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Waves

Waves. Lecture 17. OEAS-604. November 28, 2011. Outline: Types of Waves Linear Wave Theory Surface Gravity Waves Refraction, Diffraction, Shoaling Other Types of waves Homework Exam. Basic Properties of Waves. Wave Period T. Amplitude (A). (H). Wave Frequency. Wave Number.

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Waves

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  1. Waves Lecture 17 OEAS-604 November 28, 2011 • Outline: • Types of Waves • Linear Wave Theory • Surface Gravity Waves • Refraction, Diffraction, Shoaling • Other Types of waves • Homework • Exam

  2. Basic Properties of Waves Wave Period T Amplitude (A) (H) Wave Frequency Wave Number Wave Speed Note that these definitions work for all types of waves (radio waves, gamma rays, light waves, internal waves, etc…)

  3. Generalized description of a wave can be derived from “dispersion relationship”

  4. Types of Waves

  5. Linear Wave Theory (much abbreviated) continuity First we assume flow is incompressible and irrotational (rotation not important) Wave Propagation convergence divergence divergence The water at any given point simply oscillates back and forth (no water is transported), but wave form propagates (energy is transmitted)

  6. Next we define the velocity potential (ϕ) Combined with continuity: Gives the Laplace Equation: Partial Differential Equation This equation is then solved using the kinematic and dynamic boundary conditions Kinematic Boundary Condition: Dynamic Boundary Condition: From Bernoulli’s Equation assuming surface pressure equals atmospheric pressure Surface is always made up of same fluid particles

  7. Now we now have to solve the Laplace Equation using a bunch of boundary conditions and simplifying assumptions. No flow through bottom. Small amplitude waves L A w = 0 @ z= -h L>>A So kinematic boundary condition becomes: or … @ z = 0 Dynamic boundary condition can be linearized: This now becomes a second order linear differential equation, with the solution: Where h is water depth @ z = 0

  8. This is the “dispersion relationship” h = water depth Grav. Accel (9.8 m/s2) Hyperbolic Tangent Waves with a given frequency (period), must have a certain wavelength (wavenumber). Since the wave speed is defined as: Waves with longer lengths and periods travel faster. Thus waves “disperse.” http://polar.ncep.noaa.gov/waves/index2.shtml

  9. In very deep water (h > λ/2), the wave length is much smaller than the water depth. (this means kh gets really big) λ As kh gets big, tanh(kh) equals 1 h so … In deep water, wave speed only depends on wavelength (which sets the wave period bc/ of dispersion relationship)

  10. In shallow water (h < λ/20), the wave length is similar to the water depth. λ h This means that kh is small and as kh gets small… so … In shallow water, the wave speed only depends on water depth (and not wavelength). So, shallow water waves are non-dispersive.

  11. In shallow water, waves “feel” the bottom and friction starts to become important.

  12. Longer Waves travel faster

  13. 1) There will always be a range of wave periods generated by the wind. 2) Longer period waves travel faster. 3) Constructive and destructive interference leads to “Wave Trains” or “Wave Groups” or “Sets”

  14. Interaction between waves leads to “wave groups” that travel at a different speed than the individual waves. For dispersive waves the speed of the wave group is half the speed of the individual waves: This is not true for non-dispersive waves (i.e. shallow water)

  15. Summary of Key Properties of Waves deep waves (short) λ < 2h intermediate waves 2h < λ < 20h shallow waves (long) λ > 20h

  16. Winds produce waves by transferring momentum to the sea surface. Waves start out as Capillary Waves Capillary waves are small ripples (wavelengths < a few cm), that are impacted by both gravity and surface tension.

  17. Turbulence in the air, disturbs the sea surface Pressure Gradient Wind waves are gravity waves formed by the transfer of wind energy into water. Wind forces convert capillary waves to wind waves. A capillary wave interrupts the smooth sea surface, deflecting surface wind upward, slowing it, and causing some of the wind’s energy to be transferred into the water to drive the capillary wave crest forward (point a). The wind may eddy briefly downwind of the tiny crest, creating a slight partial vacuum there ( - ). Atmospheric pressure ( + ) pushes the trailing crest forward (downwind) toward the trough (point b), adding still more energy to the water surface.

  18. Factors that Influence Wind Wave Development • Wind strength - wind must be moving faster than the wave crests for energy transfer to continue • Wind duration - winds that blow for a short time will not generate large waves • Fetch - the uninterrupted distance over which the wind blows without changing direction

  19. Once produced, wind waves can travel long distances Wind waves versus Swell Locally generated wind waves Remotely-generated Swell In contrast to local wind waves, swell are waves which were produced elsewhere, but have traveled into an area.

  20. Waves slow down, shoal and break as they approach shore. In shallow water, wave speed is controlled by depth (c = √gh). So, wave begins to converge and increase in height. When the orbital velocity at the crest equals the wave speed, the wave breaks. Empirically this is shown to occur when the wave height approaches some fraction of the water depth:

  21. Waves will break in the open ocean if they get too steep. Wave Steepness = H/L In deep water wave breaking or white capping limits wave steepness:

  22. Waves Refract When They Approach a Shore at an Angle θ1 θ2 This can be explained in terms of Snells Law:

  23. Wave Refraction Refraction: change of wave speed (bending of wave rays) due to changes in bathymetry

  24. Refraction can focus wave energy.

  25. Wave Diffraction is the propagation of a wave around an obstacle:

  26. Internal Waves Internal wave speed is controlled by density difference (g’) Disturbances can lead to waves on the pycnocline. Just like surface waves, internal waves can propagate along a density interface.

  27. There is surface convergence over the internal wave troughs, where floating material will collect. Columbia River Plume

  28. Tsunamis Because the have such a long wave length, tsunami are always “shallow water waves” Almost 500 mph!

  29. Estimated Height of the 2004 Indian Ocean Tsunami Max wave > 25 m http://www.globalsecurity.org/eye/images/tsunami-indonesia2004.mov Small waves (5 cm) from the Tsunami were observed on East Coast of US.

  30. Homework #3 is posted on the class website. It is due in class on Wednesday, December 7, 2011

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