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Mathematical Description of Waves and Wave Energy1
Appendix 2 M A T H E M A T I C A L D E S C R I P T I O N OF W A V E S A N D W A V E ENERGY I Hydrodynamics of Sea Waves The creation o f waves is a complex, nonlinear process in which energy is slowly exchanged between different components (see Komen et al. 1994). However, on a scale of tens of kilometres and minutes in deep water, a stationary Gaussian random process accurately describes the local state of the sea surface. Thus the local behaviour of the waves is determined by the spectrum of the sea state S(f,0) that specifies how the wave energy, proportional to the variance of the surface elevation, is distributed in terms of frequency f a n d direction 0. This spectrum can in turn be summarised by a small number of wave parameters, namely "wave height H, period T (f- I/T), and direction. For wave height, the most widely used parameter is the significant wave height, defined as the average of the highest one third of the trough to crest wave heights, and matching reasonably well one's visual impression of wave height. It can be computed from the spectrum by H s -4m~/2
(1)
where m0 is the zero-th spectral moment, the n-th moment being defined as 2~
0o
m,,= I If"S(f,O)df 0
dO
0
(2)
For wave period, several parameters are commonly used. The most appropriate, for present purposes, are the mean (energy) period Te and the peak period Tp. The energy period is defined by (3)
Te = m-I m0
Since T-- 1/f, T e is the average value of T over the wave spectrum, Te depends mainly on the lower frequency band of the spectrum where most of the energy is contained. It is thus more stable than the traditional zero-crossing period Tmo2, the average time elapsed between two sequential crests
157
WAVE ENERGY CONVERSION computed by (mo / m2)1/2. Its dependence on m2 makes it very sensitive to the high frequency spectral tail that exhibits high variability and minute energy contents. The peak period Tp is the inverse of the peak frequency fp that corresponds to the highest spectral density 1 T =~ P ~ (4) Several wave direction parameters can be used. Taking the directional spectrum, mean wave direction is computed by
_
2i ~ S(f,O)sin(O)dOdf
0 - arcta
2i ~
0
o
0
S(f,O)cos(O)dO df (5)
o
Directional buoys have often provided only frequency spectra E(/), related to the directional spectrum by
E(f)-2~S(f ,O)dO o
(6)
in addition to the mean direction ~ ) and its spreading for each frequency. Mean direction is then computed by
arcta~!E(f) sin(O(f))df -
L E(:)
d: " (7)
Often, an oceanic sea state will include both locally generated wind sea, whose principal direction should be that of the local wind, and swell, i.e., long period, far travelled waves, generated up to several days earlier by distant weather patterns (which may have a quite different mean direction). In such cases, an adequate summary of the sea state will require separate heights, periods and mean directions of wind sea and (occasionally more than one) swell components. For a more precise description one can add standard deviations of period and direction for each component. The spectral width, generally decreasing with the age of the wave systems, can be characterised by different parameters. For wave energy studies, it is preferable to use a parameter that depends
158
APPENDIX 2: Mathematical description of waves and wave energy
on the lower frequency range, where most of the energy of the sea state occurs. The standard deviation of the period trr (Mollison 1986) defined by
o- r -
-1
(8) is an appropriate parameter for that purpose because it depends on m_~ and m_2 instead of m 2 and m4 (as other more usual spectral width parameters). When directional spectra are not available, it is generally assumed that E ( f , 0) - E ( f ) / ~ f , 0),
(9)
with D (f 0) satisfying the normalizing condition 2 ~ oo
~ l~f,O)dfdO-1
0
0
(10)
In general, no information is available about the asymmetry of energy spreading with respect to t ~ , and a symmetric function is assumed, generally of circular type. Another simplifying assumption is to take the same spreading function for the whole frequency range D (fi,O)-D(O ). This is a reasonable approach for sea states having only one system with mean direction 0. One of the most widely used expressions for D(0 ) is the cos 2Stype law (Longuet-Higgins 1961 ) defined by
D(O-O)- C(s)c~ 0 the normalizing
C(s) _
1
for
]0-0l( 2
otherwise
(11)
C(s) being given by
(1)
v(
et al.
+0
F s+?
(12)
with F the Gamma function (see Abramowitz & Stegun 1979). The value of s, which describes the directional spreading of the energy around mean direction 0, depends on the age of the sea state. For locally generated waves (wind sea), where a wide range of directions (and frequencies) are present, generally s--1 is appropriate. With the increase of the distance to the area where that sea state (swell) was generated, the directional spreading decreases, requiring for s values of 2 to
159
WAVE ENERGY CONVERSION 5 or more for its description. Different procedures for selecting the value of the spreading parameter s are adopted according to the available wave information. When the energy spectra are available, a spectral width parameter such as a r (equation 8) can be used to select the value of s. If only mean height and period parameters are known, the choice of the directional spreading parameter s for each sea state can be based on the slope H/2, 2 being a typical wave length of that sea state. For monochromatic waves, the relationship between 2 and T is obtained from the dispersion relationship
co 2 _ gk
tanh(kh).
(13)
where g is the acceleration due to gravity; o~=2af, the circular frequency; k=27r/2, the wave number; and h is the water depth. In deep water, tanh(kh) = 1, thus the dispersion relationship reads
(o2-gk
(14)
The slope fl is then given by H 2M-/
fl----~= gr---T. (15) For irregular waves, the corresponding slope parameter can be defined as po _
(16) In the case of purely wind waves (wind sea), fl --0.004 can be assumed. Then the significant wave height of such a sea state, denoted by H~.w,is approximately given by Hs. w =0.06Te 2. (17) For a sea state having fl --0.04 or equivalent, Hs=Hs, w, would correspond to a cos 2s directional distribution, with s-1. The lower Hs is with respect to H~.w, the more it is towards swell, and the narrower the directional distribution. For Hs-0.1Hs, w , the cos j~ directional distribution seems to be an appropriate choice.
Wave Energy The wave power, or flux of energy per unit crest length, is computed by: 2zr oo
P - jog ~ ~ Cg(f,h)S(f,O)df dO 0
(18)
0
160
APPENDIX 2: Mathematical description of waves and wave energy
where p is the water density. The group velocity Cg, i.e., the velocity at which the energy propagates, is defined by &o (19)
Cg = O k .
In deep water, Cg reduces to g Cg = 4 n f
(20)
thus the wave power is given by 2z oo
P - Pg I I f - ' S ( f , O) d f dO - p g m o o 4 z -~
(21)
which can be expressed in terms of H s and Te as pg2
p-
64z
H ro (22)
When H s is expressed in meters and Te in seconds, the power level is given by the equation
pw_0. 5 H~T~ 2 kW/n
(23)
The energy flux incoming from the angular sector Ok (sector centered on Ok with width A0) is computed by: A0 2 0 k 4 2 oo
Pg Po~ - 4 z
~ y f -1S ( f , O ) d f d O ao o ok 2
9
(24)
The energy flux in a given direction 00, another relevant quantity wave energy utilisation, is computed by /gg 22z
P o o-- 4- -z
! ~ i f -'S ( f , 0) c o s ( 0 - 00) df dO. o
(25)
The most favourable direction Of maximizes P ~ . The directional coefficient, defined by do=--
P, (26) is useful to characterise the energy spreading9 Values of d o close to unity indicate that the energy is concentrated around Of, which is a favourable situation for the extraction of wave energy by systems sensitive to the direction. 161
WAVE ENERGY CONVERSION References
Abramowitz, M. and Stegun, I. A. 1979 Handbook of Mathematical Functions; New York., NY: Dover. Komen, G., Cavaleri, L., Donelan, M., Hasselmann.,K., Hasselmann, S. and Janssen, P.A.E.M. 1994, Dynamics and Modelling of Ocean Waves. Cambridge Univ. Press, 532 p. Longuet-Higgins, M. S., Cartwright, D.E., and Smith, N.D. 1961 Observations of the direction spectrum of sea waves using the motions of a floating buoy. In Ocean Wave Spectra, Prentice Hall, Englewood Cliffs, N.J., 111-132. Mollison, D. 1986 Wave climate and the wave power resource. In Hydrodynamics of Ocean Wave-Energy Utilization (eds. D.V. Evans & A. F. de O. Falc~o), 133-156. I. Contributed by Dr. Teresa Pontes, INETI-Department of Renewable Energies, Lisbon, Portugal. (Dr. Pontes is a member of the ECOR Working Group on Wave Energy Conversion.)
162
(1)
where m0 is the zero-th spectral moment, the n-th moment being defined as 2~
0o
m,,= I If"S(f,O)df 0
dO
0
(2)
For wave period, several parameters are commonly used. The most appropriate, for present purposes, are the mean (energy) period Te and the peak period Tp. The energy period is defined by (3)
Te = m-I m0
Since T-- 1/f, T e is the average value of T over the wave spectrum, Te depends mainly on the lower frequency band of the spectrum where most of the energy is contained. It is thus more stable than the traditional zero-crossing period Tmo2, the average time elapsed between two sequential crests
157
WAVE ENERGY CONVERSION computed by (mo / m2)1/2. Its dependence on m2 makes it very sensitive to the high frequency spectral tail that exhibits high variability and minute energy contents. The peak period Tp is the inverse of the peak frequency fp that corresponds to the highest spectral density 1 T =~ P ~ (4) Several wave direction parameters can be used. Taking the directional spectrum, mean wave direction is computed by
_
2i ~ S(f,O)sin(O)dOdf
0 - arcta
2i ~
0
o
0
S(f,O)cos(O)dO df (5)
o
Directional buoys have often provided only frequency spectra E(/), related to the directional spectrum by
E(f)-2~S(f ,O)dO o
(6)
in addition to the mean direction ~ ) and its spreading for each frequency. Mean direction is then computed by
arcta~!E(f) sin(O(f))df -
L E(:)
d: " (7)
Often, an oceanic sea state will include both locally generated wind sea, whose principal direction should be that of the local wind, and swell, i.e., long period, far travelled waves, generated up to several days earlier by distant weather patterns (which may have a quite different mean direction). In such cases, an adequate summary of the sea state will require separate heights, periods and mean directions of wind sea and (occasionally more than one) swell components. For a more precise description one can add standard deviations of period and direction for each component. The spectral width, generally decreasing with the age of the wave systems, can be characterised by different parameters. For wave energy studies, it is preferable to use a parameter that depends
158
APPENDIX 2: Mathematical description of waves and wave energy
on the lower frequency range, where most of the energy of the sea state occurs. The standard deviation of the period trr (Mollison 1986) defined by
o- r -
-1
(8) is an appropriate parameter for that purpose because it depends on m_~ and m_2 instead of m 2 and m4 (as other more usual spectral width parameters). When directional spectra are not available, it is generally assumed that E ( f , 0) - E ( f ) / ~ f , 0),
(9)
with D (f 0) satisfying the normalizing condition 2 ~ oo
~ l~f,O)dfdO-1
0
0
(10)
In general, no information is available about the asymmetry of energy spreading with respect to t ~ , and a symmetric function is assumed, generally of circular type. Another simplifying assumption is to take the same spreading function for the whole frequency range D (fi,O)-D(O ). This is a reasonable approach for sea states having only one system with mean direction 0. One of the most widely used expressions for D(0 ) is the cos 2Stype law (Longuet-Higgins 1961 ) defined by
D(O-O)- C(s)c~ 0 the normalizing
C(s) _
1
for
]0-0l( 2
otherwise
(11)
C(s) being given by
(1)
v(
et al.
+0
F s+?
(12)
with F the Gamma function (see Abramowitz & Stegun 1979). The value of s, which describes the directional spreading of the energy around mean direction 0, depends on the age of the sea state. For locally generated waves (wind sea), where a wide range of directions (and frequencies) are present, generally s--1 is appropriate. With the increase of the distance to the area where that sea state (swell) was generated, the directional spreading decreases, requiring for s values of 2 to
159
WAVE ENERGY CONVERSION 5 or more for its description. Different procedures for selecting the value of the spreading parameter s are adopted according to the available wave information. When the energy spectra are available, a spectral width parameter such as a r (equation 8) can be used to select the value of s. If only mean height and period parameters are known, the choice of the directional spreading parameter s for each sea state can be based on the slope H/2, 2 being a typical wave length of that sea state. For monochromatic waves, the relationship between 2 and T is obtained from the dispersion relationship
co 2 _ gk
tanh(kh).
(13)
where g is the acceleration due to gravity; o~=2af, the circular frequency; k=27r/2, the wave number; and h is the water depth. In deep water, tanh(kh) = 1, thus the dispersion relationship reads
(o2-gk
(14)
The slope fl is then given by H 2M-/
fl----~= gr---T. (15) For irregular waves, the corresponding slope parameter can be defined as po _
(16) In the case of purely wind waves (wind sea), fl --0.004 can be assumed. Then the significant wave height of such a sea state, denoted by H~.w,is approximately given by Hs. w =0.06Te 2. (17) For a sea state having fl --0.04 or equivalent, Hs=Hs, w, would correspond to a cos 2s directional distribution, with s-1. The lower Hs is with respect to H~.w, the more it is towards swell, and the narrower the directional distribution. For Hs-0.1Hs, w , the cos j~ directional distribution seems to be an appropriate choice.
Wave Energy The wave power, or flux of energy per unit crest length, is computed by: 2zr oo
P - jog ~ ~ Cg(f,h)S(f,O)df dO 0
(18)
0
160
APPENDIX 2: Mathematical description of waves and wave energy
where p is the water density. The group velocity Cg, i.e., the velocity at which the energy propagates, is defined by &o (19)
Cg = O k .
In deep water, Cg reduces to g Cg = 4 n f
(20)
thus the wave power is given by 2z oo
P - Pg I I f - ' S ( f , O) d f dO - p g m o o 4 z -~
(21)
which can be expressed in terms of H s and Te as pg2
p-
64z
H ro (22)
When H s is expressed in meters and Te in seconds, the power level is given by the equation
pw_0. 5 H~T~ 2 kW/n
(23)
The energy flux incoming from the angular sector Ok (sector centered on Ok with width A0) is computed by: A0 2 0 k 4 2 oo
Pg Po~ - 4 z
~ y f -1S ( f , O ) d f d O ao o ok 2
9
(24)
The energy flux in a given direction 00, another relevant quantity wave energy utilisation, is computed by /gg 22z
P o o-- 4- -z
! ~ i f -'S ( f , 0) c o s ( 0 - 00) df dO. o
(25)
The most favourable direction Of maximizes P ~ . The directional coefficient, defined by do=--
P, (26) is useful to characterise the energy spreading9 Values of d o close to unity indicate that the energy is concentrated around Of, which is a favourable situation for the extraction of wave energy by systems sensitive to the direction. 161
WAVE ENERGY CONVERSION References
Abramowitz, M. and Stegun, I. A. 1979 Handbook of Mathematical Functions; New York., NY: Dover. Komen, G., Cavaleri, L., Donelan, M., Hasselmann.,K., Hasselmann, S. and Janssen, P.A.E.M. 1994, Dynamics and Modelling of Ocean Waves. Cambridge Univ. Press, 532 p. Longuet-Higgins, M. S., Cartwright, D.E., and Smith, N.D. 1961 Observations of the direction spectrum of sea waves using the motions of a floating buoy. In Ocean Wave Spectra, Prentice Hall, Englewood Cliffs, N.J., 111-132. Mollison, D. 1986 Wave climate and the wave power resource. In Hydrodynamics of Ocean Wave-Energy Utilization (eds. D.V. Evans & A. F. de O. Falc~o), 133-156. I. Contributed by Dr. Teresa Pontes, INETI-Department of Renewable Energies, Lisbon, Portugal. (Dr. Pontes is a member of the ECOR Working Group on Wave Energy Conversion.)
162