Wave Thermodynamics and Ideas on How to Capture Wave
Energy
Joe Ott
MANE-6540H01
04 Dec 2008
ABSTRACT
Waves are excellent sourced of energy, are a natural resource, and are
renewable. Waves in shallow water are one of the best ways to capture this
energy as this is when the wave energy is at its maximum. How much energy is
captured depends upon a number of factors: wave translation, wave height,
the number of waves in a set period, the type of converter used and its
efficiency of capturing both the potential and kinetic wave energy.
INTRODUCTION
As a topic for this paper I have chosen to review the thermodynamics behind
ocean waves and discuss their application to the generation of energy in a
limited fashion. This topic was chosen as a potential area for future thesis
work in evaluating applications that would turn this natural resource into a
usable medium; namely electricity. This paper will review the fundamentals
behind ocean waves and the energy they provide as well as briefly touch upon
some design considerations and how an apparatus could use this energy.
Ocean wave are a naturally occurring phenomenon as well as a renewable
energy source. Waves form in water when it is excited by external forces, such
as wind, vibrations, displacement, etc. The water medium translates this
energy from its point of initiation outward forming visible waves as the
environmental conditions change. This energy can be harnessed to be used in
other forms. “Exploiting the low, variable frequency motion of waves, and
coupling the power to a fixed frequency and fixed voltage grid system, is a
challenging task which device designers have tackled in different ways”8. This
paper will also focus on the visible waves that form at the shore as this is
where the most energy can be harnessed. This will be discussed further in the
Solutions section.
FORMULATION
The following explanation of wave theory is taken from (Meadows4) so we can
understand the basics behind ocean waves. Wave formation, motion, and
energy can be described as follows and viewed in Figure 1. – Wave Basics:
Figure 1. – Wave Basics4
The wave height, H, is the vertical distance between its crest and leading
trough. Wavelength, L, is the horizontal distance between any two
corresponding points on successive waves and wave period is the time required
for two successive crests or troughs to pass a given point.
The celerity of a wave C, is the speed of propagation of the waveform or how
fast it travels (phase speed), defined as C = L/T, where T = time. It is of note
that waves entering shallow-water begin to show a net displacement of water
(visible wave) in the direction of propagation and are classified as translational
in their movement. The equilibrium position used to reference surface wave
motion, (still water level, SWL) is z = 0 and the bottom is located at z = –d.
Small-Amplitude Wave — Properties
The equation for the free surface displacement of a progressive wave is
The expression relating individual wave properties and water depth, d, to the
propagation behavior of these waves is the dispersion relation, sigma2,
where g is the acceleration of gravity.
(1)
From this equation (1) and the definition of celerity (C) it can be shown that
and
(2a, 2b)
The hyperbolic function tanh kd approaches useful simplifying limits of 1 for
large values of kd (deep water) and for small values of kd (shallow water).
Applying these limits results in expressions for shallow water where
(3)
which shows that wave speed in shallow water is dependent only on water
depth. Some useful functions for calculating wave properties at any water
depth, from deep water wave properties reference in Meadows4, are
.
(4)
Values of d/L can be calculated as a function of d/Lo by successive
approximations using
.
(5)
Particle Motions
The horizontal component of particle velocity, u, beneath a wave is
.
The corresponding acceleration, ax, is
(6)
.
(7)
The vertical particle velocity, w, and acceleration, az, are respectively
(8)
and
.
(9)
It can be seen from eqns. (6) and(8) that the horizontal and vertical particle
velocities are 90 Deg out of phase. Extreme values of horizontal velocity occur
in the crest in the direction of wave propagation, and the trough in the
direction opposite to the direction of wave propagation while extreme vertical
velocities occur mid-way between the crest and trough, where water
displacement is zero.
The u and w velocity components are at a minimum at the bottom and both
increase as distance upward in the water column increases. Maximum vertical
accelerations correspond to maximum in horizontal velocity and maximum
horizontal accelerations correspond to maximum in vertical velocity. Figure1
provides a graphic summary of these relationships.
The particle displacements can be obtained by integrating the velocity with
respect to time and simplified by using the dispersion relationship to give a
horizontal displacement
(10)
and vertical displacement
(11)
where (xo, zo) is the mean position of an individual particle.
Pressure Field
The pressure distribution beneath a progressive water wave is given by the
following form of the Bernoulli Equation
(12)
where  is fluid density and Kp, the pressure response coefficient, is
which will always be less than 1, below mean still water level. gz is the
hydrostatic pressure and gKp(z)is the dynamic pressure term. This dynamic
pressure term accounts for two factors that influence pressure, the free
surface displacement and the vertical component of acceleration. A frequently
used method for measuring waves at the coast is to record pressure
fluctuations from a bottom-mounted pressure gage. Isolating the dynamic
pressure (PD) from the recorded signal by subtracting out the hydrostatic
pressure gives the relative free surface displacement
(13)
where Kp(–d) = 1/cosh kd.
It is necessary, therefore, when determining wave height from pressure records
to apply the dispersion relationship to obtain Kp from the frequency of the
measured waves. It is important to note that Kp for short period waves is very
small at the bottom (–d), which means that very short period waves may not be
measured by a pressure gage.
Wave Energy
Progressive surface water waves possess potential energy from the free surface
displacement (vertical) and kinetic energy from the water particle motions
(translational). From linear wave theory it can be shown that the average
potential energy per unit surface area for a free surface sinusoidal
displacement, restored by gravity, is
(14)
Likewise the average kinetic energy per unit surface area is
(15)
and the total average energy per unit surface area is
.
The unit surface area considered is a unit width times the wavelength L so that
the total energy per unit width is
(16)
The total energy per unit surface area in a linear progressive wave is always
equi-partitioned as one half potential and one half kinetic energy. Energy flux
is the rate of energy transfer across the sea surface in the direction of wave
propagation. The average energy flux per wave is
where
(17, 18)
and Cg is the group speed defined as the speed of energy propagation.
In deep water n = 1/2 and in shallow water n = 1 indicating that energy in deep
water travels at half the speed of the wave while in shallow water energy
propagates at the same speed as the wave.
Wave Breaking
Waves propagating into shallow water tend to experience an increase in wave
height to a point of instability at which the wave breaks, dissipating energy in
the form of turbulence and work done on the bottom. Breaking waves are
classified as: spilling breakers generally associated with low sloping bottoms
and a gradual dissipation of energy; plunging breakers generally associated
with steeper sloping bottoms and a rapid, often spectacular, “explosive”
dissipation of energy; and surging breakers generally associated with very
steep bottoms and a rapid narrow region of energy dissipation. A widely used
classic criteria applied to shoaling waves relates breaker height, Hb, to depth
of breaking, db, through the relation
.
(19)
However, this useful estimate neglects important shoaling parameters such as
bottom slope (m) and deepwater wave angle of approach (o). Meadows4 uses
the following to solve for breaker depth (db), distance from the shoreline to
the breaker line (xb) and breaker height (Hb) as
(20)
and
(21)
where m = beach slope and K = Hb /db
The author makes note that wave breaking is still not well understood and
caution is urged when dealing with engineering design in the active breaker
zone.
TABLE 1: Wave Formulas4
To summarize useful formulas when working wave problems, Meadows4 includes
the above Table 1.
SOLUTION
The above wave formulas can be used to understand wave motion and energy
and be used to help formulate ways to utilize that energy. An apparatus that
will be used for shallow water waves must take into account for wave types
and energy extraction methods, as well as the environmental conditions and
location, durability, and service life. In this paper, we will focus on the energy
extraction requirements.
In consulting (Wave Energy9) when thinking of location the author recommends,
“For designers to convert the slow movement of water to a more easily used
and transmitted form, normally electricity, they must chose a converter that
changes the mechanical wave energy. In interfacing with the waves, any
converter must be constrained so that wave forces are resisted. Ways of
constraining a converter for consideration are (shown in Figure 2):
—using the sea-bed for fixing or mooring;
—mounting several converters on a common frame or spine so that relative
motion is obtained between them;
—using the inertial force due to the gyroscopic action of a flywheel;
—relying on the mass and inertia of the device.”
Figure 2.
The next consideration is how to extract the energy. In general ocean waves
are unsteady, irregular, and directional2. To adapt to these changing
conditions, an apparatus would have to be flexible.
As discussed above, total wave energy is a combination of potential (14) and
kinetic (15) that results from the vertical and horizontal translations. As such,
an apparatus that would maximize energy transfer would have to be able to
utilize both the vertical and horizontal energy. Two ways to do this are
through the use of 2 forces, namely buoyancy and drag.
Buoyancy is historically defined by Archimedes Principle: Any object, wholly or
partly immersed in a fluid, is buoyed up by a force equal to the weight of the
fluid displaced by the object. This results from the fact that fluid pressure
increases with depth and from the fact that the increased pressure is exerted
in all directions (Pascal's principle) so that there is an unbalanced upward (Lift)
force on the bottom of a submerged object8 as shown in Example 1: Buoyancy.
The buoyant force, F, exerted on a body can be calculated by integrating the
stress tensor, s, over the surface of the body, A10,
(22)
Example 1: Buoyancy
Drag force is caused mainly by separated flow in viscous fluid. Drag force is a
function of the Reynolds number and surface roughness of a body, and the
Keulegan-Carpenter number, especially in oscillating flow2.
Drag force, Fd, can be defined as10 where in this case the velocity, v, is equal
to u, the wave horizontal velocity:
(23)
Through a free body diagram and the conservation of energy, we can relate the
vertical and horizontal forces between the wave and a partially submerged
object. We would want to choose a partially submerged object prior to the
wave breaking because this is where the energy is at maximum and prior to the
dispersion of this energy during wave breaking. “Wave breaking is associated
with the generation of high levels of turbulence, air entrainment, noise and
splash, all of which must contribute to the energy dissipation and which are
seen to increase with wave breaking intensity as waves become more plunging
in nature. A relationship between breaking intensity and initial energy
dissipation, showing that the total energy dissipated in the breaking event for
each wave case increases as the relative cavity area becomes larger, i.e. as the
intensity of breaking increases1.”
Weight
FD*L
Ek
F*H + Ep

Picture from Blenkinsopp1
= angle of the tether
As a result, an object that resists both the translation of the wave and vertical
movement of the wave would transfer a portion of this energy through the
tether to an apparatus where is could be transformed.
RESULTS
Understanding these concepts it is possible to transfer the mechanical wave
energy through a medium to generate electricity. This could happen in a
number of ways. Machines such as pistons, generators, flywheels, and piezoelectric would be ideal for generating electricity from mechanical energy.
The overall amount of energy gained would depend upon the efficiency of the
converter used and the location and intensity of waves. From (Musial6) we find
the following:
Marine Renewable Energy Resource Estimates6
Wave Resource: Generally, wave energy increases with latitude and has greater
potential on the west coast of the United States because global winds tend to
move west to east across the Pacific Ocean. The total energy contained in the
waves is dependent on the linear length of wave crest, the wave height and the
wave period. Wave energy resource assessments have been performed by the
Electric Power Research Institute (EPRI) and the methods for calculation are
well documented
Table 2 shows the gross wave energy resource by region. This is an estimate of
the energy contained in the incident waves if it were converted to electricity.
US Wave Resource Regions (>10kW/m)
New England and Mid-Atlantic States
Northern California, Oregon and Washington
Alaska (exclusive of waves from the Bering Sea)
Hawaii and Midway Islands
TWh/yr
100
440
1,250
330
Table 2 – Wave Resource by Region6
CONCLUSIONS
Waves are excellent sourced of energy, are a natural resource, and are
renewable. Waves in shallow water are one of the best ways to capture this
energy as this is when the wave energy is at its maximum.
Wave energy can be categorized in both potential and kinetic, where the
potential energy is a factor of the wave amplitude and displacement and the
kinetic energy is a factor of the translation of the particles.
These energies can be captured using opposing forces such as drag and
buoyancy.
How much energy is captured depends upon a number of factors: wave
translation, wave height, the number of waves in a set period, the type of
converter used and its efficiency of capturing both the potential and kinetic
wave energy.
BIBLIOGRAPHY
1. Blenkinsopp 1, C.E. "The effect of relative crest submergence on wave
breaking over submerged slopes." Coastal Engineering - journal
homepage: www.elsevier.com/locate/coastaleng (2008).
2. El-Hawary, Ferial. The Ocean Engineering Handbook - CH 2.- Modeling
Considerations. Boca Raton, FL: CRC P LLC., 2001.
3. Herbich, John B. Coastal and Ocean Engineering - Ch 87 – “Shallow
Water and Deep Water Engineering”. CRC P, LLC., 2005.
4. Meadows, Guy A., and William L. Woods. The Civil Engineering Handbook
- Ch 36 “Coastal Engineering”. 2nd ed. CRC P LLC, 2003.
5. Mueller, Markus, and Robin Wallace. "Enabling science and technology
for marine renewable energy." 25 Oct. 2008. Institute for Energy
Systems, Joint Research Institute for Energy, UK Energy Research
Centre, School of Engineering and Electronics, University of Edinburgh,
The King’s Buildings, Edinburgh EH93JL, UK. 03 Dec. 2008
<http://www.elsevier.com/locate/enpol>.
6. Musial, W. Status of Wave and Tidal Power Technologies for the United
States. Tech.No. NREL/TP-500-43240. Department of Energy. 2008.
7. Renewable Energy Sources, Section 13 – “Economics of Renewable
Energy Sources”. Taylor & Francis Books, Inc., 2003.
8. Nave, R. "Buoyancy." Georgia State University. Physics Department Web
Site. 04 Dec. 2008 <http://hyperphysics.phyastr.gsu.edu/hbase/pbuoy.html>.
9. Wave energy. Taylor & Francis Books, 2003. Hisaaki Maeda et al.
10. “Buoyancy” and “Drag”, Wikipedia, <http://en.wikipedia.org>