Waves. Wave Parameters. Krauss Chapter Nine


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1 Waves Krauss Chapter Nine Wave Parameters 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 one wavelength (units of time) Angular Frequency = ω = 2π/T (units of 1/time) Wave Speed = C = ω/κ Distance a wave crest travels per unit time (units of distance/time) Wave Height = 2a = Twice the wave amplitude Wave Steepness = Wave Height/Wavelength 1
2  Ideal waves Propagate Energy but not Mass Wave Equation NavierStokes Equation Ignoring viscous forces and looking just at the x and z components Expanding the terms 2
3 Eq. 1 Eq. 2 These equations used to establish boundary conditions (see Krauss) Eq. 3 This expression solved to obtain wave equation (see Krauss) 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. 5 One solution to Eq. 5 is So Eq. 6 The lower boundary condition requires that w (or dφ/dz) go to zero at z = h (h is the seafloor depth) (see Krauss) The boundary condition at the free surface must satisfy the following expression (see Krauss) The lower boundary condition requires B=0 The free surface boundary condition requires (see Krauss) Eq. 7 3
4 Eq. 7 or or Also known as the dispersion relation of Lamb (1945) or Given that the phase velocity can be written as C = ω/κ it follows that Phase velocity as a function of wave number and water depth note Therefore For h < λ/20 For h > λ/2 4
5 Wave Speeds  DeepWater Waves (Bottom Depth > λ/2) Speed is a Function of Wavelength Only Waves with Longer Wavelength move faster than Waves with Shorter Wavelength ShallowWater 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 DeepWater and ShallowWater Wave Regions 5
6 Speed of DeepWater and Shallow Water Waves as a Function of Wavelength and Depth Important Consequences of Wave Speed Dependency on Wavelength or Bottom Depth 6
7 Wave Dispersion: Self Sorting of DeepWater Waves Leaving a Storm Region based on Wavelength. It Occurs Because Longer Wavelength Waves Travel Faster than Shorter Wavelength Waves (for Deep Water). Wave Refraction: Bending of ShallowWater Wave Fronts Due to Change in Bottom Depth. The Leading Edge of a Wave Front Enters Shallower Water and Slows While the Remaining Front Continues at Higher Speed. The Net Result is a Rotation of Wave Fronts To Become Parallel with Bottom Depth Contours. 7
8 Consequence of Wave Refraction Focusing and Defocusing of Wave Energy on Headlands and Bays, Respectively Group Velocity 8
9 Group Velocity Group Velocity using a trigonometric rule recall Wave Speed = C = ω/κ for: Then by analogy In the limit 9
10 C = ω/κ or ω = C κ The Main Point: Group velocity for Deep Water Waves is 1/2 the phase velocity. Group velocity for Shallow Water Waves is equal to the phase velocity. Wave Spectra 10
11 Spectral Analysis Time Domain to Frequency Domain Transformation Spectral Analysis Two Sine Waves at 260 Hz and 525 Hz, Respectively 11
12 Spectral Analysis Time Series derived from the Summation of the Two Sine Waves Spectral Analysis Fourier Transform from Time Domain to Frequency Domain of Previous Time Series 12
13 Distribution of Wave Energy in the Ocean as a Function of Wave Frequency or Wavelength 13
14 Aliasing in Wave Sampling 14
15 Wave Generation Wave Height of WindGenerated Waves is a Function of 1. Wind Speed 2. Duration of Wind Event 3. Fetch  the distance over which wind can blow without obstruction Full Developed Waves (Unlimited by Fetch and Duration) 15
16 The Importance of Fetch Northerly/Southerly Winds Produce a Long Fetch Over Finger Lakes (A), and Easterly/Westerly Winds Produce a Short Fetch (B) A B Fetch in the Open Ocean is Limited by the Size of the Storm System 16
17 Lateral Spreading of Wave Energy from a Storm Source (95% of Energy Contained Within ±45 o of Storm Direction) 17
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