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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?

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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

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• 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