Waves
Objectives
1. To learn the terminology for describing surface
gravity waves.
2. To understand deep water waves and their
interactions with each other.
3. To understand shallow water waves and their
interactions with topography.
1. General Definitions
- Waves are caused by a generating force that moves the surface from its
undisturbed level, and a restoring force that returns it.
- Generating Forces: wind physical disturbances (earthquakes, landslides,
ships).
- Restoring Forces: surface tension, small capillary waves
gravity, larger gravity waves.
- Progressive Waves: Travel in a particular direction.
- Standing Waves: Remain at a fixed location.
- Forced Waves: Still under the influence of the forcing.
- Free Waves: Have propagated out of the region of forcing.
2. Wave Anatomy Definitions
- Equilibrium surface: undisturbed state
- Crest: highest point
- Trough: lowest point
- Wavelength (L): horizontal distance between 2 crests or troughs.
- Height (H): vertical distance between a crest and trough.
- Amplitude (A): vertical distance from equilibrium surface to crest or trough
(= H/2).
- Steepness (S): height divided by wavelength.
- Period (T): Time for 1 wavelength to propagate past an observation point.
- Phase Speed (C): speed at which individual crests and troughs propagate.
- Phase Speed: C = L/T
- Group Speed (V): speed at which a group of waves propagates.
- Particle Orbit: motion of a water particle beneath a wave.
3. Wave Height Dependence
- Factors that determine wave height:
1) Wind Speed
2) Wind Duration
3) Fetch (distance over which the wind blows)
- Increases in these increase height.
- Energy is proportional to amplitude squared.
4. Wave Steepness Limitation
- S = H/L
- For Breaking S > 1/7 - wave breaks.
5. Wave Period Constraint
- Wave period remains constant over its lifetime.
- "Conservation of Wave Crests"
6. Deep Water Waves (D > L/2)
- Particle orbits (deepwater):
1) Circular, decay exponentially with depth
2) Negligible below a depth of L/2
3) Particles move forward with crest, backward with trough.
- Wavelength: L = (g/2p) T2 = 1.56 T2
for T=10 sec, L=156 m.
- Phase Speed: C= L/T = (g/2p) T = 1.56 T
for T=10 sec, C= 15.6 m/sec.
- Deepwater waves are dispersive: Longer periods have longer wavelengths
that travel faster. At a storm center, waves of all periods are generated. Far
away from the storm, the longer faster propagating waves arrive first, the
shorter slower propagating waves arrive last.
- Group speed (deep water only): V = C/2. Individual crests propagate at C.
Wave groups, packets or trains propagate at V.
7. Deepwater Wave Interactions
- Waves can be added linearly.
- For 2 waves of equal amplitude A,
If the crests intersect, the total amplitude is 2A.
If the troughs intersect, the total depression is 2A.
If the crest and trough intersect, they cancel.
- Crossing Sea: 2 wave trains crossing at an angle => checkerboard pattern.
- Rogue Waves: large episodic waves formed by the superposition of several
wave crests and interactions with topography or strong currents. 20-30 m
high.
8. Shallow Water Waves (Water Depth D < L/20)
- Particle orbits go from circular to elliptical.
- Wavelength: L = sqrt(gd)T.
- Phase speed now only depends on water depth: C = gD. Waves slow down
as water depth decreases.
- Shallow water waves are nondispersive: in the same water depth, waves with
different periods now travel at the same speed.
- Group speed now equals phase speed: V=C.
9. Interactions with bottom topography and other features
- Refraction: as a wave crest approaches a beach at an angle, the side in
shallow water travels slower than the side in deep water. The wave crest
turns toward the beach.
- Reflection: if a wave encounters a very steep beach or wall, it can be
reflected. Reflected waves can combine with the incoming progressive wave.
- Diffraction: if part of a wave crest is blocked by an obstruction, wave energy
is transported at right angles to the propagation direction into the
undisturbed area behind the obstacle. Waves propagating through a hole in
a breakwater will spread out in semicircles in the undisturbed water beyond
the breakwater.
10. Surf Zone
- Region of wave breaking.
- Plunging Breakers: steep beach, quick release of energy.
- Spilling Breakers: wide beach, slow release of energy.
- Breakers transport water shoreward. The transported water returns to the
sea in narrow rip currents.
11. Tsunamis
- Seismic sea waves caused by earthquakes.
- Height 1-2 m, wavelength 100-200 km, period 10-20 min.
- Ocean depth is about 4000 m, so Tsunamis are shallow water waves.
- As tsunamis approach the shore, all the energy over the entire 4000 m depth
of the ocean is compressed into a shallower region, so the amplitude
increases to up to 30 m.
12. Storm Surge
- Similar to Tsunamis but storm generated.
13. Internal waves
- Similar to surface waves but occurring on a density difference interface
within the ocean.
14. Standing waves.
- Stationary waves due to the interference of a progressive wave with a
perfectly reflected wave.
- Seiches: sloshing of water that occurs in natural basins (harbors, lakes).
- Antinode: areas that alternate between a crest and a trough.
- Node: areas between antinodes where there is no change in elevation from
the equilibrium surface.