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Waves

Krauss Chapter Nine

WaveParameters

• Wavelength = λ = Length between wave crests (or troughs)

• Wave Number = κ = 2π/λ (units of 1/length)

• Wave Period = T = Time it takes a wave crest to travel onewavelength (units of time)

• Angular Frequency = ω = 2π/T (units of 1/time)

• Wave Speed = C = ω/κ Distance a wave crest travels per unittime (units of distance/time)

• Wave Height = 2a = Twice the wave amplitude

• Wave Steepness = Wave Height/Wavelength

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Ideal waves Propagate Energybut not Mass

Wave Equation

Navier-Stokes Equation

Ignoring viscous forces and looking just at the x and zcomponents…

Expanding the terms…

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Guess a solution for Eq. 3 of the form…

Eq. 4

Plug Eq. 4 into Eq. 3 to yield the following differential equation…

Eq. 5

Eq. 1

Eq. 2

These equations usedto establish boundaryconditions…(see Krauss)

Eq. 3

This expressionsolved to obtainwave equation…(see Krauss)

Eq. 5

One solution to Eq. 5 is…

Eq. 6

So…

The lower boundary condition requires that w (or… dΦ/dz) go tozero at z = h (h is the seafloor depth) (see Krauss)

The boundary condition at the free surfacemust satisfy the following expression (see Krauss)

The lower boundary condition requires B=0The free surface boundary condition requires (see Krauss)…

Eq. 7

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or…

Given that the phase velocity can be written as C = ω/κ it follows that…

Phase velocity as a function of wavenumber and water depth

Eq. 7

Also known as the dispersion relation ofLamb (1945)

or…

or…

Therefore…

For h < λ/20

For h > λ/2

note…

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– Wave Speeds -

• Deep-Water Waves (Bottom Depth > λ/2)– Speed is a Function of Wavelength Only– Waves with Longer Wavelength move faster than

Waves with Shorter Wavelength

• Shallow-Water Waves (Bottom Depth < λ /20)– Speed is a Function of Depth Only– Waves Travel Slower in Shallower Water Irrespective

of Wavelength as long as Depth < λ /20

Deep-Water and Shallow-WaterWave Regions

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Speed of Deep-Water and Shallow-Water Waves as a Function of

Wavelength and Depth

Important Consequences of WaveSpeed Dependency on Wavelength

or Bottom Depth

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Wave Dispersion: Self Sorting of Deep-Water Waves Leaving aStorm Region based on Wavelength. It Occurs Because LongerWavelength Waves Travel Faster than Shorter WavelengthWaves (for Deep Water).

Bending of Shallow-Water WaveFronts Due to Change in BottomDepth. The Leading Edge of aWave Front Enters ShallowerWater and Slows While theRemaining Front Continues atHigher Speed. The Net Result isa Rotation of Wave Fronts ToBecome Parallel with BottomDepth Contours.

Wave Refraction:

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Consequence of Wave RefractionFocusing and Defocusing of Wave Energy on Headlands and

Bays, Respectively

Group Velocity

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Group Velocity

Group Velocity

recall… Wave Speed = C = ω/κ for:Then by analogy…

In the limit…

using a trigonometric rule…

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C = ω/κ or ω = C κ

The Main Point: Group velocity for Deep Water Waves is 1/2 the phasevelocity. Group velocity for Shallow Water Waves is equal to the phasevelocity.

Wave Spectra

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Spectral AnalysisTime Domain to Frequency Domain Transformation

Spectral AnalysisTwo Sine Waves at 260 Hz and 525 Hz, Respectively

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Spectral AnalysisTime Series derived from the Summation of the Two

Sine Waves

Spectral AnalysisFourier Transform from Time Domain to Frequency

Domain of Previous Time Series

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Distribution of Wave Energy in the Ocean as a Function ofWave Frequency or Wavelength

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Aliasing in Wave Sampling

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Wave Generation

1. Wind Speed2. Duration of Wind Event3. Fetch - the distance over which wind can blow without

obstruction

Wave Height of Wind-Generated Waves is aFunction of…

Full Developed Waves(Unlimited by Fetch and Duration)

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The Importance of FetchNortherly/Southerly Winds Produce a Long Fetch OverFinger Lakes (A), and Easterly/Westerly Winds Produce aShort Fetch (B)

A B

Fetch in the Open Ocean is Limited by the Sizeof the Storm System

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(95% of Energy Contained Within ±45o of Storm Direction)

Lateral Spreading of Wave Energy from a StormSource