Transcript BEZOUT IDENTITIES WITH INEQUALITY CONSTRAINTS

The Physics of Tsunamis
Wayne M. Lawton
Department of Mathematics
National University of Singapore
2 Science Drive 2
Singapore 117543
Email [email protected]
Tel (65) 874-2749
Tsunamis in the Media
movie Impact –
U.S. President
claims tsunami
speed = 2*sound
is his claim real
or exaggerated ?
what about this
tsunami photo
from Phuket ?
Hollywood Hype
Physical Facts: tsunami speed 
gd
2
g  9.8 meters/second d  oceandepth
Pacific Ocean: 1-7 km
http://www.es.flinders.edu.au/~mattom/regoc/text/8topo.html
Atlantic Ocean: 1-7 km
http://www.es.flinders.edu.au/~mattom/regoc/text/14topo.html
Indian Ocean:
1-7 km
http://www.es.flinders.edu.au/~mattom/regoc/text/11topo.html
356 < speed (4 km) = 713km/hour < 943
speed of sound in air (0 C) = 1192 km / hour
ocean depth needs to be 7 km * 6.4 = 44.8 km
Internet Assertion
Be sure and open the picture for a shock of your life to see what the
tsunami looked like just before hitting Puket, Thailand.
This picture is not a fake. It appears to have been taken from a hi-rise
building window in downtown Phuket Thailand.
The power of nature is hard to comprehend, especially the
destructiveness of water.
We have all seen the pictures on TV but I don't think any of us really
understand how big or how bad this wave was. This may be the most
impressive picture I've seen. It gives me a better understanding of
how 150,000 people perished in this disaster. If you look at this
picture that was taken right before the wave hit, it will send a chill
down your spine. Just look at the top of the wave compared to the top
of the building it is about to hit.
Urban Legends
Origins: This is another photograph being circulated as a picture of
the tsunami that struck Indian Ocean shorelines in Asia on 26
December 2004, this one purportedly taken from atop a high-rise
building in Phuket, Thailand. This looks to be another case of any
image depicting large waves being grabbed and passed around as a
"real photograph" of the December 2004 tsunami. The image clearly
doesn't fit its accompanying description because: It looks like
something other than a photograph, such as a still frame from a movie
or a composite image. The waves that struck the coast of Thailand
after the December 2004 Indian Ocean earthquake were about 4
meters in height, but the ones pictured here look several stories tall.
The skyline depicted doesn't appear resemble the city of Phuket. The
traffic flow shown isn't right for Thailand, where motorists drive on
the left-hand side of the road. As the operator of the World City Photo
Archive helpfully informed us, this image is actually a manipulated
depiction of the skyline of Antofagasta, Chile:
Scientific Approach to Tsunamis
while lacking the entertainment appeal, provides
understanding that has proven both necessary and
useful for the prediction and control of natural events
Why are earthquakes the major source of tsunamis ?
Why do many earthquakes not cause tsunamis ?
What is the surface geometry of tsunamis ?
What is the water movement in tsunamis ?
How is energy transported in tsunamis ?
How, after a powerful earthquake, can a tsunami be
detected and how can its propagation be predicted ?
Tsunamis
are physically generated by events that result
in the rapid and coherent (in the same direction)
motion of a massive amount of sea water
which of the following energy sources can physically
generate tsunamis ? sun heating, whales, impacts
from extraterrestrial objects (meteorites, comets,
asteroids), thermonuclear explosions, storms,
landslides, earthquakes, tidal forces, evil spirits,
volcanoes
Energy of an Earthquake
http://www.seismo.unr.edu/ftp/pub/louie/class/100/magnitude.html
log E(joules) = 4.8 + 1.5*Richter-Scale-Magnitude
1 J (joule) = kg * meter^2 / second^2 = watt * second
One Ton (2000 lbs) TNT = 4.148x10^9 J
Richter TNT for Seismic Example Magnitude Energy Yield (approximate)
-1.5 6 ounces Breaking a rock on a lab table
1.0 30 pounds Large Blast at a Construction Site
2.0 1 ton Large Quarry or Mine Blast
4.0 1,000 tons Small Nuclear Weapon
4.5 5,100 tons Average Tornado (total energy)
5.0 32,000 tons
6.0 1 million tons Double Spring Flat, NV Quake, 1994
7.0 32 million tons Hyogo-Ken Nanbu, Japan Quake, 1995; Largest Therm. Weapon
7.5 160 million tons Landers, CA Quake, 1992
8.0 1 billion tons San Francisco, CA Quake, 1906
8.5 5 billion tons Anchorage, AK Quake, 1964 (THIS IS 2 X 10^19 j)
9.0 32 billion tons Chilean Quake, 1960
12.0 160 trillion tons (Fault Earth in half through center, OR Earth's daily solar energy)
Physical Properties of Water
Water is incompressible, viscous, massive
which of these properties explains the following ?
1. when the container is turned the water
does not immediately turn with it, but
eventually will start to turn as long as
the container is turning
2. the water will fall when the bottom is not held
3. when the container is raised the water is raised
raises the same distance
Energy Transmission
Earthquakes generate tsunamis when they cause
large water displacements, this requires that they
cause large movements of earth in direction normal
(perpendicular) to the earth-ocean interface
which of the motions can generate tsunamis?
ocean
ocean
ocean
earth
earth
earth
earth
Slow Rising of Ocean Floor
We consider a large area of ocean floor that raises up
air
ocean
earth
if it moves very slowly it transmits energy from the
earthquake (seismic energy) into potential energy of
the water, equivalent to lifting a certain volume of,
water up to the sea level.
the potential energy remains as potential energy
Fast Rising of Ocean Floor
air
ocean
earth
if it moves fast it transmits more energy from the
earthquake (seismic energy) into more potential
energy of the water, equivalent to lifting a certain
volume of water up to a height above sea level
why does the earth do more work when it rises fast ?
does the lifted water stay above the sea level ?
Falling of Lifted Water
As gravity works to restore equilibrium to the
incompressible water the falling water lifts water
n its sides and that water falls in a cascade
Real Picture

wavelength
of course water near the edge is rounded, after several
fall-rise cycles the picture (from far right side looks
more like) at the bottom

direction of propagation
Surface Geometry
y
wave height
 H (x  v t)
graph of
y  H (x)
x
direction of propagation
Wave Velocity
depends on both wavelength and water depth
g
v
tanh(2d /  )
2
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/watwav2.html
deep water - surface (wind generated) waves
d    v  g / 2
shallow water - tsunami waves
d    v  g d
Surface Wave Parameters on the Deep Ocean
Amplitude = maximum H(x), y = 0 sea level
Period = time one cycle of wave to go by =
 /v
http://observe.arc.nasa.gov/nasa/exhibits/tsunami/tsun_physics.html
Wind Waves A = 10ft
Tsunamis A = 1.6ft

= 300 ft
v
= 10-20mph
 = 316800 ft v = 450-650mph
convert these parameter values into civilised units !
what is ratio of periods of tsunamis to wind waves ?
why are tsunamis hard to detect ON the deep ocean ?
Water Movement in Tsunamis
Wind Waves: water particles move in near circles,
whose diameters decrease rapidly with depth
Tsunamis: water particles move in elongated ellipses
whose lengths decrease linearly to zero at the bottom
http://electron4.phys.utk.edu/141/dec8/December%208.htm
http://www.coastal.udel.edu/faculty/rad/linearplot.html
http://mmu.jcu.edu.au/coastal/wavemaker.html
would tsunamis be easy to detect by scuba divers ?
Energy Transport in Tsunamis
is through kinetic energy of horizontal water motion
in deep ocean, and in contrast to wind waves
little viscosity loss since velocity changes slowly
no dispersion since v does not depend on wavelength
as wave approaches land amplitude increases to
A > 100 ft, lambda = 5000-10000ft, v = 30-200mph
Tsunami Detection and Prediction
how can tsunamis be detected in deep oceans ?
once detected how can their future propagation
be predicted rapidly to issue effective warnings ?
http://serc.carleton.edu/NAGTWorkshops/visualization/collections/tsunami.html
http://wcatwc.arh.noaa.gov/