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 - Ideal waves Propagate Energy but not Mass Wave Equation Navier-Stokes Equation Ignoring viscous forces and looking just at the x and z components… Expanding the terms… 2 These equations used Eq. 1 to establish boundary conditions… (see Krauss) Eq. 2 This expression solved to obtain wave equation… (see Krauss) Eq. 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. 5 One solution to Eq. 5 is… Eq. 6 So… 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 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 – 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-Water Wave Regions 5 Speed of Deep-Water and Shallow- Water Waves as a Function of Wavelength and Depth Important Consequences of Wave Speed Dependency on Wavelength or Bottom Depth 6 Wave Dispersion: Self Sorting of Deep-Water 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 Shallow-Water 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 Consequence of Wave Refraction Focusing and Defocusing of Wave Energy on Headlands and Bays, Respectively Group Velocity 8 Group Velocity Group Velocity using a trigonometric rule… recall… Wave Speed = C = ω/κ for: Then by analogy… In the limit… 9 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 Spectral Analysis Time Domain to Frequency Domain Transformation Spectral Analysis Two Sine Waves at 260 Hz and 525 Hz, Respectively 11 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 Distribution of Wave Energy in the Ocean as a Function of Wave Frequency or Wavelength 13 Aliasing in Wave Sampling 14 Wave Generation Wave Height of Wind-Generated 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 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 Lateral Spreading of Wave Energy from a Storm Source (95% of Energy Contained Within ±45o of Storm Direction) 17.