Parameters  Cornell .Remaining Front Continues at Higher Speed. The Net Result is ... (Unlimited
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1
Waves
Krauss Chapter Nine
WaveParameters
Wavelength = = Length between wave crests (or troughs)
Wave Number = = 2/ (units of 1/length)
Wave Period = T = Time it takes a wave crest to travel onewavelength (units of time)
Angular Frequency = = 2/T (units of 1/time)
Wave Speed = C = / 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
NavierStokes Equation
Ignoring viscous forces and looking just at the x and zcomponents
Expanding the terms
3
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 d/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 = / 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 < /20
For h > /2
note
5
Wave Speeds 
DeepWater Waves (Bottom Depth > /2) Speed is a Function of Wavelength Only Waves with Longer Wavelength move faster than
Waves with Shorter Wavelength
ShallowWater Waves (Bottom Depth < /20) Speed is a Function of Depth Only Waves Travel Slower in Shallower Water Irrespective
of Wavelength as long as Depth < /20
DeepWater and ShallowWaterWave Regions
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Speed of DeepWater and ShallowWater 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 DeepWater Waves Leaving aStorm Region based on Wavelength. It Occurs Because LongerWavelength Waves Travel Faster than Shorter WavelengthWaves (for Deep Water).
Bending of ShallowWater 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 = / for:Then by analogy
In the limit
using a trigonometric rule
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C = / or = C
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 WindGenerated 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