Waves Part II. non-dispersive (C g =C)

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1 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)

2 In shallow waters, waves refract, diffract, reflect and/or break Refraction: Change of wave propagation (bending of rays) due to changes in bathymetry

3 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)

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5 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

6 Diffraction: Change of wave propagation due to the presence of an obstacle

7 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

8 Best for surfing! Most desired by surfers

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10 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.

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13 Another type of shallowwater wave: Tsunami

14 The Sumatra Tsunami: - 26 December, magnitude earthquake - 30 m high waves - ~300,000 dead - billions of dollars in damage - waves reach 10,000s miles away

15 2004 Tsunami in the Indian Ocean

16 Tsunami in the Indian Ocean, Dec Countries affected

17 Global reach of the tsunami waves beyond the Indian Ocean (Titov et al., Science, 2005)

18 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

19 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

20 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

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23 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

24 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

25 What happens if we have a closed basin? standing waves

26 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

27 Standing Waves Half-wave oscillator Quarter-wave oscillator Natural standing wave (lake, harbor, estuary) ---- seiche

28 Seiches in the Great Lakes

29 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

30 What happens when density is not constant? Internal Waves T = min to several hrs Wave length = m

31 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( / )

32 Propagation of internal waves: phase velocity perpendicular to group velocity C Cg

33 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)

34 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

35 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

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