Waves Part II Previously we discussed Surface Gravity Waves Deep Water Waves Shallow Water Waves C g T 2 C g h dispersive (C g =C/2) Definitions: phase speed C= /T= /k non-dispersive (C g =C) group speed C g = / k Next: What happens when deep- water waves approach the coast? What types of waves we find when our previous assumptions are not valid: waves are steep density is not constant water body has boundary (e.g., a lake)
In shallow waters, waves refract, diffract, reflect and/or break Refraction: Change of wave propagation (bending of rays) due to changes in bathymetry
What happens to waves when they reach the coast? - become shallow-water waves with slowing speed, C=(gh) ½ - period unchanged wavelength decrease height increase become steep and breaking - turn toward coastline; wave heights decrease/increase over underwater canyons/ridges - what surfers need to know to catch the best waves? offshore storm conditions and coastal topography waves converge: more energy (surfing) waves diverge: less energy (swimming)
Wavelengths affect the refraction of waves near the coast: long waves feel the bottom sooner and will refract more than short waves. long waves Area protected from short waves but not from long waves short waves
Diffraction: Change of wave propagation due to the presence of an obstacle
Waves break when steepness (H / λ) ~ 1 / 7 or H / d ~ 0.8 Breaking waves -front of wave slows down - wave get steeper - then break most common
Best for surfing! Most desired by surfers
Example of the interaction of waves with the coast: Rip Currents: strong currents (a few m/s, dangerous for swimmers), as water piled up by waves return toward the open ocean in distinct jets. Coastal topography may affect location of jets.
Another type of shallowwater wave: Tsunami
The Sumatra Tsunami: - 26 December, 2004-9.3 magnitude earthquake - 30 m high waves - ~300,000 dead - billions of dollars in damage - waves reach 10,000s miles away
2004 Tsunami in the Indian Ocean http://en.wikipedia.org/wiki/2004_indian_ocean_earthquake http://www.digitalglobe.com/tsunami_gallery.html
Tsunami in the Indian Ocean, Dec 26 2004 Countries affected
Global reach of the tsunami waves beyond the Indian Ocean (Titov et al., Science, 2005)
Tsunami in Japan: Magnitude 9 earthquake off the Pacific coast of Tohoku, Japan on 11- March-2011 Waves over 40m; reach 10km inland Meltdowns of nuclear reactors in Fukushima; evacuation of 100,000s ~20,000 deaths; $15-35B damage Before After
What is a Tsunami? -Caused by an underwater earthquake which displaces the ocean floor - Displacement of the surface of the ocean - Very long ( ~100km, T~10min) surface wave relative to ocean depth - Shallow-water wave (even in deep waters!) with speed C - wave slows down near shore: h=4000m C~200m/s = 450 mph h=50m C~20m/s = 45 mph - barely noticeable in deep water, but amplified to a huge wave when reaching shallower water: height increases from ~1m wave at h=4000m 15m at 20m! - Warning systems: seismic, buoys & models gh
Stokes Waves and Stokes Drift: -For large steep waves (close to breaking) waves are: (1) not sinusoidal ( non-linear waves ) (2) particle motion is not close circles- near surface Stokes Drift. wave steepness factor (3) Deep water Stokes waves: C 2 =(g/k)[1+ 2 (H/ ) 2 ] H/ =wave height/length (4) Shallow water Stokes waves: C 2 =(gh)[1+(h/2h)] H/h=wave height/depth so for small H/ Stokes waves speed sinusoidal wave speed
Capillary waves: - small waves of a few centimeters superimposed on larger waves or flat water - generated when wind starts blowing; due to surface tension - shorter capillary waves move faster! (opposite to gravity waves) - group speed faster than phase speed! (opposite to gravity waves) - ripples/capillary waves make the surface rough & affect wind stress
Propagation speed of Capillary waves depends on: - wave length ( =2 /k) - density ( ) deep water waves surface tension correction - surface tension ( ) 2 g k C k 1.7 cm short waves long waves 22 cm/s k wave number
What happens if we have a closed basin? standing waves
Shallow-water standing waves are called Seiches - In closed lakes, channels and harbors - generated by moving storms pile up water at one end when wind dies, water returns downslope and starts oscillating back and forth - wavelength is twice the length of the channel or lake wind 2l T C 2l gh
Standing Waves Half-wave oscillator Quarter-wave oscillator Natural standing wave (lake, harbor, estuary) ---- seiche
Seiches in the Great Lakes
Lake Erie (shallowest of the Great Lakes) Buffalo Average depth, H ~ 20m Maximum depth, H ~ 60m Length, L ~ 400km Toledo Buffalo Toledo Phase speed near center of lake: C=(gx50) ½ ~ 22m/s Time for a wave to travel the entire length of the lake and back: T= 2L/c ~ 10 hours
What happens when density is not constant? Internal Waves T = min to several hrs Wave length = 1-300 m
Characteristics of Internal Waves: Internal waves move slower than surface waves at speed c= (g h) ½ internal waves have long period and long wavelength, T~min to hrs, L~100s m Internal waves have larger amplitude than surface waves, 10s m Energy of internal waves largest near the pycnocline Propagation: in all 3 directions (horizontal and up/down) Group velocity: at 90 degree angle to phase velocity surface wave internal wave 2b 2a Energy of internal waves: E=½ρg a 2 compare to surf. Waves: larger a, but g <<g) g =g( / )
Propagation of internal waves: phase velocity perpendicular to group velocity C Cg
The direction of propagation of internal waves depends on their frequency!, as shown in this lab experiment N cos /N=0.62 /N=0.32 /N=0.9 Frequency of internal waves in the ocean must be in the range: z x f N max freq : N min freq : f g z (Brunt Vaisala) (Coriolis) Nowbrey and Rarity, 1967)
Refraction of internal waves from a boundary Because of the dependency of the propagation direction of internal waves on frequency, when the waves interact with a sloping ocean bottom: low-frequency (long waves) would be reflected backwards, while high-frequency (short waves) would be reflect forward at some critical slope angle all waves will accumulate, break, and create a lot of turbulence Bot. slope > angle of wave (2), so it is reflected back
Next Class: The longest and largest waves: TIDES wave length: up to half/one distance around the earth wave period: ~12h, ~24h important for coastal processes