We have to continually be jumping off cliffs OCNG 251: Oceanography and developing our wings on the way down Spring 2012 Kurt Vonnegut The surface of the Ocean: 1) Waves 1. Waves: Beware of the man who works hard to learn a. Anatomy of a wave something, learns it, and finds himself no b. Wave motion: speed and energy c. Deep- & Shallow-water waves wiser than before. […] He is full of murderous resentment of people who are ignorant without having come by their ignorance the hard way. Kurt Vonnegut, "Cat's Cradle" a. Anatomy of a wave a. Anatomy of a wave A wave can be regarded initially as an ideal a sinusoidal wave (though Ideal waves: If some water is lifted up and allowed to fall back the ideal regular waves bears little resemblance to real waves under the action of gravity, its inertia will cause it to overshoot observed at sea ! irregular in form and period) the equilibrium position. Pressure forces will then push it back up and oscillation will ensue. Energy can be transmitted at the interface (boundary) between A two fluids of different densities ! progressive orbital waves Characteristics: • H: Height (2 x A) • L: Wavelength (from crest to crest) • T: Period (time required for two successive crests to pass a point) • h: Depth of water column below mean surface level a. Anatomy of a wave a. Anatomy of a wave All waves can be regarded as A few waves: A case of temporal- and spatial-scales " Progressive waves, in that energy is travelling through, or across the surface of, the material (i.e. water) " Standing waves (i.e. plucked guitar strings) can be considered as the sum of two progressive waves of equal dimensions but traveling in opposite direction (later…) Waves can result from periodic and non-periodic disturbance of the water surface: In the case of non-periodic disturbances, the water particles are displaced from an equilibrium position ! restoring force acts to bring back equilibrium to the system (flat water level): # Gravitational force exerted by Earth # Surface tension (tendency of water molecules to stick together and present the smallest possible surface to the air) Water waves are affected by both of these forces: $ When L ≤ 1.7 cm ! capillary waves $ When L > 1.7 cm ! gravity waves (main interest of oceanography!) Waves are classified according to their wave period Sea level Sea level change Sea level change ! Oscillation due to geological changes in the Sea level change ! Oscillation due to geological changes in the hydrological cycle hydrological cycle a.Anatomy of a wave a. Anatomy of a wave The ideal relationship among wave properties. If one Wind waves: characteristic can be measured, the other two can be calculated. As the wind blows ! formation of ripples (capillary waves) As the wind increases in speed ! waves become larger (gravity waves) Variety and size of wind-generated waves are controlled by four principal factors: • Wind velocity • Wind duration • Unobstructed distance over which wind blows (fetch) • Original sea state a. Anatomy of a wave a. Anatomy of a wave Although the Wind as a causing factor: The relationship is a direct one… relationship is a direct one… Wind Wave Wave Wave Fetch Duration One condition will Speed Height Length Period (km) (h) always be limiting as (km/h) (m) (m) (s) to the maximum 20 25 2.8 0.33 10.6 3.2 development of wave 30 77 7 0.88 22.2 4.6 height and period! 60 660 27.5 5.1 89.2 9.9 e.g. Wind Speed: 30 kn 90 2446 65 13.9 201.6 13.9 Fetch: 200 nm Time: 40 hrs H ~ 7 m T ~ 12 sec a. Anatomy of a wave b. Wave motion: Speed and energy Although the There are mathematical relationships linking the characteristics relationship is a of wavelength (L), wave period (T) and wave height (H) to direct one… wave speed and to wave energy. One condition will Wave speed (celerity) ! speed = distance/time always be limiting as to the maximum C = L/T (time one wavelength takes to pass a certain point) development of wave Wave speed can also be represented by the equation: height and period! gL 2πh e.g. C = tanh( ) Wind Speed: 30 kn 2π L Fetch: 100 nm Where: Time: 10 hrs " g = acceleration due to gravity " L = Wavelength H ~ 3.5 m " tanh = hyperbolic tangent T ~ 7.5 sec " h = water depth c. Wave motion: Speed Approximations b. Wave motion: Speed Approximations 1) For h > L/2 ! Deep-water waves 1) For h > L/2 (L < 2h) ! Deep-water waves If x is larger than π, then tanh x ~ 1 tanh (2πh/L) = 1 (within 0.5%) gL 2πh C = tanh( ) 2π L gL or 1.25 L 2π Circular motion of particles ! Exponential decrease of path with depth! Wavelength (or period) is the only variable affecting the wave speed! The change in pressure (size of the orbit) is undetectable at a water depth ≥ L/2 (the wave base) € c. Wave motion: Speed Approximations b. Wave motion: Speed Approximations 2) For h < L/20 (L > 20h) ! Shallow-water waves 2) For h < L/20 ! Shallow-water waves If x is small, then tanh x ~ x tanh (2πh/L) = (2πh/L) (within 3%) gL 2πh C = tanh( ) 2π L gh or 3.1 h Or: C= L/T = (gh)1/2 ! L = 3.13 (h)1/2.T Elliptical motion of particles The water depth is the only variable affecting the wave speed! Exponential decrease of vertical path with depth Not of horizontal path! c. Wave motion: Speed Approximations c. Wave motion: Speed Approximations 1) For h > L/2 ! Deep-water waves (L < 2h) 2) For h < L/20 ! Shallow-water waves (L > 20h) Deep-water waves that have the longest wavelength and greatest period, will arrive first in regions distant from the storm which 3) For L/20<h<L/2 (2h < L < 20h) generated them: The separation of waves by virtue of their differing rates of travel (speed) is known as dispersion. For h > L/2 (L < 2h) ! Deep-water waves gL 2π or 1.25 L € c. Wave motion: Group Speed c. Wave motion: Group Speed Individual waves do not persist for long in the open Ocean. They usually only exist for as long as they take to pass through a group of waves. i.e. two sets of waves: in phase and out of phase Individual waves do not persist for long in the open Ocean. They usually only exist for as long as they take to pass through a group of waves. If two sets of waves are interfering to produce a succession of wave groups, the group speed (Cg) is: C1× C2 C 2 C Cg = Cg = Cg = C1+ C2 2C 2 € c. Wave motion: Speed Approximations c. Wave motion: Speed Approximations Assume You have a storm blowing at 30 knots (nautical mile = Assume You have a storm blowing at 30 knots (nautical mile = 1.853 km) over a fetch of 100 nautical miles for 10 hours. 1.853 km) over a fetch of 100 nautical miles for 10 hours. 1) What is the longest wave period generated by the storm? 2) What is the longest wavelength of the waves generated by the storm? L gL L2 gT 2 C = = = L =1.56 × T 2 T 2π L 2π ~88 m H ~ 3.5 m 3) When will the first wave reach€ a boat that is 500 km away? T ~ 7.5 sec € € ~12 hrs d. Tsunamis c. Wave motion: Group Speed Seismic waves created by sudden movements of the Earths Wave energy travels at the group speed (not at the phase crust (Earthquake), volcanic activity, or mudslide event speed of individual waves) ! disturbance of the whole water column! C = C /2 Shallow-water wave g p C = gh p For shallow-water waves the group speed (energy) is the 3) Tsunamis reaching the coast: h = 4000m; L = 200 km same as the phase speed (the energy is in the wave!) ! L > 20h C = 200 m/s p Cg = Cp (Cg = Cp ! the energy is in the wave) Examples: 1) Mid-oceanic swell: h > 3000 m; L = 120 m ! L < 2h Cp = 13.7 m/s 2) Mid-oceanic swell reaching the coast: h = 5m ! L > 20h Cp = 7 m/s Sumatra Tsunami e. Standing waves (Resonance) On December 25, 2004, a series of about 15 earthquakes occurred one Standing waves do not move horizontally but remain stationary after another along the west coast of Sumatra. The strongest (Guitar string). earthquake measured 9.0 magnitude and is the fourth strongest this century. Current estimates put the death toll at about 234,000. The water oscillates back and forth about a fixed point: Node. Millions more were at risk from disease and starvation The properties of a standing wave depend on the geometry of the Bay of Bengal/ Indian Ocean littoral regions of Indonesia, Malaysia, basin. The larger the “container”, the longer its characteristic Thailand, Burma, Bangladesh, India, Sri Lanka and Maldives. standing wave will take to oscillate. In an open basin: seiche L 4l 4l T gh C p = = = gh ←!→T = ←!→l = T T gh 4 e. Standing waves (Resonance) f. Refraction Refraction of waves in progressively shoaling waters can be described by L 4l 4l T gh C p = = = gh ←!→T = ←!→l = a relationship similar to Snell s law (refraction of light) T T gh 4 Shallow-water wave C1 gh1 h1 h1 = = = C2 gh2 h2 h2 g. Breaking waves As a wave nears the shore, only the period remains constant! For Next Time Shallow-water wave L T = Tides (Chap.