Wave Hydrodynamics (Strukpan 1)

Wave Hydrodynamics

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Beach Terminology

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.)

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.)

Ocean WavesOcean waves may be classified by the generating force (wind, seismic events, or gravitational pull of the moon), the restoring force, (surface tension, gravity, the earth’s rotation), or the frequency of the waves.

Idealized Ocean Wave Spectrum

Wind WavesA wind wave is generated by the friction of the wind over the water’s surface.

As the wind blows over the surface of the water, friction and pressure differences create small ripples in the water surface.

The wind pushes on the back side of the wave and pulls on the front, transferring energy and momentum to the water.

As the wind continues to transfer momentum to the water, the wave becomes higher.

Wave GrowthThe area where wind waves are form and grow is called the generation area.

Higher wind speeds mean more momentum to transfer to the water, resulting in higher waves.

Duration is the length of time the wind is blowing. The longer the wind blows, the higher the waves and more chaotic the seas.

The heights of the waves in the generation area are determined by three factors: wind speed, duration, and fetch.

FetchFetch is the horizontal distance that the wind blows across the water.

Fetch is important in the early stages of wave formation, and will control how large the wave will be at a given time.

SwellAs deep-water waves depart the generation area, they disperse with the long waves travel faster.

This sorting by wave speed creates long regular wave patterns called swell.

Shoaling WavesAs a wave shoals (approaches the shoreline) the wave period remains constant, causing the wavelength to decrease and the wave height to increase.

Friction slows the bottom of the wave to while the top continues at the same speed, causing the wave to tip forward.

When H/L, the ratio of the wave height to wavelength, reaches the critical value of 1/7, the wave breaks.

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

SMALL AMPLITUDE/FIRST ORDER/AIRY WAVE THEORY

1. Fluid is homogenous and incompressible, therefore, the density is a constant.

2. Surface tension is neglected.3. Coriolis effect is neglected.4. Pressure at the free surface is uniform

and constant.5. Fluid is ideal (lacks viscosity).

SMALL AMPLITUDE/FIRST ORDER/AIRY WAVE THEORY

6. The wave does not interact with any other water motion.

7. The bed is a horizontal, fixed, impermeable boundary which implies that the vertical velocity at the bed is zero.

8. The wave amplitude is small and the wave form is invariant in time and space.

9. Waves are plane or low crested (two dimensional).

Can accept 1, 2, and 3 and relax assumptions 4-9

for most practical solutions.

Can accept 1, 2, and 3 and relax assumptions 4-9

for most practical solutions.

WAVE CHARACTERISTICS

T = WAVE PERIOD

Time taken for two successive crests to pass a given point in space

Definition of TermsELEMENTARY, SINUSOIDAL,

PROGRESSIVE WAVE

=eta

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.

DEEP WATER:

Since:

Then:

Here, Since:

Then:

When d/L >0.5 = DEEP WATER

1. Longer waves travel faster than shorter waves.

2. 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.

MOTION IN A SURFACE WAVE

Local Fluid Velocities and Accelerations

(VERTICAL)

(HORIZONTAL)

Water particle displacements from mean position for shallow-water and deepwater waves.

As waves approach a shoreline the water shallows and they change from deepwater to transitional waves.

As water shallows the waves steepen and finally break to form surf which surges towards the shoreline.

When surf reaches the beach it rushes up the beach face as swash and then runs back down the slope as backwash.

Swash and backwash moves sediment up and down the beach face.

SUMMARY OF LINEAR WAVESC = Celerity = Length/Time

Relating L (Wavelength) and D (Water Depth)

Since C = L/T, then becomes:

Since C = L/T, then becomes:

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.

WAVE ENERGY AND POWERKinetic + 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)

WAVE ENERGY FLUX(Wave Power)

Rate at which energy is transmitted in the direction of progradation.

Summary of LINEAR (AIRY) WAVE THEORY:

WAVE CHARACTERISTICS

Regions of validity for various wave theories.

HIGHER ORDER THEORIES

1. Better agreement between theoretical and observed wave behavior.

2. Useful in calculating mass transport.

HIGHER ORDER WAVES ARE:

• More peaked at the crest.

• Flatter at the trough.

• Distribution is skewed above SWL.

Comparison of second-order Stokes’ profile with linear profile.

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.

HIGHER ORDER WAVESStokes

• Takes wave height to 2nd order (H ) and higher

• Useful in higher energy environments

2

2nd order approximate wave profile is:

If H/L is small, then profile can be represented by linear wave theory

For deep H2O – Eq. reduces to:

THIRD ORDER APPROX. (Wave Velocity)

NB. If (H/L) is small, use linear wave theory equation.

TERM: Peaks crests

Flattens troughs

Conforms to shallow H2O wave profile

VELOCITY OF A WAVE GROUPWAVE 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

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

Time U(T) UMassSediment Transport

Airy Wave Theory LO = 6.24 L = 5.783304

T = 2s H = 0.25m D = 1.5m

NB. Umass Symmetry

Time U(T) UMassSediment Transport

Airy Wave Theory LO = 6.24 L = 5.363072

T = 2s H = 0.25m D = 1.1m

Depth at which C.T. took place

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Deformasi Gelombang

• Breaking

• Refraction

• Diffraction

• Reflection

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Refraction

• Waves travel more slowly in shallow water (shallower than the wave base).

• This is called refraction

• This causes the wave front to bend so it is more parallel to shore.

• It focuses wave energy on headlands.

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Wave Refraction

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Orthogonal

Surf / Breaker Zone

Waves travel faster in deper water

Waves travel more slowly in shallow water

Beach

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Wave Refraction

Seabed contour

Wave Crest

Path of crests diverge and minimize impact of waves on shore

Seabed contourWave crest

Path of crests converge and maximize impact of waves on shore

Shallow

Deep

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Long shore Transport

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Wave Diffraction

50Orthogonal Wave Crest

Orthogonal

Energy Transfer

Wave Diffraction

BreakwaterHi

Hd

r

L

Shadow Zone

Wave Diffraction

Diffraction Diffraction CoeficientCoeficient( K’ )( K’ )

K’ = Hd / HiK’ = Hd / HiK’ = K’ = (r/L, (r/L, , , ))

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Refleksi Gelombang

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Refleksi Gelombang

Untuk dinding vertikal, kedap air, dgn elevasi diatas muka air, hampir seluruh energi akan dipantulkan kembali ke laut.

Hanya sebagian saja energi yang dipantulkan jika gelombang menjalar di pantai yang agak landai

Refleksi tergantung pada kelandaian pantai, kekasaran dasar laut, porositas dinding, dan Angka Irribarren (Ir) :

tanr

i

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IH

L

Kr = Hr / HiKr = Hr / HiKr = fungsi (a, Kr = fungsi (a, n, P, Ir)n, P, Ir)

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Perbedaan Gelombang

WAVES – BREAKING

Dean and Dalrymple, 2002

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SuntoyoHp. 081230988146

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