waves and wave -driven processes in the nearshore: part 1 · 12 hr 5 min 30 s 1 s 0.1 s wave type...
TRANSCRIPT
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Waves and WaveWaves and Wave--Driven Driven Processes in the Nearshore:Processes in the Nearshore:
Part 1Part 1
GEO3-4306: Coastal Morphodynamics
This lecture
• different wave types
• wave generation
• wave characteristics (anatomy)
• deep-water waves
• from deep to intermediate water
before
after
Why study waves?
2
Why study waves?
• transfer of energy (wind � breaking)
• generate other type of waves and currents
• !! determine morphology in the nearshore• erosion • sedimentation
10-6 10-5 10-4 10-3 10-2 10-1 100 101 102
Frequency (Hz)
Period12 hr 5 min 30 s 1 s 0.1 s
Wave Type
Disturbing Force
Restoring Force
TidesInfra.
Waves GravityCapillary Waves
Long Waves
Sun, Moon
Storms, Tsunami
Wind
CoriolisForce
Gravity
Surface Tension
Ene
rgy
(L2 )
f = 1/T
swell and sea
edge
and
leak
y w
aves
Wave Classification
• period (or, frequency)• gravity waves• infragravity waves• tides
• relationship with wind:• gravity waves: sea and swell
• structure• infragravity waves: edge and leaky
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Wave Measurements
• In situ (in, on, below the waves)• buoys• pressure gages• surface gages• ...
• Remote sensing (above the waves)• visual / airplane / satellites• e.g., radar
buoysbuoyssurface gagesurface gage
pressure sensorpressure sensor
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Wave Anatomy (1)Wave Anatomy (1)Still
water levelWavelength
Hei
ght
Crest Trough
Frequency (f): Number of wave crests passing point A or point B each second
Period (T): Time required for wave crest at point A to reach point B
A BDirection of
wave advance
Orbital path of individual water
particles
f = 1/T
Wave Anatomy (2)
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Wave Anatomy (3)
L/4 to L/2
6
• initial formation: turbulent eddies in wind field• capillary waves
• subsequent growth: sheltering effect• pressure difference
Wave Generation (1)
• during growth:• short waves steepen and break• energy absorbed by longer waves• with time, more and more energy at larger periods• sweeping mechanism
• wind � capillary waves � larger-period waves
Wave Generation (2)
generation and growth of wind waves is a function of
(1) wind velocity
(2) wind duration
(3) fetch
(SEA)
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Deep water, fully developed sea: prediction of H and T given U, F and t
Example: U = 15 m/s, F = 200 km, duration t = 6 hours
2.9 m5.8 s
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Example: U = 15 m/s, F = 200 km, duration t = 18 hours
4.0 m 7.4 s
Wave generation at Sylt (D)Offshore winds - JONSWAP
Wave generation revisited:
wind � capillary waves �longer-period waves
Transformation at sea
• Dispersion– Waves of different length travel at different speeds
• Wave interference– wave groups
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Dispersion
• DISPERSION RELATIONSHIP:
• Multiplying by L and reducing we can isolate the wavelength:
• Wave celerity:
)tanh(2 khgk=σT
πσ
2=
Lk
π2=
=L
hgTL
ππ
2tanh
2
2
=L
hgTC
ππ
2tanh
2
)tanh(2 khgk=σ
kh- is the relative depth
Large kh
Deep water tanh(kh) = 1
Small kh
Shallow water tanh (kh) → kh
Home exercise: h = 15 m, T = 10 s, L = ?
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Dispersion• Waves of different length travel at different speeds
• In deep water the wavelength (L) and wave celerity (C) are proportional to the wave period (T):
π2
2gTL =
π2gT
C = These equations are only valid for
deep water!!
Sea and Swell
• sea: many waves with different heights and period
• swell: waves escape from wind• waves with larger periods travel faster (c ∝T)• waves are sorted out by period• dispersion of swell
• swell dispersion
Sea and Swell
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• waves are not a single sinusoid
• waves are often grouped (5 – 10 waves in a group)
Wave Interference
•• real waves are a summation of sinusoidsreal waves are a summation of sinusoids
•• here, just 2; in practice, an awful lothere, just 2; in practice, an awful lot
Wave Interference
Wave Group Velocity• For waves to propagate into undisturbed water at large depths then some energy must
be dissipated in setting up the initial oscillation
• Group Velocity (cg = n c):– Deep water: ½ of the individual wave velocity (n = ½)– Shallow water: approaches the velocity of the individual wave (n = 1)
Undisturbed Surface
12345
23456
Undisturbed Surface
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Wave energy (Etotal) = potential wave energy (Epotential) +kinetic wave energy (Ekinetic)
Etotal = Epotential + Ekinetic
Etotal = 1/8.� .g.H2
Energy flux (P) = wave energy (Etotal) * group velocity (cg)
P = Etotal . cg
Wave Energy
Waves start to feel the sea bed!!
Changes:• orbital motion becomes an ellips• wave shape changes, from linear to non-linear theory• mass transport becomes non-zero• wave shoaling• wave refraction• wave diffraction
Deep to intermediate water
Orbital Motion
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Wave Shape changes:
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1.5
-1
-0.5
0
0.5
1
1.5
Normalized time (t/T)
Nor
mal
ized
am
plitu
de
Deep waterIntermediate waterShallow water
Airy and Stokes WaveAiry wave = simple sinusoid:
( ) ( )tkxH
tx ση −= cos2
,
η
H
Stokes wave = simple sinusoid + disturbance:
( ) [ ][ ]
[ ])(2cos)sinh(
)2cosh(2)cosh(
2cos
2 3
2
tkxkh
khkh
L
Htkx
Hσ
πση −
++−=
If H/L →0 the above equation reduces to the Airy solution
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Stokes WavesAiry wave
Stokes wave
Harmonic wave
You can think of a Stokes waves as an
Airy wave with a smaller wave of 1/2 period added to it
Onshore velocity
Offshore velocity
STOKES WAVE
More time is spent under the trough then
under the crest
Onshore velocity
Offshore velocity
AIRY WAVE
Equal time is spent under the trough as
under the crest
An important prediction of
Stokes’Theory is
that:
Water particle orbits are
NOT CLOSED
• Thus a MASS TRANSPORT or WAVE DRIFT is predicted:
[ ][ ]2
2
)sinh(
)(2cosh
2
1
kh
hzkC
L
Hu
+
=π
Downwave Mass Transport(STOKES DRIFT)
Assumes an infinitely long length of wave travel and a non-viscous fluid
Direction of Waves
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• Longuet-Higgins (1953) subsequently provided a solution of a finite length of travel and for the presence of viscosity. Thus the flow must satisfy CONTINUITY and the equations suggest:
DOWNWAVE (landward) MEAN flow at the water surface & bed
UPWAVE (seaward) return flow at mid-depthDirection of Waves
From deep to intermediate depth
What will happen to:
1. Wave period T?
2. Wave length L?
3. Wave speed c?
4. Group velocity cg?
5. Wave energy flux?
Answers:
1. Remains the same
2. Reduces
3. Reduces
4. Reduces
5. Remains the same
Shoaling
Energy flux deep water = energy flux intermediate depth
E * cg (deep) = E * cg (intermediate)
Because of the conservation of energy flux, the wave heightwill increase when a wave field enters intermediate waters.This is called shoaling.
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Refraction
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• wave velocity decrease with decreasing water depth• waves travel slowly in shallower water (Snell’s law)
• assumptions:parallel depth contoursno energy losses during wave propagationno lateral leaking of energy
Refraction
sin sinc
cα α∞
∞=
• wave energy flux between the wave rays = constant fromdeep to shallow water
• wave height?
Refraction
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Shoaling and Refraction
0.5 0.5
2H c s
H nc s∞ ∞
∞
=
when the water depth decreases:
decreasesincreases
Complex Refraction Patterns
Complex Refraction Patterns
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Complex Refraction Patterns
wave rays in shadow zones …. energy is leaking away ….
energy losses …. decreasing wave heights
Diffraction
Waves start to feel the sea bed!!
Changes:• orbital motion becomes an ellips• wave shape changes, from linear to non-linear theory• mass transport becomes non-zero• wave shoaling• wave refraction• wave diffraction
From deep to intermediate depth