Wave Hydrodynamics

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Wave Hydrodynamics Chapters 5 and 6 in Komar,
P.D. Beach Processes and Sedimentation.
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The inner shelf is a friction-dominated realm
where surface and bottom boundary layers overlap.
(From Nitrouer, C.A. and Wright,
L.D., Rev. Geophys., 32, 85, 1994. With
permission.)
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Conceptual diagram illustrating physical
transport processes on the inner shelf.
(From Nitrouer, C.A. and Wright, L.D.,
Rev. Geophys., 32, 85, 1994. With permission.)
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Approximate distribution of ocean surface wave
energy illustrating the classification of surface
waves by wave band, primary disturbance force,
and primary restoring force.
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SEAS
Waves under the
influence of winds in a generating area SWELL

Waves moved away from
the generating area and no longer influenced by
winds
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SMALL AMPLITUDE/FIRST ORDER/AIRY WAVE THEORY

• Fluid is homogenous and incompressible,
  therefore, the density is a constant.
• Surface tension is neglected.
• Coriolis effect is neglected.
• Pressure at the free surface is uniform and
  constant.
• Fluid is ideal (lacks viscosity).

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SMALL AMPLITUDE/FIRST ORDER/AIRY WAVE THEORY

• The wave does not interact with any other water
  motion.
• The bed is a horizontal, fixed, impermeable
  boundary which implies that the vertical velocity
  at the bed is zero.
• The wave amplitude is small and the wave form is
  invariant in time and space.
• Waves are plane or low crested (two dimensional).

Can accept 1, 2, and 3
and relax assumptions 4-9
for most practical
solutions.
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WAVE CHARACTERISTICS
T WAVE PERIOD Time taken for two successive
crests to pass a given point in space
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Definition of TermsELEMENTARY, SINUSOIDAL,
PROGRESSIVE WAVE
heta
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WAVE CELERITY, LENGTH, AND PERIOD
PHASE VELOCITY/WAVE CELERITY (C) speed at which
a waveform
moves.
Relating wavelength and H2O depth to celerity,
then
Since C L/T, then is
NOTE L exists on both sides of the equation.
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When d/L gt0.5 DEEP WATER
Since
Here,
Then
Since
DEEP WATER
Then
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• Longer waves travel faster than shorter waves.
• Small increases in T are associated with large
  increases in L.
• Long waves (swell) move fast and lose
  little energy.
• Short wave moves slower and loses most
  energy
  before reaching a distant coast.

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MOTION IN A SURFACE WAVE

Local Fluid Velocities and Accelerations
(HORIZONTAL)
(VERTICAL)
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Water particle displacements from mean position
for shallow-water and deepwater waves.
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SUMMARY OF LINEAR WAVES
C Celerity Length/Time
Relating L (Wavelength) and D (Water Depth)
Since C L/T, then becomes
Since C L/T, then becomes
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PROBLEMS
GIVEN A wave with a period T 10 secs. is
propagated shoreward from a depth d 200m to a
depth d 3 m. FIND C and L at d 200m and
d 3m.
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WAVE ENERGY AND POWER
Kinetic Potential Total Energy of Wave
System Kinetic due to H2O particle
velocity Potential due to part of fluid mass
being above trough. (i.e. wave crest)
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WAVE ENERGY FLUX(Wave Power)
Rate at which energy is transmitted in the
direction of progradation.
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Summary of LINEAR
(AIRY) WAVE THEORYWAVE CHARACTERISTICS
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Regions of validity for various wave theories.
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HIGHER ORDER THEORIES

• Better agreement between theoretical and observed
  wave behavior.
• Useful in calculating mass transport.

• HIGHER ORDER WAVES ARE
• More peaked at the crest.
• Flatter at the trough.
• Distribution is skewed above SWL.

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Comparison of second-order Stokes profile with
linear profile.
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USEFULNESS OF HIGHER ORDER THEORIES
MASS TRANSPORT VELOCITY U(2)
The distance a particle is
displaced during one
wave period.
NB Mass transport in the direction of
propagation.
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HIGHER ORDER WAVES

• Stokes
• Takes wave height to 2nd order (H ) and higher
• Useful in higher energy environments

2
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For deep H2O Eq. reduces to
If H/L is small, then profile can be represented
by linear wave theory
THIRD ORDER APPROX. (Wave Velocity)

NB. If (H/L) is small, use linear wave theory
equation.
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VELOCITY OF A WAVE GROUP
WAVE GROUP/WAVE TRAIN Speed not equal to wave
travel for individual waves GROUP SPEED GROUP
VELOCITY (Cg). INDIVIDUAL WAVE SPEED Phase
velocity or wave celerity. Waves in DEEP or
TRANSITIONAL WATER In SHALLOW WATER
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K .4085376 YT 1.065959

Keulegan and Patterson (1940)
Cnoidal Wave Theory

SI Units (m)
Wave Height .25 Wave Period 2 WaterDepth
1.1 Deep
Water Length 6.24 Present Length 3.757897
Elliptical Modulus .4085376
Net Onshore Displacement Umass Mass
Transport Velocity
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Airy Wave Theory LO 6.24 L
5.783304
Sediment Transport
Time
U(T)
UMass
T 2s H 0.25m D 1.5m
NB. Umass Symmetry
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Airy Wave Theory LO 6.24 L
5.363072
Sediment Transport
Time
U(T)
UMass
T 2s H 0.25m D 1.1m
Depth at which C.T. took place