universiti Kebangsaan Malaysia
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Environmental loadings KEJ AWAM MARITIM
Dept. of Civil & Structural Engineering
Faculty of Engineering 2
Universiti Kebangsaan Malaysia
1
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Contents KEJ AWAM MARITIM
• Wave characteristics
• Boundary value problem
• Small-amplitude wave theory
– Velocity potential
– Wave dispersion relations
– Wave kinematics
– Wave dynamics
• Non-linear wave theories
Dept. of Civil & Structural Engineering
Faculty of Engineering 3
Universiti Kebangsaan Malaysia
Types of Waves KH4434
KEJ AWAM MARITIM
Typical Period
• Tides • 12-24 hours
• Storm surges • 1-10 hours
• Internal Waves • 2 minutes-10 hours
• Tsunamis • 10 minutes-2 hours
• Wind Waves • 1-25 seconds
• Capillary waves • <0.1 seconds
Dept. of Civil & Structural Engineering
Faculty of Engineering
Universiti Kebangsaan Malaysia
2
KH4434
KEJ AWAM MARITIM
Dept. of Civil & Structural Engineering
Faculty of Engineering 5
Universiti Kebangsaan Malaysia
Wave Description KH4434
KEJ AWAM MARITIM
• Wave evolution near the coast is a very complex
problem
– Turbulence wave breaking
• Chaotic velocity patterns
• Air-entrainment, spray
– Current patterns riptides
– Waves can be huge
– Reefs and sand bars
– Beaches move
Dept. of Civil & Structural Engineering
Faculty of Engineering
Universiti Kebangsaan Malaysia
3
KH4434
Wave Characteristics KEJ AWAM MARITIM
Definition of terms…
Dept. of Civil & Structural Engineering
Faculty of Engineering 7
Universiti Kebangsaan Malaysia
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Boundary Value Problem KEJ AWAM MARITIM
Dept. of Civil & Structural Engineering
Faculty of Engineering 8
Universiti Kebangsaan Malaysia
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Boundary Conditions KEJ AWAM MARITIM
GE xx zz 0
BBC z 0
KFSBC z t x x
1 2
DFSBC t g
2
x z2 C ( t )
PBC ( x, z, t ) ( x L, z, t ) ( x, z, t T )
( x, z, t ) ( x L, z, t ) ( x, z, t T )
Dept. of Civil & Structural Engineering
Faculty of Engineering 9
Universiti Kebangsaan Malaysia
Wave Description KH4434
KEJ AWAM MARITIM
• To examine this complex situation mathematically,
we would need to solve the Navier-Stokes equations:
It is still very hard to do this, and nearly impossible
for large coastal regions.
Dept. of Civil & Structural Engineering
Faculty of Engineering
Universiti Kebangsaan Malaysia
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Wave Description KH4434
KEJ AWAM MARITIM
• Simplifications:
– Fluid is homogeneous & Incompressible
– Surface tension/Coriolis effect neglected
– Inviscid & irrotational flow = no turbulence
– Wave does not interact with any other water motions,
like currents
– Waves are small compared to the water depth
– Seafloor is horizontal & fixed
Linear Potential Wave Theory
or Airy’s Wave Theory
(small-amplitude theory)
Dept. of Civil & Structural Engineering
Faculty of Engineering
Universiti Kebangsaan Malaysia
KH4434
Wave theories KEJ AWAM MARITIM
Dept. of Civil & Structural Engineering
Faculty of Engineering 12
Universiti Kebangsaan Malaysia
6
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KEJ AWAM MARITIM
LINEAR WAVE THEORY
Dept. of Civil & Structural Engineering
Faculty of Engineering 13
Universiti Kebangsaan Malaysia
KH4434
Small-Amplitude Wave Theory (I) KEJ AWAM MARITIM
GE xx zz 0
BBC z 0
KFSBC z t
DFSBC t g C (t )
PBC ( x, z, t ) ( x L, z, t ) ( x, z, t T )
( x, z, t ) ( x L, z, t ) ( x, z, t T )
Dept. of Civil & Structural Engineering
Faculty of Engineering 14
Universiti Kebangsaan Malaysia
7
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Small-Amplitude Wave theory (2) KEJ AWAM MARITIM
H
cos( kx t )
2
gH cosh k ( z d )
sin( kx t )
2 cosh kd
C (t ) 0
Dept. of Civil & Structural Engineering
Faculty of Engineering 15
Universiti Kebangsaan Malaysia
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Wave dispersion relation (1) KEJ AWAM MARITIM
To determine the wave length with given wave
period and water depth
g 2 2 d
L T tanh
2 L
Dept. of Civil & Structural Engineering
Faculty of Engineering 16
Universiti Kebangsaan Malaysia
8
Small Amplitude Wave Theory
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KEJ AWAM MARITIM
All Formulas Depend on L (or d/L)
gT 2 2 d
L
2
tanh
L
Whoops ! [3]
2 d
Deep : tanh kd = tanh 1 (2.4)
L
2 d 2 d (2.5)
Shallow : tanh kd = tanh
L L
gT 2 (2.6)
Lo
2
Dept. of Civil & Structural Engineering
Faculty of Engineering 17
Universiti Kebangsaan Malaysia
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Solving the Implicit Equation KEJ AWAM MARITIM
You know H, T and d
gT 2 d
Lo Lo
2
d Wave Table d
Lo Approximation L
Dept. of Civil & Structural Engineering
Faculty of Engineering 18
Universiti Kebangsaan Malaysia
9
Small Amplitude Wave Theory KH4434
KEJ AWAM MARITIM
Wave Table
Dept. of Civil & Structural Engineering
Faculty of Engineering 19
Universiti Kebangsaan Malaysia
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Linear Wave Description KEJ AWAM MARITIM
• Use of function table
– Find d/Lo
– Find associated d/L
– Knowing d, then determine L
• Example
– Find L, when T=1.0secs and d=0.7m
– Find L, when T=7.0 secs and d = 11.9m
Dept. of Civil & Structural Engineering
Faculty of Engineering 20
Universiti Kebangsaan Malaysia
10
Linear Wave Description KH4434
KEJ AWAM MARITIM
• If we are given T (or f) to find L to make
our lives easier:
Java Applets for Coastal Engineering
http://www.coastal.udel.edu/faculty/rad/
index.html
Dept. of Civil & Structural Engineering
Faculty of Engineering
Universiti Kebangsaan Malaysia
Linear Wave Description KH4434
KEJ AWAM MARITIM
• We can also use SOLVER or GOALSEEK in excel.
– This will be the best approach
Dept. of Civil & Structural Engineering
Faculty of Engineering
Universiti Kebangsaan Malaysia
11
W a v e T a b le C a lc u la to r
T itle : E x a m p le 2 .1
IN P U T A R E A : In s e rt th e v a lu e s in th e b o x e s S u rfa c e A t z = -4 .0 B o tto m
G ra vity g (m /s 2 ) 9 .8 0 6 9 .8 0 6 9 .8 0 6
F lu id D e n s ity Rho (k g /m 3 ) 1035 1035 1035
K in e m a tic V is c o s ity Nu (m 2 /s ) 0 .0 0 0 0 0 1 0 .0 0 0 0 0 1 0 .0 0 0 0 0 1
In c id e n t W a v e P e rio d (s ) 8 .0 0 8 .0 0 8 .0 0
In c id e n t W a v e H e ig h t (m ) 1 .5 0 1 .5 0 1 .5 0
In c id e n t W a te r D e p th (m ) 6 .0 0 6 .0 0 6 .0 0
D e p th o f In te re s t (-) (-)z (m ) 0 .0 0 -4 .0 0 -6 .0 0
W A V E T A B L E C A L C U L A T IO N :
D e p th o f w a te r d (m ) 6 .0 0 0 6 .0 0 0 6 .0 0 0
W a v e P e rio d T (s ) 8 .0 0 0 8 .0 0 0 8 .0 0 0
W a v e H e ig h t H (m ) 1 .5 0 0 1 .5 0 0 1 .5 0 0
D e p th o f In te re s t (-) (-)z ! (m ) 0 .0 0 0 -4 .0 0 0 -6 .0 0 0
D e e p W a te r W a v e L e n g th (T a b le 2 .2 - E q 3 ) Lo (m ) 9 9 .8 4 0 9 9 .8 4 0 9 9 .8 4 0
D e e p W a te r W a v e S p e e d (T a b le 2 .2 - E q 2 ) Co (m /s ) 1 2 .4 8 0 1 2 .4 8 0 1 2 .4 8 0
d /L o 0 .0 6 0 0 .0 6 0 0 .0 6 0
D e e p W a te r A rg u m e n t [2 *P i*d /L o ] 0 .3 7 8 0 .3 7 8 0 .3 7 8
W a v e S p e e d (E q 2 .1 7 ) C (m /s ) 7 .1 8 9 7 .1 8 9 7 .1 8 9
W a v e L e n g th (= C T ) L (m ) 5 7 .5 1 3 5 7 .5 1 3 5 7 .5 1 3
A rg u m e n t (k d = 2 *P i*d /L ) 0 .6 5 5 0 .6 5 5 0 .6 5 5
s in h (k d ) 0 .7 0 3 0 .7 0 3 0 .7 0 3
c o s h (k d ) 1 .2 2 3 1 .2 2 3 1 .2 2 3
ta n h (k d ) 0 .5 7 5 0 .5 7 5 0 .5 7 5
G ro u p V e lo c ity P a ra m e te r (T a b le 2 .2 - E q . 1 3 ) n 0 .8 8 1 0 .8 8 1 0 .8 8 1
G ro u p V e lo c ity (= n C ) Cg (m /s ) 6 .3 3 4 6 .3 3 4 6 .3 3 4
D e p th b e lo w E le v a tio n (-z) D=d+z (m ) 6 .0 0 0 2 .0 0 0 0 .0 0 0
A rg u m e n t a t d e p th (-z ) (= k D ) 0 .6 5 5 0 .2 1 8 0 .0 0 0
s in h (k D ) 0 .7 0 3 0 .2 2 0 0 .0 0 0
c o s h (k D ) 1 .2 2 3 1 .0 2 4 1 .0 0 0
S e m i M a jo r A x is (T a b le 2 .2 - E q . 6 ) A (m ) 1 .3 0 4 1 .0 9 2 1 .0 6 6
S e m i-M in o r A x is (T a b le 2 .2 - E q . 7 ) B (m ) 0 .7 5 0 0 .2 3 5 0 .0 0 0
M a xim u m H o riz o n ta l V e lo c ity (T a b le 2 .2 - E q . 4 ) <u> (m /s ) 1 .0 2 4 0 .8 5 7 0 .8 3 7
M a xim u m V e rtic a l V e lo c ity (T a b le 2 .2 - E q . 5 ) <v> (m /s ) 0 .5 8 9 0 .1 8 4 0 .0 0 0
P re s s u re R e s p o n s e F a c to r (T a b le 2 .2 - E q 9 ) Kp 1 .0 0 0 0 .8 3 8 0 .8 1 8
M a xim u m P re s s u re F lu c tu a tio n (= K p *H ) D el p (m ) 1 .5 0 0 1 .2 5 6 1 .2 2 7
E n e rg y D e n s ity (T a b le 2 .2 - E q . 1 0 ) E (k j/m 2 ) 2 .8 5 4 2 .8 5 4 2 .8 5 4
W a v e P o w e r (T a b le 2 .2 - E q . 1 1 ) P (k w /m ) 1 8 .0 8 1 1 8 .0 8 1 1 8 .0 8 1
23
M a s s T ra n s p o rt V e lo c ity (T a b le 2 .2 - E q . 1 4 ) Ub (m /s ) 0 .1 2 2 0 .1 2 2 0 .1 2 2
KH4434
Wave characteristics KEJ AWAM MARITIM
• Wave base is 1/2 wave length
– Negligible water movement due to waves below this depth
Dept. of Civil & Structural Engineering
Faculty of Engineering
Universiti Kebangsaan Malaysia
12
KH4434
Deep-water wave KEJ AWAM MARITIM
• Depth of water is greater than 1/2 wavelength
• Speed of wave form (celerity) is proportional to
wavelength
Dept. of Civil & Structural Engineering
Faculty of Engineering
Universiti Kebangsaan Malaysia
KH4434
Shallow-water wave KEJ AWAM MARITIM
• Water depth is less than 1/20 wavelength
• Friction with seafloor retards speed
• Wave speed (celerity) is proportional to depth of water
• Orbital motion is flattened
Dept. of Civil & Structural Engineering
Faculty of Engineering
Universiti Kebangsaan Malaysia
13
Linear Wave Description KH4434
KEJ AWAM MARITIM
• It is also termed as “short” or “long” wave.
– A wave is considered short when L/d<2
– A wave is long when L/d>20
Dept. of Civil & Structural Engineering
Faculty of Engineering
Universiti Kebangsaan Malaysia
KH4434
Wave dispersion relation KEJ AWAM MARITIM
Deep water (d/L>0.5)
gT 2
L
2
Shallow Water (d/L<0.05)
L gd T
Dept. of Civil & Structural Engineering
Faculty of Engineering 28
Universiti Kebangsaan Malaysia
14
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Transitional waves KEJ AWAM MARITIM
• Water depth is 1/2 to 1/20 of wavelength
• Characteristics of deep and shallow-water waves
• Wave speed (celerity) is proportional to both wavelength
and depth of water
Dept. of Civil & Structural Engineering
Faculty of Engineering
Universiti Kebangsaan Malaysia
KH4434
Determination of wave length KEJ AWAM MARITIM
(I) Hunt’s formula (1979)
a 1 0 .666
2 d
L a 2 0 . 355
2 y
y 6
1 an yn a 3 0 . 1608465608
n 1
a 4 0 .0632098765
2d a 5 0 . 0217540484
y
g a 6 0 . 0065407983
Dept. of Civil & Structural Engineering
Faculty of Engineering 30
Universiti Kebangsaan Malaysia
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KH4434
Determination of Wave Length (2) KEJ AWAM MARITIM
(II) Eckart’s (1952) approximation
gT 2 4 2 d
L tanh 2
2 gT
Dept. of Civil & Structural Engineering
Faculty of Engineering 31
Universiti Kebangsaan Malaysia
Linear Wave Description KH4434
KEJ AWAM MARITIM
• Given L to find T
– Explicit formula by Eckart (1952)
Error in L predic tion (% ) by the Eck art (1952) formula
4
Error (%)
0
1 10 100 1000
Dim e nsionle ss W a ve Pe riod [T/sqrt(d/g)]
Dept. of Civil & Structural Engineering
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Universiti Kebangsaan Malaysia
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KH4434
Determination of Wave Length (3) KEJ AWAM MARITIM
(III) Curve method
2
L o gT / 2 C o gT / 2
Dept. of Civil & Structural Engineering
Faculty of Engineering 33
Universiti Kebangsaan Malaysia
KH4434
Determination of Wave Length (4) KEJ AWAM MARITIM
(IV) Numerical method-Newton’s method
g 2 d
f (L) L T 2 tanh
2 L
f ( Ln )
L n 1 L n
f ( L n )
Dept. of Civil & Structural Engineering
Faculty of Engineering 34
Universiti Kebangsaan Malaysia
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Example 2.1 KEJ AWAM MARITIM
A wave with a period T=10 seconds is propagated
shoreward over a uniform sloping shelf from a depth
d=200 m to a depth d=3 m. Determine the wave
celerity (C) and wavelength (L) corresponding to
depth d=200 m and d=3 m.
Dept. of Civil & Structural Engineering
Faculty of Engineering 35
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Wave kinematics (1): KH4434
Water particle velocities KEJ AWAM MARITIM
gkH cosh k ( z d )
u cos( kx t )
2 cosh kd
gkH sinh k ( z d )
w sin( kx t )
2 cosh kd
http://www.coastal.
udel.edu/faculty/ra
d/linearplot.html
Dept. of Civil & Structural Engineering
Faculty of Engineering 36
Universiti Kebangsaan Malaysia
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Wave kinematics (2): KH4434
Water particle acceleration KEJ AWAM MARITIM
2
w H sinh k ( z d )
az cos( kx t )
t 2 sinh kd
2
u H cosh k ( z d )
ax sin( kx t )
t 2 sinh kd
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Faculty of Engineering 37
Universiti Kebangsaan Malaysia
KH4434
KEJ AWAM MARITIM
Dept. of Civil & Structural Engineering
Faculty of Engineering 38
Universiti Kebangsaan Malaysia
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Example 2.2 KEJ AWAM MARITIM
A wave with a period T=8 seconds, in a water
depth d=15 m, and a height H=5.5 m. Find the
local horizontal and vertical velocities u and w
and acceleration ax and az at an elevation z=-5
m below the still water level when = kx - t =
/3 .
Dept. of Civil & Structural Engineering
Faculty of Engineering 39
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Wave Kinematics (3): KH4434
Water particle displacements KEJ AWAM MARITIM
http://cavity.ce.utexas.edu/kinnas/wow/public_html/waveroom/Applet/
WaveKinematics/WaveKinematics.html
Dept. of Civil & Structural Engineering
Faculty of Engineering 40
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Wave Kinematics (3): KH4434
Water particle displacements KEJ AWAM MARITIM
H cosh k ( z 1 d )
( x1 , z 1 , t ) u ( x1 , z 1 ) dt sin( kx 1 t )
2 sinh kd
H sinh k ( z 1 d )
( x1 , z 1 , t ) w ( x1 , z 1 ) dt cos( kx 1 t )
2 sinh kd
Dept. of Civil & Structural Engineering
Faculty of Engineering 41
Universiti Kebangsaan Malaysia
Wave Kinematics (3): KH4434
Water particle displacements KEJ AWAM MARITIM
2 2
1
A B
H cosh k ( z 1 d )
A
2 sinh kd
H sinh k ( z 1 d )
B
2 sinh kd
Dept. of Civil & Structural Engineering
Faculty of Engineering 42
Universiti Kebangsaan Malaysia
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Wave Kinematics (3): KH4434
Water particle displacements KEJ AWAM MARITIM
Deep water (d/L>0.5):
H kz 1
A B e
2
Shallow water (d/L<0.05):
H HT g
A
2 kd 4 d
H z
B (1 1 )
2 d
Dept. of Civil & Structural Engineering
Faculty of Engineering 43
Universiti Kebangsaan Malaysia
Wave Kinematics (3): KH4434
Water particle displacements KEJ AWAM MARITIM
Dept. of Civil & Structural Engineering
Faculty of Engineering 44
Universiti Kebangsaan Malaysia
22
KH4434
KEJ AWAM MARITIM
Dept. of Civil & Structural Engineering
Faculty of Engineering 45
Universiti Kebangsaan Malaysia
KH4434
Example 2.3 KEJ AWAM MARITIM
A wave in a depth d = 12 m, height H= 3 m and a period
T=1 0 seconds. The corresponding deepwater wave
height is =3.13 m. Find:
a) the maximum horizontal and vertical displacement of a
water particle from its mean position when z=0 and z=-d;
b) the maximum water particle displacement at an
elevation z=-7.5 m when the wave is in infinitely deep
water, and
c) for the deepwater condition of (b), shown the particle
displacements are small relative to the wave height,
when z=-Lo /2 .
Dept. of Civil & Structural Engineering
Faculty of Engineering 46
Universiti Kebangsaan Malaysia
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Wave dynamics (1): KH4434
Wave pressures KEJ AWAM MARITIM
gH cosh k ( z d )
p gz cos( kx t )
2 cosh kd
gz g K p (z)
K p ( z ) = pressure response factor
Dept. of Civil & Structural Engineering
Faculty of Engineering 47
Universiti Kebangsaan Malaysia
KH4434
Example 2.4 KEJ AWAM MARITIM
Two pressure sensors are located as shown in the following
figure. For an 8-second progressive wave, the dynamic pressure
amplitudes at sensors 1 and 2 are 2.07X104 N/m2 and 2.56 X104
N/m2, respectively. What are the water depth, wave height, and
wavelength?
Dept. of Civil & Structural Engineering
Faculty of Engineering 48
Universiti Kebangsaan Malaysia
24
KH4434
KEJ AWAM MARITIM
Dept. of Civil & Structural Engineering
Faculty of Engineering September, 2006 Hour 1b Water Waves
49
Universiti Kebangsaan Malaysia
KH4434
KEJ AWAM MARITIM
Dept. of Civil & Structural Engineering
Faculty of Engineering September, 2006 Hour 1b Water Waves
50
Universiti Kebangsaan Malaysia
25
Wave dynamics (2): KH4434
Wave energy KEJ AWAM MARITIM
Potential Energy: 1 2
( PE ) wave gH
16
Kinetic Energy: 1 2
( KE ) wave gH
16
1 2
Total Energy: ( E ) wave gH
8
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Faculty of Engineering 51
Universiti Kebangsaan Malaysia
Wave dynamics (3): KH4434
Wave energy flux KEJ AWAM MARITIM
1 t T gH 2 1 2kd
p D udzdt 1 Ecn
T t h 8 k 2 sinh 2kd
Dept. of Civil & Structural Engineering
Faculty of Engineering 52
Universiti Kebangsaan Malaysia
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Wave Reflection: KH4434
Standing Waves KEJ AWAM MARITIM
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Faculty of Engineering 53
Universiti Kebangsaan Malaysia
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