OCE421 Marine Structure Designs Lecture

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OCE421 Marine Structure DesignsLecture 4
(wave energy wave transformation)

• Fall, 2003

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Elementary Concepts of Waves

• A wave is defined as a disturbance propagating
  through a medium. Since the concept of a
  disturbance implies energy, the basic
  characteristic of wave propagation is the
  transfer of energy through a system.

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Why Studying Wave Energy

• Studying the energy in a wave is important to the
  understanding of several phenomena
• the generation of waves by wind,
• the changes that occur as a wave propagates from
  deep to shallow water,
• the spectral characteristics of waves.

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Total Energy in Waves

• potential energy - owing to the water surface
  displacement from the still condition (against a
  gravitational field).
• kinetic energy - due to the fact that the water
  particles throughout the fluid are moving.

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Kinetic Energy
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Water Particle Velocities
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Potential Energy
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Potential Energy
with wave present
without wave present
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Specific Energy
specific energy or energy density
Energy density is a function of the wave height
squared, independent of water depth and wave
length.
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Instantaneous Energy Flux
the rate at which the energy is transferred is
called the energy flux
Instantaneous energy flux instantaneous rate at
which work is being done by the dynamic
pressure per unit width in the direction of wave
propagation
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Pressure Field
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Average Energy Flux (Power)
Average energy flux of a wave is the average
energy per unit time and per crest width
transmitted in the direction of wave propagation
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Group Velocity
(energy travels with group velocity)
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Fraction of Energy Transmission

• n can be interpreted as the fraction of the
  energy in a wave that is transmitted forward each
  wave period.
• The term n is a function of kd or the relative
  depth d/L.
• Its values varies from 0.5 in deep water to 1.0
  in shallow water.

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Wave Shoaling
Calculate the wave height changes as a wave train
propagates toward the shore
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Energy Flux Conservation
The wave energy per unit time passing 1 is equal
to the wave energy per unit time passing 2
(neglecting the energy transfer to and from
waves owing to surface and bottom effects)
between two wave rays.
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Wave Shoaling without Refraction
When wave rays are normal to the bottom contour,
the spacing between wave rays is unchanged.
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Snells Law
Is d1 greater than d2 ?
L1
B1
d1
D
d2
L2
B2
wave ray
wave ray
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Why d1 is greater than d2
In intermediate and shallow water, the celerity
(length) of a wave depends on the local relative
depth d/L.

• If the depth varies along the crest of a wave,
  the portion of the wave in shallower water will
  have a lower celerity.
• This will cause the wave crest to reorient its
  alignment toward the alignment of the bottom
  contours.

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Summary of Snells Law
a angle between wave crest and bottom
contour
B spacing between wave rays
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Shoaling and Refraction Coefficients
total power in the wave between orthogonal lines
is constant (no energy flows along the crest)
wave height ratio
shoaling coefficient
refraction coefficient
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Wave Crest Pattern for Shoaling Waves
wave crest
parallel, straight bottom contours
d/Lo 0.5
mean shore line
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Wave propagation at parallel, straight bottom
contours
(a simplified scenario)
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Calculation for Shoaling Coefficient
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Matlab Code Shoaling Coefficient
function Ks shoal(d,T) ------------------------
---------------------------------- function
Ks shoal(d,T) To calculate shoaling
coefficient T wave period d water
depth -------------------------------------------
--------------- Lldis(d,T)
wave length k2 pi / L
wave number n1/2(12kd/ sinh(2kd)) Ks
sqrt(coth(kd)/(2n)) shoaling
coefficient
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Calculation for Refraction Coefficient
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Matlab Code Refraction Coefficient
function Kr,alpha refra(d,T,alpha_o) --------
--------------------------------------------------
------- function Kr,alpha
refra(d,T,alpha_o) Calculate refraction
coefficient and angle of incidence (between
wave crests and bottom contours) for parallel,
straight bottom contours. alpha_o angle
of incidence (in degrees) T wave
period (in seconds) d water depth
(in meters) Kr refraction
coefficient alpha wave angle at shallow
water site (in degrees) by James Hu
2/8/95 (revised) -----------------------
-------------------------------------------
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Matlab Code Refraction Coefficient
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Unrefracted Deep Water Wave Height
( the prime denotes wave shoaling without
refraction)
2-D in nature, always the case on conducting
experimental investigation using an ordinary
2-D wave tank.
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Plan view of wave refraction around an island
http//www.coastal.udel.edu/ngs/waves.html
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Wave Diffraction
(lee side)
(shadow zone)
If the sides of the island are sloping under the
water, then refraction would also be present
http//www.coastal.udel.edu/ngs/waves.html