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Wave Propagation and Shoaling

Focus on movement and natural alteration of the characteristics of waves as they travel from the source region toward shore Waves moving from deep to intermediate/shallow water change their shape and characteristics significantly. Wave velocity and wavelength decrease, while height increases to conserve wave energy flux. Period remains the same. Just offshore of the breaker zone, the waves have peaked crests and broad troughs; a very different appearance to their deep-water sinusoidal form.

Wave Groups and Dispersion

https://www.meted.ucar.edu/training_module.php?id=188

Define the term "wave group" to mean a bundle of wave energy that travels from source region to the shore. Wave groups travel with speed Cg= Cn, a.k.a. the "group velocity". Longer period waves outrun and leave behind the shorter period waves. Dispersion produces a narrowing of the energy spectrum, so the greater the travel distance, the narrower the strong frequency band in the energy density spectrum. Individual waves arise at rear of group, move through group, and die out at front. Hence, they cannot be traced across the ocean - but the groups themselves can!

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Arrival Times for Swells of Different Periods

Wave Energy Losses

Waves lose very little energy as they traverse great distances, but four sources of wave energy loss have been suggested and investigated: 1.  Viscous Damping 2.  Angular Spreading 3.  Contrary Winds 4.  Wave-wave Interactions

https://www.meted.ucar.edu/training_module.php?id=188

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Wave Shoaling Schematic

Changes that occur when a wave shoals (moves into shallow water): In deep water = profile of swell is nearly sinusoidal. Enter shallow water, waves undergo a systematic transformation. Wave velocity and wave length decrease while the wave height increases. Only wave period remains constant.

Shallow water - L, C depend only on the …

Deep water - L, C depend only on …

Wave Shoaling – L & C

Summarize regions of applications of approximations Behavior of normalized variables.

period

water depth

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Pin Pout

Outside the surf zone, wave energy flux is conserved:

Pin = Pout

Ecn = constant Ecn = E∞c∞n∞

The shoaling coefficient, Ks, is given as the ration of shallow water to deep water wave height:

Wave Shoaling – Effects on H – the shoaling coefficient, Ks

Explains why orthogonally directed waves increase height during shoaling Direct compensation for slowing of individual waves and need to maintain constant wave energy flux Waves convert a significant fraction of their kinetic energy to potential energy

Effect of Shoaling

H = 2 m T = 10 s othogonal angle of incidence

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Wave Shoaling – Steepness (H/L)

Straightforward consequence of combined shoaling behavior of H & L. Steepness initially decreases upon entry to intermediate water depth, then rapidly increases until instability condition associated with wave breaking.

Wave Refraction

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Refraction

Analogous to the bending of light rays as they pass through different media – glass of water, for example.

Wave Refraction: Distinction - Wave Crests vs. Wave Rays

Wave crests are the line segments that connect the peaks (or troughs) of a wave field. The crests are visible to the observer. Wave rays are the lines orthogonal (perpendicular) to the wave crests, which represent the direction of wave propagation

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c = (gh)1/2

)sin()sin(

)sin()sin(

2

2

1

1

∞∞

=

=

αα

αα

cc

cc

Wave Refraction - Snell’s Law

Wave Refraction: Energy flux per unit length of wave crest

Energy flux per unit length of wave crest is not necessarily conserved Can lead to a decrease in wave height during the shoaling and refraction process.

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Wave Refraction Over Canyons and Headlands

2211)()( sEnCsEnC =

2/12/1

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&=∞

∞∞

ss

cc

nHH

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Kr =s∞s

#

$ % &

' (

1/ 2

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=cos(α∞)cos(α)

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( )

1/ 2

Computing the Refraction Coefficient, Kr

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Effect of Shoaling and Refraction

H = 2 m T = 10 s compare: orthogonal wave vs. refracting wave

2/12/1

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&=∞

∞∞

ss

cc

nHH

Modeling refraction - wave rays

H = 2 m T = 10 s α = 270˚

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Modeling refraction - wave rays - double period

H = 2 m T = 20 s α = 270˚

bathymetry (in feet) focusing waves

“Jaws” Surfing Reef, Maui

Model simulations of individual waves - not time averaged.

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Wave Diffraction

Lateral translation of energy along a wave crest. Most noticeable where a barrier interrupts a wave train creating a "shadow zone". Energy leaks along wave crests into the shadow zone. Also by analogy to light, Huygen's Principle explains the physics of diffraction through a superposition of point sources along the wave crest.

Wave Diffraction- Barcelona

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Modeling with SWAN Ref/Diff numerical simulation of shoaling and refraction - monochromatic (not spectral) SWAN - used here at UF (spectral)

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