### Sinusoidal waves in deep water - Técnico Lisboa

```Specialization in Ocean Energy
MODELLING OF WAVE
ENERGY CONVERSION
António F.O. Falcão
Instituto Superior Técnico,
2014
PART 2
LINEAR THEORY OF OCEAN
SURFACE WAVES
FLUID MOTION IN WAVES
• Surface tension neglected (no ripples)
• Perfect fluid (no viscosity)
 Incompressible flow   u  0
 Irrotational flow
  u  0 or u  
Laplace equation
 2  0
Boundary conditions
• At the free-surface: p  pa (atmospheric pressure)
 At the bottom: u n  0 (normal velocity is zero)
 The
free-surface is unknown, which makes the problem non-linear.
In general the boundary condition is applied at the undisturbed free-surface
(flat surface): LINEAR THEORY.
pa  atmospheric pressure
z
y
p0  undisturbed pressure(no waves)
p0  pa  g z
x
Excess pressure due to waves
pe  p  p0
D
 derivative following the flow
Dt
Du
 accelerati on of fluid particle
Dt
Euler’s equation (perfect fluid)
Du

  p  gk
Dt
Du

  pe
Dt
Du

  pe
Dt
Du u

 (u  )u
Dt t
Linear theory: the products of small quantities are neglected
u

  pe
t
Equation of the disturbed water free-surface
z   ( x, y, t )
p  p0  pe  pa on thefreesurface  pe   g
p0  pa  g z
p  p0  pe  pa on thefreesurface  pe   g
u

  pe
t
u  
z   ( x, y, t )
pe   g

pe   
t
  
  g
 t 
z 
  
  g
 t 
z 0
Kinematic boundary condition on the free surface
D 
  

 u     
Dt
t
 z  z 
   
 
t  z  z 0
  
  g
 t 
z 0
   
 
t  z  z 0
 2



g
on z  0 (boundarycondition)
2
z
t
2  0




g
on
z

0
(boundary
condition)
z
t 2
2
Sinusoidal waves
Complex variable
 
f (t )  Re(F ei t )
F  F0 ei 
 
cost  Re ei t , sin t  Im ei t
ei t  cost  i sin t


f (t )  Re F0 ei( t  )  F0 cos(t   )
• In most cases, we will omit the notation Re( ).
• It will be assumed that, whenever a complex expression is
equated to a physical quantity, the real part of it is to be
taiken.
f (t )  Re(F ei t )
f (t )  F eit
Sinusoidal waves in deep water
1. 0

crest
Aw
0. 5
x
0. 5
1. 0
1. 5
through
0. 5
1. 0
  ( z) expi (t  x c)  ( z) expi(t  k x)
2
 2

, k 
T
c

T = period (s),
  cT .
f = 1/T = frequency (Hz or c/s),
λ = wavelength (m),
k  2 /  = wave number (m-1)
( z)  k 2( z)  0.
( z )   0 e kz ,
2. 0
Sinusoidal waves in deep water
  ( z) expi (t  x c)  ( z) expi(t  k x)
( z )   0 e kz ,
Dispersion relationship

12
g
Wave velocity c    
k k
12
 g 

 .
 2 
  30m c  6.8m/s T  4.4s
  300m c  21.6m/s T  13.9s
Sinusoidal waves in deep water

1. 0
Aw
0. 5
0. 5
x
1. 0
1. 5
2. 0
0. 5
1. 0
Particle orbits: circles of radius =  1k 0ekz
distance to free surface
Disturbance practically vanishes at
depth   2
Wave amplitude: Aw   1k 0
0 

k
Aw
In deep water, the water particles have circular orbits.
The orbit radius decreases exponentially with the distance to the
surface.

Sinusoidal waves in water of arbitrary, but
uniform, depth h

1. 0
Aw
0. 5
0. 5
x
1. 0
1. 5
2. 0
0. 5
1. 0
  ( z) expi (t  x c)  ( z) expi(t  k x)
( z)  k 2( z)  0
( z)  1ekz  2ekz
1ekh  2ekh
(h)  0 boundary condition at bottom
( z)  0 coshk ( z  h)
cosh x 
1
2
e  e 
x
x
d cosh x
sinh x 

dx
1
2
e  e   cosh x tanh x
x
x
4
3
2
cosh x
sinh x
1
tanh x
0
0.0
0.5
1.0
x
1.5
2.0
Sinusoidal waves in water of arbitrary, but uniform, depth h
 2

 g
on z  0
2
z
t
c

k
 2  gk tanhkh
 ( gk 1 tanh kh )1 2
12
Wave speed
2h 
 g
c
tanh
 
 2
c

g
 tanh
h
c
Shallow water or long-wave limit: kh small
c  ( gh)1 2 for kh  1
tanh kh  kh
Sinusoidal waves in water of arbitrary, but uniform, depth h
Sinusoidal waves in water of arbitrary, but uniform, depth h
Numerical example: T  8 s, g  9,8 m/s2

Deep water
gT 9,8  8
c

 12,5 m/s
2
2
  cT  12,5  8  100m

Shallow water h  1 m
cc  gH
gh  9,8 1  3,1 m/s
  cT  3,1 8  25,0 m/s

Intermediate water depth h  15 m
2 2


T
8

h
0,785
0,785 15
c  tanh
c
 tanh
g
c
9,8
c
  cT  10,2  8  81,8 m
c  10,2 m/s
Sinusoidal waves in water of arbitrary, but uniform, depth h
T  8s
h (m)
1
3
5
10
15
20
25
30
40
50

c (m/s)
3,10
5,25
6,63
8,86
10,22
11,09
11,65
12,00
12,33
12,44
12,48
 (m)
24,8
42,0
53,0
70,9
81,8
88,7
93,2
96,0
98,6
99,5
99,8
Sinusoidal waves in water of arbitrary, but uniform, depth h
Orbits of water particles: elipses with semi-axes:
 1k0 coshk ( z  h)
 1k0 sinhk ( z  h)
(major)
(minor)
Aw   1k0 sinh kh
h  0.16
Refraction effects due to bottom bathymetry
The propagation velocity c decreases with decreasing depth h.
As the waves propagate in decreasing depth, their crests tend to
become parallel to the shoreline
wave crests
shoreline
crests
rays
shoreline
Dispersion of energy at a bay.
shoreline
Concentration of energy at a headland.
Standing waves. Reflection on a vertical wall
Standing waves. Reflection on a vertical wall
1  ( z) expi(t  kx),

2  ( z) expi(t  kx)

  1    ( z) eit eikx  eikx  2 ( z) eit coskx
Surface elevation  ( x, t )  2 Aw coskx sin t
Horizontal velocity component:
 x  2k ( z) eit sin kx
 0 at kx  n , n  0,  1,  2, ...
Satisfies condition for reflection at vertical wall
antinodes
nodes
Wave energy and wave energy flux
Unlike wind, waves permit the transport of energy without the
need for any net transport of material.
 Kinetic energy (circular or elliptic orbits)
v
 Potential energy (sea surface is not plane)
Potential energy per unit horizontal
surface area (time averaged)
Kinetic energy per unit horizontal
surface area (time averaged)
Epotential 
Ekinetic
1
 gAw2
4
1
  gAw2
4
Total energy per unit horizontal surface area (time averaged, any
water depth)
1
E  Epotential  Ekinetic   gAw2
2
Wave energy and wave energy flux
Wave energy flux (or transmitted power)
We are more interested in the energy flux across
a vertical plane parallel to the wave crests (from
bottom to surface).

Ppotential   (  gz ) dz
h
x


Pkinetic  
( 12
h

  ) dz
x
2
Time average: Pwave  Ppotential  Pkinetic  E cg
1
2kh 
1
2kh 
cg 
1 
  c 1 

2 k  sinh 2kh 
2  sinh 2kh 
1
E
1
 gAw2
2
group velocity
The group velocity may be regarded as the velocity at which the
wave energy is propagated.
Wave energy and wave energy flux
1
2kh 
1
2kh 
cg 
1 
  c 1 

2 k  sinh 2kh 
2  sinh 2kh 
group velocity
Deep water limit  kh    sinh 2kh    cg 
c
2
Shallow water limit  kh  0  sinh 2kh  2kh  cg  c
In general, the energy travels at a velocity smaller than the wave
crests.
Deep
water
c gT

2 4
1
E   gAw2
2
cg 
1
Pwave  E cg 
 g 2 Aw2 T
8
Exercise
Consider a two-dimensional OWC subject to regular
waves. The submergence of the OWC walls is small so
that the wave diffraction they produce may be
neglected. The air pressure p(t) inside the chamber is a
sinusoidal function of time, and its amplitude and
phase may be controlled.
From the interference between the incident wave and
the radiated waves, determine the maximum fraction
of the incident wave power that can be absorbed by
the OWC.
p(t )
As above, with the back wall extending to the sea
bottom.
p(t )
Irregular waves
Real ocean waves are not regular: they are irregular
and random
1.5
1
m
0.5
0
0.5
1
100
150
200
t s
250
300
Sea surface elevation at one location as a function of time
We want to describe the sea surface as a stochastic
process, i.e. to characterize all possible observations
(time records) that could have been made under the
conditions of the actual observation.
An observation is thus formally treated as one
realization of a stochastic process.
1.5
We consider a wave record with duration D
(typically 15 to 30 min).
1
m
0.5
We can exactly reproduce that record as the
sum of a large (theoretically infinite)
number of harmonic wave components (a
Fourier series) as
0
0.5
1
100
N
 (t )   ai cos(2 fi t  i ),
i 1
150
200
t s
250
fi  i D
With a Fourier analysis, we can determine the amplitudes ai and phases i
for each frequency.
For wave records, the phases have any value between 0 and 2 without
any preference for any one value. So we will ignore the phase spectrum.
300
Only the amplitude spectrum remains to characterize the wave record.
To remove the sample character of the spectrum, we should repeat the
experiment many times (M) and take the average over all these
experiments, to find the average amplitude spectrum
1
ai 
M
M
 ai,m
m 1
for all frequencies f i
However, it is more meaningful to use the variance
1 a2  1
2 i
M
M

m 1
1 a2
2 i,m
of each wave component . An important reason is that the wave energy is
proportional to the square of the wave amplitude (not to the amplitude).
The variance spectrum is discrete, i.e., only the frequencies fi  i D
are present, whereas in fact all frequencies are present at sea.
This is resolved by letting the frequency interval f  1 D  0.
The variance
density spectrum is defined as
1
S f ( f )  lim
f 0 f
Units for S f ( f ) are m2s or m2 /Hz
1 a2
2 i
4
3
2
1
0.05
0.10
0.15
0.20
Frequency f (Hz)
Typical variance density spectrum
0.25
The variance density spectrum gives a complete description of the
surface elevation of ocean waves in statistical sense, provided that
the surface elevation can be seen as a stationary Gaussian
process.
To use this approach, a wave record needs to be divided into
segments that are each assumed to be approximately stationary (a
duration of about 30 min is commonly used).
The sea surface elevation is a random function of time. Its total
variance is

   S f ( f ) df
2
0
We recall that the time-averaged total (potential plus kinetic energy)
of a regular wave per unit horizontal surface is
E  12  gAw2
If we multiply S f ( f ) by  g we obtain the energy density
spectrum
1 2
E f ( f )   gS f ( f )   g lim
ai .
f 0 f
1
2
The variance density S f ( f ) was defined in terms of frequency f  1 T
(where T is the period of the harmonic wave).
It can also be formulated in terms of radian frequency   2 f  2 T
We may write
d  2 df


0
0
   S f ( f ) df   S () d
2
1
S ( ) 
S f ( f ).
2
The overall appearance of the waves can be inferred from
the shape of the spectrum: the narrower the spectrum,
the more regular the waves are.
When the random sea-surface elevation is treated as a stationary,
Gaussian process, then all statistical characteristics are determined
by the variance density spectrum S f ( f ) .
These characteristics will be expressed in terms of the moments of
the spectrum (moment of order m)

mn   f n S f ( f )df
0
(m  ...,3,  2,  1, 0,1, 2, 3,...).
For example, the mean-square or
variance of surface elevation:

   S f ( f ) df  m0.
2
0
Significant wave height and mean wave period
The significant wave height H s is the mean value of the
highest one-third of wave heights in the wave record.
It is given approximately by
H s  H m0  4 m0
Several different definitions of “mean” period for irregular waves
are used.
One is the peak period Tp  1 f p
Another is the energy period
m1
Te 
m0
fp
The characteristics of the frequency spectra of sea waves have been
fairly well established through analyses of a large number of wave
records taken in various oceans and seas.
The spectra of fully developed waves in deep water can be
approximated by the Pierson-Moskowitz equation


exp  1052(T  ) 
S f ( f ) 0.1688H s2 Te 4 f 5 exp  0.675(Te f )  4
S ( ) 262.6 H s2 Te 4 5
4
e
Exercise
Establish a relationship between the peak period and the energy
period for the Pierson-Mokowitz spectrum.
Energy flux of irregular waves
In regular waves:
• energy per unit horizontal surface area E  12  gAw2
• energy flux per unit wave crest length Pwave  E cg
• In deep water
cg  12 c  g ( 4f ) .
In irregular waves:
• In ( f , f  df )
g2
1
dPwave  cg E f ( f )df   gcg S f ( f ) df 
S f ( f ) df
4
f
• By integration
g2 
1
g2
Pwave 
S f ( f ) df 
m1

4 0
f
4
g2
Pwave 
m1
4
Te  m1 m0
g2 2
Pwave 
H s Te .
64
H s  4 m0
irregular
waves
Pwave in kW/m
2
g  9.8 ms
3
  1025kgm
Pwave  0.490H s2Te
H s in m
Te in s
Wave climate
So far, the statistical characteristics of the waves were considered for
short term, stationary conditions, usually for the duration of a wave
record (15 to 30 min).
• For long-term statistics, over durations of several years (possibly
tens of years) the conditions are not stationary.
• For these long time scales, each stationary condition (with a
duration of 15 to 30 min) is replaced with its values of the
significant wave height and period. This gives a long-term
sequence of these values with a time interval of typically 3 h,
which can be analysed to estimate the long-term statistical
characteristics of the waves.
• The number of observations in then presented (instead of the
probability density) in bins of size (H s , Te )
Example of annual joint relative frequency of occurrence of H s and Te
END OF PART 2
LINEAR THEORY OF OCEAN
SURFACE WAVES
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