Conference on Tsunami and Nonlinear Waves Waves in shallow water

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Conference on Tsunami and Nonlinear Waves Waves
in shallow water

• Harvey Segur
• University of Colorado, USA
• March 6, 2006

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Joe Hammack 1944-2004
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Equations of water waves

• (i) On bottom, z -h(x,y)
• (ii) In fluid, -h lt z lt ?(x,y,t)
• (iii) At free surface, z ?(x,y,t)
• (iv) Ignore viscosity, surface tension, variable
  density, fishes,

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Basic facts about wave propagation(according to
linear theory)

• Sound waves
• All travel at the same speed (speed of sound)
• Water waves
• Longer waves travel faster than shorter waves
• (for gravity-induced surface water waves)

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Basic facts about wave propagation(according to
linear theory)

• Sound waves
• All travel at the same speed (speed of sound)
• Water waves
• Longer waves travel faster than shorter waves
• (for gravity-induced surface water waves)
• But all very long waves all travel at speed
• gravity fluid depth

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Long waves travel faster than short waves
from Stokers Water Waves, 1957
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What happens to very long waves?

• D.J. Korteweg G. deVries
• (1895) derived their equation
• to describe the motion of
• long waves of moderate
• amplitude.
• Is the KdV equation relevant to the tsunami
  of 2004?

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To derive KdV or KP

• Assume

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To derive KdV or KP

• Assume
• Long waves (??gtgt h)
• (shallow water)

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To derive KdV or KP

• Assume
• Long waves (??gtgt h)
• (shallow water)
• Small amplitude (h gtgt a)

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To derive KdV or KP

• Assume
• Long waves (??gtgt h)
• (shallow water)
• Small amplitude (h gtgt a)
• Motion primarily in one direction
• if exactly so, ? KdV
• if approximately so, ? KP

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To derive KdV or KP

• Assume
• Long waves (??gtgt h)
• (shallow water)
• Small amplitude (h gtgt a)
• Motion primarily in one direction
• if exactly so, ? KdV
• if approximately so, ? KP
• All small effects are comparable

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At leading order (with constant h)

• Wave equation in 1-D
• with
• ?

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At leading order

• Wave equation in 1-D
• with
• ?
• At next order, satisfies either
• KdV
• or
• KP

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Tsunami of 2004 - approximate scales

• In the Bay of Bengal,
• h 3 km (fluid depth)
• For the tsunami,
• ? 100 km (wavelength)
• a 1 m (wave height)
• L 900 km (lateral extent)
• 620 km/hr (wave speed)

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Computer animation of 2004 tsunami
Kenji Satake, Japan http//ioc.unesco.oro/itsu/
or, see Steven Ward,
US http//www.es.ucsc.edu/ward/
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Volume of water in 2004 tsunami(per width of
shoreline)
1 m
100 km
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Wave speed

• Open ocean
• c 620 km/hr
• A wave 100 km long passes by in about 10 minutes
• Near shore
• Front of wave slows as it approaches the shore
• Back of the wave is still in deep water
• Consequence Wave compresses horizontally and
  grows vertically

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How long waves evolve(in water of uniform depth)
(Hammack Segur, 1978)
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Experimental data
(Hammack Segur, 1978)
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Tsunami summary (so far)

• A long water wave of small amplitude, like a
  tsunami, travels with approximate speed
• The tsunami changes its shape primarily in
  shallower coastal waters, where h(x,y) decreses.
• A tsunami that is barely noticeable in the open
  ocean becomes deadly in coastal waters because
• - (locally) and
• - wave volume is conserved.

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Other waves in shallow water

• Objective
• Find the natural structure(s) of ocean waves,
  including those in shallow water

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Waves caused by winds

• Most ocean surface waves are caused by storms and
  winds.
• Travel thousands of miles, over several days
• Oscillate, approximately periodically
• Long waves travel faster than short waves
• Two separate theories
• Waves in shallow water (today)
• Waves in deep water (next Friday)

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Long waves in shallow water(wave length gt water
depth)

• Simplest model
• All waves travel with speed
• For long waves of moderate amplitude, all
  traveling in approximately the same direction in
  water of uniform depth, a better approximation is

(Kadomtsev-Petviashvili, 1970)
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Uniform train of plane waves in shallow water
(National Geographic, 1933)
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Another train of plane waves in shallow
water(Lima, Peru, 2004)
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Oblique interaction of plane waves in shallow
water
(T. Toedtemeier)
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Two-phase solution of KP equation
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Experiments on two-phase waves in shallow water
(Hammack, Scheffner, Segur, 1989) (Hammack,
McCallister, Scheffner, Segur, 1995)
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Application rip currents
Rip Currents at Rosarita Beach Baja California
Periodic Array of Rip Currents Sand City, CA
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Experiments on rip currents
(Hammack, Scheffner, Segur, 1991)
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Thats all, folks
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A rogue wave
Waves measured at Draupner oil platform in the
North Sea, 1995. Peak wave 18.5 meters,
standard deviation of wave record 3 meters
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Do two-phase waves exist in the ocean?
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Tata Research Development and Design Centre Waves
in shallow water

• Harvey Segur
• University of Colorado, USA
• March 22, 2006

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SANUM Conference Waves in shallow water

• Harvey Segur
• University of Colorado, USA
• April, 2006